Research ArticleFast Stiffness Matrix Calculation for NonlinearFinite Element Method

Emir Guumlluumlmser1 ULur Guumlduumlkbay1 and Sinan Filiz2

1 Department of Computer Engineering Bilkent University 06800 Ankara Turkey2Department of Mechanical Engineering Bilkent University 06800 Ankara Turkey

Correspondence should be addressed to Ugur Gudukbay gudukbaycsbilkentedutr

Received 29 May 2014 Revised 24 July 2014 Accepted 6 August 2014 Published 28 August 2014

Academic Editor Henggui Zhang

Copyright copy 2014 Emir Gulumser et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We propose a fast stiffness matrix calculation technique for nonlinear finite element method (FEM) Nonlinear stiffness matricesare constructed usingGreen-Lagrange strains which are derived from infinitesimal strains by adding the nonlinear terms discardedfrom small deformations We implemented a linear and a nonlinear finite element method with the same material properties toexamine the differences between themWe verified our nonlinear formulationwith different applications and achieved considerablespeedups in solving the system of equations using our nonlinear FEM compared to a state-of-the-art nonlinear FEM

1 Introduction

Mesh deformations have widespread usage areas such ascomputer games computer animations fluid flow heattransfer surgical simulations cloth simulations and crashtest simulations The major goal in mesh deformations isto establish a good balance between the accuracy of thesimulation and the computational cost achieving this balancedepends on the application The speed of the simulation isfar more important than the accuracy in computer gamesThe simulation needs to be real-time in order to be used ingames so free-form deformation or fast linear FEM solverscan be used However high computation cost gives muchmore accurate results when we are working with life-criticalapplications such as car crash tests surgical simulators andconcrete analysis of buildings even linear FEM solvers arenot adequate enough for these types of applications in termsof the accuracy

For realistic and highly accurate deformations one canuse the finite element method (FEM) a numerical techniqueto find approximate solutions to engineering and mathemat-ical physics problems FEM could be used to solve problems

in areas such as structural analysis heat transfer fluid flowmass transport and electromagnetics [1 2]

We propose a fast stiffness matrix calculation techniquefor nonlinear FEM We derive nonlinear stiffness matri-ces using Green-Lagrange strains themselves derived frominfinitesimal strains by adding the nonlinear terms discardedfrom infinitesimal strain theory

We mainly focus on the construction of the stiffnessmatrices because change in material parameters and changein boundary conditions can be directly represented andapplied without choosing a proper FEM [3] Joldes et al[4] and Taylor et al [5] achieve real-time computations ofsoft tissue deformations for nonlinear FEM using GPUshowever they do not describe how they compute stiffnessmatrices thus we cannot implement their methods andcompare them with our proposed method Cerrolaza andOsorio describe a simple and efficient method to reduce theintegration time of nonlinear FEM for dynamic problemsusing hexahedral 8-noded finite elements [6] We compareour stiffness matrix calculations with Pedersenrsquos method [3]to measure performance and verify correctnessWe achieve a142 speedup in calculating the stiffness matrices and a 15

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 932314 12 pageshttpdxdoiorg1011552014932314

2 Journal of Applied Mathematics

speedup in solving the whole system on average compared toPedersenrsquos method

2 The Nonlinear FEM withGreen-Lagrange Strains

We use tetrahedral elements for modeling meshes in theexperiments Overall there are 12 unknown nodal displace-ments in a tetrahedral element They are given by [2]

119889 = 119906 (119909 119910 119911) =

1199061

V1

1199081

1199064

V4

1199084

(1)

In global coordinates we represent displacements by linearfunction by

119906119890(119909 119910 119911) = 119888

1+ 1198882119909 + 1198883119910 + 1198884119911 (2)

For all 4 vertices (2) is extended as

[[[

[

1 1199091

1199101

1199111

1 1199092

1199102

1199112

1 1199093

1199103

1199113

1 1199094

1199104

1199114

]]]

]

1198881

1198882

1198883

1198884

=

1199061

1199062

1199063

1199064

(3)

Constants 119888119899can be found as

119888119899= Vminus1119906

119899 (4)

where Vminus1 is given by

Vminus1 =1

det (V)[[[

[

1205721

1205722

1205723

1205724

1205731

1205732

1205733

1205734

1205741

1205742

1205743

1205744

1205751

1205752

1205753

1205754

]]]

]

(5)

det(V) is 6119881 where 119881 is the volume of the tetrahedron If wesubstitute (4) into (2) we obtain

119906119890(119909 119910 119911) =

1

6119881119890[1 119909 119910 119911]

[[[

[

1205721

1205722

1205723

1205724

1205731

1205732

1205733

1205734

1205741

1205742

1205743

1205744

1205751

1205752

1205753

1205754

]]]

]

1199061

1199062

1199063

1199064

(6)

120572 120573 120574 120575 and the volume 119881 are calculated by

1205721=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1199092

1199102

1199112

1199093

1199103

1199113

1199094

1199104

1199114

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205722= minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1199091

1199101

1199111

1199093

1199103

1199113

1199094

1199104

1199114

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205723=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1199091

1199101

1199111

1199092

1199102

1199112

1199094

1199104

1199114

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205724= minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1199091

1199101

1199111

1199092

1199102

1199112

1199093

1199103

1199113

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205731= minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1199102

1199112

1 1199103

1199113

1 1199104

1199114

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205732=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1199101

1199111

1 1199103

1199113

1 1199104

1199114

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205733= minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1199101

1199111

1 1199102

1199112

1 1199104

1199114

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205734=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1199101

1199111

1 1199102

1199112

1 1199103

1199113

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205741=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1199092

1199112

1 1199093

1199113

1 1199094

1199114

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205742= minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1199091

1199111

1 1199093

1199113

1 1199094

1199114

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205743=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1199091

1199111

1 1199092

1199112

1 1199094

1199114

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205744= minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1199091

1199111

1 1199092

1199112

1 1199093

1199113

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205751= minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1199092

1199102

1 1199093

1199103

1 1199094

1199104

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205752=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1199091

1199101

1 1199093

1199103

1 1199094

1199104

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205753= minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1199091

1199101

1 1199092

1199102

1 1199094

1199104

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205754=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1199091

1199101

1 1199092

1199102

1 1199093

1199103

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

6119881 =

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 119909119894

119910119894

119911119894

1 119909119895

119910119895

119911119895

1 119909119896

119910119896

119911119896

1 119909119897

119910119897

119911119897

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(7)

Because of the differentials in strain calculation 120572 is not usedin the following stages If we expand (6) we obtain

119906119890(119909 119910 119911)

=1

6119881119890

[[[

[

1205721+ 1205731119909 + 1205741119910 + 1205751119911

1205722+ 1205732119909 + 1205742119910 + 1205752119911

1205723+ 1205733119909 + 1205743119910 + 1205753119911

1205724+ 1205734119909 + 1205744119910 + 1205754119911

]]]

]

times [119906(119909 119910 119911)1

119906(119909 119910 119911)2

119906(119909 119910 119911)3

119906(119909 119910 119911)4]

(8)

For tetrahedral elements to express displacements in simplerform shape functions are introduced (120595

1 1205952 1205953 1205954) They

are given by

1205951=

1

6119881(1205721+ 1205731119909 + 1205741119910 + 1205751119911) 119906(119909 119910 119911)

1

1205952=

1

6119881(1205722+ 1205732119909 + 1205742119910 + 1205752119911) 119906(119909 119910 119911)

2

1205953=

1

6119881(1205723+ 1205733119909 + 1205743119910 + 1205753119911) 119906(119909 119910 119911)

3

1205954=

1

6119881(1205724+ 1205734119909 + 1205744119910 + 1205754119911) 119906(119909 119910 119911)

4

(9)

In our method nonlinear stiffness matrices are derivedusing Green-Lagrange strains (large deformations) which

Journal of Applied Mathematics 3

y uy

x ux

dx

C

C

998400

B

B

998400

A

A

998400 120597uy

120597xdx

dx +120597ux

120597xdx

Figure 1 2D element before and after deformation

themselves are derived directly from infinitesimal strains(small deformations) by adding the nonlinear terms dis-carded in infinitesimal strain theoryThe proposed nonlinearFEM uses the linear FEM framework but it does not requirethe explicit use of weight functions and differential equationsHence numerical integration is not needed for the solution ofthe proposed nonlinear FEM Instead of using weight func-tions and integrals we use displacement gradients and strainsto make the elemental stiffness matrices space-independentin order to discard the integral We extend the linear FEMto the nonlinear FEM by extending the linear strains to theGreen-Lagrange strains

We constructed our linear FEM by extending Loganrsquos 2Dlinear FEM to 3D [2] To understandGreen-Lagrange strainswemust first see how they differ from the infinitesimal strainsused to calculate the global stiffness matrices in a linear FEMFigure 1 shows a 2D element before and after deformationwhere the element edge 119860119861 with initial length 119889119909 becomes11986010158401198611015840 The engineering normal strain is calculated as the

change in the length of the line

120576119909=

100381610038161003816100381610038161198601015840119861101584010038161003816100381610038161003816

minus |119860119861|

|119860119861| (10)

The final length of the elemental edge can be calculated using

100381610038161003816100381610038161198601015840119861101584010038161003816100381610038161003816

2

= (119889119909 +120597119906119909

120597119909119889119909)

2

+ (

120597119906119910

120597119909119889119909)

2

100381610038161003816100381610038161198601015840119861101584010038161003816100381610038161003816

2

= 1198891199092[1 + 2(

120597119906119909

120597119909) + (

120597119906119909

120597119909)

2

+ (

120597119906119910

120597119909)

2

]

(11)

By neglecting the higher-order terms in (11) 2D infinitesimalstrains are defined by

120576119909119909

=120597119906119909

120597119909 120576

119910119910=

120597119906119910

120597119910 120576

119909119910=

1

2(

120597119906119909

120597119910+

120597119906119910

120597119909)

(12)

By the definition the nonlinear FEM differs from thelinear FEM because of the nonlinearity that arises from thehigher-order term neglected in calculation of strains Thestrain vector used in the linear FEM relies on the assumptionthat the displacements at the 119909-axis 119910-axis and 119911-axis arevery small The initial and final positions of a given particleare practically the same thus the higher-order terms areneglected [7] When the displacements are large howeverthis is no longer the case and one must distinguish betweenthe initial and final coordinates of the particles thus thehigher-order terms are added into the strain equations Byadding these high-order terms 3D strains are defined as [8]

120578119909119909

=120597119906119909

120597119909+

1

2[(

120597119906119909

120597119909

120597119906119909

120597119909) + (

120597119906119910

120597119909

120597119906119910

120597119909) + (

120597119906119911

120597119909

120597119906119911

120597119909)]

120578119910119910

=

120597119906119910

120597119910+

1

2[(

120597119906119909

120597119910

120597119906119909

120597119910) + (

120597119906119910

120597119910

120597119906119910

120597119910) + (

120597119906119911

120597119910

120597119906119911

120597119910)]

120578119911119911

=120597119906119911

120597119911+

1

2[(

120597119906119909

120597119911

120597119906119909

120597119911) + (

120597119906119910

120597119911

120597119906119910

120597119911) + (

120597119906119911

120597119911

120597119906119911

120597119911)]

120578119909119910

=1

2(

120597119906119909

120597119910+

120597119906119910

120597119909)

+1

2[(

120597119906119909

120597119909

120597119906119909

120597119910) + (

120597119906119910

120597119909

120597119906119910

120597119910) + (

120597119906119911

120597119909

120597119906119911

120597119910)]

120578119911119909

=1

2(120597119906119911

120597119909+

120597119906119909

120597119911)

+1

2[(

120597119906119909

120597119911

120597119906119909

120597119909) + (

120597119906119910

120597119911

120597119906119910

120597119909) + (

120597119906119911

120597119911

120597119906119911

120597119909)]

120578119910119911

=1

2(

120597119906119910

120597119911+

120597119906119911

120597119910)

+1

2[(

120597119906119909

120597119910

120597119906119909

120597119911) + (

120597119906119910

120597119910

120597119906119910

120597119911) + (

120597119906119911

120597119910

120597119906119911

120597119911)]

(13)

which leads to

119899 =

120578119909119909

120578119910119910

120578119911119911

2 (120578119909119910

+ 120578119910119909

)

2 (120578119909119911

+ 120578119911119909

)

2 (120578119910119911

+ 120578119911119910

)

=

120578119909119909

120578119910119910

120578119911119911

2120578119909119910

2120578119911119909

2120578119910119911

(14)

The Green-Lagrange strain tensor is represented in matrixnotation as

120578 = [119861119879

119871] 119889 +

1

2119889119879[119861NL] 119889 (15)

where 119889 is the nodal displacement [119861119871] is the linear and

[119861NL] is the nonlinear part of the [1198610]matrix [3] For a specific

4 Journal of Applied Mathematics

element [119861119871] and [119861NL] are constant as with the [119861] matrix

in the linear FEMWith the variation of 119889 [9] (15) becomes

120578 = [119861119879

119871] 119889 + 119889

119879[119861NL] 119889 (16)

Gathering the strain components together we can rewrite(15) and (16) as

120578 = ([119861119871] +

1

2119889119879 [119861NL]) 119889 = [119861

0] 119889

120578 = ([119861119871] + 119889

119879 [119861NL]) 119889 = [119861

0] 119889

(17)

The linear part of the [1198610]matrix ([119861

119871]) is the same as the

[119861]matrix in the linear FEM Calculating [1198610] becomesmore

complex with the introduction of the nonlinear terms Afterfinding the nonlinear strains these equations are combinedwith the shape functions to find matrix [119861

0]

120578 = [1198610] 119889 (18)

The most frequently used terms which are the nine dis-placement gradients for calculating the nonlinear strains are120597119906119909120597119909 120597119906

119909120597119910 120597119906

119909120597119911 120597119906

119910120597119909 120597119906

119910120597119910 120597119906

119910120597119909 120597119906

119911120597119909

120597119906119911120597119910 and 120597119906

119911120597119911 Using (8) for displacements we can con-

struct the displacement gradients using the partial derivativesof the shape functions They are represented by

119906119909119909

=1

6119881(12057311199061+ 12057321199062+ 12057331199063+ 12057341199064)

119906119910119909

=1

6119881(1205731V1+ 1205732V2+ 1205733V3+ 1205734V4)

119906119911119909

=1

6119881(12057311199081+ 12057321199082+ 12057331199083+ 12057341199084)

119906119909119910

=1

6119881(12057411199061+ 12057421199062+ 12057431199063+ 12057441199064)

119906119910119910

=1

6119881(1205741V1+ 1205742V2+ 1205743V3+ 1205744V4)

119906119911119910

=1

6119881(12057411199081+ 12057421199082+ 12057431199083+ 12057441199084)

119906119909119911

=1

6119881(12057511199061+ 12057521199062+ 12057531199063+ 12057541199064)

119906119910119911

=1

6119881(1205751V1+ 1205752V2+ 1205753V3+ 1205754V4)

119906119911119911

=1

6119881(12057511199081+ 12057521199082+ 12057531199083+ 12057541199084)

(19)

where 119906119909119909

represents 120597119906119909120597119909

We can evaluate the partial derivatives of the shapefunctions as follows (for the 1st node of [119861NL])

[(120597119906119909

120597119909

120597119906119909

120597119909) + (

120597119906119910

120597119909

120597119906119910

120597119909) + (

120597119906119911

120597119909

120597119906119911

120597119909)]

=1

6119881(1205731(119906119909119909

+ 119906119910119909

+ 119906119911119909

))

[(120597119906119909

120597119910

120597119906119909

120597119910) + (

120597119906119910

120597119910

120597119906119910

120597119910) + (

120597119906119911

120597119910

120597119906119911

120597119910)]

=1

6119881(1205741(119906119909119910

+ 119906119910119910

+ 119906119911119910

))

[(120597119906119909

120597119911

120597119906119909

120597119911) + (

120597119906119910

120597119911

120597119906119910

120597119911) + (

120597119906119911

120597119911

120597119906119911

120597119911)]

=1

6119881(1205751(119906119909119911

+ 119906119910119911

+ 119906119911119911

))

[(120597119906119909

120597119909

120597119906119909

120597119910) + (

120597119906119910

120597119909

120597119906119910

120597119910) + (

120597119906119911

120597119909

120597119906119911

120597119910)]

=1

6119881(1205741(119906119909119909

+ 119906119910119909

+ 119906119911119909

))

+1

6119881(1205731(119906119909119910

+ 119906119910119910

+ 119906119911119910

))

[(120597119906119909

120597119911

120597119906119909

120597119909) + (

120597119906119910

120597119911

120597119906119910

120597119909) + (

120597119906119911

120597119911

120597119906119911

120597119909)]

=1

6119881(1205751(119906119909119909

+ 119906119910119909

+ 119906119911119909

))

+1

6119881(1205731(119906119909119911

+ 119906119910119911

+ 119906119911119911

))

[(120597119906119909

120597119910

120597119906119909

120597119911) + (

120597119906119910

120597119910

120597119906119910

120597119911) + (

120597119906119911

120597119910

120597119906119911

120597119911)]

=1

6119881(1205741(119906119909119911

+ 119906119910119911

+ 119906119911119911

))

+1

6119881(1205751(119906119909119910

+ 119906119910119910

+ 119906119911119910

))

(20)

Using (17) and (20) we obtain [1198610] for the 1st node (21)

Similarly using (17) and (20) we obtain [1198610] for the 1st node

(22)

Journal of Applied Mathematics 5

Consider

[11986101

] =

[[[[[[[[[[[[[[

[

1205731+ 1205731(119906119909119909

) 1205731(119906119910119909

) 1205731(119906119911119909

)

1205741(119906119909119910

) 1205741+ 1205741(119906119910119910

) 1205741(119906119911119910

)

1205751(119906119909119911

) 1205751(119906119910119911

) 1205751+ 1205751(119906119911119911

)

1205741+ 1205741(119906119909119909

) + 1205731(119906119909119910

) 1205741(119906119910119909

) + 1205731+ 1205731(119906119910119910

) 1205741(119906119911119909

) + 1205731(119906119911119910

)

1205751+ 1205751(119906119909119909

) + 1205731(119906119909119911

) 1205751(119906119910119909

) + 1205731(119906119910119911

) 1205751(119906119911119909

) + 1205731+ 1205731(119906119911119911

)

1205741(119906119909119911

) + 1205751(119906119909119910

) 1205741+ 1205741(119906119910119911

) + 1205751(119906119910119910

) 1205741(119906119911119911

) + 1205751+ 1205751(119906119911119910

)

]]]]]]]]]]]]]]

]

1199061

V1

1199081

(21)

[11986101

] =

[[[[[[[[[[[[[[[[[[[[[

[

1205731+

1

21205731(119906119909119909

)1

21205731(119906119910119909

)1

21205731(119906119911119909

)

1

21205741(119906119909119910

) 1205741+

1

21205741(119906119910119910

)1

21205741(119906119911119910

)

1

21205751(119906119909119911

)1

21205751(119906119910119911

) 1205751+

1

21205751(119906119911119911

)

1205741+

1

21205741(119906119909119909

) +1

21205731(119906119909119910

)1

21205741(119906119910119909

) + 1205731+

1

21205731(119906119910119910

)1

21205741(119906119911119909

) +1

21205731(119906119911119910

)

1205751+

1

21205751(119906119909119909

) +1

21205731(119906119909119911

)1

21205751(119906119910119909

) +1

21205731(119906119910119911

)1

21205751(119906119911119909

) + 1205731+

1

21205731(119906119911119911

)

1

21205741(119906119909119911

) +1

21205751(119906119909119910

) 1205741+

1

21205741(119906119910119911

) +1

21205751(119906119910119910

)1

21205741(119906119911119911

) + 1205751+

1

21205751(119906119911119910

)

]]]]]]]]]]]]]]]]]]]]]

]

1199061

V1

1199081

(22)

The FEM is derived from conservation of the potentialenergy which is defined by

120587 = Estrain + 119882 (23)

where Estrain is the strain energy of the linear element and 119882

is the work potential They are given by

Estrain =1

2int

Ω119890

120576119879120590119889119909 119882 = 119891

119890119889119879 (24)

where the engineering strain vector 120576 is

120576 = [119861] 119889 (25)

From (24) the engineering stress vector 120591 is related to thestrain vector by

120591 = [119864] 120578 = [119864] [1198610] 119889 (26)

The secant relations are described by the matrix [119864] Wesubstitute (15) and (26) into (24) obtaining the elementstiffness matrix

[119896 (119906)] = ∭119889119879[1198610]119879

[119864] [1198610] 119889 119889119909 119889119910 119889119911 (27)

We can discard the integrals as we did for the linear FEM[1198610] [119864] and [119861

0] are constant for the four-node tetrahedral

element so (27) is rewritten as

[119896 (119906)] = 119889119879[1198610]119879

[119864] [1198610]119881 (28)

The secant stiffness matrix which is [119896119904(119889)119879] = [119861

0]119879[119864][119861

0]

is nonsymmetric because of the fact that [1198610]119879

= [1198610]

Introducing nodal forces we obtain

119891 =

1198911119909

1198911119910

1198911119911

1198914119909

1198914119910

1198914119911

119889119879 (29)

With the equilibrium equation and cancelling 119889119879 the whole

system for one element reduces to

119896119904(119889)119890119889119890= 119891119890 (30)

By substituting 119889 with 119906 we obtain

119896119904(119906)119890119906119890= 119891119890 (31)

Finally only nonlinear displacement functions remain whichare solved with Newton-Raphson to find the unknowndisplacements 119906 [10]

Element residuals are necessary for the iterative Newton-RaphsonmethodThe element residual is a 12times1 vector for aspecific elementThe residual for a specific element is definedas

119903119890= 119896119904(119906)119890minus 119891119890 (32)

6 Journal of Applied Mathematics

Having determined 119903119890 we can now express (32) in expanded

vector form as

1199031

1199032

1199033

11990312

=

[[[[[[[[[[[

[

119896119904(119906)(11)

+ 119896119904(119906)(12)

+ 119896119904(119906)(13)

+ sdot sdot sdot + 119896119904(119906)(112)

119896119904(119906)(21)

+ 119896119904(119906)(22)

+ 119896119904(119906)(23)

+ sdot sdot sdot + 119896119904(119906)(212)

119896119904(119906)(31)

+ 119896119904(119906)(32)

+ 119896119904(119906)(33)

+ sdot sdot sdot + 119896119904(119906)(312)

119896119904(119906)(121)

+ 119896119904(119906)(122)

+ 119896119904(119906)(123)

+ sdot sdot sdot + 119896119904(119906)(1212)

]]]]]]]]]]]

]

minus

1198911

1198912

1198913

11989112

(33)

The tangent stiffness matrix [119870]119890

119879(1199031015840119890) is also neces-

sary for the iterative Newton-Raphson method The tan-gent stiffness matrix is also 12 times 12 matrix like theelemental stiffness matrix However the tangent stiffnessmatrix depends on residuals unlike the elemental stiffness

matrix Elemental stiffness matrices are used to constructresiduals and the derivatives of the residuals are used toconstruct the elemental tangent stiffness matrices We canexpress the elemental tangent stiffness matrix for a specificelement as

1199031015840119890

= [119870]119890

119879=

[[[[[[[[[[[[[[[[

[

120597

1205971199061

1199031

120597

120597V1

1199031

120597

1205971199081

1199031

sdot sdot sdot120597

1205971199064

1199031

120597

120597V4

1199031

120597

1205971199084

1199031

120597

1205971199061

1199032

120597

120597V1

1199032

120597

1205971199081

1199032

sdot sdot sdot120597

1205971199064

1199032

120597

120597V4

1199032

120597

1205971199084

1199032

120597

1205971199061

1199033

120597

120597V1

1199033

120597

1205971199081

1199033

sdot sdot sdot120597

1205971199064

1199033

120597

120597V4

1199033

120597

1205971199084

1199033

120597

1205971199061

11990312

120597

120597V1

11990312

120597

1205971199081

11990312

sdot sdot sdot120597

1205971199064

11990312

120597

120597V4

11990312

120597

1205971199084

11990312

]]]]]]]]]]]]]]]]

]

(34)

Newton-Raphsonmethod is a fast and popular numericalmethod for solving nonlinear equations [10] as comparedto the other methods such as direct iteration In principlethe method works by applying the following two steps (cfAlgorithm 1) (i) check if the equilibrium is reached withinthe desired accuracy (ii) if not make a suitable adjustment tothe state of the deformation [11] An initial guess for displace-ments is needed to start the iterations The displacements areupdated according to

119909119896+1

= 119909119896minus

119891119909119896

1198911015840

119909119896

(35)

In the proposed nonlinear FEM 119906 is the vector thatkeeps the information of the nodal displacements Instead ofmaking only one assumption we make whole 119906 vector initialguess in order to start the iteration

Consider

1199061= 1199060minus

1199031199060

1199031015840

1199060

(36)

where 119903 is residual of the global stiffnessmatrix [119870] calculatedin (33) and 119903

1015840 is the tangent stiffness matrix calculated in (34)

At every step the vector 119903 and the matrix 1199031015840 are updated

for every element with the new 119906119894values Then 119903 and 119903

1015840 areassembled as we did with for the global stiffnessmatrix119870 andthe global force vector 119865 in linear FEM Boundary conditionsare applied to the global 119903 vector and the global 1199031015840 matrixUsing the global 119903 vector and the global 1199031015840 matrix we have

1199031015840(119906119894) 119901 = minus119903 (119906

119894) 119901 = minus(119903

1015840(119906119894))minus1

119903 (119906119894) (37)

119906119894is updated with the solution of (37) Consider

119906119894+1

= 119906119894+ 119901 (38)

Then we check if the equilibrium is reached within thedesired accuracy defined by 120575 as

1003816100381610038161003816119903 (119906119894)1003816100381610038161003816 le 120575 (39)

After the desired accuracy is reached the unknown nodaldisplacements are found

3 Experimental Results

The proposed nonlinear FEM and Pedersenrsquos nonlinear FEMwere implemented using MATLAB programming language

Journal of Applied Mathematics 7

Make initial guess 119891(119909)

while 1003816100381610038161003816119891 (119909)1003816100381610038161003816 le 120575 do

Compute 119901 = minus119891(119909)

1198911015840(119909)

Update 119909 = 119909 + 119901

Calculate 119891(119909)

end while

Algorithm 1 Newton-Raphson method

1

2 3

4

5

6 7

8

Figure 2 A 10 times 10 times 10m cube mesh with eight nodes and sixtetrahedra is constrained at the blue nodes and pulled downwardsfrom the green nodes

The visualizer was implemented with C++ language andconnected to the solver using the MATLAB engine [12]which allows users to call the MATLAB solver from CC++or Fortran programsThe simulation results interaction withthe 3Dmodel and the 3Dmodels themselves were visualizedusing OpenGL and the nose experiment was visualizedusing 3ds Max [13] To speed up the nonlinear FEM weused MAPLErsquos symbolic solver [14] which is integrated intoMATLAB We conducted all the experiments on a desktopcomputer with a Core i7 3930K processor overclockedat 42 GHz with 32GB of RAM We used linear materialproperties for the models in the experiments We used 1GPafor Youngrsquos modulus (120598) because polypropylene has Youngrsquosmodulus between 15 and 2GPa and polyethylene HDPEhas Youngrsquos modulus 08GPa which shows plastic propertiesand they are close to 1 GPa Because most steels and plasticmaterials undergo plastic deformation near the value of 03we used 025 for Poissonrsquos ratio (]) Our simulation is staticso we used a single load step in all experiments Multiple loadsteps are usedwhen the load forces are time-dependent or thesimulation is dynamic [15]

We conducted four experiments each having differentnumber of elements to observe the speedup for both stiff-ness matrix calculation and for the solution of the systemAs expected the proposed method and Pedersenrsquos methodproduced same amount of nodal displacements in all experi-ments

Table 1 The displacements (in m) at nodes 1 2 3 and 4 using thelinear FEM for the first experimentThe displacements of nodes 5 to8 for all axes are zero

Node Displacement-119909 Displacement-119910 Displacement-1199111 0027234 0011064 minus02899652 0004306 minus0109719 minus04407393 minus0066065 minus0056547 minus03435194 minus0107536 0070143 minus0514524

Table 2 The displacements (in m) at nodes 1 2 3 and 4 using thenonlinear FEM for the first experimentThe displacements of nodes5 to 8 for all axes are zero

Node Displacement-119909 Displacement-119910 Displacement-1199111 0029911 0012665 minus02783652 0008606 minus0103350 minus04155943 minus0058835 minus0051901 minus03241264 minus0098945 0068928 minus0478495

Table 3 The displacements (in m) at green nodes using the linearFEM for the second experiment The displacements at blue nodesare zero

Node Displacement-119909 Displacement-119910 Displacement-1199110 minus3717 4208 minus003941 4738 4208 minus0049472 4737 minus4245 0037773 minus3716 minus4246 00490220 301 minus3547 minus00542921 minus4117 minus3548 minus00614322 minus4117 3581 00434823 301 3581 005155

Table 4 The displacements (in m) at green nodes using thenonlinear FEM for the second experiment The displacements atblue nodes are zero

Node Displacement-119909 Displacement-119910 Displacement-1199110 minus2083 4102 037981 472 2298 035882 2913 minus4501 045863 minus3884 minus2699 0480920 328 minus2561 minus042421 minus2842 minus3931 minus0403222 minus4217 2194 minus0301823 1911 3565 minus03212

The first experiment was conducted for a cube mesh witheight nodes and six tetrahedral elements Figure 2 shows thatthe cube is constrained at the upper four nodes and pulleddownwards with a small amount of force (one unit force foreach of the upper four nodes) This experiment was con-ducted with a small mesh in order to carefully examine thenodal displacements and strains for each element Figure 3shows the initial and final positions of the nodes for the linearand nonlinear FEMs respectively As seen in Figure 3 the

8 Journal of Applied Mathematics

(a) (b)

Figure 3 (a) The initial and final positions of the nodes for the linear FEM (b) The initial and final positions of the nodes for the nonlinearFEM

6

5

4

3

2

1

0

Disp

lace

men

t (m

)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Force (N)

Linear FEMNonlinear FEM

Liner FEM versus nonlinear FEM

Figure 4 Force displacements (in m) at node 4 for the linear andnonlinear FEMs

100

10

1

01

001

0001

Erro

r (

)

1 2 3 4 5

Newton-Raphson convergence graphics8079

2043

118

001

000

Iteration

Figure 5 Newton-Raphson convergence graphics for the nonlinearFEMThe graph is plotted using the logarithmic scale

linear and nonlinear methods produce similar displacementswhen the force magnitude is small Tables 1 and 2 show thedisplacements at force applied nodes (first second thirdand fourth) using the linear and nonlinear FEMs respec-tively Figure 4 shows that displacement increases linearlywith force magnitude However as expected the nonlinear

Table 5 The displacements (in m) at green nodes using the linearFEM for the third experiment The displacements at blue nodes arezero

Node Displacement-119909 Displacement-119910 Displacement-1199115 5004 2305 72416 minus04613 2399 0014449 minus23 2318 627110 minus4602 2375 minus0143741 05991 2345 minus017

Table 6 The displacements (in m) at green nodes using thenonlinear FEM for the third experiment The displacements at bluenodes are zero

Node Displacement-119909 Displacement-119910 Displacement-1199115 6439 6934 50146 minus02123 7996 14589 minus3372 4471 minus478810 05483 7737 00658741 1196 8284 01512

Table 7 The displacements (in m) at green nodes using the linearFEM for the fourth experimentThe displacements at blue nodes arezero

Node Displacement-119909 Displacement-119910 Displacement-119911271 1086 minus05297 1188

Table 8 The displacements (in m) at green nodes using thenonlinear FEM for the fourth experimentThe displacements at bluenodes are zero

Node Displacement-119909 Displacement-119910 Displacement-119911271 06538 minus01851 422

FEM behaves quadratically due to the nonlinear straindefinitions Figure 5 depicts the convergence of the Newton-Raphson method for the nonlinear FEM

The second experiment was conducted on a beam with90 nodes and 216 tetrahedral elements Figures 6(a) and 6(b)show that the beam is constrained at the blue nodes and

Journal of Applied Mathematics 9

(a) (b)

Figure 6The beammesh is constrained at the blue nodes and twisted at the green nodes (a) Front view (b) side view which shows the forcedirections applied on each green node

(a) (b)

(c) (d)

Figure 7 Nonlinear FEM results for the both the proposed and Pedersenrsquos methods ((a) wireframe tetrahedra and nodes (b) nodes only (c)wireframe surface mesh and (d) shaded mesh)

Figure 8 The cross mesh is constrained at the blue nodes andpushed towards the green nodes

twisted at both ends Figure 7 shows the final shape of thebeam mesh for both the proposed and Pedersenrsquos methodsTables 3 and 4 show the displacements at force applied nodes(green nodes) for the second experiment using the linear andnonlinear FEMs respectively

The third experiment was conducted with a cross meshof 159 nodes and 244 tetrahedral elements We aimed toobserve if there is a root jump occurring when solving thesystem for a high amount of force (50N units) and its effect

10 Journal of Applied Mathematics

(a) (b)

Figure 9 Nonlinear FEM results for the both proposed and Pedersenrsquos methods (a) initial and final wireframemeshes are overlaid (b) initialand final shaded meshes are overlaid

(a) (b)

Figure 10 The liver mesh is constrained at the blue nodes and pulled from the green node ((a) initial nodes (b) initial shaded mesh andnodes)

Table 9 Computation times (in seconds) of stiffness matrices(Pedersen Pedersenrsquos nonlinear FEM Proposed proposed nonlin-ear FEM and Speed-up relative performance comparison of thestiffness matrix calculation of the proposed nonlinear FEMmethodwith Pedersenrsquos nonlinear FEMmethod using single thread)

Exp Elements Pedersen Proposed Speed-up1st 6 07322 03308 22134222nd 216 281624 111542 25248253rd 224 301094 122725 24534044th 1580 2398753 967840 2478460

Table 10 Computation times (s) of system solutions (PedersenPedersenrsquos nonlinear FEM Proposed proposed nonlinear FEMand Speed-up relative performance comparison of the proposednonlinear FEM method with Pedersenrsquos nonlinear FEM methodusing single thread)

Exp Elements Pedersen Proposed Speed-up1st 6 30144 24427 12340442nd 216 1925288 1596241 12061393rd 224 5862708 6125911 9570344th 1580 28407558 24010994 1183106

Table 11 Newton-Raphson iteration count to reach desired accu-racy (Pedersen Pedersenrsquos nonlinear FEM Proposed proposednonlinear FEM)

Exp Elements Pedersen Proposed1st 6 5 52nd 216 7 73rd 224 26 324th 1580 8 8

on the computation times for both methods Figure 8 showsthat the cross shape is constrained at the blue nodes andpushed towards the green nodes Figure 9 shows the finalshape of the beammesh for both the proposed and Pedersenrsquosmethods Tables 5 and 6 show the displacements at forceapplied nodes (green nodes) using the linear and nonlinearFEMs respectively

We conducted fourth experiment with a livermesh of 465nodes and 1560 tetrahedral elements Figure 10 shows thatthe mesh is constrained at the blue nodes and pulled fromthe green node (30N units) in the direction of the arrow Weaimed to observe the similar amount of speedup like previousexperiments for a high density mesh Figure 11 shows thefinal shape of the beam mesh for both the proposed andPedersenrsquos methods Tables 7 and 8 show the displacementsat force applied node (node number 271) using the linear andnonlinear FEMs respectively

Computation times of the finite element experiments arerequired to compare how much faster our proposed methodis than Pedersenrsquos When comparing nonlinear FEMs wecalculated the computation times to construct the stiffnessmatrices as well as the computation times of the nonlinearFEM solutions to determine how different calculations affectthem Table 9 depicts the computation times for the stiffnessmatrix calculation and Table 10 depicts the computationtimes for the system solution Table 11 shows the iterationcounts to solve the system using the Newton-Raphson pro-cedure

The speed-up columns of Tables 9 and 10 depict thespeedups of the proposed method compared to Pedersenrsquosmethod for the stiffness matrix calculation and the systemsolution using a single thread respectively The speedup iscalculated as follows

Speedup =Runtime (Pedersenrsquosmethod)

Runtime (Theproposedmethod) (40)

Journal of Applied Mathematics 11

(a) (b)

Figure 11 Nonlinear FEM results for both the proposed and Pedersenrsquos methods (a) left wireframe surface mesh right wireframe surfacemesh with nodes (b) left shaded mesh right shaded mesh with nodes

(a) (b)

(c)

Figure 12 (a) Initial misshapen nose (b) The head mesh is constrained at the blue nodes and pushed upwards at the green nodes (c) Theresult of the nonlinear FEM left wireframe surface mesh with nodes right shaded mesh with texture

Our proposed method outperforms Pedersenrsquos methodOn the average it is 142 faster at computing stiffnessmatrices because Pedersenrsquos method uses more symbolicterms However bothmethods use Newton-Raphson to solvenonlinear equations which takes approximately 90 of thecomputation time Thus the overall speedup decreases to15 on average In experiment 3 because of more iterationsdue to root jumps there was a performance loss againstPedersenrsquos method

We also applied our method for corrective operationon the misshapen nose of a head mesh The head mesh iscomposed of 6709 nodes and 25722 tetrahedral elements (seeFigure 12(a)) all the operations were performed in the nose

area of only 1458 tetrahedral elements Figure 12(b) showsthat the head mesh is constrained at the blue nodes andpushed upwards at the green nodes and Figure 12(c) showsthe result of the nonlinear FEM

4 Conclusions and Future Work

We propose a new stiffness matrix calculation method fornonlinear FEM that is easier to analyze in terms of construct-ing elemental stiffness matrices and is faster than Pedersenrsquosmethod The proposed method is approximately 24 timesfaster on average at computing stiffness matrices and 15faster at computing thewhole system thanPedersenrsquosmethod

12 Journal of Applied Mathematics

Although the proposed nonlinear FEM has significantadvantages over Pedersenrsquos nonlinear FEM there is still roomfor the following development

(1) Heuristics could be applied to avoid root jumps(2) Although we decreased system memory usage by

simplifying the solution process for the nonlinearFEM a significant amount of system memory is stillusedThe solution process could be further optimizedto decrease memory usage

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] K-J Bathe Finite Element Procedures Prentice-Hall Engle-wood Cliffs NJ USA 1996

[2] D L Logan A First Course in the Finite Element MethodCengage Learning 5th edition 2012

[3] P Pedersen ldquoAnalytical stiffness matrices for tetrahedral ele-mentsrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 196 no 1-3 pp 261ndash278 2006

[4] G R Joldes A Wittek and K Miller ldquoReal-time nonlinearfinite element computations on GPU application to neurosur-gical simulationrdquo Computer Methods in Applied Mechanics andEngineering vol 199 no 49ndash52 pp 3305ndash3314 2010

[5] Z A Taylor M Cheng and S Ourselin ldquoHigh-speed nonlinearfinite element analysis for surgical simulation using graphicsprocessing unitsrdquo IEEE Transactions on Medical Imaging vol27 no 5 pp 650ndash663 2008

[6] M Cerrolaza and J C Osorio ldquoRelations among stiffnesscoefficients of hexahedral 8-noded finite elements a simple andefficient way to reduce the integration timerdquo Finite Elements inAnalysis and Design vol 55 pp 1ndash6 2012

[7] J Bonet and R D Wood Nonlinear Continu um Mechanicsfor Finite Element Analysis Cambridge University Press Cam-bridge UK 1st edition 1997

[8] C A Felippa ldquoLecture notes in nonlinear fin ite element meth-odsrdquo Tech Rep CUCSSC-96-16 Department of AerospaceEngineering Sciences and Center for Aerospace StructuresUniversity of Colorado 1996

[9] P Pedersen The Basic Matrix Approach for Three Simple FiniteElements 2008 httpwwwtopoptdtudkfilesPauliBooksBasicMatrixApproachInFEpdf

[10] C T Kelley Iterative Methods for Linear and Nonlinear Equa-tions Frontiers in Applied Mathematics Society for IndustrialMathematics 1st edition 1987

[11] S Krenk Non-linear Modeling and Analysi s of Solids andStructures Cambridge University Press 1st edition 2009

[12] The Mathworks ldquoCalling MATLAB Engine from CC++and Fortran Programsrdquo 2014 httpwwwmathworkscomhelpmatlabmatlab external f38569html

[13] Autodesk ldquo3ds Maxmdash3D Modeling Ani mation and Render-ing Softwarerdquo 2012 httpwwwautodeskcomproductsauto-desk-3ds-max

[14] Maplesoft ldquoMATLAB ConnectivitymdashMaple Featuresrdquo 2012httpwwwmaplesoftcomproductsmaplefeaturesmatlab-connectivityaspx

[15] E Madenci and I Guven The Finite Element Method andApplications in Engineering Using ANSYS Springer BerlinGermany 1st edition 2006

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Differential EquationsInternational Journal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Discrete Dynamics in Nature and Society

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Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

speedup in solving the whole system on average compared toPedersenrsquos method

2 The Nonlinear FEM withGreen-Lagrange Strains

We use tetrahedral elements for modeling meshes in theexperiments Overall there are 12 unknown nodal displace-ments in a tetrahedral element They are given by [2]

119889 = 119906 (119909 119910 119911) =

1199061

V1

1199081

1199064

V4

1199084

(1)

In global coordinates we represent displacements by linearfunction by

119906119890(119909 119910 119911) = 119888

1+ 1198882119909 + 1198883119910 + 1198884119911 (2)

For all 4 vertices (2) is extended as

[[[

[

1 1199091

1199101

1199111

1 1199092

1199102

1199112

1 1199093

1199103

1199113

1 1199094

1199104

1199114

]]]

]

1198881

1198882

1198883

1198884

=

1199061

1199062

1199063

1199064

(3)

Constants 119888119899can be found as

119888119899= Vminus1119906

119899 (4)

where Vminus1 is given by

Vminus1 =1

det (V)[[[

[

1205721

1205722

1205723

1205724

1205731

1205732

1205733

1205734

1205741

1205742

1205743

1205744

1205751

1205752

1205753

1205754

]]]

]

(5)

det(V) is 6119881 where 119881 is the volume of the tetrahedron If wesubstitute (4) into (2) we obtain

119906119890(119909 119910 119911) =

1

6119881119890[1 119909 119910 119911]

[[[

[

1205721

1205722

1205723

1205724

1205731

1205732

1205733

1205734

1205741

1205742

1205743

1205744

1205751

1205752

1205753

1205754

]]]

]

1199061

1199062

1199063

1199064

(6)

120572 120573 120574 120575 and the volume 119881 are calculated by

1205721=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1199092

1199102

1199112

1199093

1199103

1199113

1199094

1199104

1199114

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205722= minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1199091

1199101

1199111

1199093

1199103

1199113

1199094

1199104

1199114

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205723=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1199091

1199101

1199111

1199092

1199102

1199112

1199094

1199104

1199114

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205724= minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1199091

1199101

1199111

1199092

1199102

1199112

1199093

1199103

1199113

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205731= minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1199102

1199112

1 1199103

1199113

1 1199104

1199114

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205732=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1199101

1199111

1 1199103

1199113

1 1199104

1199114

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205733= minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1199101

1199111

1 1199102

1199112

1 1199104

1199114

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205734=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1199101

1199111

1 1199102

1199112

1 1199103

1199113

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205741=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1199092

1199112

1 1199093

1199113

1 1199094

1199114

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205742= minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1199091

1199111

1 1199093

1199113

1 1199094

1199114

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205743=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1199091

1199111

1 1199092

1199112

1 1199094

1199114

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205744= minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1199091

1199111

1 1199092

1199112

1 1199093

1199113

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205751= minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1199092

1199102

1 1199093

1199103

1 1199094

1199104

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205752=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1199091

1199101

1 1199093

1199103

1 1199094

1199104

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205753= minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1199091

1199101

1 1199092

1199102

1 1199094

1199104

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205754=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1199091

1199101

1 1199092

1199102

1 1199093

1199103

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

6119881 =

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 119909119894

119910119894

119911119894

1 119909119895

119910119895

119911119895

1 119909119896

119910119896

119911119896

1 119909119897

119910119897

119911119897

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(7)

Because of the differentials in strain calculation 120572 is not usedin the following stages If we expand (6) we obtain

119906119890(119909 119910 119911)

=1

6119881119890

[[[

[

1205721+ 1205731119909 + 1205741119910 + 1205751119911

1205722+ 1205732119909 + 1205742119910 + 1205752119911

1205723+ 1205733119909 + 1205743119910 + 1205753119911

1205724+ 1205734119909 + 1205744119910 + 1205754119911

]]]

]

times [119906(119909 119910 119911)1

119906(119909 119910 119911)2

119906(119909 119910 119911)3

119906(119909 119910 119911)4]

(8)

For tetrahedral elements to express displacements in simplerform shape functions are introduced (120595

1 1205952 1205953 1205954) They

are given by

1205951=

1

6119881(1205721+ 1205731119909 + 1205741119910 + 1205751119911) 119906(119909 119910 119911)

1

1205952=

1

6119881(1205722+ 1205732119909 + 1205742119910 + 1205752119911) 119906(119909 119910 119911)

2

1205953=

1

6119881(1205723+ 1205733119909 + 1205743119910 + 1205753119911) 119906(119909 119910 119911)

3

1205954=

1

6119881(1205724+ 1205734119909 + 1205744119910 + 1205754119911) 119906(119909 119910 119911)

4

(9)

In our method nonlinear stiffness matrices are derivedusing Green-Lagrange strains (large deformations) which

Journal of Applied Mathematics 3

y uy

x ux

dx

C

C

998400

B

B

998400

A

A

998400 120597uy

120597xdx

dx +120597ux

120597xdx

Figure 1 2D element before and after deformation

themselves are derived directly from infinitesimal strains(small deformations) by adding the nonlinear terms dis-carded in infinitesimal strain theoryThe proposed nonlinearFEM uses the linear FEM framework but it does not requirethe explicit use of weight functions and differential equationsHence numerical integration is not needed for the solution ofthe proposed nonlinear FEM Instead of using weight func-tions and integrals we use displacement gradients and strainsto make the elemental stiffness matrices space-independentin order to discard the integral We extend the linear FEMto the nonlinear FEM by extending the linear strains to theGreen-Lagrange strains

We constructed our linear FEM by extending Loganrsquos 2Dlinear FEM to 3D [2] To understandGreen-Lagrange strainswemust first see how they differ from the infinitesimal strainsused to calculate the global stiffness matrices in a linear FEMFigure 1 shows a 2D element before and after deformationwhere the element edge 119860119861 with initial length 119889119909 becomes11986010158401198611015840 The engineering normal strain is calculated as the

change in the length of the line

120576119909=

100381610038161003816100381610038161198601015840119861101584010038161003816100381610038161003816

minus |119860119861|

|119860119861| (10)

The final length of the elemental edge can be calculated using

100381610038161003816100381610038161198601015840119861101584010038161003816100381610038161003816

2

= (119889119909 +120597119906119909

120597119909119889119909)

2

+ (

120597119906119910

120597119909119889119909)

2

100381610038161003816100381610038161198601015840119861101584010038161003816100381610038161003816

2

= 1198891199092[1 + 2(

120597119906119909

120597119909) + (

120597119906119909

120597119909)

2

+ (

120597119906119910

120597119909)

2

]

(11)

By neglecting the higher-order terms in (11) 2D infinitesimalstrains are defined by

120576119909119909

=120597119906119909

120597119909 120576

119910119910=

120597119906119910

120597119910 120576

119909119910=

1

2(

120597119906119909

120597119910+

120597119906119910

120597119909)

(12)

By the definition the nonlinear FEM differs from thelinear FEM because of the nonlinearity that arises from thehigher-order term neglected in calculation of strains Thestrain vector used in the linear FEM relies on the assumptionthat the displacements at the 119909-axis 119910-axis and 119911-axis arevery small The initial and final positions of a given particleare practically the same thus the higher-order terms areneglected [7] When the displacements are large howeverthis is no longer the case and one must distinguish betweenthe initial and final coordinates of the particles thus thehigher-order terms are added into the strain equations Byadding these high-order terms 3D strains are defined as [8]

120578119909119909

=120597119906119909

120597119909+

1

2[(

120597119906119909

120597119909

120597119906119909

120597119909) + (

120597119906119910

120597119909

120597119906119910

120597119909) + (

120597119906119911

120597119909

120597119906119911

120597119909)]

120578119910119910

=

120597119906119910

120597119910+

1

2[(

120597119906119909

120597119910

120597119906119909

120597119910) + (

120597119906119910

120597119910

120597119906119910

120597119910) + (

120597119906119911

120597119910

120597119906119911

120597119910)]

120578119911119911

=120597119906119911

120597119911+

1

2[(

120597119906119909

120597119911

120597119906119909

120597119911) + (

120597119906119910

120597119911

120597119906119910

120597119911) + (

120597119906119911

120597119911

120597119906119911

120597119911)]

120578119909119910

=1

2(

120597119906119909

120597119910+

120597119906119910

120597119909)

+1

2[(

120597119906119909

120597119909

120597119906119909

120597119910) + (

120597119906119910

120597119909

120597119906119910

120597119910) + (

120597119906119911

120597119909

120597119906119911

120597119910)]

120578119911119909

=1

2(120597119906119911

120597119909+

120597119906119909

120597119911)

+1

2[(

120597119906119909

120597119911

120597119906119909

120597119909) + (

120597119906119910

120597119911

120597119906119910

120597119909) + (

120597119906119911

120597119911

120597119906119911

120597119909)]

120578119910119911

=1

2(

120597119906119910

120597119911+

120597119906119911

120597119910)

+1

2[(

120597119906119909

120597119910

120597119906119909

120597119911) + (

120597119906119910

120597119910

120597119906119910

120597119911) + (

120597119906119911

120597119910

120597119906119911

120597119911)]

(13)

which leads to

119899 =

120578119909119909

120578119910119910

120578119911119911

2 (120578119909119910

+ 120578119910119909

)

2 (120578119909119911

+ 120578119911119909

)

2 (120578119910119911

+ 120578119911119910

)

=

120578119909119909

120578119910119910

120578119911119911

2120578119909119910

2120578119911119909

2120578119910119911

(14)

The Green-Lagrange strain tensor is represented in matrixnotation as

120578 = [119861119879

119871] 119889 +

1

2119889119879[119861NL] 119889 (15)

where 119889 is the nodal displacement [119861119871] is the linear and

[119861NL] is the nonlinear part of the [1198610]matrix [3] For a specific

4 Journal of Applied Mathematics

element [119861119871] and [119861NL] are constant as with the [119861] matrix

in the linear FEMWith the variation of 119889 [9] (15) becomes

120578 = [119861119879

119871] 119889 + 119889

119879[119861NL] 119889 (16)

Gathering the strain components together we can rewrite(15) and (16) as

120578 = ([119861119871] +

1

2119889119879 [119861NL]) 119889 = [119861

0] 119889

120578 = ([119861119871] + 119889

119879 [119861NL]) 119889 = [119861

0] 119889

(17)

The linear part of the [1198610]matrix ([119861

119871]) is the same as the

[119861]matrix in the linear FEM Calculating [1198610] becomesmore

complex with the introduction of the nonlinear terms Afterfinding the nonlinear strains these equations are combinedwith the shape functions to find matrix [119861

0]

120578 = [1198610] 119889 (18)

The most frequently used terms which are the nine dis-placement gradients for calculating the nonlinear strains are120597119906119909120597119909 120597119906

119909120597119910 120597119906

119909120597119911 120597119906

119910120597119909 120597119906

119910120597119910 120597119906

119910120597119909 120597119906

119911120597119909

120597119906119911120597119910 and 120597119906

119911120597119911 Using (8) for displacements we can con-

struct the displacement gradients using the partial derivativesof the shape functions They are represented by

119906119909119909

=1

6119881(12057311199061+ 12057321199062+ 12057331199063+ 12057341199064)

119906119910119909

=1

6119881(1205731V1+ 1205732V2+ 1205733V3+ 1205734V4)

119906119911119909

=1

6119881(12057311199081+ 12057321199082+ 12057331199083+ 12057341199084)

119906119909119910

=1

6119881(12057411199061+ 12057421199062+ 12057431199063+ 12057441199064)

119906119910119910

=1

6119881(1205741V1+ 1205742V2+ 1205743V3+ 1205744V4)

119906119911119910

=1

6119881(12057411199081+ 12057421199082+ 12057431199083+ 12057441199084)

119906119909119911

=1

6119881(12057511199061+ 12057521199062+ 12057531199063+ 12057541199064)

119906119910119911

=1

6119881(1205751V1+ 1205752V2+ 1205753V3+ 1205754V4)

119906119911119911

=1

6119881(12057511199081+ 12057521199082+ 12057531199083+ 12057541199084)

(19)

where 119906119909119909

represents 120597119906119909120597119909

We can evaluate the partial derivatives of the shapefunctions as follows (for the 1st node of [119861NL])

[(120597119906119909

120597119909

120597119906119909

120597119909) + (

120597119906119910

120597119909

120597119906119910

120597119909) + (

120597119906119911

120597119909

120597119906119911

120597119909)]

=1

6119881(1205731(119906119909119909

+ 119906119910119909

+ 119906119911119909

))

[(120597119906119909

120597119910

120597119906119909

120597119910) + (

120597119906119910

120597119910

120597119906119910

120597119910) + (

120597119906119911

120597119910

120597119906119911

120597119910)]

=1

6119881(1205741(119906119909119910

+ 119906119910119910

+ 119906119911119910

))

[(120597119906119909

120597119911

120597119906119909

120597119911) + (

120597119906119910

120597119911

120597119906119910

120597119911) + (

120597119906119911

120597119911

120597119906119911

120597119911)]

=1

6119881(1205751(119906119909119911

+ 119906119910119911

+ 119906119911119911

))

[(120597119906119909

120597119909

120597119906119909

120597119910) + (

120597119906119910

120597119909

120597119906119910

120597119910) + (

120597119906119911

120597119909

120597119906119911

120597119910)]

=1

6119881(1205741(119906119909119909

+ 119906119910119909

+ 119906119911119909

))

+1

6119881(1205731(119906119909119910

+ 119906119910119910

+ 119906119911119910

))

[(120597119906119909

120597119911

120597119906119909

120597119909) + (

120597119906119910

120597119911

120597119906119910

120597119909) + (

120597119906119911

120597119911

120597119906119911

120597119909)]

=1

6119881(1205751(119906119909119909

+ 119906119910119909

+ 119906119911119909

))

+1

6119881(1205731(119906119909119911

+ 119906119910119911

+ 119906119911119911

))

[(120597119906119909

120597119910

120597119906119909

120597119911) + (

120597119906119910

120597119910

120597119906119910

120597119911) + (

120597119906119911

120597119910

120597119906119911

120597119911)]

=1

6119881(1205741(119906119909119911

+ 119906119910119911

+ 119906119911119911

))

+1

6119881(1205751(119906119909119910

+ 119906119910119910

+ 119906119911119910

))

(20)

Using (17) and (20) we obtain [1198610] for the 1st node (21)

Similarly using (17) and (20) we obtain [1198610] for the 1st node

(22)

Journal of Applied Mathematics 5

Consider

[11986101

] =

[[[[[[[[[[[[[[

[

1205731+ 1205731(119906119909119909

) 1205731(119906119910119909

) 1205731(119906119911119909

)

1205741(119906119909119910

) 1205741+ 1205741(119906119910119910

) 1205741(119906119911119910

)

1205751(119906119909119911

) 1205751(119906119910119911

) 1205751+ 1205751(119906119911119911

)

1205741+ 1205741(119906119909119909

) + 1205731(119906119909119910

) 1205741(119906119910119909

) + 1205731+ 1205731(119906119910119910

) 1205741(119906119911119909

) + 1205731(119906119911119910

)

1205751+ 1205751(119906119909119909

) + 1205731(119906119909119911

) 1205751(119906119910119909

) + 1205731(119906119910119911

) 1205751(119906119911119909

) + 1205731+ 1205731(119906119911119911

)

1205741(119906119909119911

) + 1205751(119906119909119910

) 1205741+ 1205741(119906119910119911

) + 1205751(119906119910119910

) 1205741(119906119911119911

) + 1205751+ 1205751(119906119911119910

)

]]]]]]]]]]]]]]

]

1199061

V1

1199081

(21)

[11986101

] =

[[[[[[[[[[[[[[[[[[[[[

[

1205731+

1

21205731(119906119909119909

)1

21205731(119906119910119909

)1

21205731(119906119911119909

)

1

21205741(119906119909119910

) 1205741+

1

21205741(119906119910119910

)1

21205741(119906119911119910

)

1

21205751(119906119909119911

)1

21205751(119906119910119911

) 1205751+

1

21205751(119906119911119911

)

1205741+

1

21205741(119906119909119909

) +1

21205731(119906119909119910

)1

21205741(119906119910119909

) + 1205731+

1

21205731(119906119910119910

)1

21205741(119906119911119909

) +1

21205731(119906119911119910

)

1205751+

1

21205751(119906119909119909

) +1

21205731(119906119909119911

)1

21205751(119906119910119909

) +1

21205731(119906119910119911

)1

21205751(119906119911119909

) + 1205731+

1

21205731(119906119911119911

)

1

21205741(119906119909119911

) +1

21205751(119906119909119910

) 1205741+

1

21205741(119906119910119911

) +1

21205751(119906119910119910

)1

21205741(119906119911119911

) + 1205751+

1

21205751(119906119911119910

)

]]]]]]]]]]]]]]]]]]]]]

]

1199061

V1

1199081

(22)

The FEM is derived from conservation of the potentialenergy which is defined by

120587 = Estrain + 119882 (23)

where Estrain is the strain energy of the linear element and 119882

is the work potential They are given by

Estrain =1

2int

Ω119890

120576119879120590119889119909 119882 = 119891

119890119889119879 (24)

where the engineering strain vector 120576 is

120576 = [119861] 119889 (25)

From (24) the engineering stress vector 120591 is related to thestrain vector by

120591 = [119864] 120578 = [119864] [1198610] 119889 (26)

The secant relations are described by the matrix [119864] Wesubstitute (15) and (26) into (24) obtaining the elementstiffness matrix

[119896 (119906)] = ∭119889119879[1198610]119879

[119864] [1198610] 119889 119889119909 119889119910 119889119911 (27)

We can discard the integrals as we did for the linear FEM[1198610] [119864] and [119861

0] are constant for the four-node tetrahedral

element so (27) is rewritten as

[119896 (119906)] = 119889119879[1198610]119879

[119864] [1198610]119881 (28)

The secant stiffness matrix which is [119896119904(119889)119879] = [119861

0]119879[119864][119861

0]

is nonsymmetric because of the fact that [1198610]119879

= [1198610]

Introducing nodal forces we obtain

119891 =

1198911119909

1198911119910

1198911119911

1198914119909

1198914119910

1198914119911

119889119879 (29)

With the equilibrium equation and cancelling 119889119879 the whole

system for one element reduces to

119896119904(119889)119890119889119890= 119891119890 (30)

By substituting 119889 with 119906 we obtain

119896119904(119906)119890119906119890= 119891119890 (31)

Finally only nonlinear displacement functions remain whichare solved with Newton-Raphson to find the unknowndisplacements 119906 [10]

Element residuals are necessary for the iterative Newton-RaphsonmethodThe element residual is a 12times1 vector for aspecific elementThe residual for a specific element is definedas

119903119890= 119896119904(119906)119890minus 119891119890 (32)

6 Journal of Applied Mathematics

Having determined 119903119890 we can now express (32) in expanded

vector form as

1199031

1199032

1199033

11990312

=

[[[[[[[[[[[

[

119896119904(119906)(11)

+ 119896119904(119906)(12)

+ 119896119904(119906)(13)

+ sdot sdot sdot + 119896119904(119906)(112)

119896119904(119906)(21)

+ 119896119904(119906)(22)

+ 119896119904(119906)(23)

+ sdot sdot sdot + 119896119904(119906)(212)

119896119904(119906)(31)

+ 119896119904(119906)(32)

+ 119896119904(119906)(33)

+ sdot sdot sdot + 119896119904(119906)(312)

119896119904(119906)(121)

+ 119896119904(119906)(122)

+ 119896119904(119906)(123)

+ sdot sdot sdot + 119896119904(119906)(1212)

]]]]]]]]]]]

]

minus

1198911

1198912

1198913

11989112

(33)

The tangent stiffness matrix [119870]119890

119879(1199031015840119890) is also neces-

sary for the iterative Newton-Raphson method The tan-gent stiffness matrix is also 12 times 12 matrix like theelemental stiffness matrix However the tangent stiffnessmatrix depends on residuals unlike the elemental stiffness

matrix Elemental stiffness matrices are used to constructresiduals and the derivatives of the residuals are used toconstruct the elemental tangent stiffness matrices We canexpress the elemental tangent stiffness matrix for a specificelement as

1199031015840119890

= [119870]119890

119879=

[[[[[[[[[[[[[[[[

[

120597

1205971199061

1199031

120597

120597V1

1199031

120597

1205971199081

1199031

sdot sdot sdot120597

1205971199064

1199031

120597

120597V4

1199031

120597

1205971199084

1199031

120597

1205971199061

1199032

120597

120597V1

1199032

120597

1205971199081

1199032

sdot sdot sdot120597

1205971199064

1199032

120597

120597V4

1199032

120597

1205971199084

1199032

120597

1205971199061

1199033

120597

120597V1

1199033

120597

1205971199081

1199033

sdot sdot sdot120597

1205971199064

1199033

120597

120597V4

1199033

120597

1205971199084

1199033

120597

1205971199061

11990312

120597

120597V1

11990312

120597

1205971199081

11990312

sdot sdot sdot120597

1205971199064

11990312

120597

120597V4

11990312

120597

1205971199084

11990312

]]]]]]]]]]]]]]]]

]

(34)

Newton-Raphsonmethod is a fast and popular numericalmethod for solving nonlinear equations [10] as comparedto the other methods such as direct iteration In principlethe method works by applying the following two steps (cfAlgorithm 1) (i) check if the equilibrium is reached withinthe desired accuracy (ii) if not make a suitable adjustment tothe state of the deformation [11] An initial guess for displace-ments is needed to start the iterations The displacements areupdated according to

119909119896+1

= 119909119896minus

119891119909119896

1198911015840

119909119896

(35)

In the proposed nonlinear FEM 119906 is the vector thatkeeps the information of the nodal displacements Instead ofmaking only one assumption we make whole 119906 vector initialguess in order to start the iteration

Consider

1199061= 1199060minus

1199031199060

1199031015840

1199060

(36)

where 119903 is residual of the global stiffnessmatrix [119870] calculatedin (33) and 119903

1015840 is the tangent stiffness matrix calculated in (34)

At every step the vector 119903 and the matrix 1199031015840 are updated

for every element with the new 119906119894values Then 119903 and 119903

1015840 areassembled as we did with for the global stiffnessmatrix119870 andthe global force vector 119865 in linear FEM Boundary conditionsare applied to the global 119903 vector and the global 1199031015840 matrixUsing the global 119903 vector and the global 1199031015840 matrix we have

1199031015840(119906119894) 119901 = minus119903 (119906

119894) 119901 = minus(119903

1015840(119906119894))minus1

119903 (119906119894) (37)

119906119894is updated with the solution of (37) Consider

119906119894+1

= 119906119894+ 119901 (38)

Then we check if the equilibrium is reached within thedesired accuracy defined by 120575 as

1003816100381610038161003816119903 (119906119894)1003816100381610038161003816 le 120575 (39)

After the desired accuracy is reached the unknown nodaldisplacements are found

3 Experimental Results

The proposed nonlinear FEM and Pedersenrsquos nonlinear FEMwere implemented using MATLAB programming language

Journal of Applied Mathematics 7

Make initial guess 119891(119909)

while 1003816100381610038161003816119891 (119909)1003816100381610038161003816 le 120575 do

Compute 119901 = minus119891(119909)

1198911015840(119909)

Update 119909 = 119909 + 119901

Calculate 119891(119909)

end while

Algorithm 1 Newton-Raphson method

1

2 3

4

5

6 7

8

Figure 2 A 10 times 10 times 10m cube mesh with eight nodes and sixtetrahedra is constrained at the blue nodes and pulled downwardsfrom the green nodes

The visualizer was implemented with C++ language andconnected to the solver using the MATLAB engine [12]which allows users to call the MATLAB solver from CC++or Fortran programsThe simulation results interaction withthe 3Dmodel and the 3Dmodels themselves were visualizedusing OpenGL and the nose experiment was visualizedusing 3ds Max [13] To speed up the nonlinear FEM weused MAPLErsquos symbolic solver [14] which is integrated intoMATLAB We conducted all the experiments on a desktopcomputer with a Core i7 3930K processor overclockedat 42 GHz with 32GB of RAM We used linear materialproperties for the models in the experiments We used 1GPafor Youngrsquos modulus (120598) because polypropylene has Youngrsquosmodulus between 15 and 2GPa and polyethylene HDPEhas Youngrsquos modulus 08GPa which shows plastic propertiesand they are close to 1 GPa Because most steels and plasticmaterials undergo plastic deformation near the value of 03we used 025 for Poissonrsquos ratio (]) Our simulation is staticso we used a single load step in all experiments Multiple loadsteps are usedwhen the load forces are time-dependent or thesimulation is dynamic [15]

We conducted four experiments each having differentnumber of elements to observe the speedup for both stiff-ness matrix calculation and for the solution of the systemAs expected the proposed method and Pedersenrsquos methodproduced same amount of nodal displacements in all experi-ments

Table 1 The displacements (in m) at nodes 1 2 3 and 4 using thelinear FEM for the first experimentThe displacements of nodes 5 to8 for all axes are zero

Node Displacement-119909 Displacement-119910 Displacement-1199111 0027234 0011064 minus02899652 0004306 minus0109719 minus04407393 minus0066065 minus0056547 minus03435194 minus0107536 0070143 minus0514524

Table 2 The displacements (in m) at nodes 1 2 3 and 4 using thenonlinear FEM for the first experimentThe displacements of nodes5 to 8 for all axes are zero

Node Displacement-119909 Displacement-119910 Displacement-1199111 0029911 0012665 minus02783652 0008606 minus0103350 minus04155943 minus0058835 minus0051901 minus03241264 minus0098945 0068928 minus0478495

Table 3 The displacements (in m) at green nodes using the linearFEM for the second experiment The displacements at blue nodesare zero

Node Displacement-119909 Displacement-119910 Displacement-1199110 minus3717 4208 minus003941 4738 4208 minus0049472 4737 minus4245 0037773 minus3716 minus4246 00490220 301 minus3547 minus00542921 minus4117 minus3548 minus00614322 minus4117 3581 00434823 301 3581 005155

Table 4 The displacements (in m) at green nodes using thenonlinear FEM for the second experiment The displacements atblue nodes are zero

Node Displacement-119909 Displacement-119910 Displacement-1199110 minus2083 4102 037981 472 2298 035882 2913 minus4501 045863 minus3884 minus2699 0480920 328 minus2561 minus042421 minus2842 minus3931 minus0403222 minus4217 2194 minus0301823 1911 3565 minus03212

The first experiment was conducted for a cube mesh witheight nodes and six tetrahedral elements Figure 2 shows thatthe cube is constrained at the upper four nodes and pulleddownwards with a small amount of force (one unit force foreach of the upper four nodes) This experiment was con-ducted with a small mesh in order to carefully examine thenodal displacements and strains for each element Figure 3shows the initial and final positions of the nodes for the linearand nonlinear FEMs respectively As seen in Figure 3 the

8 Journal of Applied Mathematics

(a) (b)

Figure 3 (a) The initial and final positions of the nodes for the linear FEM (b) The initial and final positions of the nodes for the nonlinearFEM

6

5

4

3

2

1

0

Disp

lace

men

t (m

)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Force (N)

Linear FEMNonlinear FEM

Liner FEM versus nonlinear FEM

Figure 4 Force displacements (in m) at node 4 for the linear andnonlinear FEMs

100

10

1

01

001

0001

Erro

r (

)

1 2 3 4 5

Newton-Raphson convergence graphics8079

2043

118

001

000

Iteration

Figure 5 Newton-Raphson convergence graphics for the nonlinearFEMThe graph is plotted using the logarithmic scale

linear and nonlinear methods produce similar displacementswhen the force magnitude is small Tables 1 and 2 show thedisplacements at force applied nodes (first second thirdand fourth) using the linear and nonlinear FEMs respec-tively Figure 4 shows that displacement increases linearlywith force magnitude However as expected the nonlinear

Table 5 The displacements (in m) at green nodes using the linearFEM for the third experiment The displacements at blue nodes arezero

Node Displacement-119909 Displacement-119910 Displacement-1199115 5004 2305 72416 minus04613 2399 0014449 minus23 2318 627110 minus4602 2375 minus0143741 05991 2345 minus017

Table 6 The displacements (in m) at green nodes using thenonlinear FEM for the third experiment The displacements at bluenodes are zero

Node Displacement-119909 Displacement-119910 Displacement-1199115 6439 6934 50146 minus02123 7996 14589 minus3372 4471 minus478810 05483 7737 00658741 1196 8284 01512

Table 7 The displacements (in m) at green nodes using the linearFEM for the fourth experimentThe displacements at blue nodes arezero

Node Displacement-119909 Displacement-119910 Displacement-119911271 1086 minus05297 1188

Table 8 The displacements (in m) at green nodes using thenonlinear FEM for the fourth experimentThe displacements at bluenodes are zero

Node Displacement-119909 Displacement-119910 Displacement-119911271 06538 minus01851 422

FEM behaves quadratically due to the nonlinear straindefinitions Figure 5 depicts the convergence of the Newton-Raphson method for the nonlinear FEM

The second experiment was conducted on a beam with90 nodes and 216 tetrahedral elements Figures 6(a) and 6(b)show that the beam is constrained at the blue nodes and

Journal of Applied Mathematics 9

(a) (b)

Figure 6The beammesh is constrained at the blue nodes and twisted at the green nodes (a) Front view (b) side view which shows the forcedirections applied on each green node

(a) (b)

(c) (d)

Figure 7 Nonlinear FEM results for the both the proposed and Pedersenrsquos methods ((a) wireframe tetrahedra and nodes (b) nodes only (c)wireframe surface mesh and (d) shaded mesh)

Figure 8 The cross mesh is constrained at the blue nodes andpushed towards the green nodes

twisted at both ends Figure 7 shows the final shape of thebeam mesh for both the proposed and Pedersenrsquos methodsTables 3 and 4 show the displacements at force applied nodes(green nodes) for the second experiment using the linear andnonlinear FEMs respectively

The third experiment was conducted with a cross meshof 159 nodes and 244 tetrahedral elements We aimed toobserve if there is a root jump occurring when solving thesystem for a high amount of force (50N units) and its effect

10 Journal of Applied Mathematics

(a) (b)

Figure 9 Nonlinear FEM results for the both proposed and Pedersenrsquos methods (a) initial and final wireframemeshes are overlaid (b) initialand final shaded meshes are overlaid

(a) (b)

Figure 10 The liver mesh is constrained at the blue nodes and pulled from the green node ((a) initial nodes (b) initial shaded mesh andnodes)

Table 9 Computation times (in seconds) of stiffness matrices(Pedersen Pedersenrsquos nonlinear FEM Proposed proposed nonlin-ear FEM and Speed-up relative performance comparison of thestiffness matrix calculation of the proposed nonlinear FEMmethodwith Pedersenrsquos nonlinear FEMmethod using single thread)

Exp Elements Pedersen Proposed Speed-up1st 6 07322 03308 22134222nd 216 281624 111542 25248253rd 224 301094 122725 24534044th 1580 2398753 967840 2478460

Table 10 Computation times (s) of system solutions (PedersenPedersenrsquos nonlinear FEM Proposed proposed nonlinear FEMand Speed-up relative performance comparison of the proposednonlinear FEM method with Pedersenrsquos nonlinear FEM methodusing single thread)

Exp Elements Pedersen Proposed Speed-up1st 6 30144 24427 12340442nd 216 1925288 1596241 12061393rd 224 5862708 6125911 9570344th 1580 28407558 24010994 1183106

Table 11 Newton-Raphson iteration count to reach desired accu-racy (Pedersen Pedersenrsquos nonlinear FEM Proposed proposednonlinear FEM)

Exp Elements Pedersen Proposed1st 6 5 52nd 216 7 73rd 224 26 324th 1580 8 8

on the computation times for both methods Figure 8 showsthat the cross shape is constrained at the blue nodes andpushed towards the green nodes Figure 9 shows the finalshape of the beammesh for both the proposed and Pedersenrsquosmethods Tables 5 and 6 show the displacements at forceapplied nodes (green nodes) using the linear and nonlinearFEMs respectively

We conducted fourth experiment with a livermesh of 465nodes and 1560 tetrahedral elements Figure 10 shows thatthe mesh is constrained at the blue nodes and pulled fromthe green node (30N units) in the direction of the arrow Weaimed to observe the similar amount of speedup like previousexperiments for a high density mesh Figure 11 shows thefinal shape of the beam mesh for both the proposed andPedersenrsquos methods Tables 7 and 8 show the displacementsat force applied node (node number 271) using the linear andnonlinear FEMs respectively

Computation times of the finite element experiments arerequired to compare how much faster our proposed methodis than Pedersenrsquos When comparing nonlinear FEMs wecalculated the computation times to construct the stiffnessmatrices as well as the computation times of the nonlinearFEM solutions to determine how different calculations affectthem Table 9 depicts the computation times for the stiffnessmatrix calculation and Table 10 depicts the computationtimes for the system solution Table 11 shows the iterationcounts to solve the system using the Newton-Raphson pro-cedure

The speed-up columns of Tables 9 and 10 depict thespeedups of the proposed method compared to Pedersenrsquosmethod for the stiffness matrix calculation and the systemsolution using a single thread respectively The speedup iscalculated as follows

Speedup =Runtime (Pedersenrsquosmethod)

Runtime (Theproposedmethod) (40)

Journal of Applied Mathematics 11

(a) (b)

Figure 11 Nonlinear FEM results for both the proposed and Pedersenrsquos methods (a) left wireframe surface mesh right wireframe surfacemesh with nodes (b) left shaded mesh right shaded mesh with nodes

(a) (b)

(c)

Figure 12 (a) Initial misshapen nose (b) The head mesh is constrained at the blue nodes and pushed upwards at the green nodes (c) Theresult of the nonlinear FEM left wireframe surface mesh with nodes right shaded mesh with texture

Our proposed method outperforms Pedersenrsquos methodOn the average it is 142 faster at computing stiffnessmatrices because Pedersenrsquos method uses more symbolicterms However bothmethods use Newton-Raphson to solvenonlinear equations which takes approximately 90 of thecomputation time Thus the overall speedup decreases to15 on average In experiment 3 because of more iterationsdue to root jumps there was a performance loss againstPedersenrsquos method

We also applied our method for corrective operationon the misshapen nose of a head mesh The head mesh iscomposed of 6709 nodes and 25722 tetrahedral elements (seeFigure 12(a)) all the operations were performed in the nose

area of only 1458 tetrahedral elements Figure 12(b) showsthat the head mesh is constrained at the blue nodes andpushed upwards at the green nodes and Figure 12(c) showsthe result of the nonlinear FEM

4 Conclusions and Future Work

We propose a new stiffness matrix calculation method fornonlinear FEM that is easier to analyze in terms of construct-ing elemental stiffness matrices and is faster than Pedersenrsquosmethod The proposed method is approximately 24 timesfaster on average at computing stiffness matrices and 15faster at computing thewhole system thanPedersenrsquosmethod

12 Journal of Applied Mathematics

Although the proposed nonlinear FEM has significantadvantages over Pedersenrsquos nonlinear FEM there is still roomfor the following development

(1) Heuristics could be applied to avoid root jumps(2) Although we decreased system memory usage by

simplifying the solution process for the nonlinearFEM a significant amount of system memory is stillusedThe solution process could be further optimizedto decrease memory usage

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] K-J Bathe Finite Element Procedures Prentice-Hall Engle-wood Cliffs NJ USA 1996

[2] D L Logan A First Course in the Finite Element MethodCengage Learning 5th edition 2012

[3] P Pedersen ldquoAnalytical stiffness matrices for tetrahedral ele-mentsrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 196 no 1-3 pp 261ndash278 2006

[4] G R Joldes A Wittek and K Miller ldquoReal-time nonlinearfinite element computations on GPU application to neurosur-gical simulationrdquo Computer Methods in Applied Mechanics andEngineering vol 199 no 49ndash52 pp 3305ndash3314 2010

[5] Z A Taylor M Cheng and S Ourselin ldquoHigh-speed nonlinearfinite element analysis for surgical simulation using graphicsprocessing unitsrdquo IEEE Transactions on Medical Imaging vol27 no 5 pp 650ndash663 2008

[6] M Cerrolaza and J C Osorio ldquoRelations among stiffnesscoefficients of hexahedral 8-noded finite elements a simple andefficient way to reduce the integration timerdquo Finite Elements inAnalysis and Design vol 55 pp 1ndash6 2012

[7] J Bonet and R D Wood Nonlinear Continu um Mechanicsfor Finite Element Analysis Cambridge University Press Cam-bridge UK 1st edition 1997

[8] C A Felippa ldquoLecture notes in nonlinear fin ite element meth-odsrdquo Tech Rep CUCSSC-96-16 Department of AerospaceEngineering Sciences and Center for Aerospace StructuresUniversity of Colorado 1996

[9] P Pedersen The Basic Matrix Approach for Three Simple FiniteElements 2008 httpwwwtopoptdtudkfilesPauliBooksBasicMatrixApproachInFEpdf

[10] C T Kelley Iterative Methods for Linear and Nonlinear Equa-tions Frontiers in Applied Mathematics Society for IndustrialMathematics 1st edition 1987

[11] S Krenk Non-linear Modeling and Analysi s of Solids andStructures Cambridge University Press 1st edition 2009

[12] The Mathworks ldquoCalling MATLAB Engine from CC++and Fortran Programsrdquo 2014 httpwwwmathworkscomhelpmatlabmatlab external f38569html

[13] Autodesk ldquo3ds Maxmdash3D Modeling Ani mation and Render-ing Softwarerdquo 2012 httpwwwautodeskcomproductsauto-desk-3ds-max

[14] Maplesoft ldquoMATLAB ConnectivitymdashMaple Featuresrdquo 2012httpwwwmaplesoftcomproductsmaplefeaturesmatlab-connectivityaspx

[15] E Madenci and I Guven The Finite Element Method andApplications in Engineering Using ANSYS Springer BerlinGermany 1st edition 2006

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Differential EquationsInternational Journal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 3

y uy

x ux

dx

C

C

998400

B

B

998400

A

A

998400 120597uy

120597xdx

dx +120597ux

120597xdx

Figure 1 2D element before and after deformation

themselves are derived directly from infinitesimal strains(small deformations) by adding the nonlinear terms dis-carded in infinitesimal strain theoryThe proposed nonlinearFEM uses the linear FEM framework but it does not requirethe explicit use of weight functions and differential equationsHence numerical integration is not needed for the solution ofthe proposed nonlinear FEM Instead of using weight func-tions and integrals we use displacement gradients and strainsto make the elemental stiffness matrices space-independentin order to discard the integral We extend the linear FEMto the nonlinear FEM by extending the linear strains to theGreen-Lagrange strains

We constructed our linear FEM by extending Loganrsquos 2Dlinear FEM to 3D [2] To understandGreen-Lagrange strainswemust first see how they differ from the infinitesimal strainsused to calculate the global stiffness matrices in a linear FEMFigure 1 shows a 2D element before and after deformationwhere the element edge 119860119861 with initial length 119889119909 becomes11986010158401198611015840 The engineering normal strain is calculated as the

change in the length of the line

120576119909=

100381610038161003816100381610038161198601015840119861101584010038161003816100381610038161003816

minus |119860119861|

|119860119861| (10)

The final length of the elemental edge can be calculated using

100381610038161003816100381610038161198601015840119861101584010038161003816100381610038161003816

2

= (119889119909 +120597119906119909

120597119909119889119909)

2

+ (

120597119906119910

120597119909119889119909)

2

100381610038161003816100381610038161198601015840119861101584010038161003816100381610038161003816

2

= 1198891199092[1 + 2(

120597119906119909

120597119909) + (

120597119906119909

120597119909)

2

+ (

120597119906119910

120597119909)

2

]

(11)

By neglecting the higher-order terms in (11) 2D infinitesimalstrains are defined by

120576119909119909

=120597119906119909

120597119909 120576

119910119910=

120597119906119910

120597119910 120576

119909119910=

1

2(

120597119906119909

120597119910+

120597119906119910

120597119909)

(12)

By the definition the nonlinear FEM differs from thelinear FEM because of the nonlinearity that arises from thehigher-order term neglected in calculation of strains Thestrain vector used in the linear FEM relies on the assumptionthat the displacements at the 119909-axis 119910-axis and 119911-axis arevery small The initial and final positions of a given particleare practically the same thus the higher-order terms areneglected [7] When the displacements are large howeverthis is no longer the case and one must distinguish betweenthe initial and final coordinates of the particles thus thehigher-order terms are added into the strain equations Byadding these high-order terms 3D strains are defined as [8]

120578119909119909

=120597119906119909

120597119909+

1

2[(

120597119906119909

120597119909

120597119906119909

120597119909) + (

120597119906119910

120597119909

120597119906119910

120597119909) + (

120597119906119911

120597119909

120597119906119911

120597119909)]

120578119910119910

=

120597119906119910

120597119910+

1

2[(

120597119906119909

120597119910

120597119906119909

120597119910) + (

120597119906119910

120597119910

120597119906119910

120597119910) + (

120597119906119911

120597119910

120597119906119911

120597119910)]

120578119911119911

=120597119906119911

120597119911+

1

2[(

120597119906119909

120597119911

120597119906119909

120597119911) + (

120597119906119910

120597119911

120597119906119910

120597119911) + (

120597119906119911

120597119911

120597119906119911

120597119911)]

120578119909119910

=1

2(

120597119906119909

120597119910+

120597119906119910

120597119909)

+1

2[(

120597119906119909

120597119909

120597119906119909

120597119910) + (

120597119906119910

120597119909

120597119906119910

120597119910) + (

120597119906119911

120597119909

120597119906119911

120597119910)]

120578119911119909

=1

2(120597119906119911

120597119909+

120597119906119909

120597119911)

+1

2[(

120597119906119909

120597119911

120597119906119909

120597119909) + (

120597119906119910

120597119911

120597119906119910

120597119909) + (

120597119906119911

120597119911

120597119906119911

120597119909)]

120578119910119911

=1

2(

120597119906119910

120597119911+

120597119906119911

120597119910)

+1

2[(

120597119906119909

120597119910

120597119906119909

120597119911) + (

120597119906119910

120597119910

120597119906119910

120597119911) + (

120597119906119911

120597119910

120597119906119911

120597119911)]

(13)

which leads to

119899 =

120578119909119909

120578119910119910

120578119911119911

2 (120578119909119910

+ 120578119910119909

)

2 (120578119909119911

+ 120578119911119909

)

2 (120578119910119911

+ 120578119911119910

)

=

120578119909119909

120578119910119910

120578119911119911

2120578119909119910

2120578119911119909

2120578119910119911

(14)

The Green-Lagrange strain tensor is represented in matrixnotation as

120578 = [119861119879

119871] 119889 +

1

2119889119879[119861NL] 119889 (15)

where 119889 is the nodal displacement [119861119871] is the linear and

[119861NL] is the nonlinear part of the [1198610]matrix [3] For a specific

4 Journal of Applied Mathematics

element [119861119871] and [119861NL] are constant as with the [119861] matrix

in the linear FEMWith the variation of 119889 [9] (15) becomes

120578 = [119861119879

119871] 119889 + 119889

119879[119861NL] 119889 (16)

Gathering the strain components together we can rewrite(15) and (16) as

120578 = ([119861119871] +

1

2119889119879 [119861NL]) 119889 = [119861

0] 119889

120578 = ([119861119871] + 119889

119879 [119861NL]) 119889 = [119861

0] 119889

(17)

The linear part of the [1198610]matrix ([119861

119871]) is the same as the

[119861]matrix in the linear FEM Calculating [1198610] becomesmore

complex with the introduction of the nonlinear terms Afterfinding the nonlinear strains these equations are combinedwith the shape functions to find matrix [119861

0]

120578 = [1198610] 119889 (18)

The most frequently used terms which are the nine dis-placement gradients for calculating the nonlinear strains are120597119906119909120597119909 120597119906

119909120597119910 120597119906

119909120597119911 120597119906

119910120597119909 120597119906

119910120597119910 120597119906

119910120597119909 120597119906

119911120597119909

120597119906119911120597119910 and 120597119906

119911120597119911 Using (8) for displacements we can con-

struct the displacement gradients using the partial derivativesof the shape functions They are represented by

119906119909119909

=1

6119881(12057311199061+ 12057321199062+ 12057331199063+ 12057341199064)

119906119910119909

=1

6119881(1205731V1+ 1205732V2+ 1205733V3+ 1205734V4)

119906119911119909

=1

6119881(12057311199081+ 12057321199082+ 12057331199083+ 12057341199084)

119906119909119910

=1

6119881(12057411199061+ 12057421199062+ 12057431199063+ 12057441199064)

119906119910119910

=1

6119881(1205741V1+ 1205742V2+ 1205743V3+ 1205744V4)

119906119911119910

=1

6119881(12057411199081+ 12057421199082+ 12057431199083+ 12057441199084)

119906119909119911

=1

6119881(12057511199061+ 12057521199062+ 12057531199063+ 12057541199064)

119906119910119911

=1

6119881(1205751V1+ 1205752V2+ 1205753V3+ 1205754V4)

119906119911119911

=1

6119881(12057511199081+ 12057521199082+ 12057531199083+ 12057541199084)

(19)

where 119906119909119909

represents 120597119906119909120597119909

We can evaluate the partial derivatives of the shapefunctions as follows (for the 1st node of [119861NL])

[(120597119906119909

120597119909

120597119906119909

120597119909) + (

120597119906119910

120597119909

120597119906119910

120597119909) + (

120597119906119911

120597119909

120597119906119911

120597119909)]

=1

6119881(1205731(119906119909119909

+ 119906119910119909

+ 119906119911119909

))

[(120597119906119909

120597119910

120597119906119909

120597119910) + (

120597119906119910

120597119910

120597119906119910

120597119910) + (

120597119906119911

120597119910

120597119906119911

120597119910)]

=1

6119881(1205741(119906119909119910

+ 119906119910119910

+ 119906119911119910

))

[(120597119906119909

120597119911

120597119906119909

120597119911) + (

120597119906119910

120597119911

120597119906119910

120597119911) + (

120597119906119911

120597119911

120597119906119911

120597119911)]

=1

6119881(1205751(119906119909119911

+ 119906119910119911

+ 119906119911119911

))

[(120597119906119909

120597119909

120597119906119909

120597119910) + (

120597119906119910

120597119909

120597119906119910

120597119910) + (

120597119906119911

120597119909

120597119906119911

120597119910)]

=1

6119881(1205741(119906119909119909

+ 119906119910119909

+ 119906119911119909

))

+1

6119881(1205731(119906119909119910

+ 119906119910119910

+ 119906119911119910

))

[(120597119906119909

120597119911

120597119906119909

120597119909) + (

120597119906119910

120597119911

120597119906119910

120597119909) + (

120597119906119911

120597119911

120597119906119911

120597119909)]

=1

6119881(1205751(119906119909119909

+ 119906119910119909

+ 119906119911119909

))

+1

6119881(1205731(119906119909119911

+ 119906119910119911

+ 119906119911119911

))

[(120597119906119909

120597119910

120597119906119909

120597119911) + (

120597119906119910

120597119910

120597119906119910

120597119911) + (

120597119906119911

120597119910

120597119906119911

120597119911)]

=1

6119881(1205741(119906119909119911

+ 119906119910119911

+ 119906119911119911

))

+1

6119881(1205751(119906119909119910

+ 119906119910119910

+ 119906119911119910

))

(20)

Using (17) and (20) we obtain [1198610] for the 1st node (21)

Similarly using (17) and (20) we obtain [1198610] for the 1st node

(22)

Journal of Applied Mathematics 5

Consider

[11986101

] =

[[[[[[[[[[[[[[

[

1205731+ 1205731(119906119909119909

) 1205731(119906119910119909

) 1205731(119906119911119909

)

1205741(119906119909119910

) 1205741+ 1205741(119906119910119910

) 1205741(119906119911119910

)

1205751(119906119909119911

) 1205751(119906119910119911

) 1205751+ 1205751(119906119911119911

)

1205741+ 1205741(119906119909119909

) + 1205731(119906119909119910

) 1205741(119906119910119909

) + 1205731+ 1205731(119906119910119910

) 1205741(119906119911119909

) + 1205731(119906119911119910

)

1205751+ 1205751(119906119909119909

) + 1205731(119906119909119911

) 1205751(119906119910119909

) + 1205731(119906119910119911

) 1205751(119906119911119909

) + 1205731+ 1205731(119906119911119911

)

1205741(119906119909119911

) + 1205751(119906119909119910

) 1205741+ 1205741(119906119910119911

) + 1205751(119906119910119910

) 1205741(119906119911119911

) + 1205751+ 1205751(119906119911119910

)

]]]]]]]]]]]]]]

]

1199061

V1

1199081

(21)

[11986101

] =

[[[[[[[[[[[[[[[[[[[[[

[

1205731+

1

21205731(119906119909119909

)1

21205731(119906119910119909

)1

21205731(119906119911119909

)

1

21205741(119906119909119910

) 1205741+

1

21205741(119906119910119910

)1

21205741(119906119911119910

)

1

21205751(119906119909119911

)1

21205751(119906119910119911

) 1205751+

1

21205751(119906119911119911

)

1205741+

1

21205741(119906119909119909

) +1

21205731(119906119909119910

)1

21205741(119906119910119909

) + 1205731+

1

21205731(119906119910119910

)1

21205741(119906119911119909

) +1

21205731(119906119911119910

)

1205751+

1

21205751(119906119909119909

) +1

21205731(119906119909119911

)1

21205751(119906119910119909

) +1

21205731(119906119910119911

)1

21205751(119906119911119909

) + 1205731+

1

21205731(119906119911119911

)

1

21205741(119906119909119911

) +1

21205751(119906119909119910

) 1205741+

1

21205741(119906119910119911

) +1

21205751(119906119910119910

)1

21205741(119906119911119911

) + 1205751+

1

21205751(119906119911119910

)

]]]]]]]]]]]]]]]]]]]]]

]

1199061

V1

1199081

(22)

The FEM is derived from conservation of the potentialenergy which is defined by

120587 = Estrain + 119882 (23)

where Estrain is the strain energy of the linear element and 119882

is the work potential They are given by

Estrain =1

2int

Ω119890

120576119879120590119889119909 119882 = 119891

119890119889119879 (24)

where the engineering strain vector 120576 is

120576 = [119861] 119889 (25)

From (24) the engineering stress vector 120591 is related to thestrain vector by

120591 = [119864] 120578 = [119864] [1198610] 119889 (26)

The secant relations are described by the matrix [119864] Wesubstitute (15) and (26) into (24) obtaining the elementstiffness matrix

[119896 (119906)] = ∭119889119879[1198610]119879

[119864] [1198610] 119889 119889119909 119889119910 119889119911 (27)

We can discard the integrals as we did for the linear FEM[1198610] [119864] and [119861

0] are constant for the four-node tetrahedral

element so (27) is rewritten as

[119896 (119906)] = 119889119879[1198610]119879

[119864] [1198610]119881 (28)

The secant stiffness matrix which is [119896119904(119889)119879] = [119861

0]119879[119864][119861

0]

is nonsymmetric because of the fact that [1198610]119879

= [1198610]

Introducing nodal forces we obtain

119891 =

1198911119909

1198911119910

1198911119911

1198914119909

1198914119910

1198914119911

119889119879 (29)

With the equilibrium equation and cancelling 119889119879 the whole

system for one element reduces to

119896119904(119889)119890119889119890= 119891119890 (30)

By substituting 119889 with 119906 we obtain

119896119904(119906)119890119906119890= 119891119890 (31)

Finally only nonlinear displacement functions remain whichare solved with Newton-Raphson to find the unknowndisplacements 119906 [10]

Element residuals are necessary for the iterative Newton-RaphsonmethodThe element residual is a 12times1 vector for aspecific elementThe residual for a specific element is definedas

119903119890= 119896119904(119906)119890minus 119891119890 (32)

6 Journal of Applied Mathematics

Having determined 119903119890 we can now express (32) in expanded

vector form as

1199031

1199032

1199033

11990312

=

[[[[[[[[[[[

[

119896119904(119906)(11)

+ 119896119904(119906)(12)

+ 119896119904(119906)(13)

+ sdot sdot sdot + 119896119904(119906)(112)

119896119904(119906)(21)

+ 119896119904(119906)(22)

+ 119896119904(119906)(23)

+ sdot sdot sdot + 119896119904(119906)(212)

119896119904(119906)(31)

+ 119896119904(119906)(32)

+ 119896119904(119906)(33)

+ sdot sdot sdot + 119896119904(119906)(312)

119896119904(119906)(121)

+ 119896119904(119906)(122)

+ 119896119904(119906)(123)

+ sdot sdot sdot + 119896119904(119906)(1212)

]]]]]]]]]]]

]

minus

1198911

1198912

1198913

11989112

(33)

The tangent stiffness matrix [119870]119890

119879(1199031015840119890) is also neces-

sary for the iterative Newton-Raphson method The tan-gent stiffness matrix is also 12 times 12 matrix like theelemental stiffness matrix However the tangent stiffnessmatrix depends on residuals unlike the elemental stiffness

matrix Elemental stiffness matrices are used to constructresiduals and the derivatives of the residuals are used toconstruct the elemental tangent stiffness matrices We canexpress the elemental tangent stiffness matrix for a specificelement as

1199031015840119890

= [119870]119890

119879=

[[[[[[[[[[[[[[[[

[

120597

1205971199061

1199031

120597

120597V1

1199031

120597

1205971199081

1199031

sdot sdot sdot120597

1205971199064

1199031

120597

120597V4

1199031

120597

1205971199084

1199031

120597

1205971199061

1199032

120597

120597V1

1199032

120597

1205971199081

1199032

sdot sdot sdot120597

1205971199064

1199032

120597

120597V4

1199032

120597

1205971199084

1199032

120597

1205971199061

1199033

120597

120597V1

1199033

120597

1205971199081

1199033

sdot sdot sdot120597

1205971199064

1199033

120597

120597V4

1199033

120597

1205971199084

1199033

120597

1205971199061

11990312

120597

120597V1

11990312

120597

1205971199081

11990312

sdot sdot sdot120597

1205971199064

11990312

120597

120597V4

11990312

120597

1205971199084

11990312

]]]]]]]]]]]]]]]]

]

(34)

Newton-Raphsonmethod is a fast and popular numericalmethod for solving nonlinear equations [10] as comparedto the other methods such as direct iteration In principlethe method works by applying the following two steps (cfAlgorithm 1) (i) check if the equilibrium is reached withinthe desired accuracy (ii) if not make a suitable adjustment tothe state of the deformation [11] An initial guess for displace-ments is needed to start the iterations The displacements areupdated according to

119909119896+1

= 119909119896minus

119891119909119896

1198911015840

119909119896

(35)

In the proposed nonlinear FEM 119906 is the vector thatkeeps the information of the nodal displacements Instead ofmaking only one assumption we make whole 119906 vector initialguess in order to start the iteration

Consider

1199061= 1199060minus

1199031199060

1199031015840

1199060

(36)

where 119903 is residual of the global stiffnessmatrix [119870] calculatedin (33) and 119903

1015840 is the tangent stiffness matrix calculated in (34)

At every step the vector 119903 and the matrix 1199031015840 are updated

for every element with the new 119906119894values Then 119903 and 119903

1015840 areassembled as we did with for the global stiffnessmatrix119870 andthe global force vector 119865 in linear FEM Boundary conditionsare applied to the global 119903 vector and the global 1199031015840 matrixUsing the global 119903 vector and the global 1199031015840 matrix we have

1199031015840(119906119894) 119901 = minus119903 (119906

119894) 119901 = minus(119903

1015840(119906119894))minus1

119903 (119906119894) (37)

119906119894is updated with the solution of (37) Consider

119906119894+1

= 119906119894+ 119901 (38)

Then we check if the equilibrium is reached within thedesired accuracy defined by 120575 as

1003816100381610038161003816119903 (119906119894)1003816100381610038161003816 le 120575 (39)

After the desired accuracy is reached the unknown nodaldisplacements are found

3 Experimental Results

The proposed nonlinear FEM and Pedersenrsquos nonlinear FEMwere implemented using MATLAB programming language

Journal of Applied Mathematics 7

Make initial guess 119891(119909)

while 1003816100381610038161003816119891 (119909)1003816100381610038161003816 le 120575 do

Compute 119901 = minus119891(119909)

1198911015840(119909)

Update 119909 = 119909 + 119901

Calculate 119891(119909)

end while

Algorithm 1 Newton-Raphson method

1

2 3

4

5

6 7

8

Figure 2 A 10 times 10 times 10m cube mesh with eight nodes and sixtetrahedra is constrained at the blue nodes and pulled downwardsfrom the green nodes

The visualizer was implemented with C++ language andconnected to the solver using the MATLAB engine [12]which allows users to call the MATLAB solver from CC++or Fortran programsThe simulation results interaction withthe 3Dmodel and the 3Dmodels themselves were visualizedusing OpenGL and the nose experiment was visualizedusing 3ds Max [13] To speed up the nonlinear FEM weused MAPLErsquos symbolic solver [14] which is integrated intoMATLAB We conducted all the experiments on a desktopcomputer with a Core i7 3930K processor overclockedat 42 GHz with 32GB of RAM We used linear materialproperties for the models in the experiments We used 1GPafor Youngrsquos modulus (120598) because polypropylene has Youngrsquosmodulus between 15 and 2GPa and polyethylene HDPEhas Youngrsquos modulus 08GPa which shows plastic propertiesand they are close to 1 GPa Because most steels and plasticmaterials undergo plastic deformation near the value of 03we used 025 for Poissonrsquos ratio (]) Our simulation is staticso we used a single load step in all experiments Multiple loadsteps are usedwhen the load forces are time-dependent or thesimulation is dynamic [15]

We conducted four experiments each having differentnumber of elements to observe the speedup for both stiff-ness matrix calculation and for the solution of the systemAs expected the proposed method and Pedersenrsquos methodproduced same amount of nodal displacements in all experi-ments

Table 1 The displacements (in m) at nodes 1 2 3 and 4 using thelinear FEM for the first experimentThe displacements of nodes 5 to8 for all axes are zero

Node Displacement-119909 Displacement-119910 Displacement-1199111 0027234 0011064 minus02899652 0004306 minus0109719 minus04407393 minus0066065 minus0056547 minus03435194 minus0107536 0070143 minus0514524

Table 2 The displacements (in m) at nodes 1 2 3 and 4 using thenonlinear FEM for the first experimentThe displacements of nodes5 to 8 for all axes are zero

Node Displacement-119909 Displacement-119910 Displacement-1199111 0029911 0012665 minus02783652 0008606 minus0103350 minus04155943 minus0058835 minus0051901 minus03241264 minus0098945 0068928 minus0478495

Table 3 The displacements (in m) at green nodes using the linearFEM for the second experiment The displacements at blue nodesare zero

Node Displacement-119909 Displacement-119910 Displacement-1199110 minus3717 4208 minus003941 4738 4208 minus0049472 4737 minus4245 0037773 minus3716 minus4246 00490220 301 minus3547 minus00542921 minus4117 minus3548 minus00614322 minus4117 3581 00434823 301 3581 005155

Table 4 The displacements (in m) at green nodes using thenonlinear FEM for the second experiment The displacements atblue nodes are zero

Node Displacement-119909 Displacement-119910 Displacement-1199110 minus2083 4102 037981 472 2298 035882 2913 minus4501 045863 minus3884 minus2699 0480920 328 minus2561 minus042421 minus2842 minus3931 minus0403222 minus4217 2194 minus0301823 1911 3565 minus03212

The first experiment was conducted for a cube mesh witheight nodes and six tetrahedral elements Figure 2 shows thatthe cube is constrained at the upper four nodes and pulleddownwards with a small amount of force (one unit force foreach of the upper four nodes) This experiment was con-ducted with a small mesh in order to carefully examine thenodal displacements and strains for each element Figure 3shows the initial and final positions of the nodes for the linearand nonlinear FEMs respectively As seen in Figure 3 the

8 Journal of Applied Mathematics

(a) (b)

Figure 3 (a) The initial and final positions of the nodes for the linear FEM (b) The initial and final positions of the nodes for the nonlinearFEM

6

5

4

3

2

1

0

Disp

lace

men

t (m

)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Force (N)

Linear FEMNonlinear FEM

Liner FEM versus nonlinear FEM

Figure 4 Force displacements (in m) at node 4 for the linear andnonlinear FEMs

100

10

1

01

001

0001

Erro

r (

)

1 2 3 4 5

Newton-Raphson convergence graphics8079

2043

118

001

000

Iteration

Figure 5 Newton-Raphson convergence graphics for the nonlinearFEMThe graph is plotted using the logarithmic scale

linear and nonlinear methods produce similar displacementswhen the force magnitude is small Tables 1 and 2 show thedisplacements at force applied nodes (first second thirdand fourth) using the linear and nonlinear FEMs respec-tively Figure 4 shows that displacement increases linearlywith force magnitude However as expected the nonlinear

Table 5 The displacements (in m) at green nodes using the linearFEM for the third experiment The displacements at blue nodes arezero

Node Displacement-119909 Displacement-119910 Displacement-1199115 5004 2305 72416 minus04613 2399 0014449 minus23 2318 627110 minus4602 2375 minus0143741 05991 2345 minus017

Table 6 The displacements (in m) at green nodes using thenonlinear FEM for the third experiment The displacements at bluenodes are zero

Node Displacement-119909 Displacement-119910 Displacement-1199115 6439 6934 50146 minus02123 7996 14589 minus3372 4471 minus478810 05483 7737 00658741 1196 8284 01512

Table 7 The displacements (in m) at green nodes using the linearFEM for the fourth experimentThe displacements at blue nodes arezero

Node Displacement-119909 Displacement-119910 Displacement-119911271 1086 minus05297 1188

Table 8 The displacements (in m) at green nodes using thenonlinear FEM for the fourth experimentThe displacements at bluenodes are zero

Node Displacement-119909 Displacement-119910 Displacement-119911271 06538 minus01851 422

FEM behaves quadratically due to the nonlinear straindefinitions Figure 5 depicts the convergence of the Newton-Raphson method for the nonlinear FEM

The second experiment was conducted on a beam with90 nodes and 216 tetrahedral elements Figures 6(a) and 6(b)show that the beam is constrained at the blue nodes and

Journal of Applied Mathematics 9

(a) (b)

Figure 6The beammesh is constrained at the blue nodes and twisted at the green nodes (a) Front view (b) side view which shows the forcedirections applied on each green node

(a) (b)

(c) (d)

Figure 7 Nonlinear FEM results for the both the proposed and Pedersenrsquos methods ((a) wireframe tetrahedra and nodes (b) nodes only (c)wireframe surface mesh and (d) shaded mesh)

Figure 8 The cross mesh is constrained at the blue nodes andpushed towards the green nodes

twisted at both ends Figure 7 shows the final shape of thebeam mesh for both the proposed and Pedersenrsquos methodsTables 3 and 4 show the displacements at force applied nodes(green nodes) for the second experiment using the linear andnonlinear FEMs respectively

The third experiment was conducted with a cross meshof 159 nodes and 244 tetrahedral elements We aimed toobserve if there is a root jump occurring when solving thesystem for a high amount of force (50N units) and its effect

10 Journal of Applied Mathematics

(a) (b)

Figure 9 Nonlinear FEM results for the both proposed and Pedersenrsquos methods (a) initial and final wireframemeshes are overlaid (b) initialand final shaded meshes are overlaid

(a) (b)

Figure 10 The liver mesh is constrained at the blue nodes and pulled from the green node ((a) initial nodes (b) initial shaded mesh andnodes)

Table 9 Computation times (in seconds) of stiffness matrices(Pedersen Pedersenrsquos nonlinear FEM Proposed proposed nonlin-ear FEM and Speed-up relative performance comparison of thestiffness matrix calculation of the proposed nonlinear FEMmethodwith Pedersenrsquos nonlinear FEMmethod using single thread)

Exp Elements Pedersen Proposed Speed-up1st 6 07322 03308 22134222nd 216 281624 111542 25248253rd 224 301094 122725 24534044th 1580 2398753 967840 2478460

Table 10 Computation times (s) of system solutions (PedersenPedersenrsquos nonlinear FEM Proposed proposed nonlinear FEMand Speed-up relative performance comparison of the proposednonlinear FEM method with Pedersenrsquos nonlinear FEM methodusing single thread)

Exp Elements Pedersen Proposed Speed-up1st 6 30144 24427 12340442nd 216 1925288 1596241 12061393rd 224 5862708 6125911 9570344th 1580 28407558 24010994 1183106

Table 11 Newton-Raphson iteration count to reach desired accu-racy (Pedersen Pedersenrsquos nonlinear FEM Proposed proposednonlinear FEM)

Exp Elements Pedersen Proposed1st 6 5 52nd 216 7 73rd 224 26 324th 1580 8 8

on the computation times for both methods Figure 8 showsthat the cross shape is constrained at the blue nodes andpushed towards the green nodes Figure 9 shows the finalshape of the beammesh for both the proposed and Pedersenrsquosmethods Tables 5 and 6 show the displacements at forceapplied nodes (green nodes) using the linear and nonlinearFEMs respectively

We conducted fourth experiment with a livermesh of 465nodes and 1560 tetrahedral elements Figure 10 shows thatthe mesh is constrained at the blue nodes and pulled fromthe green node (30N units) in the direction of the arrow Weaimed to observe the similar amount of speedup like previousexperiments for a high density mesh Figure 11 shows thefinal shape of the beam mesh for both the proposed andPedersenrsquos methods Tables 7 and 8 show the displacementsat force applied node (node number 271) using the linear andnonlinear FEMs respectively

Computation times of the finite element experiments arerequired to compare how much faster our proposed methodis than Pedersenrsquos When comparing nonlinear FEMs wecalculated the computation times to construct the stiffnessmatrices as well as the computation times of the nonlinearFEM solutions to determine how different calculations affectthem Table 9 depicts the computation times for the stiffnessmatrix calculation and Table 10 depicts the computationtimes for the system solution Table 11 shows the iterationcounts to solve the system using the Newton-Raphson pro-cedure

The speed-up columns of Tables 9 and 10 depict thespeedups of the proposed method compared to Pedersenrsquosmethod for the stiffness matrix calculation and the systemsolution using a single thread respectively The speedup iscalculated as follows

Speedup =Runtime (Pedersenrsquosmethod)

Runtime (Theproposedmethod) (40)

Journal of Applied Mathematics 11

(a) (b)

Figure 11 Nonlinear FEM results for both the proposed and Pedersenrsquos methods (a) left wireframe surface mesh right wireframe surfacemesh with nodes (b) left shaded mesh right shaded mesh with nodes

(a) (b)

(c)

Figure 12 (a) Initial misshapen nose (b) The head mesh is constrained at the blue nodes and pushed upwards at the green nodes (c) Theresult of the nonlinear FEM left wireframe surface mesh with nodes right shaded mesh with texture

Our proposed method outperforms Pedersenrsquos methodOn the average it is 142 faster at computing stiffnessmatrices because Pedersenrsquos method uses more symbolicterms However bothmethods use Newton-Raphson to solvenonlinear equations which takes approximately 90 of thecomputation time Thus the overall speedup decreases to15 on average In experiment 3 because of more iterationsdue to root jumps there was a performance loss againstPedersenrsquos method

We also applied our method for corrective operationon the misshapen nose of a head mesh The head mesh iscomposed of 6709 nodes and 25722 tetrahedral elements (seeFigure 12(a)) all the operations were performed in the nose

area of only 1458 tetrahedral elements Figure 12(b) showsthat the head mesh is constrained at the blue nodes andpushed upwards at the green nodes and Figure 12(c) showsthe result of the nonlinear FEM

4 Conclusions and Future Work

We propose a new stiffness matrix calculation method fornonlinear FEM that is easier to analyze in terms of construct-ing elemental stiffness matrices and is faster than Pedersenrsquosmethod The proposed method is approximately 24 timesfaster on average at computing stiffness matrices and 15faster at computing thewhole system thanPedersenrsquosmethod

12 Journal of Applied Mathematics

Although the proposed nonlinear FEM has significantadvantages over Pedersenrsquos nonlinear FEM there is still roomfor the following development

(1) Heuristics could be applied to avoid root jumps(2) Although we decreased system memory usage by

simplifying the solution process for the nonlinearFEM a significant amount of system memory is stillusedThe solution process could be further optimizedto decrease memory usage

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] K-J Bathe Finite Element Procedures Prentice-Hall Engle-wood Cliffs NJ USA 1996

[2] D L Logan A First Course in the Finite Element MethodCengage Learning 5th edition 2012

[3] P Pedersen ldquoAnalytical stiffness matrices for tetrahedral ele-mentsrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 196 no 1-3 pp 261ndash278 2006

[4] G R Joldes A Wittek and K Miller ldquoReal-time nonlinearfinite element computations on GPU application to neurosur-gical simulationrdquo Computer Methods in Applied Mechanics andEngineering vol 199 no 49ndash52 pp 3305ndash3314 2010

[5] Z A Taylor M Cheng and S Ourselin ldquoHigh-speed nonlinearfinite element analysis for surgical simulation using graphicsprocessing unitsrdquo IEEE Transactions on Medical Imaging vol27 no 5 pp 650ndash663 2008

[6] M Cerrolaza and J C Osorio ldquoRelations among stiffnesscoefficients of hexahedral 8-noded finite elements a simple andefficient way to reduce the integration timerdquo Finite Elements inAnalysis and Design vol 55 pp 1ndash6 2012

[7] J Bonet and R D Wood Nonlinear Continu um Mechanicsfor Finite Element Analysis Cambridge University Press Cam-bridge UK 1st edition 1997

[8] C A Felippa ldquoLecture notes in nonlinear fin ite element meth-odsrdquo Tech Rep CUCSSC-96-16 Department of AerospaceEngineering Sciences and Center for Aerospace StructuresUniversity of Colorado 1996

[9] P Pedersen The Basic Matrix Approach for Three Simple FiniteElements 2008 httpwwwtopoptdtudkfilesPauliBooksBasicMatrixApproachInFEpdf

[10] C T Kelley Iterative Methods for Linear and Nonlinear Equa-tions Frontiers in Applied Mathematics Society for IndustrialMathematics 1st edition 1987

[11] S Krenk Non-linear Modeling and Analysi s of Solids andStructures Cambridge University Press 1st edition 2009

[12] The Mathworks ldquoCalling MATLAB Engine from CC++and Fortran Programsrdquo 2014 httpwwwmathworkscomhelpmatlabmatlab external f38569html

[13] Autodesk ldquo3ds Maxmdash3D Modeling Ani mation and Render-ing Softwarerdquo 2012 httpwwwautodeskcomproductsauto-desk-3ds-max

[14] Maplesoft ldquoMATLAB ConnectivitymdashMaple Featuresrdquo 2012httpwwwmaplesoftcomproductsmaplefeaturesmatlab-connectivityaspx

[15] E Madenci and I Guven The Finite Element Method andApplications in Engineering Using ANSYS Springer BerlinGermany 1st edition 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Applied Mathematics

element [119861119871] and [119861NL] are constant as with the [119861] matrix

in the linear FEMWith the variation of 119889 [9] (15) becomes

120578 = [119861119879

119871] 119889 + 119889

119879[119861NL] 119889 (16)

Gathering the strain components together we can rewrite(15) and (16) as

120578 = ([119861119871] +

1

2119889119879 [119861NL]) 119889 = [119861

0] 119889

120578 = ([119861119871] + 119889

119879 [119861NL]) 119889 = [119861

0] 119889

(17)

The linear part of the [1198610]matrix ([119861

119871]) is the same as the

[119861]matrix in the linear FEM Calculating [1198610] becomesmore

complex with the introduction of the nonlinear terms Afterfinding the nonlinear strains these equations are combinedwith the shape functions to find matrix [119861

0]

120578 = [1198610] 119889 (18)

The most frequently used terms which are the nine dis-placement gradients for calculating the nonlinear strains are120597119906119909120597119909 120597119906

119909120597119910 120597119906

119909120597119911 120597119906

119910120597119909 120597119906

119910120597119910 120597119906

119910120597119909 120597119906

119911120597119909

120597119906119911120597119910 and 120597119906

119911120597119911 Using (8) for displacements we can con-

struct the displacement gradients using the partial derivativesof the shape functions They are represented by

119906119909119909

=1

6119881(12057311199061+ 12057321199062+ 12057331199063+ 12057341199064)

119906119910119909

=1

6119881(1205731V1+ 1205732V2+ 1205733V3+ 1205734V4)

119906119911119909

=1

6119881(12057311199081+ 12057321199082+ 12057331199083+ 12057341199084)

119906119909119910

=1

6119881(12057411199061+ 12057421199062+ 12057431199063+ 12057441199064)

119906119910119910

=1

6119881(1205741V1+ 1205742V2+ 1205743V3+ 1205744V4)

119906119911119910

=1

6119881(12057411199081+ 12057421199082+ 12057431199083+ 12057441199084)

119906119909119911

=1

6119881(12057511199061+ 12057521199062+ 12057531199063+ 12057541199064)

119906119910119911

=1

6119881(1205751V1+ 1205752V2+ 1205753V3+ 1205754V4)

119906119911119911

=1

6119881(12057511199081+ 12057521199082+ 12057531199083+ 12057541199084)

(19)

where 119906119909119909

represents 120597119906119909120597119909

We can evaluate the partial derivatives of the shapefunctions as follows (for the 1st node of [119861NL])

[(120597119906119909

120597119909

120597119906119909

120597119909) + (

120597119906119910

120597119909

120597119906119910

120597119909) + (

120597119906119911

120597119909

120597119906119911

120597119909)]

=1

6119881(1205731(119906119909119909

+ 119906119910119909

+ 119906119911119909

))

[(120597119906119909

120597119910

120597119906119909

120597119910) + (

120597119906119910

120597119910

120597119906119910

120597119910) + (

120597119906119911

120597119910

120597119906119911

120597119910)]

=1

6119881(1205741(119906119909119910

+ 119906119910119910

+ 119906119911119910

))

[(120597119906119909

120597119911

120597119906119909

120597119911) + (

120597119906119910

120597119911

120597119906119910

120597119911) + (

120597119906119911

120597119911

120597119906119911

120597119911)]

=1

6119881(1205751(119906119909119911

+ 119906119910119911

+ 119906119911119911

))

[(120597119906119909

120597119909

120597119906119909

120597119910) + (

120597119906119910

120597119909

120597119906119910

120597119910) + (

120597119906119911

120597119909

120597119906119911

120597119910)]

=1

6119881(1205741(119906119909119909

+ 119906119910119909

+ 119906119911119909

))

+1

6119881(1205731(119906119909119910

+ 119906119910119910

+ 119906119911119910

))

[(120597119906119909

120597119911

120597119906119909

120597119909) + (

120597119906119910

120597119911

120597119906119910

120597119909) + (

120597119906119911

120597119911

120597119906119911

120597119909)]

=1

6119881(1205751(119906119909119909

+ 119906119910119909

+ 119906119911119909

))

+1

6119881(1205731(119906119909119911

+ 119906119910119911

+ 119906119911119911

))

[(120597119906119909

120597119910

120597119906119909

120597119911) + (

120597119906119910

120597119910

120597119906119910

120597119911) + (

120597119906119911

120597119910

120597119906119911

120597119911)]

=1

6119881(1205741(119906119909119911

+ 119906119910119911

+ 119906119911119911

))

+1

6119881(1205751(119906119909119910

+ 119906119910119910

+ 119906119911119910

))

(20)

Using (17) and (20) we obtain [1198610] for the 1st node (21)

Similarly using (17) and (20) we obtain [1198610] for the 1st node

(22)

Journal of Applied Mathematics 5

Consider

[11986101

] =

[[[[[[[[[[[[[[

[

1205731+ 1205731(119906119909119909

) 1205731(119906119910119909

) 1205731(119906119911119909

)

1205741(119906119909119910

) 1205741+ 1205741(119906119910119910

) 1205741(119906119911119910

)

1205751(119906119909119911

) 1205751(119906119910119911

) 1205751+ 1205751(119906119911119911

)

1205741+ 1205741(119906119909119909

) + 1205731(119906119909119910

) 1205741(119906119910119909

) + 1205731+ 1205731(119906119910119910

) 1205741(119906119911119909

) + 1205731(119906119911119910

)

1205751+ 1205751(119906119909119909

) + 1205731(119906119909119911

) 1205751(119906119910119909

) + 1205731(119906119910119911

) 1205751(119906119911119909

) + 1205731+ 1205731(119906119911119911

)

1205741(119906119909119911

) + 1205751(119906119909119910

) 1205741+ 1205741(119906119910119911

) + 1205751(119906119910119910

) 1205741(119906119911119911

) + 1205751+ 1205751(119906119911119910

)

]]]]]]]]]]]]]]

]

1199061

V1

1199081

(21)

[11986101

] =

[[[[[[[[[[[[[[[[[[[[[

[

1205731+

1

21205731(119906119909119909

)1

21205731(119906119910119909

)1

21205731(119906119911119909

)

1

21205741(119906119909119910

) 1205741+

1

21205741(119906119910119910

)1

21205741(119906119911119910

)

1

21205751(119906119909119911

)1

21205751(119906119910119911

) 1205751+

1

21205751(119906119911119911

)

1205741+

1

21205741(119906119909119909

) +1

21205731(119906119909119910

)1

21205741(119906119910119909

) + 1205731+

1

21205731(119906119910119910

)1

21205741(119906119911119909

) +1

21205731(119906119911119910

)

1205751+

1

21205751(119906119909119909

) +1

21205731(119906119909119911

)1

21205751(119906119910119909

) +1

21205731(119906119910119911

)1

21205751(119906119911119909

) + 1205731+

1

21205731(119906119911119911

)

1

21205741(119906119909119911

) +1

21205751(119906119909119910

) 1205741+

1

21205741(119906119910119911

) +1

21205751(119906119910119910

)1

21205741(119906119911119911

) + 1205751+

1

21205751(119906119911119910

)

]]]]]]]]]]]]]]]]]]]]]

]

1199061

V1

1199081

(22)

The FEM is derived from conservation of the potentialenergy which is defined by

120587 = Estrain + 119882 (23)

where Estrain is the strain energy of the linear element and 119882

is the work potential They are given by

Estrain =1

2int

Ω119890

120576119879120590119889119909 119882 = 119891

119890119889119879 (24)

where the engineering strain vector 120576 is

120576 = [119861] 119889 (25)

From (24) the engineering stress vector 120591 is related to thestrain vector by

120591 = [119864] 120578 = [119864] [1198610] 119889 (26)

The secant relations are described by the matrix [119864] Wesubstitute (15) and (26) into (24) obtaining the elementstiffness matrix

[119896 (119906)] = ∭119889119879[1198610]119879

[119864] [1198610] 119889 119889119909 119889119910 119889119911 (27)

We can discard the integrals as we did for the linear FEM[1198610] [119864] and [119861

0] are constant for the four-node tetrahedral

element so (27) is rewritten as

[119896 (119906)] = 119889119879[1198610]119879

[119864] [1198610]119881 (28)

The secant stiffness matrix which is [119896119904(119889)119879] = [119861

0]119879[119864][119861

0]

is nonsymmetric because of the fact that [1198610]119879

= [1198610]

Introducing nodal forces we obtain

119891 =

1198911119909

1198911119910

1198911119911

1198914119909

1198914119910

1198914119911

119889119879 (29)

With the equilibrium equation and cancelling 119889119879 the whole

system for one element reduces to

119896119904(119889)119890119889119890= 119891119890 (30)

By substituting 119889 with 119906 we obtain

119896119904(119906)119890119906119890= 119891119890 (31)

Finally only nonlinear displacement functions remain whichare solved with Newton-Raphson to find the unknowndisplacements 119906 [10]

Element residuals are necessary for the iterative Newton-RaphsonmethodThe element residual is a 12times1 vector for aspecific elementThe residual for a specific element is definedas

119903119890= 119896119904(119906)119890minus 119891119890 (32)

6 Journal of Applied Mathematics

Having determined 119903119890 we can now express (32) in expanded

vector form as

1199031

1199032

1199033

11990312

=

[[[[[[[[[[[

[

119896119904(119906)(11)

+ 119896119904(119906)(12)

+ 119896119904(119906)(13)

+ sdot sdot sdot + 119896119904(119906)(112)

119896119904(119906)(21)

+ 119896119904(119906)(22)

+ 119896119904(119906)(23)

+ sdot sdot sdot + 119896119904(119906)(212)

119896119904(119906)(31)

+ 119896119904(119906)(32)

+ 119896119904(119906)(33)

+ sdot sdot sdot + 119896119904(119906)(312)

119896119904(119906)(121)

+ 119896119904(119906)(122)

+ 119896119904(119906)(123)

+ sdot sdot sdot + 119896119904(119906)(1212)

]]]]]]]]]]]

]

minus

1198911

1198912

1198913

11989112

(33)

The tangent stiffness matrix [119870]119890

119879(1199031015840119890) is also neces-

sary for the iterative Newton-Raphson method The tan-gent stiffness matrix is also 12 times 12 matrix like theelemental stiffness matrix However the tangent stiffnessmatrix depends on residuals unlike the elemental stiffness

matrix Elemental stiffness matrices are used to constructresiduals and the derivatives of the residuals are used toconstruct the elemental tangent stiffness matrices We canexpress the elemental tangent stiffness matrix for a specificelement as

1199031015840119890

= [119870]119890

119879=

[[[[[[[[[[[[[[[[

[

120597

1205971199061

1199031

120597

120597V1

1199031

120597

1205971199081

1199031

sdot sdot sdot120597

1205971199064

1199031

120597

120597V4

1199031

120597

1205971199084

1199031

120597

1205971199061

1199032

120597

120597V1

1199032

120597

1205971199081

1199032

sdot sdot sdot120597

1205971199064

1199032

120597

120597V4

1199032

120597

1205971199084

1199032

120597

1205971199061

1199033

120597

120597V1

1199033

120597

1205971199081

1199033

sdot sdot sdot120597

1205971199064

1199033

120597

120597V4

1199033

120597

1205971199084

1199033

120597

1205971199061

11990312

120597

120597V1

11990312

120597

1205971199081

11990312

sdot sdot sdot120597

1205971199064

11990312

120597

120597V4

11990312

120597

1205971199084

11990312

]]]]]]]]]]]]]]]]

]

(34)

Newton-Raphsonmethod is a fast and popular numericalmethod for solving nonlinear equations [10] as comparedto the other methods such as direct iteration In principlethe method works by applying the following two steps (cfAlgorithm 1) (i) check if the equilibrium is reached withinthe desired accuracy (ii) if not make a suitable adjustment tothe state of the deformation [11] An initial guess for displace-ments is needed to start the iterations The displacements areupdated according to

119909119896+1

= 119909119896minus

119891119909119896

1198911015840

119909119896

(35)

In the proposed nonlinear FEM 119906 is the vector thatkeeps the information of the nodal displacements Instead ofmaking only one assumption we make whole 119906 vector initialguess in order to start the iteration

Consider

1199061= 1199060minus

1199031199060

1199031015840

1199060

(36)

where 119903 is residual of the global stiffnessmatrix [119870] calculatedin (33) and 119903

1015840 is the tangent stiffness matrix calculated in (34)

At every step the vector 119903 and the matrix 1199031015840 are updated

for every element with the new 119906119894values Then 119903 and 119903

1015840 areassembled as we did with for the global stiffnessmatrix119870 andthe global force vector 119865 in linear FEM Boundary conditionsare applied to the global 119903 vector and the global 1199031015840 matrixUsing the global 119903 vector and the global 1199031015840 matrix we have

1199031015840(119906119894) 119901 = minus119903 (119906

119894) 119901 = minus(119903

1015840(119906119894))minus1

119903 (119906119894) (37)

119906119894is updated with the solution of (37) Consider

119906119894+1

= 119906119894+ 119901 (38)

Then we check if the equilibrium is reached within thedesired accuracy defined by 120575 as

1003816100381610038161003816119903 (119906119894)1003816100381610038161003816 le 120575 (39)

After the desired accuracy is reached the unknown nodaldisplacements are found

3 Experimental Results

The proposed nonlinear FEM and Pedersenrsquos nonlinear FEMwere implemented using MATLAB programming language

Journal of Applied Mathematics 7

Make initial guess 119891(119909)

while 1003816100381610038161003816119891 (119909)1003816100381610038161003816 le 120575 do

Compute 119901 = minus119891(119909)

1198911015840(119909)

Update 119909 = 119909 + 119901

Calculate 119891(119909)

end while

Algorithm 1 Newton-Raphson method

1

2 3

4

5

6 7

8

Figure 2 A 10 times 10 times 10m cube mesh with eight nodes and sixtetrahedra is constrained at the blue nodes and pulled downwardsfrom the green nodes

The visualizer was implemented with C++ language andconnected to the solver using the MATLAB engine [12]which allows users to call the MATLAB solver from CC++or Fortran programsThe simulation results interaction withthe 3Dmodel and the 3Dmodels themselves were visualizedusing OpenGL and the nose experiment was visualizedusing 3ds Max [13] To speed up the nonlinear FEM weused MAPLErsquos symbolic solver [14] which is integrated intoMATLAB We conducted all the experiments on a desktopcomputer with a Core i7 3930K processor overclockedat 42 GHz with 32GB of RAM We used linear materialproperties for the models in the experiments We used 1GPafor Youngrsquos modulus (120598) because polypropylene has Youngrsquosmodulus between 15 and 2GPa and polyethylene HDPEhas Youngrsquos modulus 08GPa which shows plastic propertiesand they are close to 1 GPa Because most steels and plasticmaterials undergo plastic deformation near the value of 03we used 025 for Poissonrsquos ratio (]) Our simulation is staticso we used a single load step in all experiments Multiple loadsteps are usedwhen the load forces are time-dependent or thesimulation is dynamic [15]

We conducted four experiments each having differentnumber of elements to observe the speedup for both stiff-ness matrix calculation and for the solution of the systemAs expected the proposed method and Pedersenrsquos methodproduced same amount of nodal displacements in all experi-ments

Table 1 The displacements (in m) at nodes 1 2 3 and 4 using thelinear FEM for the first experimentThe displacements of nodes 5 to8 for all axes are zero

Node Displacement-119909 Displacement-119910 Displacement-1199111 0027234 0011064 minus02899652 0004306 minus0109719 minus04407393 minus0066065 minus0056547 minus03435194 minus0107536 0070143 minus0514524

Table 2 The displacements (in m) at nodes 1 2 3 and 4 using thenonlinear FEM for the first experimentThe displacements of nodes5 to 8 for all axes are zero

Node Displacement-119909 Displacement-119910 Displacement-1199111 0029911 0012665 minus02783652 0008606 minus0103350 minus04155943 minus0058835 minus0051901 minus03241264 minus0098945 0068928 minus0478495

Table 3 The displacements (in m) at green nodes using the linearFEM for the second experiment The displacements at blue nodesare zero

Node Displacement-119909 Displacement-119910 Displacement-1199110 minus3717 4208 minus003941 4738 4208 minus0049472 4737 minus4245 0037773 minus3716 minus4246 00490220 301 minus3547 minus00542921 minus4117 minus3548 minus00614322 minus4117 3581 00434823 301 3581 005155

Table 4 The displacements (in m) at green nodes using thenonlinear FEM for the second experiment The displacements atblue nodes are zero

Node Displacement-119909 Displacement-119910 Displacement-1199110 minus2083 4102 037981 472 2298 035882 2913 minus4501 045863 minus3884 minus2699 0480920 328 minus2561 minus042421 minus2842 minus3931 minus0403222 minus4217 2194 minus0301823 1911 3565 minus03212

The first experiment was conducted for a cube mesh witheight nodes and six tetrahedral elements Figure 2 shows thatthe cube is constrained at the upper four nodes and pulleddownwards with a small amount of force (one unit force foreach of the upper four nodes) This experiment was con-ducted with a small mesh in order to carefully examine thenodal displacements and strains for each element Figure 3shows the initial and final positions of the nodes for the linearand nonlinear FEMs respectively As seen in Figure 3 the

8 Journal of Applied Mathematics

(a) (b)

Figure 3 (a) The initial and final positions of the nodes for the linear FEM (b) The initial and final positions of the nodes for the nonlinearFEM

6

5

4

3

2

1

0

Disp

lace

men

t (m

)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Force (N)

Linear FEMNonlinear FEM

Liner FEM versus nonlinear FEM

Figure 4 Force displacements (in m) at node 4 for the linear andnonlinear FEMs

100

10

1

01

001

0001

Erro

r (

)

1 2 3 4 5

Newton-Raphson convergence graphics8079

2043

118

001

000

Iteration

Figure 5 Newton-Raphson convergence graphics for the nonlinearFEMThe graph is plotted using the logarithmic scale

linear and nonlinear methods produce similar displacementswhen the force magnitude is small Tables 1 and 2 show thedisplacements at force applied nodes (first second thirdand fourth) using the linear and nonlinear FEMs respec-tively Figure 4 shows that displacement increases linearlywith force magnitude However as expected the nonlinear

Table 5 The displacements (in m) at green nodes using the linearFEM for the third experiment The displacements at blue nodes arezero

Node Displacement-119909 Displacement-119910 Displacement-1199115 5004 2305 72416 minus04613 2399 0014449 minus23 2318 627110 minus4602 2375 minus0143741 05991 2345 minus017

Table 6 The displacements (in m) at green nodes using thenonlinear FEM for the third experiment The displacements at bluenodes are zero

Node Displacement-119909 Displacement-119910 Displacement-1199115 6439 6934 50146 minus02123 7996 14589 minus3372 4471 minus478810 05483 7737 00658741 1196 8284 01512

Table 7 The displacements (in m) at green nodes using the linearFEM for the fourth experimentThe displacements at blue nodes arezero

Node Displacement-119909 Displacement-119910 Displacement-119911271 1086 minus05297 1188

Table 8 The displacements (in m) at green nodes using thenonlinear FEM for the fourth experimentThe displacements at bluenodes are zero

Node Displacement-119909 Displacement-119910 Displacement-119911271 06538 minus01851 422

FEM behaves quadratically due to the nonlinear straindefinitions Figure 5 depicts the convergence of the Newton-Raphson method for the nonlinear FEM

The second experiment was conducted on a beam with90 nodes and 216 tetrahedral elements Figures 6(a) and 6(b)show that the beam is constrained at the blue nodes and

Journal of Applied Mathematics 9

(a) (b)

Figure 6The beammesh is constrained at the blue nodes and twisted at the green nodes (a) Front view (b) side view which shows the forcedirections applied on each green node

(a) (b)

(c) (d)

Figure 7 Nonlinear FEM results for the both the proposed and Pedersenrsquos methods ((a) wireframe tetrahedra and nodes (b) nodes only (c)wireframe surface mesh and (d) shaded mesh)

Figure 8 The cross mesh is constrained at the blue nodes andpushed towards the green nodes

twisted at both ends Figure 7 shows the final shape of thebeam mesh for both the proposed and Pedersenrsquos methodsTables 3 and 4 show the displacements at force applied nodes(green nodes) for the second experiment using the linear andnonlinear FEMs respectively

The third experiment was conducted with a cross meshof 159 nodes and 244 tetrahedral elements We aimed toobserve if there is a root jump occurring when solving thesystem for a high amount of force (50N units) and its effect

10 Journal of Applied Mathematics

(a) (b)

Figure 9 Nonlinear FEM results for the both proposed and Pedersenrsquos methods (a) initial and final wireframemeshes are overlaid (b) initialand final shaded meshes are overlaid

(a) (b)

Figure 10 The liver mesh is constrained at the blue nodes and pulled from the green node ((a) initial nodes (b) initial shaded mesh andnodes)

Table 9 Computation times (in seconds) of stiffness matrices(Pedersen Pedersenrsquos nonlinear FEM Proposed proposed nonlin-ear FEM and Speed-up relative performance comparison of thestiffness matrix calculation of the proposed nonlinear FEMmethodwith Pedersenrsquos nonlinear FEMmethod using single thread)

Exp Elements Pedersen Proposed Speed-up1st 6 07322 03308 22134222nd 216 281624 111542 25248253rd 224 301094 122725 24534044th 1580 2398753 967840 2478460

Table 10 Computation times (s) of system solutions (PedersenPedersenrsquos nonlinear FEM Proposed proposed nonlinear FEMand Speed-up relative performance comparison of the proposednonlinear FEM method with Pedersenrsquos nonlinear FEM methodusing single thread)

Exp Elements Pedersen Proposed Speed-up1st 6 30144 24427 12340442nd 216 1925288 1596241 12061393rd 224 5862708 6125911 9570344th 1580 28407558 24010994 1183106

Table 11 Newton-Raphson iteration count to reach desired accu-racy (Pedersen Pedersenrsquos nonlinear FEM Proposed proposednonlinear FEM)

Exp Elements Pedersen Proposed1st 6 5 52nd 216 7 73rd 224 26 324th 1580 8 8

on the computation times for both methods Figure 8 showsthat the cross shape is constrained at the blue nodes andpushed towards the green nodes Figure 9 shows the finalshape of the beammesh for both the proposed and Pedersenrsquosmethods Tables 5 and 6 show the displacements at forceapplied nodes (green nodes) using the linear and nonlinearFEMs respectively

We conducted fourth experiment with a livermesh of 465nodes and 1560 tetrahedral elements Figure 10 shows thatthe mesh is constrained at the blue nodes and pulled fromthe green node (30N units) in the direction of the arrow Weaimed to observe the similar amount of speedup like previousexperiments for a high density mesh Figure 11 shows thefinal shape of the beam mesh for both the proposed andPedersenrsquos methods Tables 7 and 8 show the displacementsat force applied node (node number 271) using the linear andnonlinear FEMs respectively

Computation times of the finite element experiments arerequired to compare how much faster our proposed methodis than Pedersenrsquos When comparing nonlinear FEMs wecalculated the computation times to construct the stiffnessmatrices as well as the computation times of the nonlinearFEM solutions to determine how different calculations affectthem Table 9 depicts the computation times for the stiffnessmatrix calculation and Table 10 depicts the computationtimes for the system solution Table 11 shows the iterationcounts to solve the system using the Newton-Raphson pro-cedure

The speed-up columns of Tables 9 and 10 depict thespeedups of the proposed method compared to Pedersenrsquosmethod for the stiffness matrix calculation and the systemsolution using a single thread respectively The speedup iscalculated as follows

Speedup =Runtime (Pedersenrsquosmethod)

Runtime (Theproposedmethod) (40)

Journal of Applied Mathematics 11

(a) (b)

Figure 11 Nonlinear FEM results for both the proposed and Pedersenrsquos methods (a) left wireframe surface mesh right wireframe surfacemesh with nodes (b) left shaded mesh right shaded mesh with nodes

(a) (b)

(c)

Figure 12 (a) Initial misshapen nose (b) The head mesh is constrained at the blue nodes and pushed upwards at the green nodes (c) Theresult of the nonlinear FEM left wireframe surface mesh with nodes right shaded mesh with texture

Our proposed method outperforms Pedersenrsquos methodOn the average it is 142 faster at computing stiffnessmatrices because Pedersenrsquos method uses more symbolicterms However bothmethods use Newton-Raphson to solvenonlinear equations which takes approximately 90 of thecomputation time Thus the overall speedup decreases to15 on average In experiment 3 because of more iterationsdue to root jumps there was a performance loss againstPedersenrsquos method

We also applied our method for corrective operationon the misshapen nose of a head mesh The head mesh iscomposed of 6709 nodes and 25722 tetrahedral elements (seeFigure 12(a)) all the operations were performed in the nose

area of only 1458 tetrahedral elements Figure 12(b) showsthat the head mesh is constrained at the blue nodes andpushed upwards at the green nodes and Figure 12(c) showsthe result of the nonlinear FEM

4 Conclusions and Future Work

We propose a new stiffness matrix calculation method fornonlinear FEM that is easier to analyze in terms of construct-ing elemental stiffness matrices and is faster than Pedersenrsquosmethod The proposed method is approximately 24 timesfaster on average at computing stiffness matrices and 15faster at computing thewhole system thanPedersenrsquosmethod

12 Journal of Applied Mathematics

Although the proposed nonlinear FEM has significantadvantages over Pedersenrsquos nonlinear FEM there is still roomfor the following development

(1) Heuristics could be applied to avoid root jumps(2) Although we decreased system memory usage by

simplifying the solution process for the nonlinearFEM a significant amount of system memory is stillusedThe solution process could be further optimizedto decrease memory usage

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] K-J Bathe Finite Element Procedures Prentice-Hall Engle-wood Cliffs NJ USA 1996

[2] D L Logan A First Course in the Finite Element MethodCengage Learning 5th edition 2012

[3] P Pedersen ldquoAnalytical stiffness matrices for tetrahedral ele-mentsrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 196 no 1-3 pp 261ndash278 2006

[4] G R Joldes A Wittek and K Miller ldquoReal-time nonlinearfinite element computations on GPU application to neurosur-gical simulationrdquo Computer Methods in Applied Mechanics andEngineering vol 199 no 49ndash52 pp 3305ndash3314 2010

[5] Z A Taylor M Cheng and S Ourselin ldquoHigh-speed nonlinearfinite element analysis for surgical simulation using graphicsprocessing unitsrdquo IEEE Transactions on Medical Imaging vol27 no 5 pp 650ndash663 2008

[6] M Cerrolaza and J C Osorio ldquoRelations among stiffnesscoefficients of hexahedral 8-noded finite elements a simple andefficient way to reduce the integration timerdquo Finite Elements inAnalysis and Design vol 55 pp 1ndash6 2012

[7] J Bonet and R D Wood Nonlinear Continu um Mechanicsfor Finite Element Analysis Cambridge University Press Cam-bridge UK 1st edition 1997

[8] C A Felippa ldquoLecture notes in nonlinear fin ite element meth-odsrdquo Tech Rep CUCSSC-96-16 Department of AerospaceEngineering Sciences and Center for Aerospace StructuresUniversity of Colorado 1996

[9] P Pedersen The Basic Matrix Approach for Three Simple FiniteElements 2008 httpwwwtopoptdtudkfilesPauliBooksBasicMatrixApproachInFEpdf

[10] C T Kelley Iterative Methods for Linear and Nonlinear Equa-tions Frontiers in Applied Mathematics Society for IndustrialMathematics 1st edition 1987

[11] S Krenk Non-linear Modeling and Analysi s of Solids andStructures Cambridge University Press 1st edition 2009

[12] The Mathworks ldquoCalling MATLAB Engine from CC++and Fortran Programsrdquo 2014 httpwwwmathworkscomhelpmatlabmatlab external f38569html

[13] Autodesk ldquo3ds Maxmdash3D Modeling Ani mation and Render-ing Softwarerdquo 2012 httpwwwautodeskcomproductsauto-desk-3ds-max

[14] Maplesoft ldquoMATLAB ConnectivitymdashMaple Featuresrdquo 2012httpwwwmaplesoftcomproductsmaplefeaturesmatlab-connectivityaspx

[15] E Madenci and I Guven The Finite Element Method andApplications in Engineering Using ANSYS Springer BerlinGermany 1st edition 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 5

Consider

[11986101

] =

[[[[[[[[[[[[[[

[

1205731+ 1205731(119906119909119909

) 1205731(119906119910119909

) 1205731(119906119911119909

)

1205741(119906119909119910

) 1205741+ 1205741(119906119910119910

) 1205741(119906119911119910

)

1205751(119906119909119911

) 1205751(119906119910119911

) 1205751+ 1205751(119906119911119911

)

1205741+ 1205741(119906119909119909

) + 1205731(119906119909119910

) 1205741(119906119910119909

) + 1205731+ 1205731(119906119910119910

) 1205741(119906119911119909

) + 1205731(119906119911119910

)

1205751+ 1205751(119906119909119909

) + 1205731(119906119909119911

) 1205751(119906119910119909

) + 1205731(119906119910119911

) 1205751(119906119911119909

) + 1205731+ 1205731(119906119911119911

)

1205741(119906119909119911

) + 1205751(119906119909119910

) 1205741+ 1205741(119906119910119911

) + 1205751(119906119910119910

) 1205741(119906119911119911

) + 1205751+ 1205751(119906119911119910

)

]]]]]]]]]]]]]]

]

1199061

V1

1199081

(21)

[11986101

] =

[[[[[[[[[[[[[[[[[[[[[

[

1205731+

1

21205731(119906119909119909

)1

21205731(119906119910119909

)1

21205731(119906119911119909

)

1

21205741(119906119909119910

) 1205741+

1

21205741(119906119910119910

)1

21205741(119906119911119910

)

1

21205751(119906119909119911

)1

21205751(119906119910119911

) 1205751+

1

21205751(119906119911119911

)

1205741+

1

21205741(119906119909119909

) +1

21205731(119906119909119910

)1

21205741(119906119910119909

) + 1205731+

1

21205731(119906119910119910

)1

21205741(119906119911119909

) +1

21205731(119906119911119910

)

1205751+

1

21205751(119906119909119909

) +1

21205731(119906119909119911

)1

21205751(119906119910119909

) +1

21205731(119906119910119911

)1

21205751(119906119911119909

) + 1205731+

1

21205731(119906119911119911

)

1

21205741(119906119909119911

) +1

21205751(119906119909119910

) 1205741+

1

21205741(119906119910119911

) +1

21205751(119906119910119910

)1

21205741(119906119911119911

) + 1205751+

1

21205751(119906119911119910

)

]]]]]]]]]]]]]]]]]]]]]

]

1199061

V1

1199081

(22)

The FEM is derived from conservation of the potentialenergy which is defined by

120587 = Estrain + 119882 (23)

where Estrain is the strain energy of the linear element and 119882

is the work potential They are given by

Estrain =1

2int

Ω119890

120576119879120590119889119909 119882 = 119891

119890119889119879 (24)

where the engineering strain vector 120576 is

120576 = [119861] 119889 (25)

From (24) the engineering stress vector 120591 is related to thestrain vector by

120591 = [119864] 120578 = [119864] [1198610] 119889 (26)

The secant relations are described by the matrix [119864] Wesubstitute (15) and (26) into (24) obtaining the elementstiffness matrix

[119896 (119906)] = ∭119889119879[1198610]119879

[119864] [1198610] 119889 119889119909 119889119910 119889119911 (27)

We can discard the integrals as we did for the linear FEM[1198610] [119864] and [119861

0] are constant for the four-node tetrahedral

element so (27) is rewritten as

[119896 (119906)] = 119889119879[1198610]119879

[119864] [1198610]119881 (28)

The secant stiffness matrix which is [119896119904(119889)119879] = [119861

0]119879[119864][119861

0]

is nonsymmetric because of the fact that [1198610]119879

= [1198610]

Introducing nodal forces we obtain

119891 =

1198911119909

1198911119910

1198911119911

1198914119909

1198914119910

1198914119911

119889119879 (29)

With the equilibrium equation and cancelling 119889119879 the whole

system for one element reduces to

119896119904(119889)119890119889119890= 119891119890 (30)

By substituting 119889 with 119906 we obtain

119896119904(119906)119890119906119890= 119891119890 (31)

Finally only nonlinear displacement functions remain whichare solved with Newton-Raphson to find the unknowndisplacements 119906 [10]

Element residuals are necessary for the iterative Newton-RaphsonmethodThe element residual is a 12times1 vector for aspecific elementThe residual for a specific element is definedas

119903119890= 119896119904(119906)119890minus 119891119890 (32)

6 Journal of Applied Mathematics

Having determined 119903119890 we can now express (32) in expanded

vector form as

1199031

1199032

1199033

11990312

=

[[[[[[[[[[[

[

119896119904(119906)(11)

+ 119896119904(119906)(12)

+ 119896119904(119906)(13)

+ sdot sdot sdot + 119896119904(119906)(112)

119896119904(119906)(21)

+ 119896119904(119906)(22)

+ 119896119904(119906)(23)

+ sdot sdot sdot + 119896119904(119906)(212)

119896119904(119906)(31)

+ 119896119904(119906)(32)

+ 119896119904(119906)(33)

+ sdot sdot sdot + 119896119904(119906)(312)

119896119904(119906)(121)

+ 119896119904(119906)(122)

+ 119896119904(119906)(123)

+ sdot sdot sdot + 119896119904(119906)(1212)

]]]]]]]]]]]

]

minus

1198911

1198912

1198913

11989112

(33)

The tangent stiffness matrix [119870]119890

119879(1199031015840119890) is also neces-

sary for the iterative Newton-Raphson method The tan-gent stiffness matrix is also 12 times 12 matrix like theelemental stiffness matrix However the tangent stiffnessmatrix depends on residuals unlike the elemental stiffness

matrix Elemental stiffness matrices are used to constructresiduals and the derivatives of the residuals are used toconstruct the elemental tangent stiffness matrices We canexpress the elemental tangent stiffness matrix for a specificelement as

1199031015840119890

= [119870]119890

119879=

[[[[[[[[[[[[[[[[

[

120597

1205971199061

1199031

120597

120597V1

1199031

120597

1205971199081

1199031

sdot sdot sdot120597

1205971199064

1199031

120597

120597V4

1199031

120597

1205971199084

1199031

120597

1205971199061

1199032

120597

120597V1

1199032

120597

1205971199081

1199032

sdot sdot sdot120597

1205971199064

1199032

120597

120597V4

1199032

120597

1205971199084

1199032

120597

1205971199061

1199033

120597

120597V1

1199033

120597

1205971199081

1199033

sdot sdot sdot120597

1205971199064

1199033

120597

120597V4

1199033

120597

1205971199084

1199033

120597

1205971199061

11990312

120597

120597V1

11990312

120597

1205971199081

11990312

sdot sdot sdot120597

1205971199064

11990312

120597

120597V4

11990312

120597

1205971199084

11990312

]]]]]]]]]]]]]]]]

]

(34)

Newton-Raphsonmethod is a fast and popular numericalmethod for solving nonlinear equations [10] as comparedto the other methods such as direct iteration In principlethe method works by applying the following two steps (cfAlgorithm 1) (i) check if the equilibrium is reached withinthe desired accuracy (ii) if not make a suitable adjustment tothe state of the deformation [11] An initial guess for displace-ments is needed to start the iterations The displacements areupdated according to

119909119896+1

= 119909119896minus

119891119909119896

1198911015840

119909119896

(35)

In the proposed nonlinear FEM 119906 is the vector thatkeeps the information of the nodal displacements Instead ofmaking only one assumption we make whole 119906 vector initialguess in order to start the iteration

Consider

1199061= 1199060minus

1199031199060

1199031015840

1199060

(36)

where 119903 is residual of the global stiffnessmatrix [119870] calculatedin (33) and 119903

1015840 is the tangent stiffness matrix calculated in (34)

At every step the vector 119903 and the matrix 1199031015840 are updated

for every element with the new 119906119894values Then 119903 and 119903

1015840 areassembled as we did with for the global stiffnessmatrix119870 andthe global force vector 119865 in linear FEM Boundary conditionsare applied to the global 119903 vector and the global 1199031015840 matrixUsing the global 119903 vector and the global 1199031015840 matrix we have

1199031015840(119906119894) 119901 = minus119903 (119906

119894) 119901 = minus(119903

1015840(119906119894))minus1

119903 (119906119894) (37)

119906119894is updated with the solution of (37) Consider

119906119894+1

= 119906119894+ 119901 (38)

Then we check if the equilibrium is reached within thedesired accuracy defined by 120575 as

1003816100381610038161003816119903 (119906119894)1003816100381610038161003816 le 120575 (39)

After the desired accuracy is reached the unknown nodaldisplacements are found

3 Experimental Results

The proposed nonlinear FEM and Pedersenrsquos nonlinear FEMwere implemented using MATLAB programming language

Journal of Applied Mathematics 7

Make initial guess 119891(119909)

while 1003816100381610038161003816119891 (119909)1003816100381610038161003816 le 120575 do

Compute 119901 = minus119891(119909)

1198911015840(119909)

Update 119909 = 119909 + 119901

Calculate 119891(119909)

end while

Algorithm 1 Newton-Raphson method

1

2 3

4

5

6 7

8

Figure 2 A 10 times 10 times 10m cube mesh with eight nodes and sixtetrahedra is constrained at the blue nodes and pulled downwardsfrom the green nodes

The visualizer was implemented with C++ language andconnected to the solver using the MATLAB engine [12]which allows users to call the MATLAB solver from CC++or Fortran programsThe simulation results interaction withthe 3Dmodel and the 3Dmodels themselves were visualizedusing OpenGL and the nose experiment was visualizedusing 3ds Max [13] To speed up the nonlinear FEM weused MAPLErsquos symbolic solver [14] which is integrated intoMATLAB We conducted all the experiments on a desktopcomputer with a Core i7 3930K processor overclockedat 42 GHz with 32GB of RAM We used linear materialproperties for the models in the experiments We used 1GPafor Youngrsquos modulus (120598) because polypropylene has Youngrsquosmodulus between 15 and 2GPa and polyethylene HDPEhas Youngrsquos modulus 08GPa which shows plastic propertiesand they are close to 1 GPa Because most steels and plasticmaterials undergo plastic deformation near the value of 03we used 025 for Poissonrsquos ratio (]) Our simulation is staticso we used a single load step in all experiments Multiple loadsteps are usedwhen the load forces are time-dependent or thesimulation is dynamic [15]

We conducted four experiments each having differentnumber of elements to observe the speedup for both stiff-ness matrix calculation and for the solution of the systemAs expected the proposed method and Pedersenrsquos methodproduced same amount of nodal displacements in all experi-ments

Table 1 The displacements (in m) at nodes 1 2 3 and 4 using thelinear FEM for the first experimentThe displacements of nodes 5 to8 for all axes are zero

Node Displacement-119909 Displacement-119910 Displacement-1199111 0027234 0011064 minus02899652 0004306 minus0109719 minus04407393 minus0066065 minus0056547 minus03435194 minus0107536 0070143 minus0514524

Table 2 The displacements (in m) at nodes 1 2 3 and 4 using thenonlinear FEM for the first experimentThe displacements of nodes5 to 8 for all axes are zero

Node Displacement-119909 Displacement-119910 Displacement-1199111 0029911 0012665 minus02783652 0008606 minus0103350 minus04155943 minus0058835 minus0051901 minus03241264 minus0098945 0068928 minus0478495

Table 3 The displacements (in m) at green nodes using the linearFEM for the second experiment The displacements at blue nodesare zero

Node Displacement-119909 Displacement-119910 Displacement-1199110 minus3717 4208 minus003941 4738 4208 minus0049472 4737 minus4245 0037773 minus3716 minus4246 00490220 301 minus3547 minus00542921 minus4117 minus3548 minus00614322 minus4117 3581 00434823 301 3581 005155

Table 4 The displacements (in m) at green nodes using thenonlinear FEM for the second experiment The displacements atblue nodes are zero

Node Displacement-119909 Displacement-119910 Displacement-1199110 minus2083 4102 037981 472 2298 035882 2913 minus4501 045863 minus3884 minus2699 0480920 328 minus2561 minus042421 minus2842 minus3931 minus0403222 minus4217 2194 minus0301823 1911 3565 minus03212

The first experiment was conducted for a cube mesh witheight nodes and six tetrahedral elements Figure 2 shows thatthe cube is constrained at the upper four nodes and pulleddownwards with a small amount of force (one unit force foreach of the upper four nodes) This experiment was con-ducted with a small mesh in order to carefully examine thenodal displacements and strains for each element Figure 3shows the initial and final positions of the nodes for the linearand nonlinear FEMs respectively As seen in Figure 3 the

8 Journal of Applied Mathematics

(a) (b)

Figure 3 (a) The initial and final positions of the nodes for the linear FEM (b) The initial and final positions of the nodes for the nonlinearFEM

6

5

4

3

2

1

0

Disp

lace

men

t (m

)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Force (N)

Linear FEMNonlinear FEM

Liner FEM versus nonlinear FEM

Figure 4 Force displacements (in m) at node 4 for the linear andnonlinear FEMs

100

10

1

01

001

0001

Erro

r (

)

1 2 3 4 5

Newton-Raphson convergence graphics8079

2043

118

001

000

Iteration

Figure 5 Newton-Raphson convergence graphics for the nonlinearFEMThe graph is plotted using the logarithmic scale

linear and nonlinear methods produce similar displacementswhen the force magnitude is small Tables 1 and 2 show thedisplacements at force applied nodes (first second thirdand fourth) using the linear and nonlinear FEMs respec-tively Figure 4 shows that displacement increases linearlywith force magnitude However as expected the nonlinear

Table 5 The displacements (in m) at green nodes using the linearFEM for the third experiment The displacements at blue nodes arezero

Node Displacement-119909 Displacement-119910 Displacement-1199115 5004 2305 72416 minus04613 2399 0014449 minus23 2318 627110 minus4602 2375 minus0143741 05991 2345 minus017

Table 6 The displacements (in m) at green nodes using thenonlinear FEM for the third experiment The displacements at bluenodes are zero

Node Displacement-119909 Displacement-119910 Displacement-1199115 6439 6934 50146 minus02123 7996 14589 minus3372 4471 minus478810 05483 7737 00658741 1196 8284 01512

Table 7 The displacements (in m) at green nodes using the linearFEM for the fourth experimentThe displacements at blue nodes arezero

Node Displacement-119909 Displacement-119910 Displacement-119911271 1086 minus05297 1188

Table 8 The displacements (in m) at green nodes using thenonlinear FEM for the fourth experimentThe displacements at bluenodes are zero

Node Displacement-119909 Displacement-119910 Displacement-119911271 06538 minus01851 422

FEM behaves quadratically due to the nonlinear straindefinitions Figure 5 depicts the convergence of the Newton-Raphson method for the nonlinear FEM

The second experiment was conducted on a beam with90 nodes and 216 tetrahedral elements Figures 6(a) and 6(b)show that the beam is constrained at the blue nodes and

Journal of Applied Mathematics 9

(a) (b)

Figure 6The beammesh is constrained at the blue nodes and twisted at the green nodes (a) Front view (b) side view which shows the forcedirections applied on each green node

(a) (b)

(c) (d)

Figure 7 Nonlinear FEM results for the both the proposed and Pedersenrsquos methods ((a) wireframe tetrahedra and nodes (b) nodes only (c)wireframe surface mesh and (d) shaded mesh)

Figure 8 The cross mesh is constrained at the blue nodes andpushed towards the green nodes

twisted at both ends Figure 7 shows the final shape of thebeam mesh for both the proposed and Pedersenrsquos methodsTables 3 and 4 show the displacements at force applied nodes(green nodes) for the second experiment using the linear andnonlinear FEMs respectively

The third experiment was conducted with a cross meshof 159 nodes and 244 tetrahedral elements We aimed toobserve if there is a root jump occurring when solving thesystem for a high amount of force (50N units) and its effect

10 Journal of Applied Mathematics

(a) (b)

Figure 9 Nonlinear FEM results for the both proposed and Pedersenrsquos methods (a) initial and final wireframemeshes are overlaid (b) initialand final shaded meshes are overlaid

(a) (b)

Figure 10 The liver mesh is constrained at the blue nodes and pulled from the green node ((a) initial nodes (b) initial shaded mesh andnodes)

Table 9 Computation times (in seconds) of stiffness matrices(Pedersen Pedersenrsquos nonlinear FEM Proposed proposed nonlin-ear FEM and Speed-up relative performance comparison of thestiffness matrix calculation of the proposed nonlinear FEMmethodwith Pedersenrsquos nonlinear FEMmethod using single thread)

Exp Elements Pedersen Proposed Speed-up1st 6 07322 03308 22134222nd 216 281624 111542 25248253rd 224 301094 122725 24534044th 1580 2398753 967840 2478460

Table 10 Computation times (s) of system solutions (PedersenPedersenrsquos nonlinear FEM Proposed proposed nonlinear FEMand Speed-up relative performance comparison of the proposednonlinear FEM method with Pedersenrsquos nonlinear FEM methodusing single thread)

Exp Elements Pedersen Proposed Speed-up1st 6 30144 24427 12340442nd 216 1925288 1596241 12061393rd 224 5862708 6125911 9570344th 1580 28407558 24010994 1183106

Table 11 Newton-Raphson iteration count to reach desired accu-racy (Pedersen Pedersenrsquos nonlinear FEM Proposed proposednonlinear FEM)

Exp Elements Pedersen Proposed1st 6 5 52nd 216 7 73rd 224 26 324th 1580 8 8

on the computation times for both methods Figure 8 showsthat the cross shape is constrained at the blue nodes andpushed towards the green nodes Figure 9 shows the finalshape of the beammesh for both the proposed and Pedersenrsquosmethods Tables 5 and 6 show the displacements at forceapplied nodes (green nodes) using the linear and nonlinearFEMs respectively

We conducted fourth experiment with a livermesh of 465nodes and 1560 tetrahedral elements Figure 10 shows thatthe mesh is constrained at the blue nodes and pulled fromthe green node (30N units) in the direction of the arrow Weaimed to observe the similar amount of speedup like previousexperiments for a high density mesh Figure 11 shows thefinal shape of the beam mesh for both the proposed andPedersenrsquos methods Tables 7 and 8 show the displacementsat force applied node (node number 271) using the linear andnonlinear FEMs respectively

Computation times of the finite element experiments arerequired to compare how much faster our proposed methodis than Pedersenrsquos When comparing nonlinear FEMs wecalculated the computation times to construct the stiffnessmatrices as well as the computation times of the nonlinearFEM solutions to determine how different calculations affectthem Table 9 depicts the computation times for the stiffnessmatrix calculation and Table 10 depicts the computationtimes for the system solution Table 11 shows the iterationcounts to solve the system using the Newton-Raphson pro-cedure

The speed-up columns of Tables 9 and 10 depict thespeedups of the proposed method compared to Pedersenrsquosmethod for the stiffness matrix calculation and the systemsolution using a single thread respectively The speedup iscalculated as follows

Speedup =Runtime (Pedersenrsquosmethod)

Runtime (Theproposedmethod) (40)

Journal of Applied Mathematics 11

(a) (b)

Figure 11 Nonlinear FEM results for both the proposed and Pedersenrsquos methods (a) left wireframe surface mesh right wireframe surfacemesh with nodes (b) left shaded mesh right shaded mesh with nodes

(a) (b)

(c)

Figure 12 (a) Initial misshapen nose (b) The head mesh is constrained at the blue nodes and pushed upwards at the green nodes (c) Theresult of the nonlinear FEM left wireframe surface mesh with nodes right shaded mesh with texture

Our proposed method outperforms Pedersenrsquos methodOn the average it is 142 faster at computing stiffnessmatrices because Pedersenrsquos method uses more symbolicterms However bothmethods use Newton-Raphson to solvenonlinear equations which takes approximately 90 of thecomputation time Thus the overall speedup decreases to15 on average In experiment 3 because of more iterationsdue to root jumps there was a performance loss againstPedersenrsquos method

We also applied our method for corrective operationon the misshapen nose of a head mesh The head mesh iscomposed of 6709 nodes and 25722 tetrahedral elements (seeFigure 12(a)) all the operations were performed in the nose

area of only 1458 tetrahedral elements Figure 12(b) showsthat the head mesh is constrained at the blue nodes andpushed upwards at the green nodes and Figure 12(c) showsthe result of the nonlinear FEM

4 Conclusions and Future Work

We propose a new stiffness matrix calculation method fornonlinear FEM that is easier to analyze in terms of construct-ing elemental stiffness matrices and is faster than Pedersenrsquosmethod The proposed method is approximately 24 timesfaster on average at computing stiffness matrices and 15faster at computing thewhole system thanPedersenrsquosmethod

12 Journal of Applied Mathematics

Although the proposed nonlinear FEM has significantadvantages over Pedersenrsquos nonlinear FEM there is still roomfor the following development

(1) Heuristics could be applied to avoid root jumps(2) Although we decreased system memory usage by

simplifying the solution process for the nonlinearFEM a significant amount of system memory is stillusedThe solution process could be further optimizedto decrease memory usage

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] K-J Bathe Finite Element Procedures Prentice-Hall Engle-wood Cliffs NJ USA 1996

[2] D L Logan A First Course in the Finite Element MethodCengage Learning 5th edition 2012

[3] P Pedersen ldquoAnalytical stiffness matrices for tetrahedral ele-mentsrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 196 no 1-3 pp 261ndash278 2006

[4] G R Joldes A Wittek and K Miller ldquoReal-time nonlinearfinite element computations on GPU application to neurosur-gical simulationrdquo Computer Methods in Applied Mechanics andEngineering vol 199 no 49ndash52 pp 3305ndash3314 2010

[5] Z A Taylor M Cheng and S Ourselin ldquoHigh-speed nonlinearfinite element analysis for surgical simulation using graphicsprocessing unitsrdquo IEEE Transactions on Medical Imaging vol27 no 5 pp 650ndash663 2008

[6] M Cerrolaza and J C Osorio ldquoRelations among stiffnesscoefficients of hexahedral 8-noded finite elements a simple andefficient way to reduce the integration timerdquo Finite Elements inAnalysis and Design vol 55 pp 1ndash6 2012

[7] J Bonet and R D Wood Nonlinear Continu um Mechanicsfor Finite Element Analysis Cambridge University Press Cam-bridge UK 1st edition 1997

[8] C A Felippa ldquoLecture notes in nonlinear fin ite element meth-odsrdquo Tech Rep CUCSSC-96-16 Department of AerospaceEngineering Sciences and Center for Aerospace StructuresUniversity of Colorado 1996

[9] P Pedersen The Basic Matrix Approach for Three Simple FiniteElements 2008 httpwwwtopoptdtudkfilesPauliBooksBasicMatrixApproachInFEpdf

[10] C T Kelley Iterative Methods for Linear and Nonlinear Equa-tions Frontiers in Applied Mathematics Society for IndustrialMathematics 1st edition 1987

[11] S Krenk Non-linear Modeling and Analysi s of Solids andStructures Cambridge University Press 1st edition 2009

[12] The Mathworks ldquoCalling MATLAB Engine from CC++and Fortran Programsrdquo 2014 httpwwwmathworkscomhelpmatlabmatlab external f38569html

[13] Autodesk ldquo3ds Maxmdash3D Modeling Ani mation and Render-ing Softwarerdquo 2012 httpwwwautodeskcomproductsauto-desk-3ds-max

[14] Maplesoft ldquoMATLAB ConnectivitymdashMaple Featuresrdquo 2012httpwwwmaplesoftcomproductsmaplefeaturesmatlab-connectivityaspx

[15] E Madenci and I Guven The Finite Element Method andApplications in Engineering Using ANSYS Springer BerlinGermany 1st edition 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Applied Mathematics

Having determined 119903119890 we can now express (32) in expanded

vector form as

1199031

1199032

1199033

11990312

=

[[[[[[[[[[[

[

119896119904(119906)(11)

+ 119896119904(119906)(12)

+ 119896119904(119906)(13)

+ sdot sdot sdot + 119896119904(119906)(112)

119896119904(119906)(21)

+ 119896119904(119906)(22)

+ 119896119904(119906)(23)

+ sdot sdot sdot + 119896119904(119906)(212)

119896119904(119906)(31)

+ 119896119904(119906)(32)

+ 119896119904(119906)(33)

+ sdot sdot sdot + 119896119904(119906)(312)

119896119904(119906)(121)

+ 119896119904(119906)(122)

+ 119896119904(119906)(123)

+ sdot sdot sdot + 119896119904(119906)(1212)

]]]]]]]]]]]

]

minus

1198911

1198912

1198913

11989112

(33)

The tangent stiffness matrix [119870]119890

119879(1199031015840119890) is also neces-

sary for the iterative Newton-Raphson method The tan-gent stiffness matrix is also 12 times 12 matrix like theelemental stiffness matrix However the tangent stiffnessmatrix depends on residuals unlike the elemental stiffness

matrix Elemental stiffness matrices are used to constructresiduals and the derivatives of the residuals are used toconstruct the elemental tangent stiffness matrices We canexpress the elemental tangent stiffness matrix for a specificelement as

1199031015840119890

= [119870]119890

119879=

[[[[[[[[[[[[[[[[

[

120597

1205971199061

1199031

120597

120597V1

1199031

120597

1205971199081

1199031

sdot sdot sdot120597

1205971199064

1199031

120597

120597V4

1199031

120597

1205971199084

1199031

120597

1205971199061

1199032

120597

120597V1

1199032

120597

1205971199081

1199032

sdot sdot sdot120597

1205971199064

1199032

120597

120597V4

1199032

120597

1205971199084

1199032

120597

1205971199061

1199033

120597

120597V1

1199033

120597

1205971199081

1199033

sdot sdot sdot120597

1205971199064

1199033

120597

120597V4

1199033

120597

1205971199084

1199033

120597

1205971199061

11990312

120597

120597V1

11990312

120597

1205971199081

11990312

sdot sdot sdot120597

1205971199064

11990312

120597

120597V4

11990312

120597

1205971199084

11990312

]]]]]]]]]]]]]]]]

]

(34)

Newton-Raphsonmethod is a fast and popular numericalmethod for solving nonlinear equations [10] as comparedto the other methods such as direct iteration In principlethe method works by applying the following two steps (cfAlgorithm 1) (i) check if the equilibrium is reached withinthe desired accuracy (ii) if not make a suitable adjustment tothe state of the deformation [11] An initial guess for displace-ments is needed to start the iterations The displacements areupdated according to

119909119896+1

= 119909119896minus

119891119909119896

1198911015840

119909119896

(35)

In the proposed nonlinear FEM 119906 is the vector thatkeeps the information of the nodal displacements Instead ofmaking only one assumption we make whole 119906 vector initialguess in order to start the iteration

Consider

1199061= 1199060minus

1199031199060

1199031015840

1199060

(36)

where 119903 is residual of the global stiffnessmatrix [119870] calculatedin (33) and 119903

1015840 is the tangent stiffness matrix calculated in (34)

At every step the vector 119903 and the matrix 1199031015840 are updated

for every element with the new 119906119894values Then 119903 and 119903

1015840 areassembled as we did with for the global stiffnessmatrix119870 andthe global force vector 119865 in linear FEM Boundary conditionsare applied to the global 119903 vector and the global 1199031015840 matrixUsing the global 119903 vector and the global 1199031015840 matrix we have

1199031015840(119906119894) 119901 = minus119903 (119906

119894) 119901 = minus(119903

1015840(119906119894))minus1

119903 (119906119894) (37)

119906119894is updated with the solution of (37) Consider

119906119894+1

= 119906119894+ 119901 (38)

Then we check if the equilibrium is reached within thedesired accuracy defined by 120575 as

1003816100381610038161003816119903 (119906119894)1003816100381610038161003816 le 120575 (39)

After the desired accuracy is reached the unknown nodaldisplacements are found

3 Experimental Results

The proposed nonlinear FEM and Pedersenrsquos nonlinear FEMwere implemented using MATLAB programming language

Journal of Applied Mathematics 7

Make initial guess 119891(119909)

while 1003816100381610038161003816119891 (119909)1003816100381610038161003816 le 120575 do

Compute 119901 = minus119891(119909)

1198911015840(119909)

Update 119909 = 119909 + 119901

Calculate 119891(119909)

end while

Algorithm 1 Newton-Raphson method

1

2 3

4

5

6 7

8

Figure 2 A 10 times 10 times 10m cube mesh with eight nodes and sixtetrahedra is constrained at the blue nodes and pulled downwardsfrom the green nodes

The visualizer was implemented with C++ language andconnected to the solver using the MATLAB engine [12]which allows users to call the MATLAB solver from CC++or Fortran programsThe simulation results interaction withthe 3Dmodel and the 3Dmodels themselves were visualizedusing OpenGL and the nose experiment was visualizedusing 3ds Max [13] To speed up the nonlinear FEM weused MAPLErsquos symbolic solver [14] which is integrated intoMATLAB We conducted all the experiments on a desktopcomputer with a Core i7 3930K processor overclockedat 42 GHz with 32GB of RAM We used linear materialproperties for the models in the experiments We used 1GPafor Youngrsquos modulus (120598) because polypropylene has Youngrsquosmodulus between 15 and 2GPa and polyethylene HDPEhas Youngrsquos modulus 08GPa which shows plastic propertiesand they are close to 1 GPa Because most steels and plasticmaterials undergo plastic deformation near the value of 03we used 025 for Poissonrsquos ratio (]) Our simulation is staticso we used a single load step in all experiments Multiple loadsteps are usedwhen the load forces are time-dependent or thesimulation is dynamic [15]

We conducted four experiments each having differentnumber of elements to observe the speedup for both stiff-ness matrix calculation and for the solution of the systemAs expected the proposed method and Pedersenrsquos methodproduced same amount of nodal displacements in all experi-ments

Table 1 The displacements (in m) at nodes 1 2 3 and 4 using thelinear FEM for the first experimentThe displacements of nodes 5 to8 for all axes are zero

Node Displacement-119909 Displacement-119910 Displacement-1199111 0027234 0011064 minus02899652 0004306 minus0109719 minus04407393 minus0066065 minus0056547 minus03435194 minus0107536 0070143 minus0514524

Table 2 The displacements (in m) at nodes 1 2 3 and 4 using thenonlinear FEM for the first experimentThe displacements of nodes5 to 8 for all axes are zero

Node Displacement-119909 Displacement-119910 Displacement-1199111 0029911 0012665 minus02783652 0008606 minus0103350 minus04155943 minus0058835 minus0051901 minus03241264 minus0098945 0068928 minus0478495

Table 3 The displacements (in m) at green nodes using the linearFEM for the second experiment The displacements at blue nodesare zero

Node Displacement-119909 Displacement-119910 Displacement-1199110 minus3717 4208 minus003941 4738 4208 minus0049472 4737 minus4245 0037773 minus3716 minus4246 00490220 301 minus3547 minus00542921 minus4117 minus3548 minus00614322 minus4117 3581 00434823 301 3581 005155

Table 4 The displacements (in m) at green nodes using thenonlinear FEM for the second experiment The displacements atblue nodes are zero

Node Displacement-119909 Displacement-119910 Displacement-1199110 minus2083 4102 037981 472 2298 035882 2913 minus4501 045863 minus3884 minus2699 0480920 328 minus2561 minus042421 minus2842 minus3931 minus0403222 minus4217 2194 minus0301823 1911 3565 minus03212

The first experiment was conducted for a cube mesh witheight nodes and six tetrahedral elements Figure 2 shows thatthe cube is constrained at the upper four nodes and pulleddownwards with a small amount of force (one unit force foreach of the upper four nodes) This experiment was con-ducted with a small mesh in order to carefully examine thenodal displacements and strains for each element Figure 3shows the initial and final positions of the nodes for the linearand nonlinear FEMs respectively As seen in Figure 3 the

8 Journal of Applied Mathematics

(a) (b)

Figure 3 (a) The initial and final positions of the nodes for the linear FEM (b) The initial and final positions of the nodes for the nonlinearFEM

6

5

4

3

2

1

0

Disp

lace

men

t (m

)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Force (N)

Linear FEMNonlinear FEM

Liner FEM versus nonlinear FEM

Figure 4 Force displacements (in m) at node 4 for the linear andnonlinear FEMs

100

10

1

01

001

0001

Erro

r (

)

1 2 3 4 5

Newton-Raphson convergence graphics8079

2043

118

001

000

Iteration

Figure 5 Newton-Raphson convergence graphics for the nonlinearFEMThe graph is plotted using the logarithmic scale

linear and nonlinear methods produce similar displacementswhen the force magnitude is small Tables 1 and 2 show thedisplacements at force applied nodes (first second thirdand fourth) using the linear and nonlinear FEMs respec-tively Figure 4 shows that displacement increases linearlywith force magnitude However as expected the nonlinear

Table 5 The displacements (in m) at green nodes using the linearFEM for the third experiment The displacements at blue nodes arezero

Node Displacement-119909 Displacement-119910 Displacement-1199115 5004 2305 72416 minus04613 2399 0014449 minus23 2318 627110 minus4602 2375 minus0143741 05991 2345 minus017

Table 6 The displacements (in m) at green nodes using thenonlinear FEM for the third experiment The displacements at bluenodes are zero

Node Displacement-119909 Displacement-119910 Displacement-1199115 6439 6934 50146 minus02123 7996 14589 minus3372 4471 minus478810 05483 7737 00658741 1196 8284 01512

Table 7 The displacements (in m) at green nodes using the linearFEM for the fourth experimentThe displacements at blue nodes arezero

Node Displacement-119909 Displacement-119910 Displacement-119911271 1086 minus05297 1188

Table 8 The displacements (in m) at green nodes using thenonlinear FEM for the fourth experimentThe displacements at bluenodes are zero

Node Displacement-119909 Displacement-119910 Displacement-119911271 06538 minus01851 422

FEM behaves quadratically due to the nonlinear straindefinitions Figure 5 depicts the convergence of the Newton-Raphson method for the nonlinear FEM

The second experiment was conducted on a beam with90 nodes and 216 tetrahedral elements Figures 6(a) and 6(b)show that the beam is constrained at the blue nodes and

Journal of Applied Mathematics 9

(a) (b)

Figure 6The beammesh is constrained at the blue nodes and twisted at the green nodes (a) Front view (b) side view which shows the forcedirections applied on each green node

(a) (b)

(c) (d)

Figure 7 Nonlinear FEM results for the both the proposed and Pedersenrsquos methods ((a) wireframe tetrahedra and nodes (b) nodes only (c)wireframe surface mesh and (d) shaded mesh)

Figure 8 The cross mesh is constrained at the blue nodes andpushed towards the green nodes

twisted at both ends Figure 7 shows the final shape of thebeam mesh for both the proposed and Pedersenrsquos methodsTables 3 and 4 show the displacements at force applied nodes(green nodes) for the second experiment using the linear andnonlinear FEMs respectively

The third experiment was conducted with a cross meshof 159 nodes and 244 tetrahedral elements We aimed toobserve if there is a root jump occurring when solving thesystem for a high amount of force (50N units) and its effect

10 Journal of Applied Mathematics

(a) (b)

Figure 9 Nonlinear FEM results for the both proposed and Pedersenrsquos methods (a) initial and final wireframemeshes are overlaid (b) initialand final shaded meshes are overlaid

(a) (b)

Figure 10 The liver mesh is constrained at the blue nodes and pulled from the green node ((a) initial nodes (b) initial shaded mesh andnodes)

Table 9 Computation times (in seconds) of stiffness matrices(Pedersen Pedersenrsquos nonlinear FEM Proposed proposed nonlin-ear FEM and Speed-up relative performance comparison of thestiffness matrix calculation of the proposed nonlinear FEMmethodwith Pedersenrsquos nonlinear FEMmethod using single thread)

Exp Elements Pedersen Proposed Speed-up1st 6 07322 03308 22134222nd 216 281624 111542 25248253rd 224 301094 122725 24534044th 1580 2398753 967840 2478460

Table 10 Computation times (s) of system solutions (PedersenPedersenrsquos nonlinear FEM Proposed proposed nonlinear FEMand Speed-up relative performance comparison of the proposednonlinear FEM method with Pedersenrsquos nonlinear FEM methodusing single thread)

Exp Elements Pedersen Proposed Speed-up1st 6 30144 24427 12340442nd 216 1925288 1596241 12061393rd 224 5862708 6125911 9570344th 1580 28407558 24010994 1183106

Table 11 Newton-Raphson iteration count to reach desired accu-racy (Pedersen Pedersenrsquos nonlinear FEM Proposed proposednonlinear FEM)

Exp Elements Pedersen Proposed1st 6 5 52nd 216 7 73rd 224 26 324th 1580 8 8

on the computation times for both methods Figure 8 showsthat the cross shape is constrained at the blue nodes andpushed towards the green nodes Figure 9 shows the finalshape of the beammesh for both the proposed and Pedersenrsquosmethods Tables 5 and 6 show the displacements at forceapplied nodes (green nodes) using the linear and nonlinearFEMs respectively

We conducted fourth experiment with a livermesh of 465nodes and 1560 tetrahedral elements Figure 10 shows thatthe mesh is constrained at the blue nodes and pulled fromthe green node (30N units) in the direction of the arrow Weaimed to observe the similar amount of speedup like previousexperiments for a high density mesh Figure 11 shows thefinal shape of the beam mesh for both the proposed andPedersenrsquos methods Tables 7 and 8 show the displacementsat force applied node (node number 271) using the linear andnonlinear FEMs respectively

Computation times of the finite element experiments arerequired to compare how much faster our proposed methodis than Pedersenrsquos When comparing nonlinear FEMs wecalculated the computation times to construct the stiffnessmatrices as well as the computation times of the nonlinearFEM solutions to determine how different calculations affectthem Table 9 depicts the computation times for the stiffnessmatrix calculation and Table 10 depicts the computationtimes for the system solution Table 11 shows the iterationcounts to solve the system using the Newton-Raphson pro-cedure

The speed-up columns of Tables 9 and 10 depict thespeedups of the proposed method compared to Pedersenrsquosmethod for the stiffness matrix calculation and the systemsolution using a single thread respectively The speedup iscalculated as follows

Speedup =Runtime (Pedersenrsquosmethod)

Runtime (Theproposedmethod) (40)

Journal of Applied Mathematics 11

(a) (b)

Figure 11 Nonlinear FEM results for both the proposed and Pedersenrsquos methods (a) left wireframe surface mesh right wireframe surfacemesh with nodes (b) left shaded mesh right shaded mesh with nodes

(a) (b)

(c)

Figure 12 (a) Initial misshapen nose (b) The head mesh is constrained at the blue nodes and pushed upwards at the green nodes (c) Theresult of the nonlinear FEM left wireframe surface mesh with nodes right shaded mesh with texture

Our proposed method outperforms Pedersenrsquos methodOn the average it is 142 faster at computing stiffnessmatrices because Pedersenrsquos method uses more symbolicterms However bothmethods use Newton-Raphson to solvenonlinear equations which takes approximately 90 of thecomputation time Thus the overall speedup decreases to15 on average In experiment 3 because of more iterationsdue to root jumps there was a performance loss againstPedersenrsquos method

We also applied our method for corrective operationon the misshapen nose of a head mesh The head mesh iscomposed of 6709 nodes and 25722 tetrahedral elements (seeFigure 12(a)) all the operations were performed in the nose

area of only 1458 tetrahedral elements Figure 12(b) showsthat the head mesh is constrained at the blue nodes andpushed upwards at the green nodes and Figure 12(c) showsthe result of the nonlinear FEM

4 Conclusions and Future Work

We propose a new stiffness matrix calculation method fornonlinear FEM that is easier to analyze in terms of construct-ing elemental stiffness matrices and is faster than Pedersenrsquosmethod The proposed method is approximately 24 timesfaster on average at computing stiffness matrices and 15faster at computing thewhole system thanPedersenrsquosmethod

12 Journal of Applied Mathematics

Although the proposed nonlinear FEM has significantadvantages over Pedersenrsquos nonlinear FEM there is still roomfor the following development

(1) Heuristics could be applied to avoid root jumps(2) Although we decreased system memory usage by

simplifying the solution process for the nonlinearFEM a significant amount of system memory is stillusedThe solution process could be further optimizedto decrease memory usage

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] K-J Bathe Finite Element Procedures Prentice-Hall Engle-wood Cliffs NJ USA 1996

[2] D L Logan A First Course in the Finite Element MethodCengage Learning 5th edition 2012

[3] P Pedersen ldquoAnalytical stiffness matrices for tetrahedral ele-mentsrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 196 no 1-3 pp 261ndash278 2006

[4] G R Joldes A Wittek and K Miller ldquoReal-time nonlinearfinite element computations on GPU application to neurosur-gical simulationrdquo Computer Methods in Applied Mechanics andEngineering vol 199 no 49ndash52 pp 3305ndash3314 2010

[5] Z A Taylor M Cheng and S Ourselin ldquoHigh-speed nonlinearfinite element analysis for surgical simulation using graphicsprocessing unitsrdquo IEEE Transactions on Medical Imaging vol27 no 5 pp 650ndash663 2008

[6] M Cerrolaza and J C Osorio ldquoRelations among stiffnesscoefficients of hexahedral 8-noded finite elements a simple andefficient way to reduce the integration timerdquo Finite Elements inAnalysis and Design vol 55 pp 1ndash6 2012

[7] J Bonet and R D Wood Nonlinear Continu um Mechanicsfor Finite Element Analysis Cambridge University Press Cam-bridge UK 1st edition 1997

[8] C A Felippa ldquoLecture notes in nonlinear fin ite element meth-odsrdquo Tech Rep CUCSSC-96-16 Department of AerospaceEngineering Sciences and Center for Aerospace StructuresUniversity of Colorado 1996

[9] P Pedersen The Basic Matrix Approach for Three Simple FiniteElements 2008 httpwwwtopoptdtudkfilesPauliBooksBasicMatrixApproachInFEpdf

[10] C T Kelley Iterative Methods for Linear and Nonlinear Equa-tions Frontiers in Applied Mathematics Society for IndustrialMathematics 1st edition 1987

[11] S Krenk Non-linear Modeling and Analysi s of Solids andStructures Cambridge University Press 1st edition 2009

[12] The Mathworks ldquoCalling MATLAB Engine from CC++and Fortran Programsrdquo 2014 httpwwwmathworkscomhelpmatlabmatlab external f38569html

[13] Autodesk ldquo3ds Maxmdash3D Modeling Ani mation and Render-ing Softwarerdquo 2012 httpwwwautodeskcomproductsauto-desk-3ds-max

[14] Maplesoft ldquoMATLAB ConnectivitymdashMaple Featuresrdquo 2012httpwwwmaplesoftcomproductsmaplefeaturesmatlab-connectivityaspx

[15] E Madenci and I Guven The Finite Element Method andApplications in Engineering Using ANSYS Springer BerlinGermany 1st edition 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 7

Make initial guess 119891(119909)

while 1003816100381610038161003816119891 (119909)1003816100381610038161003816 le 120575 do

Compute 119901 = minus119891(119909)

1198911015840(119909)

Update 119909 = 119909 + 119901

Calculate 119891(119909)

end while

Algorithm 1 Newton-Raphson method

1

2 3

4

5

6 7

8

Figure 2 A 10 times 10 times 10m cube mesh with eight nodes and sixtetrahedra is constrained at the blue nodes and pulled downwardsfrom the green nodes

The visualizer was implemented with C++ language andconnected to the solver using the MATLAB engine [12]which allows users to call the MATLAB solver from CC++or Fortran programsThe simulation results interaction withthe 3Dmodel and the 3Dmodels themselves were visualizedusing OpenGL and the nose experiment was visualizedusing 3ds Max [13] To speed up the nonlinear FEM weused MAPLErsquos symbolic solver [14] which is integrated intoMATLAB We conducted all the experiments on a desktopcomputer with a Core i7 3930K processor overclockedat 42 GHz with 32GB of RAM We used linear materialproperties for the models in the experiments We used 1GPafor Youngrsquos modulus (120598) because polypropylene has Youngrsquosmodulus between 15 and 2GPa and polyethylene HDPEhas Youngrsquos modulus 08GPa which shows plastic propertiesand they are close to 1 GPa Because most steels and plasticmaterials undergo plastic deformation near the value of 03we used 025 for Poissonrsquos ratio (]) Our simulation is staticso we used a single load step in all experiments Multiple loadsteps are usedwhen the load forces are time-dependent or thesimulation is dynamic [15]

We conducted four experiments each having differentnumber of elements to observe the speedup for both stiff-ness matrix calculation and for the solution of the systemAs expected the proposed method and Pedersenrsquos methodproduced same amount of nodal displacements in all experi-ments

Table 1 The displacements (in m) at nodes 1 2 3 and 4 using thelinear FEM for the first experimentThe displacements of nodes 5 to8 for all axes are zero

Node Displacement-119909 Displacement-119910 Displacement-1199111 0027234 0011064 minus02899652 0004306 minus0109719 minus04407393 minus0066065 minus0056547 minus03435194 minus0107536 0070143 minus0514524

Table 2 The displacements (in m) at nodes 1 2 3 and 4 using thenonlinear FEM for the first experimentThe displacements of nodes5 to 8 for all axes are zero

Node Displacement-119909 Displacement-119910 Displacement-1199111 0029911 0012665 minus02783652 0008606 minus0103350 minus04155943 minus0058835 minus0051901 minus03241264 minus0098945 0068928 minus0478495

Table 3 The displacements (in m) at green nodes using the linearFEM for the second experiment The displacements at blue nodesare zero

Node Displacement-119909 Displacement-119910 Displacement-1199110 minus3717 4208 minus003941 4738 4208 minus0049472 4737 minus4245 0037773 minus3716 minus4246 00490220 301 minus3547 minus00542921 minus4117 minus3548 minus00614322 minus4117 3581 00434823 301 3581 005155

Table 4 The displacements (in m) at green nodes using thenonlinear FEM for the second experiment The displacements atblue nodes are zero

Node Displacement-119909 Displacement-119910 Displacement-1199110 minus2083 4102 037981 472 2298 035882 2913 minus4501 045863 minus3884 minus2699 0480920 328 minus2561 minus042421 minus2842 minus3931 minus0403222 minus4217 2194 minus0301823 1911 3565 minus03212

The first experiment was conducted for a cube mesh witheight nodes and six tetrahedral elements Figure 2 shows thatthe cube is constrained at the upper four nodes and pulleddownwards with a small amount of force (one unit force foreach of the upper four nodes) This experiment was con-ducted with a small mesh in order to carefully examine thenodal displacements and strains for each element Figure 3shows the initial and final positions of the nodes for the linearand nonlinear FEMs respectively As seen in Figure 3 the

8 Journal of Applied Mathematics

(a) (b)

Figure 3 (a) The initial and final positions of the nodes for the linear FEM (b) The initial and final positions of the nodes for the nonlinearFEM

6

5

4

3

2

1

0

Disp

lace

men

t (m

)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Force (N)

Linear FEMNonlinear FEM

Liner FEM versus nonlinear FEM

Figure 4 Force displacements (in m) at node 4 for the linear andnonlinear FEMs

100

10

1

01

001

0001

Erro

r (

)

1 2 3 4 5

Newton-Raphson convergence graphics8079

2043

118

001

000

Iteration

Figure 5 Newton-Raphson convergence graphics for the nonlinearFEMThe graph is plotted using the logarithmic scale

linear and nonlinear methods produce similar displacementswhen the force magnitude is small Tables 1 and 2 show thedisplacements at force applied nodes (first second thirdand fourth) using the linear and nonlinear FEMs respec-tively Figure 4 shows that displacement increases linearlywith force magnitude However as expected the nonlinear

Table 5 The displacements (in m) at green nodes using the linearFEM for the third experiment The displacements at blue nodes arezero

Node Displacement-119909 Displacement-119910 Displacement-1199115 5004 2305 72416 minus04613 2399 0014449 minus23 2318 627110 minus4602 2375 minus0143741 05991 2345 minus017

Table 6 The displacements (in m) at green nodes using thenonlinear FEM for the third experiment The displacements at bluenodes are zero

Node Displacement-119909 Displacement-119910 Displacement-1199115 6439 6934 50146 minus02123 7996 14589 minus3372 4471 minus478810 05483 7737 00658741 1196 8284 01512

Table 7 The displacements (in m) at green nodes using the linearFEM for the fourth experimentThe displacements at blue nodes arezero

Node Displacement-119909 Displacement-119910 Displacement-119911271 1086 minus05297 1188

Table 8 The displacements (in m) at green nodes using thenonlinear FEM for the fourth experimentThe displacements at bluenodes are zero

Node Displacement-119909 Displacement-119910 Displacement-119911271 06538 minus01851 422

FEM behaves quadratically due to the nonlinear straindefinitions Figure 5 depicts the convergence of the Newton-Raphson method for the nonlinear FEM

The second experiment was conducted on a beam with90 nodes and 216 tetrahedral elements Figures 6(a) and 6(b)show that the beam is constrained at the blue nodes and

Journal of Applied Mathematics 9

(a) (b)

Figure 6The beammesh is constrained at the blue nodes and twisted at the green nodes (a) Front view (b) side view which shows the forcedirections applied on each green node

(a) (b)

(c) (d)

Figure 7 Nonlinear FEM results for the both the proposed and Pedersenrsquos methods ((a) wireframe tetrahedra and nodes (b) nodes only (c)wireframe surface mesh and (d) shaded mesh)

Figure 8 The cross mesh is constrained at the blue nodes andpushed towards the green nodes

twisted at both ends Figure 7 shows the final shape of thebeam mesh for both the proposed and Pedersenrsquos methodsTables 3 and 4 show the displacements at force applied nodes(green nodes) for the second experiment using the linear andnonlinear FEMs respectively

The third experiment was conducted with a cross meshof 159 nodes and 244 tetrahedral elements We aimed toobserve if there is a root jump occurring when solving thesystem for a high amount of force (50N units) and its effect

10 Journal of Applied Mathematics

(a) (b)

Figure 9 Nonlinear FEM results for the both proposed and Pedersenrsquos methods (a) initial and final wireframemeshes are overlaid (b) initialand final shaded meshes are overlaid

(a) (b)

Figure 10 The liver mesh is constrained at the blue nodes and pulled from the green node ((a) initial nodes (b) initial shaded mesh andnodes)

Table 9 Computation times (in seconds) of stiffness matrices(Pedersen Pedersenrsquos nonlinear FEM Proposed proposed nonlin-ear FEM and Speed-up relative performance comparison of thestiffness matrix calculation of the proposed nonlinear FEMmethodwith Pedersenrsquos nonlinear FEMmethod using single thread)

Exp Elements Pedersen Proposed Speed-up1st 6 07322 03308 22134222nd 216 281624 111542 25248253rd 224 301094 122725 24534044th 1580 2398753 967840 2478460

Table 10 Computation times (s) of system solutions (PedersenPedersenrsquos nonlinear FEM Proposed proposed nonlinear FEMand Speed-up relative performance comparison of the proposednonlinear FEM method with Pedersenrsquos nonlinear FEM methodusing single thread)

Exp Elements Pedersen Proposed Speed-up1st 6 30144 24427 12340442nd 216 1925288 1596241 12061393rd 224 5862708 6125911 9570344th 1580 28407558 24010994 1183106

Table 11 Newton-Raphson iteration count to reach desired accu-racy (Pedersen Pedersenrsquos nonlinear FEM Proposed proposednonlinear FEM)

Exp Elements Pedersen Proposed1st 6 5 52nd 216 7 73rd 224 26 324th 1580 8 8

on the computation times for both methods Figure 8 showsthat the cross shape is constrained at the blue nodes andpushed towards the green nodes Figure 9 shows the finalshape of the beammesh for both the proposed and Pedersenrsquosmethods Tables 5 and 6 show the displacements at forceapplied nodes (green nodes) using the linear and nonlinearFEMs respectively

We conducted fourth experiment with a livermesh of 465nodes and 1560 tetrahedral elements Figure 10 shows thatthe mesh is constrained at the blue nodes and pulled fromthe green node (30N units) in the direction of the arrow Weaimed to observe the similar amount of speedup like previousexperiments for a high density mesh Figure 11 shows thefinal shape of the beam mesh for both the proposed andPedersenrsquos methods Tables 7 and 8 show the displacementsat force applied node (node number 271) using the linear andnonlinear FEMs respectively

Computation times of the finite element experiments arerequired to compare how much faster our proposed methodis than Pedersenrsquos When comparing nonlinear FEMs wecalculated the computation times to construct the stiffnessmatrices as well as the computation times of the nonlinearFEM solutions to determine how different calculations affectthem Table 9 depicts the computation times for the stiffnessmatrix calculation and Table 10 depicts the computationtimes for the system solution Table 11 shows the iterationcounts to solve the system using the Newton-Raphson pro-cedure

The speed-up columns of Tables 9 and 10 depict thespeedups of the proposed method compared to Pedersenrsquosmethod for the stiffness matrix calculation and the systemsolution using a single thread respectively The speedup iscalculated as follows

Speedup =Runtime (Pedersenrsquosmethod)

Runtime (Theproposedmethod) (40)

Journal of Applied Mathematics 11

(a) (b)

Figure 11 Nonlinear FEM results for both the proposed and Pedersenrsquos methods (a) left wireframe surface mesh right wireframe surfacemesh with nodes (b) left shaded mesh right shaded mesh with nodes

(a) (b)

(c)

Figure 12 (a) Initial misshapen nose (b) The head mesh is constrained at the blue nodes and pushed upwards at the green nodes (c) Theresult of the nonlinear FEM left wireframe surface mesh with nodes right shaded mesh with texture

Our proposed method outperforms Pedersenrsquos methodOn the average it is 142 faster at computing stiffnessmatrices because Pedersenrsquos method uses more symbolicterms However bothmethods use Newton-Raphson to solvenonlinear equations which takes approximately 90 of thecomputation time Thus the overall speedup decreases to15 on average In experiment 3 because of more iterationsdue to root jumps there was a performance loss againstPedersenrsquos method

We also applied our method for corrective operationon the misshapen nose of a head mesh The head mesh iscomposed of 6709 nodes and 25722 tetrahedral elements (seeFigure 12(a)) all the operations were performed in the nose

area of only 1458 tetrahedral elements Figure 12(b) showsthat the head mesh is constrained at the blue nodes andpushed upwards at the green nodes and Figure 12(c) showsthe result of the nonlinear FEM

4 Conclusions and Future Work

We propose a new stiffness matrix calculation method fornonlinear FEM that is easier to analyze in terms of construct-ing elemental stiffness matrices and is faster than Pedersenrsquosmethod The proposed method is approximately 24 timesfaster on average at computing stiffness matrices and 15faster at computing thewhole system thanPedersenrsquosmethod

12 Journal of Applied Mathematics

Although the proposed nonlinear FEM has significantadvantages over Pedersenrsquos nonlinear FEM there is still roomfor the following development

(1) Heuristics could be applied to avoid root jumps(2) Although we decreased system memory usage by

simplifying the solution process for the nonlinearFEM a significant amount of system memory is stillusedThe solution process could be further optimizedto decrease memory usage

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] K-J Bathe Finite Element Procedures Prentice-Hall Engle-wood Cliffs NJ USA 1996

[2] D L Logan A First Course in the Finite Element MethodCengage Learning 5th edition 2012

[3] P Pedersen ldquoAnalytical stiffness matrices for tetrahedral ele-mentsrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 196 no 1-3 pp 261ndash278 2006

[4] G R Joldes A Wittek and K Miller ldquoReal-time nonlinearfinite element computations on GPU application to neurosur-gical simulationrdquo Computer Methods in Applied Mechanics andEngineering vol 199 no 49ndash52 pp 3305ndash3314 2010

[5] Z A Taylor M Cheng and S Ourselin ldquoHigh-speed nonlinearfinite element analysis for surgical simulation using graphicsprocessing unitsrdquo IEEE Transactions on Medical Imaging vol27 no 5 pp 650ndash663 2008

[6] M Cerrolaza and J C Osorio ldquoRelations among stiffnesscoefficients of hexahedral 8-noded finite elements a simple andefficient way to reduce the integration timerdquo Finite Elements inAnalysis and Design vol 55 pp 1ndash6 2012

[7] J Bonet and R D Wood Nonlinear Continu um Mechanicsfor Finite Element Analysis Cambridge University Press Cam-bridge UK 1st edition 1997

[8] C A Felippa ldquoLecture notes in nonlinear fin ite element meth-odsrdquo Tech Rep CUCSSC-96-16 Department of AerospaceEngineering Sciences and Center for Aerospace StructuresUniversity of Colorado 1996

[9] P Pedersen The Basic Matrix Approach for Three Simple FiniteElements 2008 httpwwwtopoptdtudkfilesPauliBooksBasicMatrixApproachInFEpdf

[10] C T Kelley Iterative Methods for Linear and Nonlinear Equa-tions Frontiers in Applied Mathematics Society for IndustrialMathematics 1st edition 1987

[11] S Krenk Non-linear Modeling and Analysi s of Solids andStructures Cambridge University Press 1st edition 2009

[12] The Mathworks ldquoCalling MATLAB Engine from CC++and Fortran Programsrdquo 2014 httpwwwmathworkscomhelpmatlabmatlab external f38569html

[13] Autodesk ldquo3ds Maxmdash3D Modeling Ani mation and Render-ing Softwarerdquo 2012 httpwwwautodeskcomproductsauto-desk-3ds-max

[14] Maplesoft ldquoMATLAB ConnectivitymdashMaple Featuresrdquo 2012httpwwwmaplesoftcomproductsmaplefeaturesmatlab-connectivityaspx

[15] E Madenci and I Guven The Finite Element Method andApplications in Engineering Using ANSYS Springer BerlinGermany 1st edition 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Applied Mathematics

(a) (b)

Figure 3 (a) The initial and final positions of the nodes for the linear FEM (b) The initial and final positions of the nodes for the nonlinearFEM

6

5

4

3

2

1

0

Disp

lace

men

t (m

)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Force (N)

Linear FEMNonlinear FEM

Liner FEM versus nonlinear FEM

Figure 4 Force displacements (in m) at node 4 for the linear andnonlinear FEMs

100

10

1

01

001

0001

Erro

r (

)

1 2 3 4 5

Newton-Raphson convergence graphics8079

2043

118

001

000

Iteration

Figure 5 Newton-Raphson convergence graphics for the nonlinearFEMThe graph is plotted using the logarithmic scale

linear and nonlinear methods produce similar displacementswhen the force magnitude is small Tables 1 and 2 show thedisplacements at force applied nodes (first second thirdand fourth) using the linear and nonlinear FEMs respec-tively Figure 4 shows that displacement increases linearlywith force magnitude However as expected the nonlinear

Table 5 The displacements (in m) at green nodes using the linearFEM for the third experiment The displacements at blue nodes arezero

Node Displacement-119909 Displacement-119910 Displacement-1199115 5004 2305 72416 minus04613 2399 0014449 minus23 2318 627110 minus4602 2375 minus0143741 05991 2345 minus017

Table 6 The displacements (in m) at green nodes using thenonlinear FEM for the third experiment The displacements at bluenodes are zero

Node Displacement-119909 Displacement-119910 Displacement-1199115 6439 6934 50146 minus02123 7996 14589 minus3372 4471 minus478810 05483 7737 00658741 1196 8284 01512

Table 7 The displacements (in m) at green nodes using the linearFEM for the fourth experimentThe displacements at blue nodes arezero

Node Displacement-119909 Displacement-119910 Displacement-119911271 1086 minus05297 1188

Table 8 The displacements (in m) at green nodes using thenonlinear FEM for the fourth experimentThe displacements at bluenodes are zero

Node Displacement-119909 Displacement-119910 Displacement-119911271 06538 minus01851 422

FEM behaves quadratically due to the nonlinear straindefinitions Figure 5 depicts the convergence of the Newton-Raphson method for the nonlinear FEM

The second experiment was conducted on a beam with90 nodes and 216 tetrahedral elements Figures 6(a) and 6(b)show that the beam is constrained at the blue nodes and

Journal of Applied Mathematics 9

(a) (b)

Figure 6The beammesh is constrained at the blue nodes and twisted at the green nodes (a) Front view (b) side view which shows the forcedirections applied on each green node

(a) (b)

(c) (d)

Figure 7 Nonlinear FEM results for the both the proposed and Pedersenrsquos methods ((a) wireframe tetrahedra and nodes (b) nodes only (c)wireframe surface mesh and (d) shaded mesh)

Figure 8 The cross mesh is constrained at the blue nodes andpushed towards the green nodes

twisted at both ends Figure 7 shows the final shape of thebeam mesh for both the proposed and Pedersenrsquos methodsTables 3 and 4 show the displacements at force applied nodes(green nodes) for the second experiment using the linear andnonlinear FEMs respectively

The third experiment was conducted with a cross meshof 159 nodes and 244 tetrahedral elements We aimed toobserve if there is a root jump occurring when solving thesystem for a high amount of force (50N units) and its effect

10 Journal of Applied Mathematics

(a) (b)

Figure 9 Nonlinear FEM results for the both proposed and Pedersenrsquos methods (a) initial and final wireframemeshes are overlaid (b) initialand final shaded meshes are overlaid

(a) (b)

Figure 10 The liver mesh is constrained at the blue nodes and pulled from the green node ((a) initial nodes (b) initial shaded mesh andnodes)

Table 9 Computation times (in seconds) of stiffness matrices(Pedersen Pedersenrsquos nonlinear FEM Proposed proposed nonlin-ear FEM and Speed-up relative performance comparison of thestiffness matrix calculation of the proposed nonlinear FEMmethodwith Pedersenrsquos nonlinear FEMmethod using single thread)

Exp Elements Pedersen Proposed Speed-up1st 6 07322 03308 22134222nd 216 281624 111542 25248253rd 224 301094 122725 24534044th 1580 2398753 967840 2478460

Table 10 Computation times (s) of system solutions (PedersenPedersenrsquos nonlinear FEM Proposed proposed nonlinear FEMand Speed-up relative performance comparison of the proposednonlinear FEM method with Pedersenrsquos nonlinear FEM methodusing single thread)

Exp Elements Pedersen Proposed Speed-up1st 6 30144 24427 12340442nd 216 1925288 1596241 12061393rd 224 5862708 6125911 9570344th 1580 28407558 24010994 1183106

Table 11 Newton-Raphson iteration count to reach desired accu-racy (Pedersen Pedersenrsquos nonlinear FEM Proposed proposednonlinear FEM)

Exp Elements Pedersen Proposed1st 6 5 52nd 216 7 73rd 224 26 324th 1580 8 8

on the computation times for both methods Figure 8 showsthat the cross shape is constrained at the blue nodes andpushed towards the green nodes Figure 9 shows the finalshape of the beammesh for both the proposed and Pedersenrsquosmethods Tables 5 and 6 show the displacements at forceapplied nodes (green nodes) using the linear and nonlinearFEMs respectively

We conducted fourth experiment with a livermesh of 465nodes and 1560 tetrahedral elements Figure 10 shows thatthe mesh is constrained at the blue nodes and pulled fromthe green node (30N units) in the direction of the arrow Weaimed to observe the similar amount of speedup like previousexperiments for a high density mesh Figure 11 shows thefinal shape of the beam mesh for both the proposed andPedersenrsquos methods Tables 7 and 8 show the displacementsat force applied node (node number 271) using the linear andnonlinear FEMs respectively

Computation times of the finite element experiments arerequired to compare how much faster our proposed methodis than Pedersenrsquos When comparing nonlinear FEMs wecalculated the computation times to construct the stiffnessmatrices as well as the computation times of the nonlinearFEM solutions to determine how different calculations affectthem Table 9 depicts the computation times for the stiffnessmatrix calculation and Table 10 depicts the computationtimes for the system solution Table 11 shows the iterationcounts to solve the system using the Newton-Raphson pro-cedure

The speed-up columns of Tables 9 and 10 depict thespeedups of the proposed method compared to Pedersenrsquosmethod for the stiffness matrix calculation and the systemsolution using a single thread respectively The speedup iscalculated as follows

Speedup =Runtime (Pedersenrsquosmethod)

Runtime (Theproposedmethod) (40)

Journal of Applied Mathematics 11

(a) (b)

Figure 11 Nonlinear FEM results for both the proposed and Pedersenrsquos methods (a) left wireframe surface mesh right wireframe surfacemesh with nodes (b) left shaded mesh right shaded mesh with nodes

(a) (b)

(c)

Figure 12 (a) Initial misshapen nose (b) The head mesh is constrained at the blue nodes and pushed upwards at the green nodes (c) Theresult of the nonlinear FEM left wireframe surface mesh with nodes right shaded mesh with texture

Our proposed method outperforms Pedersenrsquos methodOn the average it is 142 faster at computing stiffnessmatrices because Pedersenrsquos method uses more symbolicterms However bothmethods use Newton-Raphson to solvenonlinear equations which takes approximately 90 of thecomputation time Thus the overall speedup decreases to15 on average In experiment 3 because of more iterationsdue to root jumps there was a performance loss againstPedersenrsquos method

We also applied our method for corrective operationon the misshapen nose of a head mesh The head mesh iscomposed of 6709 nodes and 25722 tetrahedral elements (seeFigure 12(a)) all the operations were performed in the nose

area of only 1458 tetrahedral elements Figure 12(b) showsthat the head mesh is constrained at the blue nodes andpushed upwards at the green nodes and Figure 12(c) showsthe result of the nonlinear FEM

4 Conclusions and Future Work

We propose a new stiffness matrix calculation method fornonlinear FEM that is easier to analyze in terms of construct-ing elemental stiffness matrices and is faster than Pedersenrsquosmethod The proposed method is approximately 24 timesfaster on average at computing stiffness matrices and 15faster at computing thewhole system thanPedersenrsquosmethod

12 Journal of Applied Mathematics

Although the proposed nonlinear FEM has significantadvantages over Pedersenrsquos nonlinear FEM there is still roomfor the following development

(1) Heuristics could be applied to avoid root jumps(2) Although we decreased system memory usage by

simplifying the solution process for the nonlinearFEM a significant amount of system memory is stillusedThe solution process could be further optimizedto decrease memory usage

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] K-J Bathe Finite Element Procedures Prentice-Hall Engle-wood Cliffs NJ USA 1996

[2] D L Logan A First Course in the Finite Element MethodCengage Learning 5th edition 2012

[3] P Pedersen ldquoAnalytical stiffness matrices for tetrahedral ele-mentsrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 196 no 1-3 pp 261ndash278 2006

[4] G R Joldes A Wittek and K Miller ldquoReal-time nonlinearfinite element computations on GPU application to neurosur-gical simulationrdquo Computer Methods in Applied Mechanics andEngineering vol 199 no 49ndash52 pp 3305ndash3314 2010

[5] Z A Taylor M Cheng and S Ourselin ldquoHigh-speed nonlinearfinite element analysis for surgical simulation using graphicsprocessing unitsrdquo IEEE Transactions on Medical Imaging vol27 no 5 pp 650ndash663 2008

[6] M Cerrolaza and J C Osorio ldquoRelations among stiffnesscoefficients of hexahedral 8-noded finite elements a simple andefficient way to reduce the integration timerdquo Finite Elements inAnalysis and Design vol 55 pp 1ndash6 2012

[7] J Bonet and R D Wood Nonlinear Continu um Mechanicsfor Finite Element Analysis Cambridge University Press Cam-bridge UK 1st edition 1997

[8] C A Felippa ldquoLecture notes in nonlinear fin ite element meth-odsrdquo Tech Rep CUCSSC-96-16 Department of AerospaceEngineering Sciences and Center for Aerospace StructuresUniversity of Colorado 1996

[9] P Pedersen The Basic Matrix Approach for Three Simple FiniteElements 2008 httpwwwtopoptdtudkfilesPauliBooksBasicMatrixApproachInFEpdf

[10] C T Kelley Iterative Methods for Linear and Nonlinear Equa-tions Frontiers in Applied Mathematics Society for IndustrialMathematics 1st edition 1987

[11] S Krenk Non-linear Modeling and Analysi s of Solids andStructures Cambridge University Press 1st edition 2009

[12] The Mathworks ldquoCalling MATLAB Engine from CC++and Fortran Programsrdquo 2014 httpwwwmathworkscomhelpmatlabmatlab external f38569html

[13] Autodesk ldquo3ds Maxmdash3D Modeling Ani mation and Render-ing Softwarerdquo 2012 httpwwwautodeskcomproductsauto-desk-3ds-max

[14] Maplesoft ldquoMATLAB ConnectivitymdashMaple Featuresrdquo 2012httpwwwmaplesoftcomproductsmaplefeaturesmatlab-connectivityaspx

[15] E Madenci and I Guven The Finite Element Method andApplications in Engineering Using ANSYS Springer BerlinGermany 1st edition 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 9

(a) (b)

Figure 6The beammesh is constrained at the blue nodes and twisted at the green nodes (a) Front view (b) side view which shows the forcedirections applied on each green node

(a) (b)

(c) (d)

Figure 7 Nonlinear FEM results for the both the proposed and Pedersenrsquos methods ((a) wireframe tetrahedra and nodes (b) nodes only (c)wireframe surface mesh and (d) shaded mesh)

Figure 8 The cross mesh is constrained at the blue nodes andpushed towards the green nodes

twisted at both ends Figure 7 shows the final shape of thebeam mesh for both the proposed and Pedersenrsquos methodsTables 3 and 4 show the displacements at force applied nodes(green nodes) for the second experiment using the linear andnonlinear FEMs respectively

The third experiment was conducted with a cross meshof 159 nodes and 244 tetrahedral elements We aimed toobserve if there is a root jump occurring when solving thesystem for a high amount of force (50N units) and its effect

10 Journal of Applied Mathematics

(a) (b)

Figure 9 Nonlinear FEM results for the both proposed and Pedersenrsquos methods (a) initial and final wireframemeshes are overlaid (b) initialand final shaded meshes are overlaid

(a) (b)

Figure 10 The liver mesh is constrained at the blue nodes and pulled from the green node ((a) initial nodes (b) initial shaded mesh andnodes)

Table 9 Computation times (in seconds) of stiffness matrices(Pedersen Pedersenrsquos nonlinear FEM Proposed proposed nonlin-ear FEM and Speed-up relative performance comparison of thestiffness matrix calculation of the proposed nonlinear FEMmethodwith Pedersenrsquos nonlinear FEMmethod using single thread)

Exp Elements Pedersen Proposed Speed-up1st 6 07322 03308 22134222nd 216 281624 111542 25248253rd 224 301094 122725 24534044th 1580 2398753 967840 2478460

Table 10 Computation times (s) of system solutions (PedersenPedersenrsquos nonlinear FEM Proposed proposed nonlinear FEMand Speed-up relative performance comparison of the proposednonlinear FEM method with Pedersenrsquos nonlinear FEM methodusing single thread)

Exp Elements Pedersen Proposed Speed-up1st 6 30144 24427 12340442nd 216 1925288 1596241 12061393rd 224 5862708 6125911 9570344th 1580 28407558 24010994 1183106

Table 11 Newton-Raphson iteration count to reach desired accu-racy (Pedersen Pedersenrsquos nonlinear FEM Proposed proposednonlinear FEM)

Exp Elements Pedersen Proposed1st 6 5 52nd 216 7 73rd 224 26 324th 1580 8 8

on the computation times for both methods Figure 8 showsthat the cross shape is constrained at the blue nodes andpushed towards the green nodes Figure 9 shows the finalshape of the beammesh for both the proposed and Pedersenrsquosmethods Tables 5 and 6 show the displacements at forceapplied nodes (green nodes) using the linear and nonlinearFEMs respectively

We conducted fourth experiment with a livermesh of 465nodes and 1560 tetrahedral elements Figure 10 shows thatthe mesh is constrained at the blue nodes and pulled fromthe green node (30N units) in the direction of the arrow Weaimed to observe the similar amount of speedup like previousexperiments for a high density mesh Figure 11 shows thefinal shape of the beam mesh for both the proposed andPedersenrsquos methods Tables 7 and 8 show the displacementsat force applied node (node number 271) using the linear andnonlinear FEMs respectively

Computation times of the finite element experiments arerequired to compare how much faster our proposed methodis than Pedersenrsquos When comparing nonlinear FEMs wecalculated the computation times to construct the stiffnessmatrices as well as the computation times of the nonlinearFEM solutions to determine how different calculations affectthem Table 9 depicts the computation times for the stiffnessmatrix calculation and Table 10 depicts the computationtimes for the system solution Table 11 shows the iterationcounts to solve the system using the Newton-Raphson pro-cedure

The speed-up columns of Tables 9 and 10 depict thespeedups of the proposed method compared to Pedersenrsquosmethod for the stiffness matrix calculation and the systemsolution using a single thread respectively The speedup iscalculated as follows

Speedup =Runtime (Pedersenrsquosmethod)

Runtime (Theproposedmethod) (40)

Journal of Applied Mathematics 11

(a) (b)

Figure 11 Nonlinear FEM results for both the proposed and Pedersenrsquos methods (a) left wireframe surface mesh right wireframe surfacemesh with nodes (b) left shaded mesh right shaded mesh with nodes

(a) (b)

(c)

Figure 12 (a) Initial misshapen nose (b) The head mesh is constrained at the blue nodes and pushed upwards at the green nodes (c) Theresult of the nonlinear FEM left wireframe surface mesh with nodes right shaded mesh with texture

Our proposed method outperforms Pedersenrsquos methodOn the average it is 142 faster at computing stiffnessmatrices because Pedersenrsquos method uses more symbolicterms However bothmethods use Newton-Raphson to solvenonlinear equations which takes approximately 90 of thecomputation time Thus the overall speedup decreases to15 on average In experiment 3 because of more iterationsdue to root jumps there was a performance loss againstPedersenrsquos method

We also applied our method for corrective operationon the misshapen nose of a head mesh The head mesh iscomposed of 6709 nodes and 25722 tetrahedral elements (seeFigure 12(a)) all the operations were performed in the nose

area of only 1458 tetrahedral elements Figure 12(b) showsthat the head mesh is constrained at the blue nodes andpushed upwards at the green nodes and Figure 12(c) showsthe result of the nonlinear FEM

4 Conclusions and Future Work

We propose a new stiffness matrix calculation method fornonlinear FEM that is easier to analyze in terms of construct-ing elemental stiffness matrices and is faster than Pedersenrsquosmethod The proposed method is approximately 24 timesfaster on average at computing stiffness matrices and 15faster at computing thewhole system thanPedersenrsquosmethod

12 Journal of Applied Mathematics

Although the proposed nonlinear FEM has significantadvantages over Pedersenrsquos nonlinear FEM there is still roomfor the following development

(1) Heuristics could be applied to avoid root jumps(2) Although we decreased system memory usage by

simplifying the solution process for the nonlinearFEM a significant amount of system memory is stillusedThe solution process could be further optimizedto decrease memory usage

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] K-J Bathe Finite Element Procedures Prentice-Hall Engle-wood Cliffs NJ USA 1996

[2] D L Logan A First Course in the Finite Element MethodCengage Learning 5th edition 2012

[3] P Pedersen ldquoAnalytical stiffness matrices for tetrahedral ele-mentsrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 196 no 1-3 pp 261ndash278 2006

[4] G R Joldes A Wittek and K Miller ldquoReal-time nonlinearfinite element computations on GPU application to neurosur-gical simulationrdquo Computer Methods in Applied Mechanics andEngineering vol 199 no 49ndash52 pp 3305ndash3314 2010

[5] Z A Taylor M Cheng and S Ourselin ldquoHigh-speed nonlinearfinite element analysis for surgical simulation using graphicsprocessing unitsrdquo IEEE Transactions on Medical Imaging vol27 no 5 pp 650ndash663 2008

[6] M Cerrolaza and J C Osorio ldquoRelations among stiffnesscoefficients of hexahedral 8-noded finite elements a simple andefficient way to reduce the integration timerdquo Finite Elements inAnalysis and Design vol 55 pp 1ndash6 2012

[7] J Bonet and R D Wood Nonlinear Continu um Mechanicsfor Finite Element Analysis Cambridge University Press Cam-bridge UK 1st edition 1997

[8] C A Felippa ldquoLecture notes in nonlinear fin ite element meth-odsrdquo Tech Rep CUCSSC-96-16 Department of AerospaceEngineering Sciences and Center for Aerospace StructuresUniversity of Colorado 1996

[9] P Pedersen The Basic Matrix Approach for Three Simple FiniteElements 2008 httpwwwtopoptdtudkfilesPauliBooksBasicMatrixApproachInFEpdf

[10] C T Kelley Iterative Methods for Linear and Nonlinear Equa-tions Frontiers in Applied Mathematics Society for IndustrialMathematics 1st edition 1987

[11] S Krenk Non-linear Modeling and Analysi s of Solids andStructures Cambridge University Press 1st edition 2009

[12] The Mathworks ldquoCalling MATLAB Engine from CC++and Fortran Programsrdquo 2014 httpwwwmathworkscomhelpmatlabmatlab external f38569html

[13] Autodesk ldquo3ds Maxmdash3D Modeling Ani mation and Render-ing Softwarerdquo 2012 httpwwwautodeskcomproductsauto-desk-3ds-max

[14] Maplesoft ldquoMATLAB ConnectivitymdashMaple Featuresrdquo 2012httpwwwmaplesoftcomproductsmaplefeaturesmatlab-connectivityaspx

[15] E Madenci and I Guven The Finite Element Method andApplications in Engineering Using ANSYS Springer BerlinGermany 1st edition 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Journal of Applied Mathematics

(a) (b)

Figure 9 Nonlinear FEM results for the both proposed and Pedersenrsquos methods (a) initial and final wireframemeshes are overlaid (b) initialand final shaded meshes are overlaid

(a) (b)

Figure 10 The liver mesh is constrained at the blue nodes and pulled from the green node ((a) initial nodes (b) initial shaded mesh andnodes)

Table 9 Computation times (in seconds) of stiffness matrices(Pedersen Pedersenrsquos nonlinear FEM Proposed proposed nonlin-ear FEM and Speed-up relative performance comparison of thestiffness matrix calculation of the proposed nonlinear FEMmethodwith Pedersenrsquos nonlinear FEMmethod using single thread)

Exp Elements Pedersen Proposed Speed-up1st 6 07322 03308 22134222nd 216 281624 111542 25248253rd 224 301094 122725 24534044th 1580 2398753 967840 2478460

Table 10 Computation times (s) of system solutions (PedersenPedersenrsquos nonlinear FEM Proposed proposed nonlinear FEMand Speed-up relative performance comparison of the proposednonlinear FEM method with Pedersenrsquos nonlinear FEM methodusing single thread)

Exp Elements Pedersen Proposed Speed-up1st 6 30144 24427 12340442nd 216 1925288 1596241 12061393rd 224 5862708 6125911 9570344th 1580 28407558 24010994 1183106

Table 11 Newton-Raphson iteration count to reach desired accu-racy (Pedersen Pedersenrsquos nonlinear FEM Proposed proposednonlinear FEM)

Exp Elements Pedersen Proposed1st 6 5 52nd 216 7 73rd 224 26 324th 1580 8 8

on the computation times for both methods Figure 8 showsthat the cross shape is constrained at the blue nodes andpushed towards the green nodes Figure 9 shows the finalshape of the beammesh for both the proposed and Pedersenrsquosmethods Tables 5 and 6 show the displacements at forceapplied nodes (green nodes) using the linear and nonlinearFEMs respectively

We conducted fourth experiment with a livermesh of 465nodes and 1560 tetrahedral elements Figure 10 shows thatthe mesh is constrained at the blue nodes and pulled fromthe green node (30N units) in the direction of the arrow Weaimed to observe the similar amount of speedup like previousexperiments for a high density mesh Figure 11 shows thefinal shape of the beam mesh for both the proposed andPedersenrsquos methods Tables 7 and 8 show the displacementsat force applied node (node number 271) using the linear andnonlinear FEMs respectively

Computation times of the finite element experiments arerequired to compare how much faster our proposed methodis than Pedersenrsquos When comparing nonlinear FEMs wecalculated the computation times to construct the stiffnessmatrices as well as the computation times of the nonlinearFEM solutions to determine how different calculations affectthem Table 9 depicts the computation times for the stiffnessmatrix calculation and Table 10 depicts the computationtimes for the system solution Table 11 shows the iterationcounts to solve the system using the Newton-Raphson pro-cedure

The speed-up columns of Tables 9 and 10 depict thespeedups of the proposed method compared to Pedersenrsquosmethod for the stiffness matrix calculation and the systemsolution using a single thread respectively The speedup iscalculated as follows

Speedup =Runtime (Pedersenrsquosmethod)

Runtime (Theproposedmethod) (40)

Journal of Applied Mathematics 11

(a) (b)

Figure 11 Nonlinear FEM results for both the proposed and Pedersenrsquos methods (a) left wireframe surface mesh right wireframe surfacemesh with nodes (b) left shaded mesh right shaded mesh with nodes

(a) (b)

(c)

Figure 12 (a) Initial misshapen nose (b) The head mesh is constrained at the blue nodes and pushed upwards at the green nodes (c) Theresult of the nonlinear FEM left wireframe surface mesh with nodes right shaded mesh with texture

Our proposed method outperforms Pedersenrsquos methodOn the average it is 142 faster at computing stiffnessmatrices because Pedersenrsquos method uses more symbolicterms However bothmethods use Newton-Raphson to solvenonlinear equations which takes approximately 90 of thecomputation time Thus the overall speedup decreases to15 on average In experiment 3 because of more iterationsdue to root jumps there was a performance loss againstPedersenrsquos method

We also applied our method for corrective operationon the misshapen nose of a head mesh The head mesh iscomposed of 6709 nodes and 25722 tetrahedral elements (seeFigure 12(a)) all the operations were performed in the nose

area of only 1458 tetrahedral elements Figure 12(b) showsthat the head mesh is constrained at the blue nodes andpushed upwards at the green nodes and Figure 12(c) showsthe result of the nonlinear FEM

4 Conclusions and Future Work

We propose a new stiffness matrix calculation method fornonlinear FEM that is easier to analyze in terms of construct-ing elemental stiffness matrices and is faster than Pedersenrsquosmethod The proposed method is approximately 24 timesfaster on average at computing stiffness matrices and 15faster at computing thewhole system thanPedersenrsquosmethod

12 Journal of Applied Mathematics

Although the proposed nonlinear FEM has significantadvantages over Pedersenrsquos nonlinear FEM there is still roomfor the following development

(1) Heuristics could be applied to avoid root jumps(2) Although we decreased system memory usage by

simplifying the solution process for the nonlinearFEM a significant amount of system memory is stillusedThe solution process could be further optimizedto decrease memory usage

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] K-J Bathe Finite Element Procedures Prentice-Hall Engle-wood Cliffs NJ USA 1996

[2] D L Logan A First Course in the Finite Element MethodCengage Learning 5th edition 2012

[3] P Pedersen ldquoAnalytical stiffness matrices for tetrahedral ele-mentsrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 196 no 1-3 pp 261ndash278 2006

[4] G R Joldes A Wittek and K Miller ldquoReal-time nonlinearfinite element computations on GPU application to neurosur-gical simulationrdquo Computer Methods in Applied Mechanics andEngineering vol 199 no 49ndash52 pp 3305ndash3314 2010

[5] Z A Taylor M Cheng and S Ourselin ldquoHigh-speed nonlinearfinite element analysis for surgical simulation using graphicsprocessing unitsrdquo IEEE Transactions on Medical Imaging vol27 no 5 pp 650ndash663 2008

[6] M Cerrolaza and J C Osorio ldquoRelations among stiffnesscoefficients of hexahedral 8-noded finite elements a simple andefficient way to reduce the integration timerdquo Finite Elements inAnalysis and Design vol 55 pp 1ndash6 2012

[7] J Bonet and R D Wood Nonlinear Continu um Mechanicsfor Finite Element Analysis Cambridge University Press Cam-bridge UK 1st edition 1997

[8] C A Felippa ldquoLecture notes in nonlinear fin ite element meth-odsrdquo Tech Rep CUCSSC-96-16 Department of AerospaceEngineering Sciences and Center for Aerospace StructuresUniversity of Colorado 1996

[9] P Pedersen The Basic Matrix Approach for Three Simple FiniteElements 2008 httpwwwtopoptdtudkfilesPauliBooksBasicMatrixApproachInFEpdf

[10] C T Kelley Iterative Methods for Linear and Nonlinear Equa-tions Frontiers in Applied Mathematics Society for IndustrialMathematics 1st edition 1987

[11] S Krenk Non-linear Modeling and Analysi s of Solids andStructures Cambridge University Press 1st edition 2009

[12] The Mathworks ldquoCalling MATLAB Engine from CC++and Fortran Programsrdquo 2014 httpwwwmathworkscomhelpmatlabmatlab external f38569html

[13] Autodesk ldquo3ds Maxmdash3D Modeling Ani mation and Render-ing Softwarerdquo 2012 httpwwwautodeskcomproductsauto-desk-3ds-max

[14] Maplesoft ldquoMATLAB ConnectivitymdashMaple Featuresrdquo 2012httpwwwmaplesoftcomproductsmaplefeaturesmatlab-connectivityaspx

[15] E Madenci and I Guven The Finite Element Method andApplications in Engineering Using ANSYS Springer BerlinGermany 1st edition 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 11

(a) (b)

Figure 11 Nonlinear FEM results for both the proposed and Pedersenrsquos methods (a) left wireframe surface mesh right wireframe surfacemesh with nodes (b) left shaded mesh right shaded mesh with nodes

(a) (b)

(c)

Figure 12 (a) Initial misshapen nose (b) The head mesh is constrained at the blue nodes and pushed upwards at the green nodes (c) Theresult of the nonlinear FEM left wireframe surface mesh with nodes right shaded mesh with texture

Our proposed method outperforms Pedersenrsquos methodOn the average it is 142 faster at computing stiffnessmatrices because Pedersenrsquos method uses more symbolicterms However bothmethods use Newton-Raphson to solvenonlinear equations which takes approximately 90 of thecomputation time Thus the overall speedup decreases to15 on average In experiment 3 because of more iterationsdue to root jumps there was a performance loss againstPedersenrsquos method

We also applied our method for corrective operationon the misshapen nose of a head mesh The head mesh iscomposed of 6709 nodes and 25722 tetrahedral elements (seeFigure 12(a)) all the operations were performed in the nose

area of only 1458 tetrahedral elements Figure 12(b) showsthat the head mesh is constrained at the blue nodes andpushed upwards at the green nodes and Figure 12(c) showsthe result of the nonlinear FEM

4 Conclusions and Future Work

We propose a new stiffness matrix calculation method fornonlinear FEM that is easier to analyze in terms of construct-ing elemental stiffness matrices and is faster than Pedersenrsquosmethod The proposed method is approximately 24 timesfaster on average at computing stiffness matrices and 15faster at computing thewhole system thanPedersenrsquosmethod

12 Journal of Applied Mathematics

Although the proposed nonlinear FEM has significantadvantages over Pedersenrsquos nonlinear FEM there is still roomfor the following development

(1) Heuristics could be applied to avoid root jumps(2) Although we decreased system memory usage by

simplifying the solution process for the nonlinearFEM a significant amount of system memory is stillusedThe solution process could be further optimizedto decrease memory usage

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] K-J Bathe Finite Element Procedures Prentice-Hall Engle-wood Cliffs NJ USA 1996

[2] D L Logan A First Course in the Finite Element MethodCengage Learning 5th edition 2012

[3] P Pedersen ldquoAnalytical stiffness matrices for tetrahedral ele-mentsrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 196 no 1-3 pp 261ndash278 2006

[4] G R Joldes A Wittek and K Miller ldquoReal-time nonlinearfinite element computations on GPU application to neurosur-gical simulationrdquo Computer Methods in Applied Mechanics andEngineering vol 199 no 49ndash52 pp 3305ndash3314 2010

[5] Z A Taylor M Cheng and S Ourselin ldquoHigh-speed nonlinearfinite element analysis for surgical simulation using graphicsprocessing unitsrdquo IEEE Transactions on Medical Imaging vol27 no 5 pp 650ndash663 2008

[6] M Cerrolaza and J C Osorio ldquoRelations among stiffnesscoefficients of hexahedral 8-noded finite elements a simple andefficient way to reduce the integration timerdquo Finite Elements inAnalysis and Design vol 55 pp 1ndash6 2012

[7] J Bonet and R D Wood Nonlinear Continu um Mechanicsfor Finite Element Analysis Cambridge University Press Cam-bridge UK 1st edition 1997

[8] C A Felippa ldquoLecture notes in nonlinear fin ite element meth-odsrdquo Tech Rep CUCSSC-96-16 Department of AerospaceEngineering Sciences and Center for Aerospace StructuresUniversity of Colorado 1996

[9] P Pedersen The Basic Matrix Approach for Three Simple FiniteElements 2008 httpwwwtopoptdtudkfilesPauliBooksBasicMatrixApproachInFEpdf

[10] C T Kelley Iterative Methods for Linear and Nonlinear Equa-tions Frontiers in Applied Mathematics Society for IndustrialMathematics 1st edition 1987

[11] S Krenk Non-linear Modeling and Analysi s of Solids andStructures Cambridge University Press 1st edition 2009

[12] The Mathworks ldquoCalling MATLAB Engine from CC++and Fortran Programsrdquo 2014 httpwwwmathworkscomhelpmatlabmatlab external f38569html

[13] Autodesk ldquo3ds Maxmdash3D Modeling Ani mation and Render-ing Softwarerdquo 2012 httpwwwautodeskcomproductsauto-desk-3ds-max

[14] Maplesoft ldquoMATLAB ConnectivitymdashMaple Featuresrdquo 2012httpwwwmaplesoftcomproductsmaplefeaturesmatlab-connectivityaspx

[15] E Madenci and I Guven The Finite Element Method andApplications in Engineering Using ANSYS Springer BerlinGermany 1st edition 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Journal of Applied Mathematics

Although the proposed nonlinear FEM has significantadvantages over Pedersenrsquos nonlinear FEM there is still roomfor the following development

(1) Heuristics could be applied to avoid root jumps(2) Although we decreased system memory usage by

simplifying the solution process for the nonlinearFEM a significant amount of system memory is stillusedThe solution process could be further optimizedto decrease memory usage

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] K-J Bathe Finite Element Procedures Prentice-Hall Engle-wood Cliffs NJ USA 1996

[2] D L Logan A First Course in the Finite Element MethodCengage Learning 5th edition 2012

[3] P Pedersen ldquoAnalytical stiffness matrices for tetrahedral ele-mentsrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 196 no 1-3 pp 261ndash278 2006

[4] G R Joldes A Wittek and K Miller ldquoReal-time nonlinearfinite element computations on GPU application to neurosur-gical simulationrdquo Computer Methods in Applied Mechanics andEngineering vol 199 no 49ndash52 pp 3305ndash3314 2010

[5] Z A Taylor M Cheng and S Ourselin ldquoHigh-speed nonlinearfinite element analysis for surgical simulation using graphicsprocessing unitsrdquo IEEE Transactions on Medical Imaging vol27 no 5 pp 650ndash663 2008

[6] M Cerrolaza and J C Osorio ldquoRelations among stiffnesscoefficients of hexahedral 8-noded finite elements a simple andefficient way to reduce the integration timerdquo Finite Elements inAnalysis and Design vol 55 pp 1ndash6 2012

[7] J Bonet and R D Wood Nonlinear Continu um Mechanicsfor Finite Element Analysis Cambridge University Press Cam-bridge UK 1st edition 1997

[8] C A Felippa ldquoLecture notes in nonlinear fin ite element meth-odsrdquo Tech Rep CUCSSC-96-16 Department of AerospaceEngineering Sciences and Center for Aerospace StructuresUniversity of Colorado 1996

[9] P Pedersen The Basic Matrix Approach for Three Simple FiniteElements 2008 httpwwwtopoptdtudkfilesPauliBooksBasicMatrixApproachInFEpdf

[10] C T Kelley Iterative Methods for Linear and Nonlinear Equa-tions Frontiers in Applied Mathematics Society for IndustrialMathematics 1st edition 1987

[11] S Krenk Non-linear Modeling and Analysi s of Solids andStructures Cambridge University Press 1st edition 2009

[12] The Mathworks ldquoCalling MATLAB Engine from CC++and Fortran Programsrdquo 2014 httpwwwmathworkscomhelpmatlabmatlab external f38569html

[13] Autodesk ldquo3ds Maxmdash3D Modeling Ani mation and Render-ing Softwarerdquo 2012 httpwwwautodeskcomproductsauto-desk-3ds-max

[14] Maplesoft ldquoMATLAB ConnectivitymdashMaple Featuresrdquo 2012httpwwwmaplesoftcomproductsmaplefeaturesmatlab-connectivityaspx

[15] E Madenci and I Guven The Finite Element Method andApplications in Engineering Using ANSYS Springer BerlinGermany 1st edition 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of