00 Nonlinear Mean Strain Hex - Revision

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING

Int. J. Numer. Meth. Engng2014;00:1–24

Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme

Mean-strain 8-node Hexahedron with OptimizedEnergy-Sampling Stabilization for Large-strain Deformation

P. Krysl1,∗

1University of California, San Diego, 9500 Gilman Dr., #0085, La Jolla, CA 92093

SUMMARY

A method for stabilizing the mean-strain hexahedron for applications to anisotropic elasticity was describedby Krysl (in IJNME 2014). The technique relied on a sampling of the stabilization energy using themean-strain quadrature and the full Gaussian integration rule. This combination was shown to guaranteeconsistency and stability. The stabilization energy was expressed in terms of input parameters of the realmaterial, and the value of the stabilization parameter was fixed in a quasi-optimal manner by linking thestabilization to the bending behavior of the hexahedral element (Krysl, submitted). Here the formulation isextended to large-strain hyperelasticity (as an example, the formulation allows for inelastic behavior to bemodeled). The stabilization energy is expressed through a stored-energy function, and contact with inputparameters in the small-strain regime is made. As for small-strain elasticity, the stabilization parameter isdetermined to optimize bending performance. The accuracy and convergence characteristics of the presentformulations for both solid and thin-walled structures (shells) compare favorably with the capabilities ofmean-strain and other high-performance hexahedral elements described in the open literature and also witha number of successful shell elements. Copyrightc© 2014 John Wiley & Sons, Ltd.

Received . . .

KEY WORDS: anisotropic; hyperelasticity; nearly incompressible; mean-strain hexahedron; uniformstrain hexahedron; finite element; shell

INTRODUCTION

Eight-node mean-strain hexahedra have been shown to be efficient and robust for large strain 3-

D analysis [1, 2]. The formulation must accomplish at the same time (i) locking-free response,

volumetric and in shear, (ii) good coarse-mesh accuracy, and (iii) stability. Strictly mean-strain

hexahedra achieve locking-free response, but lose stability. Adding stability, for instance by treating

the so-called hourglassing modes, tends to reintroduce locking. Coarse-mesh accuracy poses another

demand on stabilization: namely that the stabilization should not deteriorate the response of

the element, but in fact enhance the ability of the elements to respond to deformations in the

hourglassing modes such as bending or torsion accurately. Puso’s hexahedron is a good example

of a successful approach [3].

∗Correspondence to: University of California, San Diego, 9500 Gilman Dr., #0085, La Jolla, CA 92093

Contract/grant sponsor: Michael Weise at the Office of Naval Research; contract/grant number: N00014-09-1-0611

Copyright c© 2014 John Wiley & Sons, Ltd.

Prepared usingnmeauth.cls [Version: 2010/05/13 v3.00]

2 KRYSL

A method for stabilizing the mean-strain hexahedron that differed from the then-known

approaches was described by Krysl [4]. The technique relied on a sampling of the stabilization

energy using two quadrature rules, the mean-strain quadrature and the full Gaussian integration

rule. The use of two quadrature rules was shown to guarantee both consistency and stability. The

stabilization energy was assumed to be generated by a modified constitutive matrix based on the

spectral decomposition. The stabilization required user-selected values of stabilization parameters,

which is in general undesirable.

In a subsequent work, again with application to anisotropic elasticity, the arbitrariness of the

stabilization parameters was eliminated (Krysl, under review). The parameters of the stabilization

material were expressed in terms of the input parameters of the real material with suitable

modifications to avoid locking due to volumetric and other constraints (for instance due to presence

of stiff reinforcing fibers – anisotropy induced locking). The value of the remaining stabilization

parameter was fixed in a quasi-optimal manner by linking the stabilization to the bending behavior

of the hexahedral element.

In the present work we extend these results to the large-strain hyperelasticity regime. In Section 1

the formulation is derived from a variational principle and a stored-energy function is used for

the stabilization energy. Picking a suitable stored-energy function, such as the neo-Hookean form,

allows for the input parameters of the real material to be related to the properties of the stabilization

energy. Volumetric locking can be thereby eliminated from the start. The bending response in the

small-strain regime is again used as the source of the remaining stabilization parameter value.

The stabilization is then linked to the shape of the element, allowing for the shear locking to be

eliminated, which improves coarse-mesh accuracy.

Section 2 illustrates the performance of the proposed approaches on a variety of benchmark

problems, for isotropic and anisotropic material models. Importantly, the coarse-mesh response is

significantly improved by the choice of the stabilization parameter. This proves valuable especially

in applications to thin shells, where the present element is shown to match the performance of

specialized shell and plate elements. The performance of the stabilization is also tested for highly

distorted elements in a large-strain problem. The present stabilization technique is also shown to

work for anisotropic materials. Practically important linear buckling problems are analyzed to assess

the effectiveness of the stabilization in eigenvalue analyses. Significantly, no modifications of the

stabilization scheme are necessary to address instability problems.

The accuracy and convergence characteristics of the present formulations compare favorably

with the capabilities of mean-strain and other high-performance hexahedral elements as described

in the open literature. In addition, we compare with a number of successful shell elements

and we demonstrate that the present element performs very well for thin structures. Crucially,

the hexahedron formulation presented here eliminates the need for user-selected values of the

stabilization parameters. Together with excellent performance this makes the present element a good

general-purpose hexahedron.

Copyright c© 2014 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng(2014)Prepared usingnmeauth.cls DOI: 10.1002/nme

MEAN-STRAIN 8-NODE STABILIZED HEX FOR LARGE-STRAIN DEFORMATION 3

1. METHODS

1.1. Assumed-gradient formulation

For simplicity we will restrict ourselves in this work to models of large-strain hyperelasticity.

(Nothing prevents the same approach from working for inelastic material models, the present choice

is made to limit the length of the manuscript.) We will re-derive the formulation of the mean-strain

eight-node hexahedra from the variational standpoint. The present formulation will be at variance

with the original references [5, 6] in that the kinematic equations will be used to derive the assumed

deformation gradient, which then immediately leads to a single-field variational principle.

We shall take as the starting point the functional

Π(F ,Λ, u) =∫

Ω0

[U(F ) + tr

(ΛT

(∇0φ − F

))]dΩ −W . (1)

Here φ(X, t) = X + u is the motion map, whereX is the particle reference-coordinate label,

∇0 = ∂/∂XjEj is the partial differentiation operator with the respect to coordinate curves in

the reference configuration (assumed Cartesian, without any loss of generality);u(X, t) is the

displacement; andΩ0 is the undeformed domain.Furthermore,F is the assumed deformation

gradient, andΛ is a second-order tensor of Lagrange multipliers. The deformation gradient∇0φ

is evaluated from the finite elements kinematics (interpolation), and therefore we refer to∇0φ

as driven by the element-kinematics. Contrariwise, the assumed deformationgradientF is at this

point an independent variable of the functional, and the Lagrange multiplier tensorΛ is used to

weakly enforce the subsidiary condition that these two deformation gradients should be related for

convergence to occur.

The strain energy is generated by the right-stretch tensor or the Cauchy-Green deformation tensor.

However, these quantities are computed from the assumed deformationgradientF , so to keep the

argument general we use the notationU(F ).

The first variation of the functional (1) can be worked out as

δΠ(F ,Λ, u) =∫

Ω0

[∂U(F )

∂F∙ δF + tr

(δΛT

(∇0φ − F

))+ tr

(ΛT

(∇0δφ − δF

))]

dΩ − δW .

(2)

A key step taken at this point consists of eliminating the Lagrange multiplierΛ from further

consideration. We will accomplish this by requiring the second and third term to vanish identically

∫

Ω0

[tr(δΛT

(∇0φ − F

))+ tr

(ΛT

(∇0δφ − δF

))]dΩ = 0 . (3)

In this way we will obtain a condition from which the assumed deformation gradient can be

obtained. First, the Lagrange multiplier field will be chosen to be piecewise constant over each


4 KRYSL

hexahedral finite element. Therefore, in the discrete version, where the integral is evaluated element-

by-element, Equation (3) will read

∑

e

∫

Ω(e)0

[tr(δΛT

(∇0φ − F

))+ tr

(ΛT

(∇0δφ − δF

))]dΩ =

∑

e

tr

(

δΛT

∫

Ω(e)0

(∇0φ − F

)dΩ

)

+∑

e

tr

(

ΛT

∫

Ω(e)0

(∇0δφ − δF

)dΩ

)

= 0 (4)

where the sum ranges over all the elementse in the mesh. One possible (and local, hence

inexpensive) way in which (4) can be satisfied is based upon satisfying

∫

Ω(e)0

(∇0φ − F

)dΩ = 0 ∀e , (5)

and ∫

Ω(e)0

(∇0δφ − δF

)dΩ = 0 ∀e . (6)

In this way, when we choose to construct the assumed deformation gradient as piecewise constant

over each element, we obtain from(5)

F(e)

= V(e)0

−1∫

Ω(e)0

∇0φ dΩ ∀e (7)

and

δF(e)

= V(e)0

−1∫

Ω0

∇0δu dΩ . (8)

Here and for future reference we define the volume of the hexahedral element in the reference

configuration

V(e)0 =

∫

Ω(e)0

dΩ . (9)

A couple of remarks:

• In the interest of precision, here we write the elementwise deformation gradientasF(e)

and

δF(e)

. Usually, when there is no possibility of creating confusion, we will simply use the

notationF andδF , where the association of these quantities with an element is implied.

• Also, we may define the assumed variation of displacement gradientas∇0δu = δF .

The above procedure effectively whittled the number of independent fields in (1) from three to

one. The first variation of the functional was reduced to the balance equation

δΠ(u) =∫

Ω0

tr(∇0δu

TP)

dΩ − δW . (10)

where the first Piola-Kirchhoff stress was introducedas

P =∂U(F )

∂F. (11)

Nota bene that equation (10) looks as if springing from a single-field functional; however, the

quantities are barred as they are derived by application of theassumedgradient operators.



The development in this section is an application of the multi-field methodology. We may

recall a related derivation by Bonet and Bhargava [7], where the original element of Flanagan

and Belytschko [5] is re-derived for the hyper-elastic, total deformation as opposed to strain-rate

deformation, setting. The Bonet and Bhargava (BB) hexahedron is formulated by postulating a

volumetric/deviatoric split. Consequently, their application of the multi-field methodology requires

five independent variables, which are reduced by linking the discrete volumetric deformation and

the discrete deviatoric part of the deformation gradient to the element kinematics through volume

averaging. The BB methodology claims to improve the representation of near incompressibility by

expressing the volumetric deformation as the ratio of the volumes of the element in the deformed

and undeformed configuration. This approach is not taken here, but we may note that convergence

is not adversely affected. As can be easily demonstrated, the determinant of the mean deformation

gradient is equal to the mean of the determinant of the deformation gradient over the volume of the

element

det F(e)

= V(e)0

−1∫

Ω(e)0

det F dΩ =V (e)

V(e)0

, (12)

for deformations that result in a uniform deformation gradient. Since a uniform deformation gradient

is a precondition for convergence to occur in the limit of the element sizeh approaching zero, the

two conditions that are linked to incompressibility (the BB condition of the discrete volumetric

change being equal to zero [7], and the condition in the present work ofdet F = 1) will have an

identical effect ash → 0.

1.2. Stabilization

It is understood that the mean-strain hexahedron (as derived above from the functional (1))

lacks stability [5]. In order to stabilize the element we will consider the energy-sampling

method [4]. Our approach to the stabilization differs from the well-known approaches of Flanagan

and Belytschko [5, 2] and Bonet and Bhargava [7] (perturbation hourglassing stabilization); of

Belytschko and Bindemann [6] (assumed-strain stabilization), of Puso [3] and Reese [8] and of

Reese [9] (enhanced-strain (incompatible mode) stabilization); and of Belytschko and Bachrach [10]

and Liu et al. [11] (physical stabilization based on the Taylor-series expansion of the strains).

The stabilization strain-energy density to be sampled is assumed in the form of a hyperelastic

constitutive equation. For small to moderate strains, the St. Venant-Kirchhoff material model would

suffice. Here we adopt the more realistic neo-Hookean material model with the strain-energy density

per unit undeformed volume expressed as [12]

U(C) =μ

2(tr(C) − 3) − μ ln(J) +

λ

2(ln(J))2 (13)

whereC is the right Cauchy-Green deformation tensor, andλ and μ are the small-strain Lame

parameters. For the sake of brevity we will refer to the stabilization strain-energy density as a

function of the deformation gradient, using the same letter

U(F ) = U(C) (14)

whereC = F T F is implied.


6 KRYSL

Following the ideas of [4] where the domain of study was small-strain elasticity, the functional (1)

will be modified by the simultaneous addition and subtraction of the stabilization energy to read

Π(F ,Λ, u) =∫

Ω0

[U(F ) + tr

(ΛT

(∇0φ − F

))]dΩ +

∫

Ω0

U(∇0φ) dΩ −∫

Ω0

U(F ) dΩ −W ,

(15)

where, importantly, the stabilization energy is generated by either the element-kinematics-driven

∇0φ or by the assumed deformationgradientF .

As described above, the second term in (15) is again going to be used to derive the assumed

deformation gradient, and the balance equation then follows from the first variation of (15) as

δΠ(u) =∫

Ω0

tr(∇0δu

TP)

dΩ +∫

Ω0

tr(∇0δu

T P)

dΩ −∫

Ω0

tr(∇0δu

TP)

dΩ − δW . (16)

where the first Piola-Kirchhoff stress due to the stabilization energy was introduced as

P =∂U(∇0φ)

∂∇0φand P =

∂U(F )

∂F. (17)

The balance equation can be expressed in terms of the Cauchy stress using thedefinition

σ = (det F )−1P FT

, (18)

as

δΠ(u) =∫

Ω0

tr(∇δu

Tσ)

det F dΩ

+∫

Ω0

tr(∇δuT σ

)det∇φ dΩ −

∫

Ω0

tr(∇δu

Tσ)

det F dΩ − δW . (19)

Here we have defined the assumed gradient with respect to the currentconfiguration

∇δu = ∇0δu ∙ F−1

. (20)

1.3. Numerical integration

The terms in the balance equation (19) are evaluated with numerical quadrature. The quadrature is

selected to provide both convergence and stability.

Thus, the integrands of the first and third term in (19) are seen to be constants and hence we can

write∫

Ω0

tr(∇δu

Tσ)

det F dΩ −∫

Ω0

tr(∇δu

Tσ)

det F dΩ = tr[∇δu

T(σ − σ

)]det F V

(e)0 .

(21)

The second term in (19) is evaluated with2 × 2 × 2 Gauss quadrature.

∫

Ω0

tr(∇δuT σ

)det∇φ dΩ ≈

8∑

k=1

tr(∇δu(ξk)T

σ(ξk))

det∇φ(ξk)JkWk . (22)



This is the same construction as for small-strain analysis (anisotropic elasticity) as discussed

in [4]. As a result we are guaranteed

Stability The stress produced by hourglassing deformations is nonzero when evaluated by (22)

as the measures of strain at the quadrature points of the Gauss rule are nonzero for such

deformations.

ConvergenceWhen the displacement-related deformation gradient becomes uniform over the

element volume, the second and third integral in (19) will cancel. Hence, the response of

the element will be entirely due to the first integral, as expected for convergence to the correct

solution to occur. (In the limit of infinite refinement when the deformation gradient becomes

uniform over each element, the correct state of stress is uniform.)

In summary, sampling the two stabilization energy contributions,U(∇0φ), and U(F ), by two

suitably selected quadrature rules we simultaneously achieve guaranteed stability and convergence

of the mean-strain hexahedron to the correct solution.

1.4. Stabilization energy

Krysl [4] proposed a quasi-optimal expression for the stabilization energy for the stress analysis of

linearly elastic anisotropic solids. There were two main ideas:

• Match the properties of the stabilization energy to the properties of the actual material, while

ensuring that the2 × 2 × 2 Gauss quadrature rule would not cause locking.

• Adjust the properties of the stabilization energy to endow the element with good flexural

response.

As pointed out in the previous section, we are adopting as the stabilization strain-energy function

a hyperelastic constitutive equation of the neo-Hookean solid. The Lame parameters that define the

small-strain response can be expressed in terms of the Young’s modulus and the Poisson ratio

λ =Eν

(1 + ν)(1 − 2ν), μ =

E

2(1 + ν). (23)

It is suggested in [4] to take as the catch-all values either

E = EΦ

1 + Φ, ν = 0.3 (24)

for an isotropic material with the Young’s modulusE and the Poisson ratioν, or, for an orthotropic

material with moduliEj , j = 1, 2, 3, the expression

E = min[Ej ]Φ

1 + Φ, ν = 0.3 . (25)

to take into account the possibility of extensibility constraints due to distributed stiff fibers that are

sometimes modeled using this type of material. Note that the Poisson ratio of the “stabilization

material” is taken at a value independent of the properties of the actual material. Volumetric locking

is thereby eliminated from consideration.


8 KRYSL

The correction parameter that adjusts the stiffness of the stabilization material to reflect the

geometry of the element in an attempt to improve its bending response is taken as [4]

Φ = 2(1 + ν)min[h2

x, h2y, h2

z]

max[h2x, h2

y, h2z]

(26)

wherehx, hy, hz are the norms of the columns of the reference-configuration Jacobian matrix of the

element evaluated at the centroid of the parametric shape (i.e. atξ = η = ζ = 0 of the tri-unit cube).

The role of this parameter is to produce the correct “bending” energy when the element is deformed

into a pure-bending pattern.

Note that Equations (24), (25), and (26) define the stabilization material propertieswithout any

user input. All that is required are the properties of the actual material and measures of the geometry

of the element (the length of the sides of the equivalent brick [13]). The element formulation

presented above will be labeledH8MSGSO(Hexahedral 8-node Mean-Strain Gauss-rule Stabilized

with Optimized performance).

1.5. Implementation

We will briefly describe the evaluation of the balance equation (16), and the solution process using

Newton’s method with a consistent tangent stiffness. The second term in (16) is entirely standard,

and conforms fully to the isoparametric displacement-based finite element format. The first and

third term are evaluated the same way, and therefore we will describe the procedure only for the

first term.

The directional derivative of the term∫

Ω0

tr[∇0δu

TP]

dΩ (27)

will lead to the tangent stiffness. First, in order to simplify the derivations, we will define the second

Piola-Kirchhoff stress tensoras

S = F−1

P . (28)

Hence, we obtain ∫

Ω0

tr[∇0δu

TP]

dΩ =∫

Ω0

tr[∇0δu

TF S

]dΩ . (29)

The linearization of this last expression in the direction ofΔu leads to

D

(∫

Ω0

tr[∇0δu

TF S

]dΩ

)

[Δu] =∫

Ω0

tr[∇0δu

TF DS[Δu]

]dΩ

+∫

Ω0

tr[D(∇0δu

TF)

[Δu]S]

dΩ . (30)

Next, analogously to (8) we define the linearization of the assumed displacement gradient as

D(F)[Δu] = D

(∇0u

)[Δu] = V

(e)0

−1∫

Ω(e)0

∇0Δu dΩ . (31)



1.5.1. Geometrical tangent stiffnessThe second term on the right-hand side of (30) may be

rewritten as∫

Ω0

tr[D(∇0δu

TF)

[Δu]S]

dΩ =∫

Ω0

tr[∇0δu

T∇0Δu S

]dΩ , (32)

which is a standard form for the geometrical stiffness operator computed in the reference

configuration. The only difference with respect to the classical formulation is the use of the assumed-

strain operators. Using the definition of the Cauchy stress tensor (18) and the definition of the

assumed gradient of displacement increment in the currentconfiguration∇Δu analogous to(20)

∇Δu = ∇0Δu ∙ F −1 , (33)

we can write the “Eulerian” form of the geometrical tangent stiffness

∫

Ω0

tr[S∇0Δu ∇0δu

T]

dΩ =∫

Ω0

tr[σ∇Δu ∇η

T]det F dΩ . (34)

Note that all quantities underneath the integral sign are elementwise constant.

1.5.2. Material tangent stiffnessUsing the definition of the first PKtensorP from the strain-energy

density function (11), and the definitionof S (28), we can derive the material tangent stiffness by

assumingS = S(E) where we define the Green-Lagrangetensor

E =12

(F

TF − 1

)(35)

and linearization proceeds as

D(S)[Δu] =

∂S

∂E: D(E)[Δu] . (36)

and

D(E)[Δu] =

12

(∇0Δu

TF + F

T∇0Δu

). (37)

We introduce the fourth order tensor of elasticmoduli

C =∂S

∂E. (38)

Thus we write∫

Ω0

tr[∇0δu

TF DS(F )[Δu]

]dΩ =

∫

Ω0

tr

[

∇0δuT

F C :12

(∇0Δu

TF + F

T∇0Δu

)]

dΩ . (39)


10 KRYSL

Finally, we may introduce the assumed gradients with respect to the current configuration and write

∫

Ω0

tr

[

∇0δuT

F C :12

(∇0Δu

TF + F

T∇0Δu

)]

dΩ =

∫

Ω0

tr

[

FT∇δu

TF C :

12

(F

T∇Δu

TF + F

T∇Δu F

)]

dΩ . (40)

Introducing thedefinition

ε(η) =12

(∇η

T+ ∇η

), (41)

where

∇η = ∇0η ∙ F −1 , (42)

and the push forwardof C

cijkl = (det F )−1CIJKLF iIF jJF kKF lL . (43)

we write the material stiffness on the current configuration as

∫

Ω0

tr

[

FT∇δu

TF C :

12

(F

T∇Δu

TF + F

T∇Δu F

)]

dΩ =

∫

Ω0

tr [ε(δu) c : ε(Δu)] det F dΩ . (44)

The last expression corresponds exactly to the classical form of the material stiffness operator on the

current configuration. The difference with respect to the classical displacement-based finite element

formulation is that wherever the displacement-derived deformation gradient used to be used it is

replaced with the assumed deformationgradientF . Is also worthwhile to point out that again the

integrand is piecewise constant over each hexahedral element.

Continuing the comparison with the Bonet and Bhargava formulation [7], the constitutive

equation in the present work is not split into volumetric and deviatoric part as in the BB formulation.

In our estimation that is an advantage as it allows for very straightforward incorporation of

anisotropic materials. Also, all integrals and all the gradients of the basis functions in the present

formulation are evaluated in the reference configuration as opposed to the BB formulation where

the volumetric part requires evaluation of these quantities in the current, deformed, configuration.

1.5.3. DiscretizationThe finite element approximation is introduced asu =∑

i Niui, whereui

are the column vectors of nodal displacements. The matrix of the Cartesian components of the

displacement gradient on each element is computed as

[∇0u(ξk)] =∑

i

[ui][∇0Ni(ξk)] , (45)

where we indicate the use of components by brackets. Note that the gradient of the basis function

[∇0Ni(ξk)] at the quadrature pointξk is a row matrix. The matrix of the assumed deformation



gradient of (7) is then computed as

[F ] = V(e)0

−1∫

Ω(e)0

[1] + [∇0u] dΩ (46)

= [1] +∑

i

[ui]V(e)0

−1∫

Ω(e)0

[∇0Ni(ξk)] dΩ (47)

= [1] +∑

i

[ui][∇0Ni] (48)

where we have introduced the assumed basis function gradient[∇0Ni]. The assumed basis function

gradient with respect to the current coordinates is expressed analogously to (20) as

[∇Ni] = [∇0Ni][F ]−1 . (49)

This is again at variance with the respect to the BB formulation [7], where the gradient of the basis

functions in the current configuration is calculated from the coordinates in the current configuration.

1.5.4. Implementation of the material stiffness matrixWe may conveniently switch to the Voigt

notation and express the matrices corresponding to the tensor (41) in terms of the six-vectors

(holding components 11, 22, 33, 12, 13, 23 ofε) using the “strain-displacement” matrix operator

B(Ni) =

[∇Ni]1 0 0

0 [∇Ni]2 0

0 0 [∇Ni]3[∇Ni]2 [∇Ni]1 0

[∇Ni]3 0 [∇Ni]10 [∇Ni]3 [∇Ni]2

. (50)

This operator has the same structure as in the classical implementation, the only difference is that

the derivatives with the respect to the current coordinates are replaced with the components of the

assumed gradient of the basis function (49) on the current configuration.

Importantly, if the tangent moduli tensor (38) (or (43)) displays the major and minor symmetries

expected in hyperelasticity, the6 × 6 matrix of tangent moduli will be symmetric, and the material

tangent stiffness matrix will be also symmetric.

1.5.5. Implementation of the geometrical stiffness matrixNext we will point out that also the

geometrical stiffness matrix has the well-known structure of the classical formulation, and again the

only difference is the replacement of the derivatives of the basis functions by the components of

the assumed gradient of the basis functions. So the3 × 3 contribution to the geometrical stiffness

matrix due to the interaction of the two nodesI, J of the hexahedral elemente is

[∇NI ][σ][∇NJ ]T [ 13×3

] det F V(e)0 , (51)

where[σ] is a3 × 3 matrix of the components of the Cauchy stress tensor.


12 KRYSL

1.5.6. Constitutive update and material tangent moduliFinally, we point out an important and

appealing characteristic of the presented method: the constitutive update, namely the calculation

of the Cauchy stresstensorσ and the computation of the tangentmoduli c within a given finite

element is entirely standard, even to the point that existing software routines may be usedwithout

any modification, and the only distinguishing characteristic is the consistent use of the assumed

deformation gradient in place of the element kinematics-determined deformationgradientF .

2. NUMERICAL EXAMPLES

In the numerical examples in this section we compare the performance of the present formulation

with current high-performance hexahedral solid elements (and in some cases tetrahedral elements).

As some of our examples are for thin-walled (shell) structures, for some comparisons we will also

rely on shell elements.

2.1. Cook’s tapered panel

Plane-strain tapered panel with geometrical nonlinearity is a standard benchmark problem [14, 15].

Here it is modeled with one layer of 3-D elements, with zero through-the-thickness displacement

boundary conditions applied. The panel is clamped at one end and loaded with a total shear force

P = 1600 N at the other end. The material properties are: Young’s modulus240.565 MPa, Poisson

ratio ν = 0.4999 (nearly incompressible), and the material model is neo-Hookean. Figure 1 shows

the convergence behavior of the present formulation compared to the 2-D patch-basedF technique

of de Souza Neto et al [16, 14]: F-Bar SN Q (quadrilateral), and F-Bar SN T (triangle), and to the

mixed element Q1E4 of Wriggers [17]. In accordance with the present stabilization approach, the

amount of stabilization depends on the shape of the element. Thin element is stabilized to enable

good bending response, that is with the parameterΦ relatively small, stocky element is stabilized

with the parameterΦ larger as dictated by the dimensions of the element being of similar magnitude.

In the presence model the thickness of the tapered panel is adjusted to make the hexahedral element

faces squares on the face loaded by the traction. Figure 1 shows the absolute accuracy and the

convergence rate being comparable with the mixed element Q1E4 [17].

2.2. Curved cantilever

The structure is a circular-arc cantilever, clamped at one end and loaded with a vertical force

F = 600 at the other end. The material properties are: Young’s modulus107, zero Poisson ratio.

The radius isR = 100, and a cross-section of1 × 1 units. The reference solution is reported for a

mesh of eight B31 beam elements asux = 13.62, uy = −23.78, uz = 53.58 [18]; see also [19].

Figure 2 shows the model and one computational mesh in the undeformed and final deformed

state. Mesh H8MSGSO with eight elements lengthwise and2 × 2 elements in the cross-section

yielded ux = 14.03, uy = −23.40, uz = 52.98. For comparison we also show the results for an

extremely coarse mesh, two elements lengthwise and a single element in the cross-section.



100

101

10210

-3

10-2

10-1

100

Number of elements per sideE

st. T

rue

Err

or o

f Def

lect

ion

F-Bar SN Q

F-Bar SN T

Q1E4

H8MSGSO

Figure 1. Cook’s tapered panel. Comparison of the error of the corner displacement in dependence on thenumber of elements per side. H8MSGSO – present hexahedron.

(a) (b)

Figure 2. Curved cantilever beam. Mesh with 8 elements lengthwise, undeformed and deformed. (a)Schematic. (b) Load-deflection curves for, left-to-rightuy, ux, uz . H8MSGSO – present hexahedron, mesh

with 2 × 1 × 1 elements (©), mesh with8 × 2 × 2 elements (×). Reference curves from [19].

2.3. Twisted beam

The twisted clamped beam is a standard benchmark from the MacNeal-Harder set [20]. Here we

consider the thinner-shell (thickness 0.05) variation of this benchmark in the nonlinear range for

the “out of plane” loading [21]. The uniformly distributed loading on the free end cross-section is

multiplied with the loading parameterλ and the solution is calculated in 10 increments.

The coarse mesh is4 × 2 × 2 (four elements lengthwise); uniform refinement is used to produce

a finer mesh. The undeformed and deformed finer mesh is shown in Figure 3(a). The displacements

of the centroid of the loaded cross-section is shown in Figure 3(b) compared to the nonlinear shell

model results of [21].


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(a) (b)

Figure 3. Twisted beam. (a) Finer mesh (8 × 4 × 4) before and after deformation. (b) Load-deflectioncurves for−ux, uy, −uz . H8MSGSO – present hexahedron, mesh with4 × 2 × 2 elements (♦), mesh with

8 × 4 × 4 elements (∇). Reference solution with96 × 8 quadrilateral shell elements [21].

(a) (b)

Figure 4. Hemispherical shell with pinching forces. (a) The coarsest mesh used in the present study, with thedeflected shape (no magnification) shown. (b) Load-deflection curves for points under the concentratedforces. H8MSGSO hexahedron solution obtained with two meshes,4 × 4 × 2 and 8 × 8 × 3 elements.

Reference solution was obtained with a16 × 16 quadrilateral shell element model [22].

2.4. Hemispherical shell

The hemispherical shell of18o opening at the pole and with two pairs of pinching forces at the

large circle is a well-known benchmark for shell elements [20]; the reference solution was provided

in [22]. The coarsest mesh used in the present study is illustrated in Figure 4(a). Figure 4(b) provides

the load-deflection curves, where the reference solution obtained with shell elements is compared

with the present hexahedral formulation used with two relatively coarse meshes. Especially the

coarsest mesh, shown with the deformed and original configuration in Figure 4(a), is remarkably

capable of capturing the deformation given the large aspect ratios of the solid elements.

2.5. Cylinder with line load

This benchmark was proposed to study the viability of modeling of shells with solid hexahedral

elements [23]. The structure is a cylindrical shell with open end-sections, with a line load at the top



and supported along a knife edge at the bottom (refer to Figure 5). The output is the displacement

at pointA. The mid–surface radius isR = 9.0 mm and the length of the cylinder isL = 30.0 mm.

The shell is investigated for two different thicknesses,t = 2.0 mm (thick shell), andt = 0.2 mm

(thin shell). Two symmetry planes are used to simplify the geometry as shown in Figure 5. The

material has Lame parametersλ = 24000 MPa andμ = 6000 MPa (corresponding toE = 16800,

andν = 2/5). The line loadq has magnitude of7500/15 N/m (thick shell), and8.5/15 N/m (thin

shell). The output is the magnitude of the vertical displacement of pointA.

Figure 6(a) shows the evolution of the displacement of pointA with mesh refinement for the thick

shell. The comparison is made with the Q1SP hexahedral element of Reese [23], the one-quadrature-

point Q1 element with hourglass stabilization based on the equivalent parallelogram. As Reese et

al. noted [9] when describing the performance of Q1SP:

Thus, we can conclude that Q1SP shows very convincing results and a surprising

robustness in this extreme bending situation. The performance is comparable if not

better than that of the QM1/E12 element which behaves usually extraordinarily well for

such kind of problems.

It appears that such an assessment would be equally appropriate for the present H8MSGSO.

Figure 6(b) shows the evolution of the displacement of pointA with mesh refinement for the

thin shell. The present element converges towards the true solution remarkably quickly, performing

better than the nonlinear shell model.

Reese et al. [9] study the cylindrical shells with a single element through the thickness. In order

to assess the effect of the discretization through the thickness of the shell, we employ models with

one, two, or four elements through the thickness. For the thin shell, the present formulation with a

single element through the thickness is somewhat too flexible. However with two or more elements

through the thickness the accuracy is excellent, matching the accuracy of the shell formulation with

a fraction of the number of elements. For the thick shell, the present formulation is with a single

element through the thickness somewhat stiff; models with more than two elements through the

thickness are quite accurate and match Q1SP and QM1/E12 closely. (Note that strictly speaking

the line load and the knife-edge support are boundary conditions inadmissible in 3-D elasticity, and

hence speaking of convergence must be qualified.)

2.6. Annular slit plate

The annular slit plate was selected by Sze [22] as a challenging benchmark with a long history. The

data of the problem are given in consistent units as: Material parametersE = 21 × 106, andν = 0,

internal radius of 6.0 units, external radius of 10.0 units, and thickness of 0.03 units. The distributed

loadq on the free face of the slit is 0.8 in units of force/length.

Figure 7(a) illustrates the problem and its solution with a relatively coarse mesh. The vertical

deflection of the pointsA,B is shown in the two load-deflection curves in Figure 7(b).

For very thin structures, such as the annular slit plate, the number of iterations required for

convergence may be substantial. It is then natural to inquire whether that is related to the smallness

of the stabilization parameter: since the stabilization parameterΦ depends on the ratio of the squares

of the smallest and largest dimension of the element, for very thin elements this ratio may be quite

small, and hence the stabilization energy may be determined by elastic constants comparatively


16 KRYSL

Figure 5. Cylinder with line load. Left: Thin shell. Mesh with 20 element edges circumferentially. Right:Thick shell. Mesh with 8 element edges circumferentially. The output is the magnitude of the vertical

displacement of pointA.

(a) (b)

Figure 6. Cylinder with line load. Deflection at pointA. (a) Thin shell. (b) Thick shell. Key: SHELL – shellelement [23], Q1SP – Q1 element with hourglass stabilization based on the equivalent parallelogram [23],QM1/E12 – enhanced assumed strain element with 12 internal variables [9], 1,2,4 el – H8MSGSO

hexahedron with 1, 2, and 4 elements per thickness.

small with respect to the actual material constants. Figure 8 shows the total of required full-Newton

iterations to achieve convergence for 10 load increments. The present formulation is compared with

the serendipity 20-node hexahedron. The present formulation requires somewhat larger number of

iterations, but the pattern is very similar. Hence we might conclude that the reason for the large

number of iterations is probably the very large ratio between the through-the-thickness stiffness and

the bending stiffness of the very thin elements and not the lack of stabilization.

2.7. Nearly incompressible block

The present example was studied for instance in [24, 15], with the conclusion that the B-Bar method

and the Q1SP were the only reliable methods tested. In particular the enhanced assumed strain

elements experienced incurable instabilities. Here we test whether the new element formulation

can deliver a competitive performance when used with anunstructuredmesh with poorly shaped



0 10 200

0.2

0.4

0.6

0.8

1

Vertical deflection

Load

ing

para

met

er λ

(a) (b)

Figure 7. Annular slit plate. (a) Mesh of20 × 3 × 2 elements. Undeformed and derformed shape. (b) Load-deflection curves at pointsA, B (left to right). Mesh of hexahedra with20 × 8 × 2 elements (∇), and mesh

of 40 × 8 × 2 (♦). Reference solution with80 × 10 shell elements [22].

1 2 3 4 5 6 7 8 9 100

50

100

150

200

Increment number

Num

ber

of it

erat

ions

Figure 8. Annular slit plate. Total number of required iterations as function of load increment. Presentformulation (◦) compared with serendipity 20-node hexahedron (♦).

elements. In the present case the meshes were generated by dividing a tetrahedral mesh into

hexahedra, one tetrahedron into four hexahedra each. Refer to Figure 9. One quarter of the system

is discretized, i.e. symmetry conditions are considered. The nodes on the top of the block are

constrained in the horizontal direction. The vertical surface loadq is applied on the top of the

block. The material model is a compressible neo-Hookean model with the material parameters

μ = 80.194 N/mm2 andλ = 400889.806 N/mm2.

In addition to the Q1SP hexahedron in a regular grid [24], we also compare with advanced

tetrahedra used with unstructured meshes, the mixed tetrahedron of Caylak and Mahnken [25], and

the variational multiscale tetrahedron of Masud and Truster [26]. Figure 10 shows the convergence

of the deflection of the center of the block at the top face for different load levels. The present


18 KRYSL

Figure 9. Nearly incompressible block. Sketch of the geometry. The unstructured mesh for4 × 4 × 4elements shown.

2 4 6 8 10 12 14 1620

30

40

50

60

70

80

Number of elements per side

Com

pres

sion

[%

]

Reference

Cay

MasudQ1SP

H8MSGSO

Figure 10. Nearly incompressible block. Convergence of the vertical displacement of the center of the topface for the loading parameter of 25%, 50%, 75% and 100%. Present formulation used with unstructuredmeshes (as shown in Figure 9), with4 × 4 × 4, 8 × 8 × 8, and12 × 12 × 12 edges along the side. Key: Q1SP– rectangular mesh of Q1 elements with hourglass stabilization based on the equivalent parallelogram [23],Cay - mixed tetrahedron [25], Masud – variational multiscale tetrahedron [26], H8MSGSO – present

hexahedron.

hexahedron is less accurate (with the unstructured grid) than Q1SP (with a rectangular structured

grid), but competitive with the tetrahedra. In addition, we visualize the stresses in Figure 11.

2.8. Buckling of nearly incompressible block

In this example, introduced by [27], we investigate linear buckling of an elastic52 × 16 mm

block in plane strain (see Figure 12). The problem is modeled with 3-D elements. Displacement

perpendicularly to the plane of the paper is prevented to simulate plane strain conditions, the

block sits on rollers along the vertical sides and the bottom, and vertical pressure corresponding

to compressive unit force is applied on the top surface. The material is isotropic and very nearly

incompressible,E = 448.88 MPa, ν = 0.4999999. The first two instability modes are found by

linear buckling analysis. Figure 12 shows results for the buckling modes for variable number of

elements along the width and height (a single element through the thickness is always used). The

buckling modes are entirely physical, there is no indication that the present formulation would

buckle in hourglassing low-energy deformation modes.



(a)

(b)

(c)

Figure 11. Nearly incompressible block. Mesh12 × 12 × 12 for the final value of the loading parameter.(a) Nodal values of mechanical pressure. (b) Elementwise values of mechanical pressure. (c) Element wise

values of the von Mises stress.

2.9. Buckling of Thin-Walled Right-angle frame

We consider a linear stability analysis of an L-shaped flat frame which is clamped at one end and

subjected to an in-plane shear load at the other [28, 29, 27]. The coarsest and finest mesh used,

together with the buckling modes, are shown in Figure 13(a). We may note that the ratio of the

width to the thickness is 50: the frame legs are distinctly thin-walled beams. The load direction

F1 puts the clamped leg of the frame under tension, whileF2 compresses the clamped leg. The

present formulation is compared with solutions obtained with the 64-node (cubic) hexahedron and

the 27-node (quadratic) hexahedron. Figure 13(b) shows the convergence to the reference solution


20 KRYSL

Figure 12. Buckling of an elastic plane strain block: H8MSGSO mesh with variable number of elementshorizontally. Top to bottom: 3, 9, 18, 27, and 45 elements horizontally. (Proportionate number of elementsvertically, single element through the thickness.) The first two buckling modes (mode 1 in the left column,

mode 2 in the right column), loading parameter indicated.

obtained with cubic nonlinear beam elements. (The reason for the solid elements converging for the

orientationF1 to a buckling load higher than the reference value is at present not understood.)

It is remarkable that the present formulation provides accurate buckling loads with a single

element along the leg of the frame, albeit with two elements through the thickness. It is in fact

almost able to match the performance of the cubic isoparametric element in the ability to converge

quickly.

Even though the action for the buckling frame prior to the loss of stability is in the plane of the

thin-walled section, this is not taken into consideration when applying the stabilization. The general

formula described above is used without change.

2.10. Two-layer ribbon of transversely-isotropic material under finite strain

The ribbon consists of two layers of equal thickness with unidirectional fiber reinforcement parallel

to thexy plane (+45o/ − 45o). Similar structure was analyzed by Kaliske [30], but results were

reported only qualitatively and no comparison is therefore possible. Here we strive to discern the

efficacy of the stabilization procedure for an anisotropic material under large strains.



2 4 6 8-2

-1

0

1

2x 105

Number of elements per side [ND]

Buc

klin

g lo

ad [N

]

RefH8MSGSOH64H27

(a) (b)

Figure 13. Right-angle frame instability. (a) Coarsest and finest mesh used, two elements through thethickness. (b) Convergence of the buckling loads for two opposite orientations. H8MSGSO – present mean-

strain formulation, H27 and H64 – quadratic and cubic isoparametric element.

The bottom layer is reinforced with fibers that bisect thex > 0 ∧ y > 0 quadrant, the fibers in

the top layer bisect thex > 0 ∧ y < 0 quadrant. The dimensions in thex × y × z directions are

0.4 × 0.1 × 0.01 m.

The material model is the transversely isotropic hyperelastic formulation of Bonet and

Burton [31]. The properties of the orthotropic (transversely isotropic) material are: longitudinal

elastic modulusE1 = 5 MPa, transverse elastic modulusE2 = 1 MPa, Poisson ratio for loading

along the axis of transverse isotropyν12 = 0.25, Poisson ratio for loading in the plane orthogonal

to the axis of transverse isotropyν23 = 0.25, and modulus for shear between the axis of transverse

isotropy and the plane orthogonal to itG12 = 0.5 MPa.

The ribbon is clamped at one cross-section and tensile loadq = 10 kPa is applied at the other.

The ribbon twists under the tensile loading as the fibers attempt to align themselves with the tensile

axis. Figure 14 shows the load-deflection diagrams for the pointA. No independent comparison

is possible, but at least we may note the fast convergence and the satisfactory performance of the

stabilization scheme.

CONCLUSIONS

We have described a technique for stabilizing the mean-strain hexahedron in the regime of arbitrarily

large displacements and large strains. The starting point is the energy-sampling approach of

Krysl [4]. The hourglassing modes of deformation are never explicitly separated from the rigid-

body and mean-strain modes. Instead, stabilizing energy is postulated in the form of a stored-

energy function that is both added to and subtracted from the strain energy of the element. The

stabilization stored-energy function is expressed in the first case as a function of the displacement-

driven deformation gradient, and in the second case as a function of the assumed deformation

gradient. The sampling of these two contributions is carried out by two different quadrature rules,

the mean-strain quadrature combined with the full Gauss quadrature rule, and as a result, both

consistency and stability (and hence convergence) are guaranteed.


22 KRYSL

(a) (b)

Figure 14. Two-layer ribbon of transversely-isotropic material under finite strain. (a) Finest mesh used. Finaldeformed shape shown, axial stretchΔL/L ≈ 0.066. (b) Load-deflection curves for pointA. H8MSGSO

used with mesh8 × 3 × 1 (o), mesh16 × 6 × 2 (♦), mesh32 × 12 × 4 (2).

For simplicity the developments are limited to hyperelasticity, and the formulation allows for

an arbitrarily anisotropic strain-energy function. The stabilization energy is here taken in the neo-

Hookean form, and hence there are two input parameters to define. We develop an argument

that avoids volumetric locking by offsetting the value of the Poisson ratio away from1/2, and

links the Young’s modulus of the stabilization material to the input properties of the real material

and (crucially) also to the geometry of the element. Utilizing the matching of the energy of the

hexahedron in pure bending to the desirable value obtained from the analytical solution of pure

bending, we come up with a reduced value of the Young’s modulus that substantially increases the

accuracy of the element in coarse-mesh bending applications by avoiding shear locking.

For orthotropic materials, a simple modification that considers the softest mode of deformation is

found to work well. In this way we are able to eliminate volumetric locking for isotropic materials as

well as locking due to strongly anisotropic material properties, for instance for soft-matrix materials

with unidirectional or bidirectional stiff fibers.

The present element requires multiple elements through the thickness of bent structures

to represent bending stresses accurately, as would be needed for instance for plasticity. This

characteristic is shared with the original perturbation-stiffness stabilization [32] and other mean-

strain hexahedron formulations. The cost of the present formulation is only slightly higher than

that of fully-integrated isoparametric eight node hexahedron. This is amply offset by the superior

accuracy.

The accuracy and convergence characteristics of the present formulations compare favorably

with the performance of uniform-strain elements (for instance Q1SP [33, 23]) and enhanced-strain

elements. The present formulation is also competitive with specialized shell elements for thin

structures. In summary

1. The present formulation does not require an arbitrary stabilization parameter to be set by the

user.

2. The stabilization also works for arbitrarily anisotropic materials.



3. Arbitrarily complex inelastic constitutive equation is accommodated (but not studied in this

manuscript);

4. Finally, the present stabilization improved coarse-mesh accuracy (bending) response to the

point where the element is a contender in the analysis of thin-walled (shell) structures, while

possessing excellent accuracy for solids.

We have a reason to believe the present formulation to be a good general purpose tool for the analysis

of solids and thin-walled structures. Application of the present formulation to inelastic simulations

is being studied and reports of the findings are forthcoming.

ACKNOWLEDGMENTS

Partial support from U.S. Navy CNO-N45, project management Frank Stone and Ernie Young, and

continued support of Mike Weise (Office of Naval Research), is gratefully acknowledged.

REFERENCES

1. Belytschko T, Liu W, Moran B.Nonlinear Finite Elements for Continua and Structures. Wiley: Chichester, WestSussex, 2000.

2. Daniel WJT, Belytschko T. Suppression of spurious intermediate frequency modes in under-integrated elementsby combined stiffness/viscous stabilization.International Journal for Numerical Methods in Engineering2005;64(3):335–353.

3. Puso MA. A highly efficient enhanced assumed strain physically stabilized hexahedral element.InternationalJournal for Numerical Methods in Engineering2000;49(8):1029–1064.

4. Krysl P. Mean-strain eight-node hexahedron with stabilization by energy sampling.International Journal forNumerical Methods in Engineering2014; doi:10.1002/nme.4721.

5. Flanagan DP, Belytschko T. A uniform strain hexahedron and quadrilateral with orthogonal hourglass control.International Journal for Numerical Methods in Engineering1981;17(5):679–706.

6. Belytschko T, Bindeman LP. Assumed strain stabilization of the 8 node hexahedral element.Computer Methods inApplied Mechanics and Engineering1993;105(2):225–260.

7. Bonet J, Bhargava P. A uniform deformation gradient hexahedron element with artificial hourglass control.International Journal for Numerical Methods in Engineering1995;38:2809–2828.

8. Reese S. On a physically stabilized one point finite element formulation for three-dimensional finite elasto-plasticity.Computer Methods in Applied Mechanics and Engineering2005;194(45-47):4685–4715.

9. Reese S, Wriggers P. A stabilization technique to avoid hourglassing in finite elasticity.International Journal forNumerical Methods in Engineering2000;48(1):79–109.

10. Belytschko T, Bachrach WE. Efficient implementation of quadrilaterals with high coarse-mesh accuracy.ComputerMethods in Applied Mechanics and Engineering1986;54(3):279–301.

11. Liu WK, Ong JSJ, Uras RA. Finite-element stabilization matrices - a unification approach.Computer Methods inApplied Mechanics and Engineering1985;53(1):13–46.

12. Bonet J, Wood R.Nonlinear Continuum Mechanics for Finite Element Analysis. Cambridge University Press, 2008.13. Kussner M, Reddy BD. The equivalent parallelogram and parallelepiped, and their application to stabilized finite

elements in two and three dimensions.Computer Methods in Applied Mechanics and Engineering2001;190(15-17):1967–1983, doi:10.1016/s0045-7825(00)00217-6. Kussner, M Reddy, BD.

14. Neto E, Pires F, Owen D. F-bar-based linear triangles and tetrahedra for finite strain analysis of nearlyincompressible solids. Part I: formulation and benchmarking.International Journal for Numerical Methods inEngineering2005;62(3):353–383.

15. Elguedj T, Bazilevs Y, Calo V, Hughes T.B andF projection methods for nearly incompressible linear and non-linear elasticity and plasticity using higher-order NURBS elements.Computer Methods in Applied Mechanics andEngineering2008;doi:10.1016/j.cma.2008.01.012.


24 KRYSL

16. de Souza Neto E, Peric D, Dutko M, Owen D. Design of simple low order finite elements for large strain analysisof nearly incompressible solids.International Journal Of Solids And Structures1996;33(20-22):3277 – 3296.

17. Wriggers P.Nonlinear Finite Element Methods. Springer, 2008.18. Abaqus, Inc. ..Abaqus Verification Manual. Version 6.7 edn. 2007.19. Izzuddin BA, Elnashai AS. Eulerian formulation for large-displacement analysis of space frames.Journal of

Engineering Mechanics-Asce1993;119(3):549–569, doi:10.1061/(asce)0733-9399(1993)119:3(549).20. MacNeal RH, Harder RL. A proposed standard set of problems to test finite element accuracy.Finite Elements in

Analysis and Design1985;1:3 – 20.21. Mostafa M, Sivaselvan MV, Felippa CA. A solid-shell corotational element based on ANDES, ANS and EAS for

geometrically nonlinear structural analysis.International Journal for Numerical Methods in Engineering2013;95(2):145–180, doi:10.1002/nme.4504. Mostafa, M. Sivaselvan, M. V. Felippa, C. A.

22. Sze K, Chen J, Sheng N, Liu X. Stabilized conforming nodal integration: exactness and variational justification.Finite Elements In Analysis And Design2004;41(2):147 – 171.

23. Reese S. On the equivalence of mixed element formulations and the concept of reduced integration in largedeformation problems.International Journal Of Nonlinear Sciences And Numerical Simulation2002; 3(1):1 –33.

24. Reese S, Wriggers P. A stabilization technique to avoid hourglassing in finite elasticity.International Journal ForNumerical Methods In Engineering2000;48(1):79 – 109.

25. Caylak I, Mahnken R. Stabilization of mixed tetrahedral elements at large deformations.International Journal forNumerical Methods in Engineering2012;90(2):218–242, doi:10.1002/nme.3320.

26. Masud A, Truster TJ. A framework for residual-based stabilization of incompressible finite elasticity: Stabilizedformulations and (f)over-bar methods for linear triangles and tetrahedra.Computer Methods in Applied Mechanicsand Engineering2013;267:359–399, doi:10.1016/j.cma.2013.08.010. Masud, Arif Truster, Timothy J.

27. Broccardo M, Micheloni M, Krysl P. Assumed-deformation gradient finite elements with nodal integration fornearly incompressible large deformation analysis.International Journal for Numerical Methods in Engineering2009;78(9):1113–1134.

28. Argyris J, Balmer H, Doltsinis J, Dunne P, Haase M, Kleiber M, Malejannakis G, Mlejnek HP, Moller M, ScharpfD. Finite element method - the natural approach.Computer Methods in Applied Mechanics and Engineering1979;17/18:1–106.

29. JC Simo DF, Rifai M. On a stress resultant geometrically exact shell model. Part III: Computational Aspects of thenonlinear theory.Computer Methods in Applied Mechanics and Engineering1990;79:21–70.

30. Kaliske M. A formulation of elasticity and viscoelasticity for fibre reinforced material at small and finitestrains.Computer Methods in Applied Mechanics and Engineering2000;185(2-4):225–243, doi:10.1016/s0045-7825(99)00261-3.

31. Bonet J, Burton AJ. A simple average nodal pressure tetrahedral element for incompressible and nearlyincompressible dynamic explicit applications.Communications in Numerical Methods in Engineering1998;14(5):437–449.

32. Flanagan D, Belytschko T. A uniform strain hexahedron and quadrilateral with orthogonal hourglass control.International Journal For Numerical Methods In Engineering1981;17(5):679 – 706.

33. Reese S, Wriggers P, Reddy B. A new locking-free brick element technique for large deformation problems inelasticity.Computers & Structures2000;75(3):291 – 304.


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