A uniform nodal strain tetrahedron with isochoric stabilization

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2009; 78:429–443Published online 13 November 2008 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.2493

A uniform nodal strain tetrahedron with isochoric stabilization

M. W. Gee1,∗,†, C. R. Dohrmann2, S. W. Key3 and W. A. Wall1

1Technische Universitat Munchen, Chair for Computational Mechanics, Boltzmannstrasse 15, D-85747 Garching b.Munchen, Germany

2Sandia National Laboratories, Structural Dynamics Department, PO Box 5800, Mail Stop 0346, Albuquerque,NM 87185-0346, U.S.A.

3Sandia National Laboratories, Engineering and Manufacturing Department, PO Box 0847, MS 0847,Albuquerque, NM 87185-0847, U.S.A.

SUMMARY

A stabilized node-based uniform strain tetrahedral element is presented and analyzed for finite deformationelasticity. The element is based on linear interpolation of a classical displacement-based tetrahedral elementformulation but applies nodal averaging of the deformation gradient to improve mechanical behavior,especially in the regime of near-incompressibility where classical linear tetrahedral elements performvery poorly. This uniform strain approach adopted here exhibits spurious modes as has been previouslyreported in the literature. We present a new type of stabilization exploiting the circumstance that theinstability in the formulation is related to the isochoric strain energy contribution only and we thereforepresent a stabilization based on an isochoric–volumetric splitting of the stress tensor. We demonstrate thatby stabilizing the isochoric energy contributions only, reintroduction of volumetric locking through thestabilization can be avoided. The isochoric–volumetric splitting can be applied for all types of materialswith only minor restrictions and leads to a formulation that demonstrates impressive performance inexamples provided. Copyright q 2008 John Wiley & Sons, Ltd.

Received 6 August 2008; Revised 23 September 2008; Accepted 24 September 2008

KEY WORDS: uniform strain; tetrahedra elements; finite elements; stabilization; finite elasticity

1. INTRODUCTION

The application of the finite element method to problems that involve highly complex three-dimensional structures very often necessitates the use of a low-order tetrahedral finite elementformulation as reliable meshing resulting in pure hexahedral grids for arbitrary domains is still an

∗Correspondence to: M. W. Gee, Technische Universitat Munchen, Chair for Computational Mechanics, Boltz-mannstrasse 15, D-85747 Garching b. Munchen, Germany.

†E-mail: [email protected]

Contract/grant sponsor: United States Department of Energy; contract/grant number: DE-AC04-94-AL85000

Copyright q 2008 John Wiley & Sons, Ltd.

430 M. W. GEE ET AL.

open field of research. Classical displacement-based linear tetrahedral elements have significantshortcomings for problems in solid mechanics. Besides other undesirable phenomena, they sufferfrom significant volumetric locking effects [1] when materials with nonzero or even large Poissonratio are applied.

Therefore, significant efforts have been undertaken to develop tetrahedral element formulationsthat exhibit good mechanical properties, see for example [2, 3] and the literature cited therein.While Taylor [2] proposes a mixed variational principle with nodal pressure degrees of freedom,the authors in [4] first introduce a finite element approach where pressure is nodally averagedamong adjacent low-order elements to overcome volumetric locking, where nodal averaging andnodal integration have previously been widely used in the field of meshless methods, see [5] andthe literature cited therein.

Considering ideas from [4], a triangle and tetrahedral formulation based on a nodally averagedstrain field has been first presented in [6] for the small-strain case. This approach is especiallyappealing due to the simplicity of the formulation while resulting in very good mechanical prop-erties. The formulation has been analyzed and extended in [7, 8] and various similar and relatedapproaches exist, e.g. [9–12].

It was discovered that the node-based uniform strain approach [6] suffers from spurious modes inthe low eigenfrequency regime. While Bonet et al. [7] propose a stream upwind Petrov–Galerkin-type of stabilization, Puso and Solberg [8] present a simple stabilization that uses a standarddisplacement-based tetrahedral element formulation. In [8], a modified material law with lowPoisson ratio is used in the stabilization terms to avoid introduction of volumetric locking bythe stabilization. This results in a variational inconsistency. A near-incompressible material beingconsistently applied in the stabilization as well would result in significant volumetric locking ofthe stabilization terms as is demonstrated in this contribution.

Here, we build upon the approach given in [8], but demonstrate that reintroduction of volumetriclocking by the stabilization can be avoided. We introduce a general splitting of the stresses intoisochoric and volumetric components in a variational consistent way. It is shown that stabilityof the uniform nodal strain method can be achieved when stabilization is applied to isochoriccomponents only while maintaining the full benefits of the nodally averaged approach with respectto volumetric stress components. We stress that a volumetric–isochoric split of the stress tensorcan always be performed and that an abstract linearization independent of the material law canbe performed under certain conditions. The stability of the new isochoric uniform strain elementwith isochoric stabilization is demonstrated by means of an eigenvalue analyses and behavior iscompared with the approach in [8] and other finite element formulations on two examples.

Besides exhibiting very good behavior with respect to the displacement response, nodal strainelements in general tend to exhibit pressure fluctuations as shown in [12]. The authors in [12]conclude that this class of elements can well be used in a wide variety of applications, but shouldnot be trusted to predict accurate pressure distributions. We would like to note that the approachin [8] as well as the approach taken here also exhibit pressure fluctuations while both formulationspredict deformations and reaction forces near the incompressible limit very well. For more details,see Section 6.3.

In Section 2 we define the problem of finite deformation elasticity. In Section 3 the classicaldisplacement-based tetrahedral element formulation is briefly reviewed, mainly to introduce nota-tion utilized in the following. The uniform nodal strain approach together with a stabilizationas proposed in [8] is given in Section 4 for the finite deformation case. The formulation of ouruniform nodal strain approach with isochoric stabilization is given in Section 5. An eigenvalue

Copyright q 2008 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2009; 78:429–443DOI: 10.1002/nme

A UNIFORM NODAL STRAIN TETRAHEDRON WITH ISOCHORIC STABILIZATION 431

analysis demonstrating stability of the proposed method as well as two example applications inwhich the influence of the stabilization is studied and compared are given in Section 6.

2. PROBLEM DEFINITION

Consider the boundary value problem of finite deformation elasticity on a domain �⊂R3 of theform

Div(FS)+b0 = 0 in �

u = uD on �D

PN� = t0 on �N

(1)

where

F= �x�X

=1+ �u�X

(2)

is the deformation gradient that depends on the displacement field u. P,S are first and secondPiola–Kirchhoff stresses, respectively, uD and t0 are boundary conditions on the Dirichlet andNeumann boundary �D and �N, respectively. N� is the unit outward normal on �N in the materialframe.

We assume a material behavior of the form

C= �S�E

=2�S�C

(3)

with the right Cauchy Green tensor C and Green–Lagrange strains E

C=FTF, E= 12 (F

TF−1) (4)

We stress that (3) is the general definition of a material tangent operator not imposing anylimitations on the choice of material applied.

The method of weighted residual applied to the balance Equation (1) leads to

��=∫

��u·[Div(FS)+b0]d�+

∫�N

�u·[t0−FSN]d�=0 (5)

where

�ui ∈V = {�ui ∈H1(�) , i=1,2,3|�ui |�D =0}ui ∈S = {ui ∈H1(�), i=1,2,3|ui |�D =uiD}

(6)

This results in the weak form

��=��int−��ext=∫

��E:Sd�−��ext=0 (7)



after applying Gauss’ divergence theorem. Note that E and S non-linearly depend on the displace-ment field u. For ease of notation ��ext is considered to be independent of u and we therefore donot further detail its expression.

3. DISPLACEMENT-BASED DISCRETIZATION AND NOTATION

A finite element approximation to the weak form (7) is constructed by introducing the usual finiteelement spaces uhj ∈Sh ⊂S and �uhj ∈Vh ⊂V, j =1,2,3, based on a discretization with 4-nodedlinear tetrahedral elements. Accordingly, the position and displacement field in each element isexpressed as

X(n)=4∑

i=1Ni (n)Xi , uh(n)=

4∑i=1

Ni (n)di (8)

with Xi as nodal coordinates in the material frame and di as total displacements of the i th nodeof the element. Shape functions Ni (n)=�i are linear with respect to natural coordinates

n=[�1 �2 �3 �4], 0��i�1 (9)

subject to the constraint

�1+�2+�3+�4=1 (10)

Hence, the left equation in (8) can also be expressed as

⎡⎢⎢⎢⎢⎣X1

X2

X3

1

⎤⎥⎥⎥⎥⎦(n)=

⎡⎢⎢⎢⎢⎢⎣

X11 X2

1 X31 X4

1

X12 X2

2 X32 X4

2

X13 X2

3 X33 X4

3

1 1 1 1

⎤⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎣

�1

�2

�3

�4

⎤⎥⎥⎥⎥⎦ (11)

and derivatives of the shape functions with respect to cartesian coordinates �Ni/�X�, �=1,2,3,can be derived from (11) in a standard way.

The discretized form of the internal energy in (7) per tetrahedral element e is

��e,int=∫

�e�E:Sd�e=(�de)T

∫�e

(Be)TSd�e (12)

with �de as vector of nodal test function values of the elements’ nodes i=1, . . . ,4 and S=[S11 S22 S33 S12 S23 S13]T as second Piola–Kirchhoff stresses in Voigt notation. In the following,notation ¯(·) is used to indicate vectors and matrices in Voigt notation to be distinguished fromtensor-valued quantities.



The total strain–displacement operator Be(d)=[Be1 Be

2 Be3 Be

4] is constructed as

Bei (d)=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

F11Ni,1 F21Ni,1 F31Ni,1

F12Ni,2 F22Ni,2 F32Ni,2

F13Ni,3 F23Ni,3 F33Ni,3

F11Ni,2+F12Ni,1 F21Ni,2+F22Ni,1 F31Ni,2+F32Ni,1



⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

, i=1, . . . ,4 (13)

with Ni,� =�Ni/�X�,�=1,2,3, and the deformation gradient F from (2). Note that Be dependson the nodal displacements di through (2), but does not depend on the position within the elementas all quantities are spatially constant within element e.

The usual linearization of (12) yields

Lin��e,int(d) = ��e,int+(�de)T∫

�e(Be)TS,d+(Be)T,dSd�e�de

= ��e,int+(�de)T∫

�e(Be)TCBe+∇N:S∇Nd�e�de (14)

with material tangent C(C) in Voigt notation and ∇N as gradient operator with components�Ni/�X�, i=1, . . . ,4,�=1,2,3.

4. NODE-BASED UNIFORM STRAIN FORMULATION

The nodally integrated tetrahedron according to [6] with a stabilization as in [8] in the finitedeformation case is briefly summarized. For a more detailed description, we refer to the originalliterature.

A nodally averaged deformation gradient FI at node I is introduced as

FI = 1

V I

∑e∈Sel

I

V e

4Fe (15)

withSelI the patch of elements adjacent to node I , V e the volume of element e, Fe the element-wise

deformation gradient from (2) being constant within element e, and

V I = ∑e∈Sel

I

V e

4(16)

as volume assigned to node I . The factor 14 in (15) and (16) accounts for the portion of the volume

per node where one element has four nodes. It cancels out when (16) is inserted into (15). In [6],a more sophisticated approach to assign nodal volumes than (16) has been studied, but could notexhibit considerably better results in the examples provided than this simple approach adoptedhere.



The internal part of the weak form (7) is now discretized as sum of a nodal and an element-wisecontribution as

��int=��int,nd+��int,el=Nnd∑IV I�EI :(1−�)SI +

N el∑e

∫�e

�Ee:�Se d�e (17)

where 0<��1 is a stabilization parameter assumed to be constant throughout the domain and EI

are nodally averaged Green–Lagrange strains derived from (15) with (4). N nd and N el are the totalnumber of nodes and elements, respectively. Stresses SI , Se result from the material stress–strainrelationship (3).

Choosing �=0, the contribution ��int,nd corresponds to the unstabilized nodal strain approachthat is instable as was observed in [8]. The choice �>0 blends in the element-based stabilization��int,el. When applying a material law that is close to incompressible, this stabilization contributionintroduces volumetric locking as it is based on the standard displacement formulation reviewedin Section 3. For this reason, the authors in [8] apply a modified material law with low or noPoisson ratio in ��int,el to avoid this obvious reintroduction of volumetric locking. This leads toan inconsistency between the contributions ��int,nd and ��int,el and leaves the arbitrary choice ofthe material law for the latter part.

Equation (17) defers from the approach in [8] by choosing the material model to be equal inthe nodal and stabilization part of the integration for consistency.

��int,nd is evaluated as sum over nodes I with the element set SelI of elements adjacent to I

and node set SndI of nodes adjacent to these elements including node I as

��int,nd=Nnd∑IV I�EI :(1−�)SI =�dT

Nnd∑IV I (BI )T(1−�)SI (18)

where

BI = 1

V IAe∈Sel

IJ∈Snd

I

V e

4[Be

1�1J Be2�2J Be

3�3J Be4�4J ] (19)

with BeJ from (13) and Kronecker symbol �i J , i=1, . . . ,4, indicating whether node J is adjacent

to element e. For a geometric representation, see Figure 1.Linearization of (18) yields

Lin��int,nd(d) = ��int,nd+�dTNnd∑IV I [(BI )TSI ,d+BI ,d SI ]�d

= ��int,nd+�dTNnd∑IV I [(BI )TCIBI +∇NI :SI∇NI ]�d (20)

with

∇NI = 1

V IA

e∈SelI

V e

4∇Ne (21)

The ansatz (17) using a displacement-based tetrahedral element formulation (T4) for stabilizationalso introduces all locking phenomena inherent to the T4 element in an amount depending on �.



Figure 1. Two-dimensional visualization of element set SelI and nodal set Snd

I around node Ias used in Equations (18) and (19).

Therefore, � should be chosen as small as possible but large enough to achieve a stable solution.The authors in [8] use �=0.05 throughout their examples.

5. NODE-BASED UNIFORM STRAIN ELEMENT WITH ISOCHORIC STABILIZATION

In a T4 element formulation volumetric locking is especially severe along with nearly incom-pressible material behavior. As the instability of the uniform nodal strain approach described inSection 4 is related to isochoric components of the stress state only, it is however not necessaryto use a full T4 formulation to stabilize it as given in the previous section. Hence, volumetriclocking induced by the stabilization with T4 elements can be avoided. Following this idea toavoid volumetric locking induced to the formulation (17) through the stabilization, the followingmodified stabilization ansatz is proposed

��int =Nnd∑IV I�EI :(SI −�SI

iso)+N el∑e

∫�e

�Ee:� Seiso d�e

= �dTNnd∑IV I (BI )T(SI −�SI

iso)+�dTN el∑e

∫�e

(Be)T� Seiso d�e (22)

Here, a split of stress components into isochoric and volumetric components

S=Svol+Siso=−pJC−1+Siso (23)

is utilized and the stabilization is applied to the isochoric components only. The pressure p iscomputed from

p=− 13 tr(r)=− 1

3 tr(J−1F S FT)=− 1

3 J−1S :C (24)

where J =det(F) and C is the right Cauchy Green stretch tensor introduced in (4). It is important tonote that the splitting (23) does not introduce limitations on the material law applied. The isochoric



stress components SIiso utilized in (22) for stabilization are obtained from (23) once S,Svol are

computed.Linearization of (22) yields

Lin��int(d) = ��int+�dT(Nnd∑IV I (BI )T(CI −�CI

iso)BI +∇NI :(SI −�SI

iso)∇NI

+N el∑e

∫�e

�(Be)TCeisoB

e+�∇Ne:Siso∇Ne d�e

)�d (25)

With (3) and (23) at hand, any material tangent can be split such that

C=Cvol+Ciso=2�Svol�C

+2�Siso�C

(26)

The volumetric tangent is

Cvol=−2J p(C−1�C−1)+ 13C

−1⊗S+ 13C

−1⊗(C :S) (27)

with (·)⊗(·) as dyadic (or outer) tensor product and

(C−1�C−1)= 1

2(C−1

ac C−1bd +C−1

ad C−1bc )=−�C−1

ab

�Ccd(28)

with summation over equal indices {a,b,c,d}=1,2,3. With C given and insertion of (27) in (26),the isochoric tangent Ciso used in (25) is obtained. We stress that the splittings (23) and (26)are abstract in the sense that once S and C are at hand, the splitting can always be performedindependent of the type of material model applied.

6. EXAMPLES

6.1. Stability

To demonstrate stability of the isochoric stabilization presented in Section 5 and to compare withthe results given in [8], we perform an eigenvalue analysis of a unit cube. The cube is discretizedwith 163 hexahedral elements and with 5×163 tetrahedral elements, respectively. As materialmodel, a Neo-Hookean hyperelastic material with E=1 and �=0.499 is chosen to demonstratethe near-incompressible case. Element formulations considered are given in Table I. Table II giveslowest nonzero eigenvalues �0 for several discretizations; corresponding eigenmodes are visualizedin Figure 2. Results of the stabilized uniform strain tetrahedra UT 4PS(�) and UT 4iso(�) are givenfor �=0.05 as proposed in [8] as well as for �=0.2. The latter is an unpractical large amountof stabilization applied only to study the sensitivity of the formulations to the amount ofstabilization added.

Eigenvalues and eigenmodes given in [8] for UT 4PS(0.05) can be reproduced with very goodagreement. We also observe the instability of the unstabilized uniform strain tetrahedron UT 40 aswas observed in [8], see Figure 2(b).



Table I. Element formulations considered in the examples.

Name Description

Hex8E AS21 8-noded trilinear hexahedron with enhanced assumed strains approach with 21element—internal strain degrees of freedom

T 4 Displacement-based linear tetrahedronUT 40 Uniform strain tetrahedron without stabilization as described in [6]UT 4PS(�) Same uniform strain tetrahedron as UT 40 but with stabilization as described in

Section 4 and [8]UT 4iso(�) New uniform strain tetrahedron with isochoric stabilization as proposed in

Section 5

Table II. Lowest nonzero eigenvalues of unit cube.Comparison of several discretizations.

Discretization � �0[10−4]Hex8E AS21 — 4.5301T 4 — 5.0566UT 40 0.0 0.3201UT 4PS(0.05) 0.05 4.4623UT 4iso(0.05) 0.05 4.3565UT 4PS(0.2) 0.2 4.6574UT 4iso(0.2) 0.2 4.3981

Figure 2. Eigenmodes of unit cube with E=1,�=0.499 matching eigenvalues �0 in Table II: (a)Hex8E AS21; (b) UT 40; (c) UT 4PS(0.2); (d) UT 4iso(0.2); (e) UT 4PS(0.05); and (f) UT 4iso(0.05).



Note that the presented analysis refers to the small-strain regime. An elaborate analysis of thestability properties on the large-strain regime as e.g. performed in [3] for various types of mixedelements is subject to further research.

Eigenmodes and eigenvalues demonstrate that the new UT 4iso(�) is stable as is the UT 4PS.Furthermore, comparing low eigenvalues in Table II for UT 4iso and UT 4PS for different values of� indicates that the new UT 4iso is less sensitive to the amount of stabilization added—an aspectstudied in more detail in the example in Section 6.3.

6.2. Bending of a nearly incompressible cantilever beam

The finite deformation bending of a cantilever beam with dimensions 2×2×16 is studied.The beam is fully clamped on one end and loaded with a tip area load of 2000 per unitarea on the other, see Figure 4. A Neo-Hookean material with E=106,�=0.499 is chosen.The tetrahedral discretizations applied are depicted in Figure 3. Three discretizations withhexahedral elements were also used for comparison of which the finest is given in Figure 3.Comparisons are made with respect to the maximum tip displacement in Figure 5, wheremax(uz)=6.435, resulting from the finest hexahedral mesh, serves as the reference value. InFigure 5, the Hex8E AS21 and T 4 meshes demonstrate expected behavior as best and worstdiscretization, respectively. While the UT 4PS(0.2) suffers from the large amount of volumetriclocking induced by the stabilization, the UT 4iso(0.2) performs very well even for the unpracticallarge �, as it does not inherit severe locking from the stabilization. UT 4PS(0.05) approachesthe hexahedral reference solution rather quickly considering the near-incompressibility of theproblem and the usual performance of classical linear tetrahedral elements in this case. WhileUT 4PS(0.05) still exhibits significant volumetric locking behavior for coarse discretizationsinduced by the stabilization, the UT 4iso(0.05) does not. It approaches the correct solution fromabove, a circumstance also observed in the original work in [6]. Approximation quality is verysatisfactory for the UT 4iso element even for coarse discretizations, see Figure 4, where thedeformation of the UT 4iso(0.05) meshes is visually compared with the hexahedral referencesolution.

6.3. Cook’s membrane problem

Utilizing a geometrically non-linear variant of the well known Cook’s membrane problem, westudy the influence of the stabilization parameter � on the solution. Again, the near-incompressiblecase is considered where volumetric locking effects can be severe. The geometry of the problemis given in Figure 6(a) together with our reference solution obtained with a 64 Hex8E AS21elements discretization. Thickness orthogonal to the paper-plane is 1.0 and one layer of elementsis used in this direction. The structure is loaded on the free end with a vertical surface tipload of 50 per unit area. The tetrahedral discretization in Figure 6(b) is obtained throughsubdivision of the hexahedrals into five tetrahedra each resulting in 320 elements in total.A simple Neo-Hookean hyperelastic material law with E=240.565,�=0.499 is applied. Themaximum tip displacements for Hex8E AS21, T 4, UT 4PS(�) and UT 4iso(�) are studied inFigure 7, where � is varied through its valid range 0<��1 with �=0.05 the smallest valuetested.

The UT 4PS(�) recovers the T 4 solution for �=1 as is trivially expected considering Equation(17). The UT 4iso(1.0) still is a mixed approach as it considers all volumetric stress componentsin the nodally averaged contribution independent of the choice of �, see Equation (23). For �→0,



Figure 3. Discretizations used in beam bending example. From top to bottom: hexahedra discretizationwith 4096 elements, tetrahedral discretizations with 528,1431 and 4721, respectively.

Figure 4. Three-dimensional view of beam bending example. Discretizations with 528, 1431 and4721 UT 4iso(0.05) elements are shown. Material configuration and reference solution with 4096

hexahedral elements is shown at the back for comparison.

behavior of both UT 4PS(�) and UT 4iso(�) improves though not being capable to fully recover thehexahedral solution. Though UT 4iso(�) does not inherit volumetric locking from the stabilization,there are still other locking effects such as shear locking present in both formulations. ComparingUT 4iso(�) and UT 4PS(�), it is though remarkable how performance is improved by avoiding thevolumetric stress components in the stabilization.



Figure 5. Tip displacement of cantilever beam bending problem for several refinements of meshes.Applied element formulations are described in Table II. Reference solution from finest hexahedral mesh

is max(uz)=6.435 (dotted line).

Figure 6. (a) Discretization with 64 hexahedral elements and reference solution and (b) discretization with320 tetrahedral elements and solution with UT 4iso(0.05) discretization.

While displacement response is excellent, it has to be mentioned that our isochoric stabilizationdoes not eliminate the pressure fluctuations that have been observed for various nodally averagedelement formulations in [12]. The UT 4PS(�) we compared to also exhibits these fluctuationsin pressure but less pronounced as they are damped by the volumetric locking induced by the



Figure 7. Tip displacement of Cook’s beam depending on stabilization parameter �.

stabilization. Appropriate locking-free pressure stabilization for this class of element formulationsis subject to current research, see for example [13].

7. COMPUTATIONAL EFFICIENCY AND IMPLEMENTATION

Owing to the nodal averaging in the uniform strain approaches in (19), there is coupling betweennodal degrees of freedom that are a distance of up to two elements apart. This leads to an increaseof the number of nonzero entries in the assembled tangent matrix as depicted in Figure 8 for theCook’s membrane problem from Section 6.3. However, the total number of degrees of freedomis the same as in the standard linear element case as opposed to an increase in problem size thatwould be obtained when considering higher-order element formulations. Also, the additional timerequired for system solution and assembly of nodal matrix contributions is easily outweighed bythe improvement in accuracy achieved. The uniform nodal strain approach presented in [6, 8] andhere are appealingly simple to implement once a standard displacement element formulation asoutlined in Section 3 is at hand. Nodal contributions to internal forces and tangent Jacobian aswell as the element-based stabilization in Equations (18) and (20) or (22) and (25), respectively,can be computed efficiently in a combined integration and assembly process. Special care must betaken in a parallel implementation based on domain decomposition approaches. Here, assemblyof nodal contributions (with its increased connectivity) might necessitate extra communication inthe assembly process depending on the specific type of domain decomposition chosen.

When applying the UT 4iso formulation presented here, the splitting of the material tangent (26)through (28) involves minor extra computational effort. Very often, the specific material law appliedis formulated in an additive isochoric–volumetric fashion such that the material implementationmight be able to provide separate isochoric–volumetric contributions to the internal forces andmaterial tangent to start with. This e.g. would be the case for most hyperelastic materials. However,in all examples given here the abstract splitting (26)–(28) has been applied explicitly.



Figure 8. Sparsity pattern of stiffness matrix of Cook’s membrane problem given in Section 6.3: (a) showspattern of uniform nodal strain formulations and (b) indicates matrix pattern of standard displacement-

based formulation for comparison.

8. CONCLUSIONS

A uniform nodal strain tetrahedral finite element formulation for finite deformations is presentedand investigated. Based on the formulation given in [6] and the stabilization introduced in [8],an isochoric stabilization for the nodal strain tetrahedron is developed. In contrast to [8], thenew stabilization does not reintroduce volumetric locking by means of the stabilization, in casea consistent material law is applied. It is shown that isochoric stabilization is sufficient to obtaina stable formulation that exhibits very good deformation approximation properties in the near-incompressible limit. It is demonstrated that the isochoric–volumetric splitting of the stressesand the material tangent can be performed in an abstract fashion such that it can be appliedto a wide range of material formulations. Examples comparing an enhanced strain hexahedrafinite element that serves as a reference solution with the tetrahedron given in [8] and the newtetrahedron presented here are supplied. It is demonstrated that our new tetrahedron can achievesimilar deformation predictions as the hexahedral reference even for very coarse discretizations ofnon-slender structures.

ACKNOWLEDGEMENTS

Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, forthe United States Department of Energy under contract DE-AC04-94-AL85000.

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A uniform nodal strain tetrahedron with isochoric stabilization

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