Chemnitz Scientific Computing Preprints

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1 Martina Balg Arnd Meyer Fast simulation of nearly incompressible nonlinear elastic material at large strain via adaptive mixed FEM CSC/-03 Chemnitz Scientific Computing Preprints ISSN Chemnitz Scientific Computing Preprints

2 Impressum: Chemnitz Scientific Computing Preprints ISSN : Preprintreihe des Chemnitzer SFB393 Herausgeber: Professuren für Numerische und Angewandte Mathematik an der Fakultät für Mathematik der Technischen Universität Chemnitz Postanschrift: TU Chemnitz, Fakultät für Mathematik 0907 Chemnitz Sitz: Reichenhainer Str. 4, 096 Chemnitz

3 Chemnitz Scientific Computing Preprints Martina Balg Arnd Meyer Fast simulation of nearly incompressible nonlinear elastic material at large strain via adaptive mixed FEM CSC/-03 Abstract The main focus of this work lies on the simulation of the deformation of mechanical components that consist of nonlinear elastic, incompressible material and that are subject to large deformations. Starting from a nonlinear formulation a discrete problem can be derived by using linearisation techniques and an adaptive mixed finite element method. It turns out to be a saddle point problem that can be solved via a Bramble-Pasciak conjugate gradient method. With some modifications the simulation can be improved. This work is part of the cluster of excellence "Energy-efficient product and process innovations in production engineering" eniprod. eniprod is funded by the European Union European regional development fund - EFRE and the free state of Saxony. 34 CSC/-03 ISSN July 0

4 Contents Introduction Basics 3 3 Mixed variational formulation 7 4 Solution method 4 5 Error estimation 6 LBB conditions 5 7 Improvement suggestions 9 Authors address: Martina Balg, Arnd Meyer Chemnitz University of Technology Department of Mathematics Chair of Numerical Mathematics Numerical Analysis Reichenhainer Strasse 4 D-096 Chemnitz References [] D. Braess. Finite Elemente; Theorie, schnelle Löser und Anwendungen in der Elastizitätstheorie. Springer, Berlin, , 5, 5 [] A. Meyer. Grundgleichungen und adaptive Finite-Elemente-Simulation bei großen Deformationen. Preprint CSC 07-0, Chemnitz, , [3] A. Bucher, U.-J. Görke, P. Steinhorst, R. Kreißig, and A. Meyer. Ein Beitrag zur adaptiven gemischten Finite-Elemente-Formulierung der nahezu inkompressiblen Elastizität bei großen Verzerrungen. Preprint CSC 07-06, Chemnitz, , 9 [4] B.D. Coleman and M.E.Gurtin. Thermodynamics with internal state variables. J. Chem. Phys., 47:597 63, [5] M. Rüter and E. Stein. Analysis, finite element computation and error estimation in transversely isotropic nearly incompressible finite elasticity. Comput. Method. Appl. M., 90:59 54, , 0, 5 [6] P. J. Flory. Thermodynamic relations for high elastic materials. Transactions of the Faraday Society, 57:89 838, [7] A. Meyer. Error estimators and the adaptive finite element method on large strain deformation problems. Math. Meth. Appl. Sci., 3:48 59, 009. [8] M. Meyer. Parameter identifcation problems for elastic large deformations - part : model and solution of the inverse problem. Preprint CSC 09-05, Chemnitz, 009., [9] J. H. Bramble and J. E. Pasciak. A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems. Math. Comput., 508: 7, [0] A. Meyer and T. Steidten. Improvements and experiments on the bramblepsaciak type cg for mixed problems in elasticity. Preprint SFB , Chemnitz, [] P. Steinhorst. Anwendung adaptiver FEM für piezoelektrische und spezielle mechanische Probleme. PhD thesis, TU Chemnitz, Chemnitz Germany, [] P. Neff. Mathematische Analyse multiplikativer Viskoplastizitat. PhD thesis, TU Darmstadt, Darmstadt Germany, [3] P. Neff. On korn s first inequality with nonconstant coefficients. Proc. Roy. Soc. Edinb. A, 3: 43,

5 Because of Q Q ζ one can replace Q ζ by Q and this leads to the desired saddle point problem for all Q Q and V V 0. t f 0 V a D U ; V a V U, P ζ, ; V 7.7 au, P ζ, ; δu, V + a V U, δp ζ, ; V c ζ U ; Pζ,, Q b ζ U ; Q av U ; δu, Q c ζ U ; δp ζ,, Q 7.8 Introduction. Content The object of this work is the numerical simulation of mechanical components, that consist of nonlinear elastic and nearly incompressible material and which are subject to a large deformation. As a special case that also includes linear elastic material behaviour with small deformations. The special property of incompressible material, namely the constant volume during any shape changing deformation, needs special treatment in the mathematical formulation. Therefore a mixed ansatz plays an important role.. Notation In order to describe the considered problem of deformation, we need several mechanical quantities and the corresponding operators. These operators mostly are defined as tensors of order n and the space of these tensors is denoted with T n. By choosing a fixed basis the notation of Voigt can be used. That allows the representation of the tensors as matrices or vectors. For distinction we use different typefaces for different types of values. This is shown in table., only few exceptions can occur. Q, φ scalar function, tensor of order zero V, v vector field, tensor of order one T, σ tensor of order two M tensor of order four a, R n-vector of coefficients w.r.t. a basis of a function space A matrix Table.: types of notation Throughout this paper, pairs of vectors, such as UV, form a nd order tensor and in general a nd order tensor is any linear combination of such pairs. In the same way, a pair of nd order tensors, such as EF, defines a 4 th order tensor. Obviously any nd order tensor E is a linear operator with E : T T, as well as any 4 th order tensor M is a linear operator with M : T T. E : U E U CD U D U C. As usual we define a dot product V U R for all first order tensors U, V T. 3

6 The double dot products are defined as follows. CD : UV D U C V R U, V, C, D T. ABCD : UV D U C V AB T U, V, C, D, A, B T.3 For later use we also define the symmetric part of a second order tensor. Sym Y Y + Y T.4 With [,, ] we denote the usual scalar triple product and with we denote the usual outer vector product. Furthermore we use the L -scalar products.5-.7 and norms.8-. for all Q, P T 0, V, U T and E, F T. Q, P : V, U : E, F : Ω Ω Ω Q P dω.5 V U dω.6 E : F T dω.7 Q : Q, Q.8 V : V, V.9 V,Ω Grad V : Grad V T dω.0 Ω V,Ω : V + V,Ω. Next to the tensors itself, its derivatives are also of importance, especially the derivatives of scalar tensors or tensors of second order. The derivative of a scalar tensor ζy : T R is denoted with ζ ζy. This is a tensor of second Y order that fulfils equation. for all applied directions δy T. ζy + δy ζy + ζy Y : δy + O δy. The derivative T T Y of a tensor T Y T Y with Y T is of order four and is defined via.3. T Y + δy T Y + T Y Y : δy + O δy.3 We adapt our notation from and define a new bilinear form. b ζ U ; δu, Q : ZU S V,ζ U : EU ; δu, Q L Ω 7.7 ZU : ζi 3 U φ V I 3 U φv I 3 ζi 3 ζi 3 U I 3 I y lni 3 Then we can recompute the linearisation, which results in a modified linear problem for all Q Q and V V 0 t f 0 V a D U ; V a V U, P ζ, ; V 7.9 au, P ζ, ; δu, V + a V U, δp ζ, ; V cp ζ,, Q b 0 U ; Q b ζ U ; δu, Q cδp ζ,, Q 7.0 with a slightly changed stress and material tenor in a.... T U, P ζ, T D U + K D φ V I 3 S V U + P ζ, ζi 3 S V U 7. MU, P ζ, T D + K DS V S V + K D φ V I 3 S V + P ζ, ζc S V C + ζi 3 S V C 7. In order to achieve a saddle point problem again we have to match the bilinear forms b ζ U ; δu, Q and a V U, δp ζ, ; V. In 7.0 we rewrite Q as Z Q ζ with Q ζ Q ζ : { Q ζ Z Q : Q Q } in brief it is Z : ZU and reformulate the scalar products with equivalent equations In the last step 7.6 we define new bilinear forms. cp ζ,, Z Q ζ b 0 U ; Z Q ζ b ζ U ; δu, Z Q ζ cδp ζ,, Z Q ζ 7.3 κ P ζ,, Z Q ζ L Ω φ V, Z Q ζ κ Z P ζ,, Q ζ L Ω Z S V,ζ : EU ; δu, Z Q ζ κ δp ζ,, Z Q ζ Z φ V, Q ζ S V,ζ : EU ; δu, Q ζ κ Z δp ζ,, Q ζ L Ω L Ω c ζ U ; Pζ,, Q ζ bζ U ; Qζ av U ; δu, Q ζ c ζ U ; δp ζ,, Q ζ 7.6 3

7 With this ansatz we receive a new formulation of the stress tensor and the material tensor in the initial configuration. 7.3 Scaling function T T D U + K D φ V I 3 U S V +P S V 7.8 }{{} : T U We change the volumetric part of the stress tensor T by introducing a suitable scaling function ζi 3 R for all I 3 I 3 CU which leads to 7.0. P S V K φ V I 3 φ V I 3 C P ζ S V,ζ K φ V I 3 ζi 3 φ V I 3 C ζi Together with the decomposition of the bulk modulus we also could think of a change like 7.. P ζ, S V,ζ K φ V I 3 ζi 3 ζi 3 φ V I 3 C 7. A reasonable property of ζi 3 seems to be given by ζ and that is why 7. could be a good choice. ζi 3 I 3 y, y 0 7. Now the substitution with the hydrostatic pressure P ζ requires a slightly changed side condition 7.3. Note: Formula 7.0 yields S V,ζ ζi 3 C. κp ζ ζi 3 φ V I or κ P ζ, ζi 3 φ V I Instead of 3.5 we now have to consider a modified system of equations. t f0 V ad U ; V + a V U, P ζ, ; V b 0 U ; Q cp ζ,, Q T U, EU ; V ζu + P ζ, S V,ζ U, EU ; V φ V U, Q κ P ζ,, Q 7.6 Basics The following subsections deal with the basic terms, that are needed to describe an elastic deformation [] chap. 6 and []. In doing so geometrical and kinematic terms are introduced first and the basic equations of motion are shown later on.. Geometry We consider an arbitrary elastic body K as a representation of any given mechanical component. We assume that K has the initial configuration. prior to the deformation. Ω Ω 0 { Xη R 3 : η P R 3}. Via the influences of various external loads a deformation process occurs that transfers K into a current configuration. Ω τ { xη R 3 : η P R 3} with τ 0. We note that for all τ the parametrising set P remains the same. Hence the material point Xη in Ω turns over to the spacial point xη during the deformation. Because of the fixed parametrisation it is possible to define a covariant and contravariant tensor basis in Ω that is denoted with G i and G i respectively. Analogously one can define these bases in Ω τ. G i Xη, ηi.3 G j G i δ j i.4 g i xη, ηi.5 g j g i δ j i.6 Note: With these tensors of order one and by using the summation convention of Einstein the unit tensor of second order I T can be written as shown in.7. 3 G i G i G i G i I g i g i.7 i 30 3

8 We use the summation convention of Einstein and introduce two differential operators in each configuration, i. e. the gradient or the divergence. Grad G i in Ω.8 η i grad g i in Ω η i τ.9 Div Grad G i in Ω.0 η i div grad g i in Ω η i τ. Note: For vector fields U we have Grad U G i η U i Gi U,i which implies a correct Taylor expansion U X + V U X + V Grad U + O V.. Kinematics The deformation of the body K shall be described by a sufficiently smooth function Φ such that. holds. Φ : Ω Ω τ, Xη xη. Introducing the first order displacement tensor U one can rewrite Φ as a sum. ΦX x X + U X.3 Using the derivatives of U the covariant tensor basis of Ω τ decomposes with g i η X + U G i i + η U G i i + U,i..4 We need some important tensors to measure the deformation. First of all we get the deformation gradient F that is defined via.5. F dx dx.5 With a few transformation steps use the deformation, the displacement and the tensor basis the explicit representation.6 follows. F Grad Φ T I + Grad U T g i G i.6 7 Improvement suggestions Especially in cases of very large deformations we notice that the numerical realisation sometimes exhibits a bad behaviour. One problem is the decreasing minimum of JU h. It tents to zero because sometimes given rectangular angles are deformed into a nearly straight line. On the other hand after a certain adaptive refinement we witness a jump in the estimated error and a negative residual w T o A w o A 0 r o T A A 0 r o or even a negative determinant J occurs. May be the problem is that we use all terms of a... to build the system matrix A although a... is only coercive on the kernel of b.... Or maybe in that case the initial value U h, P h contains a high error and that influences A. The next sections show some improvements. 7. Specific energy density function Instead of 3.7 we can use 7. to formulate the deviatoric part of φc. φ D C φ D a, I 3 c 0 a 3 lni 3 7. This leads to an easier formulation of T D and M D. T D CU c 0 I c 0 C U 7. T D C { c 0 C C I 3 a II IC + CI a I + C } 7.3 c Ĉ Simple split of the bulk modulus The bulk modulus can be decomposed into two parts. K D + K : K such that 0 K D K 7.5 Thereby we predefine a new structure of φc. φc φ D I, I 3 + K Dφ V I 3 + }{{} K φ V I : φ I, I 3 The hydrostatic pressure comes in with the substitution 7.7 but in the case of incompressibility with K K the new variable P equals P. P K φ V C

9 and in general it follows 6.0 or 6. as a pull back onto Ω τ. au, P ; δu, V 4 πi 3, P Sym Grad V T F, Sym Grad δu T F + πi 3, P Grad V T F, Grad δu T F + Grad V, φ D Grad δu a J πi 3, p Sym grad v, Sym grad δu τ + J πi 3, p grad v, grad δu τ + Grad V, φ D Grad δu a We could sum up au, P ; δu, V φ D Grad V T : Grad δu dω Ω a πi 3, P F T Grad V : F T Grad δu dω 6. Ω To show 6. we use 6.0 with the argument V, V. Then, the term Grad V, φ D a Grad V would give us the needed norm estimation if the rest vanishes. We get the feeling that Korn s inequality see [], [3] thm. 3 or 4 is applicable with 6.3 for v 0. Γτ,D Note: It is c + if v Ωτ 0. Sym grad v τ grad v + grad v T c + grad v τ τ 6.3 A positive coefficient Jπi 3, p would not destroy this inequality, but yet we have no knowledge about its sign in 6.. Furthermore if c + is smaller than then a remainder is left. To guarantee that ΦX is a feasible deformation the determinant of F, denoted by J, has to fulfil condition.7. det F : J > 0.7 Note: There exists a unique polar decomposition of F with F P V U P, where V and U resp. is a unitary tensor of rotation and P is a symmetric stretch tensor. Due to F being invertible P is even positive definite. Furthermore we can derive the right Cauchy-Green strain tensor C which describes the local change of length and the Green-Lagrange strain tensor E. C : F T F.8 E : C I.9 E Grad U + Grad U T + Grad U Grad U T.0 For later use we additionally need the directional derivative EU ; V of the strain. EU ; V Grad V + Grad V T.3 Equilibrium of forces + Grad U Grad V T + Grad V Grad U T. Grad V FU + FU T Grad V T We assume that the deformation Φ of the body K is caused by the influences of external loads. These loads can be of different types: a given displacement. on the Dirichlet boundary, a deformation independent force density per unit mass.4, i.e. an acceleration field, or a force density per unit surface.3 on the Neumann boundary, a so called surface force. U 0 T on Γ D,τ Ω τ. g R 3 on Γ N,τ Ω τ.3 f R 3 with fxη fxη η P.4 After the deformation, K shall be in a state of equilibrium of forces. With these assumptions we can derive integral equilibriums and by using the theorem of Cauchy see [], p. 75 we can prove the existence of a displacement U Xη in Ω and even the existence of a symmetric, second order stress tensor σu, xη C Ω τ with the properties divσ + ρ τ f 0 in Ω τ.5 n τ σ g τ on Γ τ,n Ω τ.6 U U 0 on Γ D s. t. Γ τ,d Γ D + U

10 For a certain parameter τ the scalar material density in Ω τ is given by ρ τ and n τ T describes the outer normal vector in a boundary point x Ω τ. The boundary conditions can also be stated component wise. Note: On parts of the boundary without any given loads it is n τ σ 0. Our goal is now to simulate the displacement U..4 Incompressibility Incompressibility means that there is no change in volume by any deformation and change of shape. To ensure this J has to fulfil condition.8. det F J..8 Because J gives the ratio of the volume of K before and after the deformation see [3], the condition above yields a constant volume. In case of almost incompressiblity the condition.8 needs to be fulfilled only approximately. Furthermore one has a restriction to the material parameter K, which is called the bulk modulus. This number describes how much pressure needs to be applied to produce a relative change of volume and to compress a body K. Because we want to consider nearly or complete incompressible material we have to include the limit K.9 in our calculations. 6.3 Coercivity We plug in MU, P from 4.8 and the stress tensor from EU ; V, MU, P : EU ; δu φd EU ; V, 4I 3 + P φ V Ĉ : EU ; δu I 3 I 3 φd + EU ; V, 4I φ C D 3 + I 3 C : EU ; δu I 3 I 3 φ D + EU ; V, 4I 3 C I + IC : EU ; δu a I Since δu and V should be in the kernel of B we are allowed to omit the two last terms because it is Q, C : EU ; V 0 for Q L Ω. Furthermore it is 6.6. Grad V, T U, P Grad δu Grad V, φ D a Grad δu + F T φd Grad V, I 3 + P φ V I 3 I 3 After direct computing from 3.7 we define πi 3, P. πi 3, P : I 3 φd I 3 F T Grad δu P φ V a c 0 I 3 3 I 3 /3 + P 6.7 φ D a c 0 I 3 /3 > Note: In πi 3, P it clearly is c 0 a 3 I 3 /3 > 0, but the second addend depends on the problem. By a short computation with interchanging tensor factors it is 4 EU ; δu : Ĉ : EU ; V 4 EU ; δu C : EU ; V C Sym Grad δu T F : Sym Grad V T F grad δu + grad δu T : grad V + grad V T : 4 εδu : εv

11 The scalar product gives Q, C : F T Grad V T Q, F : Grad V T Q, F : grad V T F Q I, grad V QX divv X dx qx divvx dx. 6.9 Ω Ω τ Jx We can apply the theorem of the surjectivity of the divergence on L : For all q L Ω τ there is a v H Ω τ 3 with the properties We can choose such a pair q, v and use it in 6.9. q div v, 6.0 v,ωτ c q τ. 6. Q C, Grad V F J q, divv τ With 6. we estimate the norms. Hence the inf-sup condition follows. J q, q τ Q, Q Q 6. c 0 V V v,ωτ c q τ c Q 6.3 bu ; Q, V Q, div V sup sup V V 0 V V0 V V 0 V V0 Q Q c 0 Q V V0 c c 0 Q c Mixed variational formulation In this chapter we formulate the nonlinear problem of deformation for incompressible material. To that it is necessary to introduce a new process variable. Additionally the representation of T as a derivative of the specific energy density function φc becomes important, which is discussed in a later subsection. 3. Variational formulation of nonlinear elasticity To solve the problem by means of the finite element method the week formulation is needed. As usual we multiply with test functions V V 0 V D : { V H Ω 3 : V ΓD U 0 } 3. { v H Ω τ 3 : v Φ ΓD U 0 } V 0 : { V H Ω 3 : V ΓD 0 } 3. { v H Ω τ 3 : v Φ ΓD 0 } and integrate over the domain Ω τ. This yields the integral equation divσ, v τ + ρ τ f, v τ 0 v V After the application of the integral theorem of Gauss we get 3.4. σ, grad v τ ρ τ f, v τ + n τ σ, v 0,Γτ,N v V Contrary to the case of small deformations we need to distinguish between the values on Ω and Ω τ. But there are some quite handsome transformation rules available. dω τ [g, g, g 3 ] dη dη dη 3 [F G, F G, F G 3 ] dη det F [G, G, G 3 ] dη J dω 3.5 ρ τ Φ J ρ With the introduction of two new second order tensors, namely the first and second Piola-Kirchhoff stress tensor T and T respectively we can also generate a pull back of the stress σ from Ω τ to Ω. σ J F T 3.7 J F T F T

12 Differently from σ these new terms are living on the initial configuration and at least T keeps the symmetry property of σ. With 3.8 and the symmetry of T we transform the left hand side of 3.4. σ, grad v τ J J F T, grad V T, Grad V T : F T Grad V T dω Ω T, EU ; V 3.9 W.l.o.g. we assume ρ 0. Applying ρ τ x J ρ 0 X from 3.6 on ρ τ f, v τ from 3.4 we obtain Again w.l.o.g., we assume that ρ τ f, v τ f, V. 3.0 Γ N,τ : { xη, η, η 3 : η 3 η 0 constant, η, η P N }. 3. Then it is ds τ g g dη dη and the boundary integral in 3.4 can be transformed. n τ σ, v 0,ΓN,τ n τ σ v ds τ Γ N,τ g g P N g g σ v g g dη dη [ g, g, σ v ] dη dη P N [ ] F G P, F G, J F T V dη dη N [ ] det F P N J G, G, T V dη dη G G P N G G T V G G dη dη n 0 T, V Setting g : n 0 T we finally get the weak formulation on Ω. ΓN 3. 6 LBB conditions It is known that a saddle point problem is uniquely solvable, if the three bilinear forms au, P ; δu, V, bu ; Q, V and cp, Q are continuous and fulfil the conditions see [] sec. III.4 theorem 4. or [5] sec. 3.. au, P ; V, V α V V 0 V N 0 B; α > 0 6. bu ; Q, V sup β Q V V 0 V Q Q Q; β > 0 6. V0 cq, Q 0 Q Q 6.3 Here we use the kernel of B, which is an associate operator to b.... N 0 B { V V 0 : bu ; Q, V Q, BU V 0 Q Q } Instead of showing the coercivity 6. it suffices to show the stability on the kernel. au, P ; δu, V sup α δu V N 0B V V0 δu N 0 B; α > V0 sup δu N 0B 6. First steps au, P ; δu, V δu V0 α V V0 V N 0 B; α > Condition 6.3 can be shown easily with 4.3. cq, Q cq, Q 6. Inf-sup condition Due to the symmetry of S V U C U 6.8 holds. bu ; Q, V sup V V 0 V V0 Ω Ω κ Q dω κ Q κ Q dω max κ Q < 6.7 }{{}}{{} :γ < < sup V H Ω 3 V V0 Q, S V U : EU ; V sup V H Ω 3 V V0 Q, C : F T Grad V T

13 With de U, e P : γt e U,Ω + e P we get the element wise error indicator η T,comb with 5.. η T,comb : r P 0,T + h T γ R T 0,T + h T R T γ F T T 0,F F T 5.0 J { } / e U, e P c η T,comb T Υ h 5. However, numerical tests indicate that the r P part can be left out because it is dominated by the other ones. There are also reasonable arguments why it is possible to leave out the element residual too. To prevent the case that there is no mesh refinement if the estimated error is zero e. g. if Ω Γ D without any inner faces we set the refinement condition as follows. { } refine element T i if η Ti p max ηti, p 0, 5. T i Find U V D such that 3.3 is fulfilled. T, EU ; V Note: 0 f + Div T U, V f, V + g, V } {{ 0,Γ N } : f 0V V V 0 3. Stress tensor of nonlinear elasticity V V The tensor T from 3.8 can be derived from the Clausius-Duhem inequality see [4] sec. -5 or [3] p. 5. T : Ċ ρ φc Here φc stands for the free Helmholtz energy density per mass unit and the dot symbolises the usual time derivative. For simplification we define the specific strain energy density function per volume unit φc. φc : ρ 0 φc 3.5 As it is shown in [4], sec. hyperelasticity. 5 the inequality 3.4 yields the so called law of T φc Decomposition ansatz of Flory We consider the so called Flory split of the deformation gradient F and later on of the Cauchy-Green strain tensor C into a deviatoric and a volumetric isochoric part see [3], [5] or [6]. F F D F V 3.7 The part F D shall describe the change of shape whereas F V shall describe the change of volume during the deformation Φ. That is why one can postulate det F D and 3.8 det F V J κ

14 These conditions can be fulfilled easily by setting F D to J /3 F and F V to J /3 I. Analogously one can multiplicatively decompose C with C C D C V. Both quantities have to fulfil 3.0 and 3.. C D F DT F D J /3 F T J /3 F J /3 C 3.0 C V F V T F V J /3 I T J /3 I J /3 I 3. This decomposition also affects the specific strain energy density, in such a way that φc splits into two additive parts. Again we get a deviatoric part φ D C which is dedicated to the energy of the changing shape and a volumetric part φ V C which only describes the energy of the changing volume during a deformation. φc φ D C D + UC V φ D C D + K φ V C V 3. As we consider isotropic and objective material we can choose an energy density that only depends on the three tensor invariants 3.5 instead of the tensor itself. φc φ D I C D, I C D, I 3 C D + K φ V I C V, I C V, I 3 C V 3.3 with I Y try 3.4 I Y try tr Y 3.5 I 3 Y dety Y T With 3.0 and 3. the identity I 3 C D follows. In addition we can show that I k C V depends only on J for all k,, 3. Therefore we can omit the dependencies on I 3 C D and replace I k C V by J. Additionally we also decide to omit all terms depending on I C D. φc φ D I C D + K φ V J 3.6 We also want to achieve that φ V J is convex and is zero in J. In this case a suitable material function is the Neo-Hooke material see [5]. Plugging in our values this yields 3.7. φc c 0 I C D 3 + K lnj c 0 I CI 3 C /3 3 + K }{{} lni 3C }{{} : φ DI C,I 3C : K φv I3C Note: As an abbreviation we use I k : I k C from now on. 3.7 The sum of all error indicators gives an upper bound on our error functional. J e U a U, P ; e U a U h, P h ; e U 5.3 γt e U,Ω { } / c T Υ h η T Following the derivation for the case of linear elasticity, where we usually choose γt to represent the smallest eigenvalue of M, we take γt c 0. Note: Analogously one can treat the second equation of 5.. We denote the error with e P : P P h in 0, Ω and define a test function Q : I I h ep in 0, Ω. Subsequent we consider the error functional 5.7 and the corresponding denominator D. J e P b 0 U ; e P cp ; e P b 0 U h ; e P + cp h ; e P d P e P, e P / 5.5 D b 0 U ; e P cp ; e P b 0 U h ; e P + cp h ; e P b 0 U h ; Q + cp h ; Q κ P h φ V U h, Q 5.6 With κ P h φ V U h : r P 0, T we get the following estimate. We should set 5.8. D C r P / 0,T ep 5.7 T d P e P, e P / e P 5.8 If we combine these two results, we would get the following. de U, e P / J e U, e P a U, P ; e U a U h, P h ; e U b 0 U ; e P + b 0 U h ; e P + cp ; e P cp h ; e P c { T h T γ T R T 0,T + T,F T h T R γ F T + T 0,F T r P 0,T } / { γt e U,Ω + e P } /

15 We introduce the abbreviation T h T U h, P h and the operator F,ΓN that is the identity if F Γ N and zero elsewhere. Again we set ρ 0 equal to one, w.l.o.g. The integral theorem of Gauss yields 5.7. D f, V 0,T + g, V 0,F T Υ h F Γ N + Div T h, V T Υ h T Υ h f + Div T h, V We define the residuals R T and R F T. 0,T F T + 0,T F T n T h, V 0,F F,ΓN g n T h, V 0,F 5.7 R T : f + Div T h 5.8 n T h n T h if F T T T T R F T : 5.9 g n T h if F Γ N 0 else Note: n n R F T T T n T T h T h T T With the new notation, 5.7 can be reformulated as follows. D RT, V 0,T + RF T, V 0,F T Υ h F T 5.0 This can be estimated from above by using Cauchy-Schwarz inequalities, interpolation estimates and the Clément interpolation operator as well as the patches ˆT and ˆF around T and F respectively and a scalar material function γt γ T T that is constant on every element T. Finally this approach yields 5.. { D C γ R T 0,T + T h T T Υ h T,F T h } / T R γ F T γt e T 0,F U,Ω 5. From 5. we can define the element wise error indicator η T as shown in 5. and d U e U, e U with d U e U, e U / : γt e U,Ω. η T : h T γ T R T 0,T + F T h T γ T R F T 0,F Hydrostatic pressure With 3.6 and 3.7 we can define the second Piola-Kirchhoff stress tensor more precisely. T C φc φ D φ V +K φ V 3.8 }{{ }}{{ } :T D :S V However the product term K φ V cannot be determined explicitly, because K diverges according to.9 and φ V vanishes for J. One way out is given by the substitution 3.9, which eliminates the problematic term. P : K φ V I 3 κ φ V I The new material parameter κ in 3.9 is defined as the reciprocal of the bulk modulus and it is called the compressibility of the material. Obviously, we have κ 0 for incompressible material and κ 0 for nearly incompressible material. After plugging in the substitution 3.9 in 3.8 we obtain a new formulation of the stress T T U, P T CU, P T D C + P S V C 3.30 together with the side condition 3.3, which is later used in a weak sense. 3.5 Piola-Kirchhoff stress φ V I 3 κ P The derivative φc in 3.8 can be build with the help of the pseudo invariants a k see [7], [8]. a i Y : i tr Y i i,, 3, Y T 3.3 a a a a 3 T with a i : a i C 3.33 They permit the reformulation of the tensor invariants I Y a Y I Y a Y a Y 3.34 I 3 Y a Y 3 6a Ya Y + 6a 3 Y 6

16 and they have a simple derivative w. r. t. any second order tensor Y see [8], use Taylor expansion. a i Y Y Y i 3.35 With 3.34 the specific energy density function can be reformulated in terms of a, instead of 3.7. φc φ V a, I 3 a + K φ V I 3 a 3.36 φ V a + K φ V a 3.37 Note: We keep the notation φ D and φ V although the dependencies of the functions has been changed. By applying the chain rule to 3.37 the derivatives of φ D C and φ V C with respect to C can be determined as the second order tensors T D φ DC 3 φd a a i, 3.38 i a i S V φ V C 3 φv a a i a i In general we get i T D φ D a I + φ D a C + φ D a 3 C, 3.40 S V φ V a I + φ V a C + φ V a 3 C. 3.4 If the dependency on I and I 3 see 3.7 is used directly we obtain in the same way /3 T D c 0 I 3 I a, C S V C The equivalence of 3.4, 3.43 with 3.40, 3.4 would follow from the theorem of Cayley-Hamilton as well. 3.6 Mixed formulation With the decomposition 3.30 of T we get the equation 3.44 instead of 3.3. T D U, EU ; V + P S V U, EU ; V f 0 V 3.44 V V 0 5 Error estimation In order to control the adaptive mesh refinement we need an appropriate error estimator of the approximated solution U h, P h of Reliability a U h, P h ; V f 0 V V V h,0 b 0 U h ; Q cp h ; Q 0 Q Q h 5. s. t. a U, P ; V : a D U ; V + a V U, P ; V T U, P, Grad V 5. We use e U V 0 with e U : U U h to denote the error and we define a test function V V 0 with V Ie U I h e U and the projection I h : V 0 V h,0. As it is shown in [] the form a U, P ; V does not lead to a proper energy norm in a sense of U, P A and the Galerkin orthogonality becomes like 5.3. a V : a U, P ; V a U h, P h ; V a U, P ; V f 0 V 0 V V Now we consider 5.4 as a functional to meassure the error, whereas d U V, V denots some kind of norm. J e U a U, P ; e U a U h, P h ; e U d U e U, e U / 5.4 After exploiting the Galerkin orthogonality the denominator D : a e U yields 5.5. D a U, P ; V a U h, P h ; V T U, P, EU ; V T Uh, P h, EU h ; V ρ 0 f, V + g, V 0,ΓN T Uh, P h, Grad V 5.5 Using the discretisation Υ h we get 5.6. D ρ 0 f, V 0,T T Uh, P h, Grad V T Υ h 0,T + g, V 0,F 5.6 F Γ N

17 Since the matices are self adjoint A0 A x o, y o o x o, A 0 A y o C0 K x u, y u u x u, C 0 K y u we can estimate the eigenvalues in 4.36 with the Rayleigh quotient Θ x. We choose Θ o,a0 Ax o : λ min C 0 K A0 A x o, x o o u o λ max A 0 A 4.39 x o, x o o C0 K x u, x u u : Θ x u, x u u,c0 Kx u 4.40 u δ Θ o,a 0 Ax o Θ u,c0 Kx u λ maxa 0 A λ min C 0 K which yields the estimate 4.4 after applying this choice to κs h,0 S h c A κm c A κa 0 A κc 0 K 4.4 Note: With the exact choice of δ we would get κm max { κa 0 A, κc 0 K } for δ λ maxa 0 A λ max C 0 K, κm κa 0 A κc 0 K for δ λ maxa 0 A λ min C 0 K. Additionally we have to guarantee condition 3.3 in a weak sense. Therefor we choose suitable test functions Q Q : L Ω and get φ V I 3 U, Q κ P, Q 0 Q Q We introduce a new notation a D U ; V T D U, EU ; V 3.46 a V U, P ; V P S V U, EU ; V 3.47 b 0 U ; Q φ V I 3 U, Q 3.48 cp ; Q κ P, Q 3.49 f 0 V f, V + g, V 0,ΓN 3.50 that allows us to formulate the mixed boundary value problem of nearly incompressible nonlinear elasticity with large deformations via an nonlinear system of equations. This yields the nonlinear saddle-point like problem. Find the displacement U V D and the hydrostatic pressure P Q such that a D U ; V + a V U, P ; V f 0 V b 0 U ; Q cp ; Q 0 holds for all test functions V, Q V 0 Q

18 4 Solution method To solve problem 3.5 we use Newton s method to linearise the equations and mixed finite elements to discretise them afterwards. The resulting system can be solved with a method of conjugate gradients due to Bramble and Pasciak see [9] or [0]. Altogether this leads to a nested iteration method. 4. Newton s method First of all we transform 3.5 into a homogeneous problem. 0 ad U ; V + a V U, P ; V f 0 V : SU, P ; V, Q 4. 0 b 0 U ; Q cp ; Q Even though 4. seems to be nonlinear in U only, we have to perform a linearisation in both U and P. This can be done with Newton s method. We choose a initial solution U 0, P 0, solve equation 4. several times for i 0 S U i, P i ; V, Q, δu, δp SU i, P i ; V, Q V, Q V 0 Q 4. and update the current initial solution U i, P i with the increment δu, δp in each step. U i+ U i δu P i+ P i δp Because this method has only local convergence we additionally use incremental load steps. Instead of 4. and 4. we consider the operator 4.4 with t 0, ] : t and equation 4.5 for all test functions V V 0 und Q Q. ad U ; V + a V U, P ; V t f 0 V SU, P, t; V, Q 4.4 b 0 U ; Q cp ; Q S U, P ; V, Q, δu, δp SU, P, t; V, Q 4.5 For any fixed t we choose δu < ε U tol as the stopping criteria. For t, this yields the approximate solution of the problem 3.5. Note: We abbreviate S : SU, P, t; V, Q and S : S U, P ; V, Q, δu, δp. 4. Newton s equation We need the representation of the linear operator S as the first derivative of S applyied to δu, δp. This follows from the Taylor expansion 4.6 of S with omitting the sufficient small nonlinear terms O δp, O δu and O δu δp δu Find such that 4.3 holds. δp δu A B δu S h : δp B T C δp Rf R g 4.3 At first glance this system 4.3 is indefinite, however it can be solved with the conjugate gradient method by Bramble and Pasciak. The idea is to use the matrix S h,0 in 4.3 as a preconditioner on the system and to define a new suitable duality product [ ] A 0 0 δc S h,0 : 0 B T A 0 γ δc0 x, y : xo, y o o + x u, y u u : A γa 0 x o, y o + δ C 0 x u, y u 4.34 Here we choose A 0 to be a preconditioner of A e. g. with BPX and C 0 to be a preconditioner of the Schur complement K : B T A 0 B + γc e. g. by taking the diagonal of C. The positive parameters δ and γ shall guarantee A γa 0 0 and decrease the condition number of the given system see next subsection. To the resulting system we apply the usual conjugate gradient method. But we use a matrix free method, i. e. the stiffness matrix is never assembled completely but used element wise. According to that the occurring matrix-vector products are determined on an element wise basis. 4.5 Optimal parameter The described conjugate gradient method by Bramble and Pasciak is especially good if the right parameters γ and δ are used. To determine γ we use a rather simple method. We consider the value of wt A w γ w T A 0 w. If it is negative we reduce γ by a fixed factor until the condition is fulfilled again. To determine δ we can use the eigenvalues of the matrices see [], section 4.. First we consider the diagonal of S h,0 S h that is denoted by M. [ ] A0 A 0 M 0 δc 0 K The corresponding condition number fulfils κs h,0 S h c A γ, A, A 0 κm 4.35 max { λ max A 0 A, δλ max C 0 K } c A min { λ min A 0 A, δλ min C 0 K }

19 a more detailed representation. MU, P 4I φ D 3 + P φ V C C I 3 φd P φ V I 3 I 3 I 3 a II IC CI a I + C φ D + 4I 3 IC + C I 4.7 a I 3 φd MU, P 4I 3 + P φ V Ĉ I 3 I 3 + 4I 3 φd I 3 + I 3 φ D I 3 C C + 4I 3 φ D a I 3 C I + IC Mixed finite element method The given linear saddle-point problem 4.8 shall be solved by means of a mixed FE method. Hence we assume that there is a regular triangulation Υ h of the domain Ω available with n T hexahedral elements T i and n N nodal points. We choose the stable Taylor-Hood element which implies the ansatz of an element wise triquadratic displacement and an element wise trilinear pressure. The computations can be made with the help of the reference element T [, ] 3 and the associated transformation map 4.9. As suitable function spaces we take B i : T T i, ˆX X 4.9 V h { V CΩ 3 H Ω 3 : V Ti B i Q T 3 T i Υ h } V h,0 { V V h : V ΓD 0 } V h,d { V V h : V ΓD U 0 } Q h { Q CΩ L Ω : Q Ti B i Q T T i Υ h } 4.30 as well as the reordering of the remaining terms w. r. t. S and S. SU +δu, P +δp, t; V, Q ad U +δu ; V + a V U +δu, P +δp ; V t f 0 V b 0 U +δu ; Q cp +δp, Q SU, P, t; V, Q + S U, P ; V, Q, δu, δp O δu + O δp + O δu δp 4.7 Therefor we consider the Taylor expansions of the integral forms and of the integrands EU, EU ; V, T D U, S V U and φ V U. Althogether this yields 4.8 and 4.9. a D U +δu ; V + a V U +δu, P +δp ; V t f 0 V a D U ; V + a V U, P ; V t f 0 V + a V U, δp ; V + EU ; V, P S V C + T DC : EU ; δu + Grad V, P S V U + T D U Grad δu + O δu + O δp + O δu δp 4.8 b 0 U +δu ; Q cp +δp, Q b 0 U ; Q cp, Q + Q S V U, EU ; δu cδp, Q Using the material tensor + O δu 4.9 We choose suitable triquadratic and trilinear nodal ansatz functions Φ i and Ψ j resp. to represent the functions of V h, and Q h as a linear combination of these ansatz functions with certain coefficients δu j and δp j. That leads to a discrete saddle point problem which can be rewritten in matrix-vector form. MU, P : T DC }{{} : M DU +P S V C }{{} : M V U

20 we introduce a new notation. au, P ; δu, V EU ; V, MU, P : EU ; δu 4. + Grad V, T U, P Grad δu bu ; δp, δu δp S V U, EU ; δu 4. cδp, Q κ δp, Q 4.3 ft, U, P ; V t f 0 V a D U ; V a V U, P ; V 4.4 gu, P ; Q cp, Q b 0 U ; Q 4.5 Finally by omitting the nonlinear terms in 4.8 and 4.9 resp. the operators S und S are built. au, P ; δu, V + bu ; δp, V S U, P ; V, Q, δu, δp, 4.6 bu ; Q, δu cδp, Q ft, U, P ; V SU, P, t; V, Q. 4.7 gu, P ; Q This formulation together with the linear Newton s equation leads to the linear saddle-point problem in each Newton s step. Let t, U, P be a given tripel in 0, ] V D Q. Find the solution pair δu, δp V 0 Q that fulfils the system 4.8 au, P ; δu, V + bu ; δp, V ft, U, P ; V for all test functions V, Q V 0 Q. 4.3 Material tensor bu ; Q, δu cδp, Q gu, P ; Q 4.8 We want to take a closer look on the material tensor 4.0. If we plug in the representations 3.38 and 3.39 of the stress tensors we can give a more detailed description of M. M 3 φ a C i i a i for D/V M 4 C i φ a + φ a i i a i a i 3 4 C i φ aj 3 φ + 4 i a i,j j a i i a i 3 φ 3 4 C i C j φ + 4 i i,j a i a j i a i The new tensors C i C j and i i are of order four with I 0 for i I for i C for i Thereby it is 0 the forth order null tensor and I the forth order unit tensor. C is the tensor that realises definition 4.3. For later use we introduce another forth order tensor Ĉ with 4.4. I : Y Y Y T 4. C : Y Y C + C Y Y T 4.3 Ĉ : Y C Y C Y T 4.4 Reformulating both parts of the material tensor from 4.0 yields M φ 4 C j C i i,j a i a j + 4 φ a I + 4 φ a 3 C 4.5 { 3 φ D } φ V MU, P 4 + P C j C i i,j a i a j a i a j φd P φ V φd I P φ V C 4.6 a a a 3 a 3 By means of the material function 3.7 that is under consideration we can find 6 7