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Research ArticleFast Stiffness Matrix Calculation for NonlinearFinite Element Method

Emir Glmser,1 ULur Gdkbay,1 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; [email protected]

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

Academic Editor: Henggui Zhang

Copyright 2014 Emir Gulumser et al.This is an open access article distributed under the Creative CommonsAttribution License,which 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 them.We 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 games.The 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 flow,mass 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 GPUs;however, 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 Pedersens method [3]to measure performance and verify correctness.We achieve a142% speedup in calculating the stiffness matrices and a 15%

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014, Article ID 932314, 12 pageshttp://dx.doi.org/10.1155/2014/932314

2 Journal of Applied Mathematics

speedup in solving the whole system on average, compared toPedersens 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]

{} = { (, , )} =

{{{{{{{{{{

{{{{{{{{{{

{

1

V1

1

...4

V4

4

}}}}}}}}}}

}}}}}}}}}}

}

. (1)

In global coordinates, we represent displacements by linearfunction by

(, , ) =

1+ 2 + 3 + 4. (2)

For all 4 vertices, (2) is extended as

[[[

[

1 1

1

1

1 2

2

2

1 3

3

3

1 4

4

4

]]]

]

{{{

{{{

{

1

2

3

4

}}}

}}}

}

=

{{{

{{{

{

1

2

3

4

}}}

}}}

}

. (3)

Constants can be found as

= V1

, (4)

where V1 is given by

V1 =1

det (V)[[[

[

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

]]]

]

. (5)

det(V) is 6, where is the volume of the tetrahedron. If wesubstitute (4) into (2), we obtain

(, , ) =

1

6[1 ]

[[[

[

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

]]]

]

{{{

{{{

{

1

2

3

4

}}}

}}}

}

.

(6)

, , , , and the volume are calculated by

1=

2

2

2

3

3

3

4

4

4

, 2=

1

1

1

3

3

3

4

4

4

,

3=

1

1

1

2

2

2

4

4

4

, 4=

1

1

1

2

2

2

3

3

3

,

1=

1 2

2

1 3

3

1 4

4

, 2=

1 1

1

1 3

3

1 4

4

,

3=

1 1

1

1 2

2

1 4

4

, 4=

1 1

1

1 2

2

1 3

3

,

1=

1 2

2

1 3

3

1 4

4

, 2=

1 1

1

1 3

3

1 4

4

,

3=

1 1

1

1 2

2

1 4

4

, 4=

1 1

1

1 2

2

1 3

3

,

1=

1 2

2

1 3

3

1 4

4

, 2=

1 1

1

1 3

3

1 4

4

,

3=

1 1

1

1 2

2

1 4

4

, 4=

1 1

1

1 2

2

1 3

3

,

6 =

1

1

1

1

.

(7)

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

(, , )

=1

6

[[[

[

1+ 1 + 1 + 1

2+ 2 + 2 + 2

3+ 3 + 3 + 3

4+ 4 + 4 + 4

]]]

]

[(, , )1

(, , )2

(, , )3

(, , )4] .

(8)

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

1, 2, 3, 4). They

are given by

1=

1

6(1+ 1 + 1 + 1) (, , )

1,

2=

1

6(2+ 2 + 2 + 2) (, , )

2,

3=

1

6(3+ 3 + 3 + 3) (, , )

3,

4=

1

6(4+ 4 + 4 + 4) (, , )

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

B

B

A

A

uy

xdx

dx +ux

xdx

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 theory.The proposed nonlinearFEM uses the linear FEM framework but it does not requirethe explicit use of weight functions and differential equations.Hence, 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 Logans 2Dlinear FEM to 3D [2]. To understandGreen-Lagrange strains,wemust first see how they differ from the infinitesimal strainsused to calculate the global stiffness matrices in a linear FEM.Figure 1 shows a 2D element before and after deformation,where the element edge with initial length becomes. The engineering normal strain is calculated as the

change in the length of the line

=

||

||. (10)

The final length of the elemental edge can be calculated using

2

= ( +

)

2

+ (

)

2

,

2

= 2[1 + 2(

) + (

)

2

+ (

)

2

] .

(11)

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

=

,

=

,

=

1

2(

+

) .

(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 -axis, -axis, and -axis arevery small. The initial and final positions of a given particleare practically the same; thus, the higher-order terms areneglected [7]. When the