Stress-based finite element methods in linear and nonlinear solid mechanics

Transcription

1 Stress-based finite element metods in linear and nonlinear solid mecanics Benjamin Müller and Gerard Starke Fakultät für Matematik, Universität Duisburg-Essen, Essen, Germany Abstract A comparison of stress-based finite element metods is given for te prototype problem of linear elasticity and ten extended to finite-strain yperelasticity. Of particular interest is te accuracy of traction forces in reasonable Sobolev norms wit an empasis on uniform approximation beavior in te incompressible limit. Te mixed formulation of Hellinger-Reissner type leading to a saddle-point problem as well as a first-order system least squares approac are investigated and te strong connections between tese two metods are studied. In addition, we also discuss stress reconstruction tecniques based on displacement approximations by nonconforming finite elements. 1 Introduction Te accurate resolution of stresses associated wit numerical simulations in solid mecanics is of paramount importance in many applications. Large stress components may cause plastic beavior or even damage and need terefore to be approximated well. In particular, if surface traction forces are of interest, finite element approximations in spaces wic allow te safe evaluation of boundary traces need to be used. For standard displacementbased or in te incompressible case displacement-pressure approaces, te associated stresses are only contained in L 2 wic means tat te normal component of te boundary traces are not defined. Variational principles wic involve stresses in Hdiv-like saddle point formulations of Hellinger- Reissner-type or first-order system least squares approaces overcome tis problem directly. Anoter option is to reconstruct stresses in Hdiv from sufficiently accurate L 2 approximations in analogy to te flux reconstruction procedures described, e.g., in Luce and Wolmut 2004; Nicaise et al. 2008; Braess et al. 2009; Cai and Zang 2012; Ern and Voralík For an equilibration approac to stress reconstruction in two-dimensional linear elasticity see also Parés et al Particularly attractive in te 1

2 context of incompressible elasticity are te quadratic nonconforming elements introduced by Fortin and Soulie 1983 and, in tree space dimensions, Fortin Flux and stress reconstruction procedures working in an element-wise way were studied for tese elements by Kim Te istory of mixed finite element metods of saddle-point type for te approximation of stresses in Hdiv in linear elasticity models goes back for at least 30 years wit early contributions by Arnold et al. 1984a, Arnold et al. 1984b and Stenberg 1988 among oters, see Boffi et al., 2013, Cp. 9 for more details. Later, tis approac also received muc attention in te engineering community, see e.g. Klaas et al For te class of first-order system least squares metods te state-of-te-art is presented in Bocev and Gunzburger 2009 wit a focus of fluid rater tan solid mecanics. Te Hdiv-based stress formulation wic will be our starting point in tis contribution was studied in Cai and Starke 2004 for te linear elasticity case and extended to yperelastic material models in Müller et al Te investigation of yperelastic models in Section 5 will be presented in detail for te specific example of a neo-hookean material law. For background on te analytical and numerical treatment of yperelasticity, we refer to Ciarlet 1988 and LeTallec Concerning a priori finite element error estimates associated wit suc models, see Carstensen and Dolzmann Our focus in Section 5 of tis contribution will again be on approaces wic remain robust in te incompressible limit. Similar to te linear elasticity case tis may be acieved eiter by adding an auxiliary pressure variable cf. Auriccio et al or by inverting te stress-strain relation, cf. Wriggers, 2008, Sect Te elasticity problems under our consideration are based on an open, bounded and connected domain Ω IR d d = 2, 3 wit Lipscitz-continuous boundary wic constitutes te reference configuration of te undeformed state. Te boundary is divided into two disjoint subsets Γ D and Γ N, for simplicity, bot assumed to be non-empty. On Γ D, omogeneous displacement boundary conditions u = 0 are imposed, wile surface traction forces σ n = g are prescribed on Γ N. Te linear elasticity model may ten be written as te first-order system div σ + f = 0 σ Cεu = 0 1 in Ω subject to te above boundary conditions wit εu = u+ u T /2 and Cε = 2µε + λtr εi. 2 2

3 Te system 1 may be derived from minimizing te energy associated wit te deformed system given by ψεv dx f v dx g v ds, 3 Ω Ω Γ N were te stored energy function is given by ψε = µ ε 2 + λ 2 tr ε2. 4 Te necessary conditions for a stationary point of 3 are ten equivalent to 1. Wile we can always scale te units suc tat µ is on te order of 1, an important issue is te beavior of te formulations in te incompressible limit λ. It is already apparent from 2 tat a naive numerical approac to te above minimization problem will cause problems for incompressible or nearly incompressible materials. One possible remedy consists in replacing λtrε by a new variable p wic as te pysical interpretation of a pressure. Anoter option is to use te inverse C 1 instead of C in te variational formulation. A straigtforward calculation sows tat C 1 σ = 1 2µ σ λ 2µ + dλ trσi λ 1 2µ σ 1d trσi = 1 dev σ, 2µ i.e. te operator C 1 remains well-defined in te incompressible limit, were it constitutes te ortogonal projection onto te trace-free matrices dev. Since C 1 itself is not invertible any more in te incompressible limit, we also write A instead of C 1 in order to avoid missunderstandings. Te first-order system 1 turns into div σ + f = 0 Aσ εu = 0. 5 Of course, any variational approac based on 5 needs to use te stress σ as an independent variable in te formulation. Suc approaces will be presented in te next sections. We will make use of norms and inner products associated wit different spaces trougout tis paper. Since L 2 Ω and its vector and matrix variants L 2 Ω d and L 2 Ω d d, respectively occurs most often, we abbreviate te associated norm simply by and te corresponding inner product by,. Since we assume Γ D more precisely, a subset of Ω of positive measure, Korn s inequality is valid in te form v C K εv for all v H 1 Γ D Ω d. 6 3

4 Our general regularity assumption is tat Ω IR d, Γ N Ω and Γ D Ω are suc tat, for any f L 2 Ω d te solution of 5 satisfies σ, u H α Ω d d H 1+α Ω d suc tat σ Hα Ω + u H 1+α Ω C R f 7 olds for some constant C R > 0 and some α > 0. 2 Stress-based mixed formulation based on te Hellinger-Reissner principle Tis section is focussed on te approximation of stresses in te Sobolev space Hdiv, Ω d. Te subspaces H ΓN div, Ω d = {τ Hdiv, Ω d : τ n = 0 on Γ N }, H 0 Γ N div, Ω d = {τ H ΓN div, Ω d : div τ = 0} will also be used. From te stress-strain relation εu = C 1 σ, integration by parts leads to C 1 σ, τ + u, div τ + γ, as τ = 0, 8 for all τ H ΓN div, Ω d, were as τ = τ τ T /2 denotes te asymmetric part and γ is a new variable introduced for as u. Togeter wit te two equations div σ + f, v = 0 for all v L 2 Ω d, as σ, θ = 0 for all θ L 2 Ω d d,as, were L 2 Ω d d,as denotes te subspace of L 2 Ω d d wit vanising symmetric part, te mixed variational formulation of Hellinger-Reissner consists in finding σ, u, γ σ N +H ΓN div, Ω d L 2 Ω d L 2 Ω d d,as suc tat 8 and 9 old. An alternative way of deriving tis mixed variational formulation consists in viewing it as te KKT conditions for te minimization of te energy C 1 σ, σ/2 subject to te constraints 9. In tis context, u and γ are Lagrange parameters for te momentum balance and symmetry conditions, respectively, in 9. For te well-posedness of te system 8, 9, te following result is of crucial importance. Teorem 2.1. Assume tat Γ N Ω consists of a finite number of connected components eac of wic as positive d 1-dimensional measure. Ten, τ dev τ 10 olds for all τ H 0 Γ N div, Ω d. 9 4

5 Teorem 2.1 follows from te more general result of Teorem 3.1 in Section 3. A direct and simple proof for te two-dimensional case is given in te following. Proof. for d = 2. Witout loss of generality assume tat Γ N Ω and tat Γ N is connected wit Γ N > 0 just replace Γ N by one of its connected components, cut someting off if Γ N = Ω. Using Girault and Raviart, 1986, Tm. I.3.1, we can write τ = curl φ wit φ H 1 Ω 2. Te boundary conditions curl φ n = 0 imply φ to be constant on Γ N, wic we may coose to be zero, i.e., φ H 1 Γ N Ω 2. Terefore, φ satisfies Korn s inequality 6 and we obtain τ = curl φ = φ εφ 1 = 1 φ 1 2 2φ φ φ φ 2 2 φ 2 1 = 2 2φ φ 2 1 φ 1 2 φ φ φ 2 2 φ = dev 1 1 φ 1 = dev curl φ = dev τ. 2 φ 2 1 φ 2 Teorem 2.1 implies C 1 τ, τ 1 2µ dev τ 2 τ 2 for all τ H 0 Γ N div, Ω d. 11 Since HΓ 0 N div, Ω d contains te null space of te constraints 9, te required coercivity condition is satisfied. As a second ingredient to te wellposedness, te inf-sup condition as to be establised for 9, see Boffi et al., 2013, Prop For te discretization of 8, 9, finite element spaces Π H ΓN div, Ω d, Z L 2 Ω d and Θ L 2 Ω d d,as are inserted into 8, 9 leading to a mixed finite element approximation σ HR, zhr, γhr. To tis end, various finite element combinations wic satisfy te discrete inf-sup condition ave been proposed, starting wit te famous PEERS element Arnold et al. 1984a. For a systematic treatment of tis topic see Boffi et al., 2013, Cp. 9. It is interesting to note tat for k 1, te triple of finite element spaces Π, Z, Θ = RT k T d DP k T d P k T d d,as 5

6 is inf-sup stable see Boffi et al and Boffi et al., 2013, Expl An important property of tis approac is tat te momentum balance is best possible, i.e., div σ HR + f = f π f = inf z Z f z. From te ellipticity and te inf-sup conditions, optimal order accuracy also follows for te stress approximation wit respect to te L 2 Ω-norm, i.e, σ σ HR inf τ Π σ τ α σ H α Ω Stress-displacement first-order system least squares In tis section, we consider te first-order system least squares approac based on div σ + f Rσ, u :=, 13 Aσ εu i.e., te minimization of Fτ, v := Rτ, v 2 = div τ + f 2 + Aτ εv 2 14 among all τ σ N + H ΓN div, Ω 3 and v H 1 Γ D Ω 3. Te minimizer σ, u of 14 satisfies div σ, div τ + Aσ εu, Aτ = f, div τ, Aσ εu, εv = 0 15 for all τ, v H ΓN div, Ω 3 H 1 Γ D Ω 3 wic constitutes a linear variational problem. Te well-posedness of 15 follows from te coercivity and continuity of te bilinear form Bσ, u; τ, v := div σ, div τ + Aσ εu, Aτ εv 16 wit respect to H ΓN div, Ω 3 H 1 Γ D Ω 3 uniformly in te incompressible limit. Tis property was sown under our assumptions on Ω, Γ N and Γ D in Cai and Starke A consequence of its validity in te incompressible limit is te following result. Teorem 3.1. Assume tat Γ N Ω consists of a finite number of connected components eac of wic as positive d 1-dimensional measure. Ten, τ dev τ + div τ 17 olds for all τ H ΓN Ω d. 6

7 Te result of Teorem 3.1 was proved in Arnold et al. 1984b for te case Γ N = under te additional constraint tr τ, 1 = 0 see also Boffi et al., 2013, Prop and in te general two-dimensional case in Carstensen and Dolzmann Te discrete first-order system least squares approximation is obtained by minimizing 14 among all τ = σ N + Π and v V, were Π H ΓN div, Ω 3 and V HΓ 1 D Ω 3 are suitable finite element spaces. Te approximate solution σ LS σ N + Π, u LS V is determined by div σ LS, div τ + Aσ LS εu LS, Aτ = f, div τ, Aσ LS εu LS, εv = 0 18 for all τ, v Π V. Due to te coercivity and continuity of te underlying bilinear form we obtain a quasi-optimal approximation, i.e., σ σ LS div,ω inf τ Π σ τ div,ω, u u LS 1,Ω inf v V u v 1,Ω. 19 In particular, using, for some l 1, Raviart-Tomas spaces of degree l 1 for Π combined wit standard conforming finite elements of degree l for V, one gets σ σ LS div,ω l σ l,ω + div σ l,ω, u u LS 1,Ω l u l+1,ω, if σ H l Ω 3 wit div σ= f H l Ω and u H l+1 Ω 3 is satisfied. It may also be wort noting tat witin te first-order least squares approac, piecewise linear conforming displacement approximations are of optimal order uniformly in te incompressible limit. Of course, tis requires te simultaneous computation of stress approximations in te lowest-order Raviart- Tomas spaces wic may be considered too costly if tese quantities are not of particular interest. If te solution is less regular, ten te optimal approximation order may be retained wit adaptively refined triangulations based on using te local evaluation of te functional as an a posteriori error estimator, cf. Cai et al For domains wit curved boundaries in association wit te iger-order case l > 1, parametric finite element spaces would be needed in order to retain te optimal approximation order. Tis would involve te parametric Raviart-Tomas spaces studied in Bertrand et al for Π in combination wit standard isoparametric elements, cf. Brenner and Scott, 2008, Sect

8 It is important to keep in mind tat te two terms in te functional defined by 14 need to be scaled appropriately in order to get reasonable approximations. Tis is due to te fact tat te constants involved in te above estimates must not become exceedingly large. Te two main ingredients wic influence tese constants are te Lamé parameter µ in te material law 2 and C K in Korn s inequality 6. If bot are on te order of one, ten te scaling in 14 is adequate. Tis can be acieved by te coice of suitable units for measuring forces and lengts. Our computational experience suggests tat it is generally less armful to weigt te momentum balance term too strong tan too weak wit respect to te above rules. In contrast to te mixed approximation σ HR, our least-squares approximation σ LS does not satisfy te momentum balance exactly if f div Π. We will now sow tat, in fact, te momentum balance term in te functional 14 converges faster tan te overall functional. Te proof is inspired by te tecniques used in Brandts et al for te investigation of te relations between saddle point and least-squares formulations for te firstorder system formulation of te Poisson equation. Teorem 3.2. Under our regularity assumptions, te momentum balance accuracy associated wit te first-order system least squares approximation satisfies divσ LS +f α σ σ LS + εu εu LS + inf f z. 20 z Z Proof. Wit f = π f Z, te triangle inequality leads to divσ LS +f divσ LS +f + f f = divσ LS +f + f π f. 21 Te first term on te rigt and side in 21 can be written as div σ LS div σ LS + f = sup + f, z z Z z div σ LS = sup + f, z div σ LS = sup σ, z. z Z z z Z z 22 For any z Z, te following auxiliary boundary value problem may be defined: Find Ξ H ΓN div, Ω 3 and η H 1 Γ D Ω 3 suc tat div Ξ = z, AΞ εη =

9 olds. Let Ξ HR Π be te mixed finite element approximation of Hellinger- Reissner type to 23 and let η V be any approximation to η, ten div σ σ LS, z = div σ σ LS, div Ξ = div σ σ LS, div Ξ + Aσ σ LS = div σ σ LS, div Ξ Ξ HR + Aσ σ LS = Aσ σ LS εu u LS, AΞ εη εu u LS, AΞ Ξ HR εη η εu u LS, AΞ Ξ HR εη η olds due to 15, 18 and te fact tat div Ξ HR Combining tis wit 22 leads to = div Ξ = z is satisfied. div σ LS + f Aσ σ LS σ σ LS sup Ξ α σ σ LS εu u LS AΞ Ξ HR εη η div Ξ sup Ξ + εu u LS Ξ Ξ HR + εη η div Ξ + εu u LS 24 due to 12 and our general regularity assumption from Section 1. Teorem 3.2 states tat te error associated wit momentum balance converges of iger order. In particular, if f H α Ω d is assumed for te rigt-and side, ten div σ σ LS α σ σ LS + εu εu LS + f 25 olds. One implication of 25 is concerned wit te approximation of boundary traces. For te approximation of te resultant traction forces, σ σ LS n, e L 2 Ω = div σ σ LS, e div σ σ LS e 26 olds for any constant displacement field e IR d. A furter implication, wic is seen best directly in 24, is tat te second term in te least squares functional 14 dominates if σ LS, uls is inserted. Tis property 9

10 can be of use, in particular, in te study of te functional as an a posteriori error estimator. 4 Stress reconstruction for displacement-pressure approaces Te most commonly used approac to compute finite element approximations for te linear elasticity model is based on minimizing te energy in 3 among all v H 1 Γ D Ω d. Te solution u H 1 Γ D Ω d satisfies Ω 2µ εu : εv + λ div u div v dx = or, in sort notation, Ω f v dx + Γ N g v ds 2µ εu, εv + λ div u, div v = f, v + g, v L2 Γ N 27 for all v H 1 Γ D Ω d. Obviously, tis formulation becomes problematic as te Lamé parameter λ tends to wic is te case for incompressible materials. One possible remedy is to introduce a new pressure-like variable p = λ div u wic leads to te saddle-point problem 2µ εu, εv + p, div v = f, v + g, v L 2 Γ N div u, q 1 λ p, q = 0 28 for all v H 1 Γ D Ω d and q L 2 Ω. Tis saddle-point problem is a regular perturbation of te Stokes problem modelling incompressible fluid flow wic coincides wit te limiting case λ = and as suc can be treated wit any inf-sup stable finite element pair V, Q for te Stokes equations, cf. Boffi et al., 2013, sect Te resulting finite-dimensional saddle-point problem is ten to find u V and p Q suc tat 2µ εu, εv + p, div v = f, v + g, v L2 Γ N div u, q 1 λ p, q = 0 29 olds for all v V and q Q. Since we are interested in te approximation quality of te stresses σu, p = 2µεu + p I computed from approximations to u and p, combinations seem favourable, were te error εu u converges at te same order as p p. Suc a combination is given, for example, by te Taylor-Hood elements continuously quadratic 10

11 for V wit continuously linear for Q and teir iger-order generalizations, cf. Boffi et al., 2013, sect Anoter possibility is te use of te quadratic nonconforming elements introduced in Fortin and Soulie 1983 for V combined wit discontinuous piecewise linears. Since tese elements ave some favourable properties wit respect to te associated derived stresses σu, p, we will investigate tem more closely. Te quadratic nonconforming elements by Fortin-Soulie Wit respect to a triangulation T of Ω wit te corresponding set of sides edges for d = 2, faces for d = 3 denoted by S, te quadratic nonconforming finite element space is defined by V F S = {v L 2 Ω d : v T P 2 T d for all T T, v S, s L2 S = 0 for all s P 1 S, S S Ω, v, s L2 S = 0 for all s P 1 S d, S S Γ D }, 30 were S denotes te jump across te side S. It is necessary to go to quadratic nonconforming elements since te linear nonconforming elements by Crouzeix-Raviart do not satisfy te discrete Korn s inequality, in general, if Γ N. Tat suc an inequality, wic reads v 2 L 2 T εv 2 L 2 T for all v V, 31 T T T T olds for V = V F S under our assumption tat Γ D is a consequence of Brenner, 2003, Tm Te validity of 31 is required for te well-posedness of te variational formulation wic now consists in finding u V F S and p Q F S := {q L 2 Ω : q T P 1 T } suc tat 2µ εu, εv L2 T T T + p, div v L 2 T = f, v + g, v 0,ΓN 32 T T div u, q L2 T 1 λ p, q = 0 T T is valid for all v V F S, q Q F S. As was already described in te original papers by Fortin and Soulie 1983 and Fortin 1985, te quadratic nonconforming space can be written as V F S = V T H + B NC H, were VT is component-wise te standard space of conforming quadratic elements and 11

12 B NC denotes again component-wise a suitable space of non-conforming bubble functions. In te two-dimensional case, tis non-conforming bubble space is given by B NC,2 = {b L 2 Ω 2 : b T P 2 T 2 for all T T, v, s L2 S = 0 for all s P 1 S 2, S S }, i.e., tere is exactly one non-conforming bubble function in B NC,2 per triangle. We denote te corresponding one-dimensional space by B NC,2 T. In te tree-dimensional case, te non-conforming bubble space is given by B NC,3 = {b L 2 Ω 3 : b T P 2 T 3 for all T T, v, s L 2 S = 0 for all s P 1 S 3, S S } + {b L 2 Ω 3 : b T P 2 T 3 for all T T, v S B NC,2 S and v S, s L 2 S = 0 for all s P 1 S 3, S S }. Te first part of B NC,3 consists of exactly one non-conforming bubble function per tetraedra, again denoted by B NC,3 T. Te second part is made up of two-dimensional non-conforming bubble functions B NC,2 S for eac face S S extended suitably into te two neigboring tetraedra. It sould be kept in mind tat te representation V F S = V T H + B NC is not a direct sum. Globally constant functions can be expressed in two different ways in tese subspaces, in general. Moreover, in te tree-dimensional case, te representation of conforming piecewise linear functions is not unique. Te following result was also already contained in te original papers by Fortin and Soulie 1983 and Fortin 1985 including te proof given below. Proposition 4.1. Assume tat f L 2 Ω is piecewise constant wit respect to te triangulation T of Ω IR d, d = 2 or 3 and tat g L 2 Γ N is piecewise linear wit respect to te subset of sides S,N = S Γ N associated wit te Neumann boundary. If we denote by S,i = S Ω te subset of sides interior to te domain, ten te piecewise linear stresses σ u, p = 2µεu + p I computed from te solution u, p V F S Q F S of 29 satisfy f + div σ u, p = 0 piecewise for all T T, 33 g σ u, p n, e i L 2 S = 0 for all S S,N, σ u, p n S, e i L 2 S = 0 for all S S,i, 34 were e i IR d denotes te i-t unit vector. 12

13 Proof. Inserting a nonconforming bubble function b T B NC wit support restricted to T as test function into 32 leads to 0 = g, b T 0,ΓN T + f, b T L 2 T σ u, p, εb T L 2 T = g, b T 0,ΓN T σ u, p n, b T 0, T + f + div σ u, p, b T L 2 T = f + div σ u, p, b T L2 T, were te fact was used tat s, b T L 2 S = 0 for all s P 1 S d, S T. Te term f + div σ u, p, constant on T, must terefore vanis. For all test functions v V F S, we terefore get from 32 tat 0 = g, v L2 Γ N + f, v σ u, p, εv L2 T T T = g σ u, p n, v L 2 S σ u, p n S, v L 2 S S S,N S S,i olds. We pick one of te sides S S and coose te test function v V F S in suc a way tat in te sum above only te term associated wit tis particular side does not vanis. In two dimensions, tis is acieved using a conforming piecewise quadratic function tat vanises on all edges besides S. In tree dimensions, te non-conforming bubble function corresponding to te face S as te desired properties note tat σ u, p n S is of degree 1 on all faces. Te symmetry properties of te cosen test functions wit respect to S finally implies 34. Te properties of σ u, p proven in Proposition 4.1 can be used to get an efficient stress reconstruction σ R Hdiv, Ωd by Raviart-Tomas elements of next-to-lowest order Π 1. We will now explain ow suc a construction can be done in an element-wise fasion. In te two-dimensional case tis is equivalent to te tecnique described in Kim Te stress reconstruction σ R Π1 is determined on eac element T T by te following conditions: σ R T n = {{ σ u, p }} S n for all S T, div σ R T = π 1 f 35 T, were {{ }} S stands for te average value on S between te two adjacent elements set {{ σ u, p }} S n = g on all sides S Γ N and π 1 denotes te L 2 Ω projection onto te piecewise linear possibly discontinuous functions on T. Te first line in 35 coincides wit te standard interpolation conditions on te sides S T for next-to-lowest order Raviart-Tomas elements, cf. Boffi et al., 2013, Example It remains to be sown tat, 13

14 in te situation encountered ere, te remaining d interpolation conditions in Boffi et al., 2013, Example are equivalent to te second line in 35. To tis end, note tat div σ R, e i L2 T = σ R n, e i L2 T = σ u, p n, e i L2 T = div σ u, p, e i L 2 T = π 0 f, e i L 2 T olds, were 34 is used in te first line and 33 in te second line. Tis means tat π 0 div σr T = π 0 f T and te second condition of 35 consists of only d linear equations at most wic may be used to satisfy te remaining interpolation conditions. Te construction is rater simple and consists of te following two steps: i Compute, on eac element T, an affine function σ R,0 P 1 T d wic T satisfies te first set of conditions in 35. Tese are dd + 1 conditions for dd + 1 coefficients and amounts to te assignment of te appropriate degrees of freedom depending on te finite element basis used. Tis results in an approximation σ R,0 Hdiv, Ω d wit σ R,0 n = g on Γ N and piecewise constant divσ R,0 wic may also be interpreted as approximation in te BDM 1 space, cf. Kim ii Update for better divergence approximation if f is not constant on T by adjusting te coefficients associated wit te interior degrees of freedom. Te above reconstruction results in a stress approximation wit similar properties as for te Hellinger-Reissner formulation using te Boffi-Brezzi- Fortin elements studied at te end of Section 2. In particular, te momentum balance error is minimized and optimal order approximation of te stress is acieved wit respect to te L 2 Ω norm. Computational comparison for incompressible linear elasticity We close te part of tis contribution associated wit linear elasticity by some two-dimensional computational results in order to provide some insigt on te actual beavior of te metods introduced above. Example 1. Te underlying domain is a quadrilateral wit vertices at 0, 0, 0.48, 0.44, 0.48, 0.6 and 0, 0.44, commonly known as Cook s membrane. It is fixed u = 0 at te left edge of te boundary x 1 = 0 wile a uniform traction force pointing upwards σ n = 0, 1 is applied at te rigt edge x 1 = At te remaining part of te boundary it is kept in equilibrium σ n = 0. All te computations are done for te incompressible limit λ = wile µ is set to 1. Figure 1 sows te initial triangulation wit 44 elements. Te results on a sequence of uniform refinements starting from tis initial triangulation are compared for different metods. 14

15 Figure 1. Initial triangulation for Cook s membrane Table 1 sows te resultant traction force n σ n ds Γ D in normal direction acting on te fixed left boundary part calculated from different finite element approximations of displacement-pressure type. Due to te divergence teorem te exact value is 0. Obviously, te evaluation of te Taylor-Hood and P2/P0 piecewise constant pressure approximations on te boundary does not reproduce tis resultant traction force exactly wile tis is te case for te Fortin-Soulie approximations in accordance wit Proposition 4.1. Anoter interpretation of tese results is tat te piecewise linear stress approximations in L 2 Ω d d are not suitable for teir evaluation on te boundary, in general. For te non-conforming Fortin- 15

16 l T Taylor-Hood P2/P0 Fortin-Soulie Table 1. Resultant normal traction for displacement-pressure metods Soulie approximations, te trace on te Diriclet boundary coincides wit tose associated wit te recovered stress in Hdiv, Ω d and does terefore produce a good approximation of te boundary tractions. Figure 2. Approximation of normal traction near singularity Figure 2 sows te quality of te normal traction approximation for different displacement-pressure elements after six uniform refinements in te neigborood of te singularity at te left upper vertex of Cook s membrane. Te saded grap is associated wit te Taylor-Hood element pair, te dotted lines are for P2/P0 and te solid straigt lines for te Fortin- Soulie elements. Te black curve represents te correct traction force distribution and was computed on an adaptively refined triangulation by te 16

17 least squares approac of Section 3. Away from te singularity all approximations are quite accurate wile severe differences are visible in te quality ow well te singular beavior is resolved. Te Fortin-Soulie does perform muc better and terefore justifies its larger number of degrees of freedom. l T HR BBF LS RT1/P2 recov. from FS Table 2. Resultant normal traction for stress-based metods Table 2 lists te same quantities as Table 1 but tis time compares te different metods from Sections 2, 3 and 4. Due to te exact momentum conservation, te approximations wit te Boffi-Brezzi-Fortin elements based on te Hellinger-Reissner principle produce te resultant traction forces perfectly up to roundoff errors. Te first-order system least squares approac does not compute te resultant traction force exactly but to quite acceptable accuracy wile te numbers associated wit te recovered stresses from te Fortin-Soulie elements are again correct up to working precision. Figure 3 sows te distribution of te normal traction at te left boundary near te singularity for te tree different approaces of Table 2 after five uniform refinements. Te saded grap belongs to te first-order system least squares approac wic performs sligtly worse tan te two alternatives. Te dotted lines are associated wit te Hellinger-Reissner principle using te finite element combination of Boffi-Brezzi-Fortin and seem to resolve te singularity sligtly better tan te stresses recovered from te Fortin-Soulie elements solid straigt lines. Considering linear elasticity computations, te stress reconstruction approac is quite attractive since te global system tat needs to be solved involves fewer unknowns and te reconstructed stresses are of a similar accuracy as tose obtained wit a mixed metod of saddle-point or least-squares type. Te situation may, owever, be different for more complicated models were te stress is involved more directly. Tis is te case, for instance, in te context of inelastic beavior caused by stress components exceeding a certain limit were te direct treatment of stresses in te variational formulation is advantageous cf. Reddy 1992 for a mixed approac of 17

18 Figure 3. Approximation of normal traction near singularity saddle-point type, Starke 2007; Scwarz et al for a least-squares type approac. A comparison of te different approaces for te nonlinear problems arising in association wit yperelastic material models will be given in Section 5. 5 Extension to finite-strain Hyperelasticity In te previous sections, te linear elasticity model was considered wic is derived under te assumption of small strains. Now we switc to te more general case of finite strains wit yperelastic material models. More details on te validity of tese models and teir matematical aspects can be found, e.g. in Ciarlet, 1988, Cp. 4. Based on te deformation gradient given by F = F u := I + u, te left and rigt Caucy-Green strain tensors are defined as B = Bu := F uf u T and C = Cu := F u T F u, respectively. Tis nonlinear dependence of strains to displacements constitutes te geometrically nonlinear nature of tis model. In addition, tere is also a nonlinearity in te material law describing te relation between stresses and strains. Tis originates from a stored energy function ψ : IR 3 3 sym IR wic generalizes 4 and is no longer quadratic. Again, we restrict ourselves to a omogeneous material wic means tat ψ does not explicitly depend on te location x Ω. 18

19 Minimizing te total energy Iv := ψcv dx Ω Ω f v dx g v ds Γ N 36 among all admissible displacements v V for some suitable space V is again equivalent to finding a solution u V of te variational problem P u, v = f, v + g, v 0,ΓN for all v V, 37 were P u := F ψcu denotes te first Piola-Kircoff stress tensor. We assume tat our problem is sufficiently regular so tat we can coose V = W 1,p Γ D Ω 3 for p > 2 as our solution space for 37. In tat case, we may also write 37 as a first-order system as div P = f in Ω P = F ψc in Ω P n = g on Γ N, u = 0 on Γ D. 38 Te first equation in 38 is an immediate consequence of te pysically necessary conservation of linear momentum for a static problem. Conservation of angular momentum for a static problem leads additionally to te symmetry of P uf u T wic is implicitly contained in te formulations 37 and 38. For omogeneous isotropic materials it is possible to express te stored energy function ψ by a function ψ : R 3 R, depending on tree terms I 1, I 2, I 3 : R 3 3 R, i.e., ψc = ψi 1 C, I 2 C, I 3 C, C = F T F, 39 wit te principal invariants I 1 C := trc, I 2 C := trcof C and I 3 C := det C cf. Simo, Tm and Ex Introducing te so-called Kircoff stress tensor τ := P F T, a simple calculation ten leads to τ = 2 ψ I 3 BI + 2 ψ + ψ I 1 B B 2 ψ B 2 =: GB 40 I 3 I 1 I 2 I 2 or, equivalently, for te second Piola-Kircoff stress tensor Σ := F 1 P : Σ = 2 ψ + ψ I 1 C I 2 ψ C + 2 ψ I 3 CC 1 := I 1 I 2 I 2 I GC,

20 were G and G are mappings from strains into stresses, similar as te fourtorder elasticity tensor C in linear elasticity. In te following we assume tat GI = 0 = GI, i.e. te reference configuration is stress-free, and tat G, G are continuously differentiable in te identity matrix wit G I[E] = 1 2 CE = G I[E], i.e. consistency of te nonlinear model wit te model of linear elasticity cf. Ciarlet, 1988, Sect Since te elasticity tensor C itself is an isomorpism, te mappings G I = 1 2 C and G I = 1 2C are also isomorpisms. Tus te local inversion teorem cf. Ciarlet, 1988, Tm is applicable and guarantees tat te inverse mappings G 1 τ and G 1 Σ are well-defined in a neigborood of τ = 0 and Σ = 0, respectively. Using tese considerations we can modify te strong formulation 38 into using te representation in B or into div P + f = 0 in Ω, G 1 P F u T Bu = 0 in Ω, P n = g on Γ N, u = 0 on Γ D 42 div P + f = 0 in Ω, G 1 F u 1 P Cu = 0 in Ω, P n = g on Γ N, u = 0 on Γ D 43 using te representation in C. Bot systems are at least well-defined for small stresses. 5.1 A least squares finite element metod for isotropic yperelastic materials Since G I = G I = 1 2C, te implicit function teorem tells us tat G 1 0 = G 1 0 = 2C 1. Tis means tat we encounter te same problem as in te linear case, namely, tat G 1 and G 1 are not invertible anymore in te incompressible limit. Due to tis observation we use te notation A and à instead of G 1 and G 1 in 42 and 43. Wit tis in mind we introduce for P = P N + ˆP W q div; Ω 3 + W q Γ N div; Ω 3 wit P N n = g on Γ N, u W 1,p Γ D Ω 3 and f L q Ω 3, p, q 4 sufficiently large, te nonlinear operators div P + f RP, u := AP F u T, Bu RP, u := div P + f ÃF u 1 P Cu 44 20

21 for 42 and 43, respectively. Based on tese operators we define nonlinear least squares functionals FP, u := RP, u 2 = div P + f 2 + AP F u T Bu 2 45 for te formulation in B and FP, u := RP, u 2 = div P + f 2 + ÃF u 1 P Cu 2 46 for te formulation in C. 5.2 Gauss-Newton iterative metod In te following we restrict ourselves to te minimization problem 45 corresponding to te B-formulation. All furter steps below can be andled similarly for te C-formulation. Te minimization of 45 is carried out iteratively solving a sequence of linearized least squares problems. Since te operator RP, u is continuously differentiable wit respect to P, u, we can linearize it around a given approximation P k, u k P N +W q Γ N div; Ω 3 W 1,p Γ D Ω 3. Te resulting linearized least squares functional depending on P k, u k is given by F lin Q, v = F lin Q, v; RP k, u k := RP k, u k + R P k, u k [Q, v] Te minimizer Q k, u k of 47 is ten sougt in a suitable normed function space Π ΓN V ΓD, provided tat te values q and p are cosen sufficiently large suc tat RP k, u k and also R P k, u k [Q, v] are contained in L 2 Ω 3 L 2 Ω 3 3. Te linearized minimization problem 47 is equivalent to te variational problem of finding Q k, v k Π ΓN V ΓD suc tat BQ k, v k, ˆQ, ˆv = RP k, u k, R P k, u k [ ˆQ, ˆv] 48 olds for all ˆQ, ˆv Π ΓN V ΓD. Te bilinear form in 48 is defined on Π ΓN V ΓD and given by BQ, v, ˆQ, ˆv := R P k, u k [Q, v], R P k, u k [ ˆQ, ˆv]. 49 For te numerical implementation, a finite dimensional space Π V Π ΓN V ΓD is cosen. Starting wit an initial guess P 0, u0 P N + W q Γ N div; Ω 3 W 1,p Γ D Ω 3 and setting k = 0, te discrete analogue of 48 21

22 is ten solved in Π V to obtain te correction term Q k, vk. Afterwards te new iterate is set to P k+1, u k+1 = P k, uk + αk Q k, vk were α k > 0 describes te step lengt in te direction Q k, vk. Te described approac is te well-known Gauss-Newton metod combined wit a line searc strategy, cf. Nocedal and Wrigt, 2006, Cp. 3 and Sect For instance, for te determination of a suitable step lengt one can use a backtracking line searc approac, cf. Nocedal and Wrigt, 2006, Alg Te new iterate P k+1, u k+1 automatically satisfies P k+1 n = g on Γ N and u k+1 = 0 on Γ D. An alternative approac for minimizing te linearized problems of te form 47 in finite dimensional spaces is te usage of te Levenberg-Marquardt metod wic replaces te line searc wit a trust-region metod, cf. Nocedal and Wrigt, 2006, Cp. 4 and Sect Considering 48 and 49 one as to evaluate te nonlinear operator R locally for given P, u at eac quadrature point for a numerical implementation. Due to te representations in 44 te problematical part is te evaluation of A. A remedy, wic works independently of te used stored energy function, is to solve te problem GB = τ for given τ := P F u T wit te elp of Newton s metod. Assuming a finite λ and a sufficiently small τ, te sequence of Newton iterations is given by were j R 3 3 is te solution of B j+1 = B j + j, G B j [ j ] = τ GB j. 50 Te initial guess B 0 = I R 3 3 is at least for small P, u reasonable, since for P, u = 0, 0 te solution is given by B = I. Te equation 50 can be solved wit te elp of a linear system of equations wit nine unknowns, were te matrix on te left-and side depends on te old approximation B j and te rigt-and side depends on P, u and B j. Applying Newton s metod on eac quadrature point and on eac element of te triangulation is numerically expensive. For a special neo-hookean material wic we consider in te next section it is possible to evaluate te operator A locally witout using Newton s metod. Moreover, one can take te limit λ in te operator A and can even set λ = in tis model. 22

23 5.3 Least squares formulation for neo-hookean model Te metod described in Section 5.1 works generally for an arbitrary isotropic stored energy function ψ, provided tat te stresses are not too large suc tat invertibility of te operators G and G in 40 and 41 is ensured. In tis section we consider an isotropic material of neo-hookean type described by ψ NH I 1, I 3 = αi 1 + βi 3 γ 2 lni 3, α, β, γ > 0, wit stored energy function ψ NH C = α trc + β detc γ lndet C 51 2 via 39 cf. Ciarlet, 1988, sect Wit te derivatives ψ NH I 1 = α, ψ NH I 2 = 0, ψ NH I 3 = β γ 2I 3 and equations 40 and 41 we acieve G NH B = 2αB + 2β det B γi, G NH C = 2αI + 2β det C γc G 1 Wit A NH = G 1 NH and ÃNH = NH denoting te corresponding inverses, we end up wit te nonlinear operators div P + f R NH P, u := A NH P F u T, Bu 53 div P + f R NH P, u := Ã NH F u 1 P Cu in te Neo-Hooke case. Te derivatives of 52 are given by G NHB[E] = 2αE + 2βCof B : EI, G NHC[E] = 2βCof C : EC 1 2β det C γc 1 EC Te conditions G NH I = 0 and G NH I[E] = 1 2 CE or G NH I = 0 and G NH I[E] = 1 2CE, respectively lead to a linear system of equations for te determination of α, β, γ wic is uniquely solvable troug α = µ 2, β = λ 4, γ = µ + λ

24 Te derivatives in 54 can be directly inverted. After inserting te coefficients 55 in 54, te inverses are given by G NHB 1 [Σ] = 1 λ Σ Cof B : ΣI, µ 2µ + λ trcof B G NHC 1 1 [Σ] = 56 C µ + λ 2 1 det C Σ λdet C 2 2µ + λ1 + 2 det C Cof C 1 : ΣC 1 Te inverses are very elpful for te direct calculation of C. R NHP, u[q, v] div Q = A NH P F ut [QF u T + P v T ] vf u T F u v T and R NH P, u[q, v], respectively. Inserting B = I = C in 56 for finite λ leads to G NH I 1 [Σ] = 2C 1 Σ = G NH I 1 [Σ], as expected. For λ te first equation in 56 becomes G NHB 1 [Σ] = 1 1 Σ Cof B : ΣI µ trcof B and coincides for B = I wit 2 AΣ from te linear elasticity case. Te wellposedness of G NH C 1 for given strain C := ÃNHΣ and te identity G NH C 1 = 2A for λ will be discussed later. Local evaluation of A NH and ÃNH We ave seen at te end of Section 5.1 tat we must evaluate A NH P F u T in te B-formulation in eac quadrature point. Analogously we ave to evaluate ÃNHF u 1 P locally using te formulation in C. For bot formulations one can evaluate A NH P F u T and ÃNHF u 1 P directly witout using Newton s metod as described in te sequel: For te B-formulation on te one and, given any stress tensor τ R 3 3, te corresponding strain B := A NH τ R 3 3 can be determined via wit trb solution of B = dev B + 1 dev τ trbi = 3 µ + 1 trbi 57 3 trb 3 + S trb + T = 0, 58 24

25 depending on te coefficients S = 9 18µ trcof dev τ + µ 2 λ, 1 T = 27 µ 3 detdev τ 1 2µ λ 2 3λ trτ. 59 A detailed derivation of 57 and 58 can be found in Müller et al If te discriminant D := S T 2 2 is positive, te cubic equation 58 as only one real solution and we obtain exactly one reasonable strain wic corresponds to te given stress. For te C-formulation on te oter and, given any stress tensor Σ R 3 3, te corresponding strain C := ÃNHΣ R 3 3 can be determined via C = ρσ µi 1, 60 provided tat Σ µi is invertible. Te parameter ρ in 60 is solution of wit coefficients ρ 3 + S ρ + T = 0 61 S := 2 λ detσ µi, T := 1 + 2µ λ detσ µi, 62 cf. Wriggers, 2008, Sect Provided tat te discriminant of 61 is positive, we get again one real solution for ρ and a unique real strain tensor C corresponding to te given stress tensor Σ can be easily computed. One remarkable fact in te cubic equations 58 and 61 is tat we can take also te limit λ ere. In fact, we can also set λ =. In tis case all fractions wit λ in te denominator in te coefficients 59 and 62 vanis. Wit tis in mind, we can come back to te discussion of te well-posedness of G NH C 1 for given strain C := ÃNHΣ in te incompressible limit λ. Inserting 62 into 61 we obtain ρ 3 = 2 λ detσ µi ρ µ detσ µi λ and wit te elp of 60 we can conclude tat det C = det ÃNH Σ = ρ 3 det Σ µi 1 = 2 λ ρ µ λ olds. Due to µ + λ 2 1 det C = µ + λ 2 2 λ ρ 2µ = ρ λ 3 detσ µi, λ 25

26 te second equation of 56 remains well-posed for λ wit G NHÃNHΣ 1 [Q] = G NHC 1 [Q] 1 = Q 3 detσ µi C det C det C Cof C 1 : QC 1 C. In te case Σ = 0 corresponding to C = ÃNH0 = I tis leads to 3 det0 µi = µ and ence G NH I 1 = 2A from te linear elasticity case for λ. Combining te neo-hookean model 51 wit te firstorder systems 42 and 43 we ave tus establised a formulation wic allows us to consider fully incompressible materials. Analysis of te formulation in B Based on te convex sets Π := {Q L Ω 3 : Q L Ω δ} P N + WΓ 4 N div; Ω 3, V := {u W 1, Ω 3 : u L Ω δ} W 1,4 63 Γ D Ω 3 again wit P N W div; Ω 3 satisfying P N n = g on Γ N, one can prove for te first-order system operator R NH cf. 44 wit A = A NH locally defined by 57, 58 and 59 te estimates R NH ˆQ, ˆv R NH Q, v 2 ˆQ Q 2 Hdiv; Ω + ˆv v 2 H 1 Ω R NH ˆQ, ˆv R NH Q, v 2 ˆQ Q 2 Hdiv; Ω + ˆv v 2 H 1 Ω, 64 provided tat ˆQ, ˆv, Q, v Π V wit sufficient small δ, cf. Müller et al., 2014, Tm In particular, 64 olds uniformly for λ. Inserting te exact solution P, u Π V wit R NH P, u = 0 as Q, v and an approximation P, u Π V as ˆQ, ˆv in 64 directly leads to P P, u u 2 V F NH P, u P P, u u 2 V 65 wit V := H ΓN div; Ω 3 H 1 Γ D Ω 3. Te coercivity and continuity of te nonlinear least squares functional F NH P, u = R NH P, u 2 in 65 justifies its use as an a posteriori error estimator. Te left inequality in 65 implies te reliability wile te rigt inequality stands for te efficiency. Since te constants in 65 are independent of λ, te approac 45 combined wit te neo-hookean stored energy function 51 is Poisson locking-free. Equation 65 also leads to a priori error estimates. For instance, if we combine, for some l 1, Raviart-Tomas elements Π l := RT l 1 T 3 26

27 Π Hdiv; Ω 3 for te approximation of P wit continuous elements V l := P lt 3 V H 1 Ω 3 for te approximation of u and let P, u be te minimizer of F NH Q, v among all Q, v Π l V l Hdiv; Ω 3 H 1 Ω 3, ten we obtain P P, u u V F NH P, u 1 2 { } = inf F NH Q, v 1 2 : Q, v Π l V l P ρ P, u r u V l P 2 H l Ω + div P 2 H l Ω + u 2 H l+1 Ω under te assumption tat P Π H l Ω 3 3 wit div P H l Ω 3, u V H l+1 Ω 3 and P, u Π V olds. Here ρ and r denote te usual interpolation operators for te Raviart-Tomas and te standard conforming finite elements, respectively, cf. Boffi et al., 2013, Sects. 2.2 and 2.5. Under tese assumptions te square of te error P P, u u 2 V and te least squares functional F NH bot are proportional to 2l n 2l d t as 0, were n t denotes te number of elements in te triangulation T. Accuracy of balance of momentum in te nonlinear case We investigate te generalization of Teorem 3.2 to te yperelastic situation. To tis end, we assume tat te linearized problem div σ + f = 0, R 2P, u[σ, υ] = 0= R 2 P, u 67 as te following regularity properties, similar to tose stated at te end of Section 1 for te linear elasticity problem: For any f L 2 Ω 3, te solution σ, υ H ΓN div, Ω d H 1 Γ D Ω d of 67 satisfies σ, υ H α Ω d d H 1+α Ω d wit σ H α Ω + υ H 1+α Ω C R f 68 for some constant C R > 0 and α > 0. Teorem 5.1. Under our regularity assumptions, te momentum balance accuracy associated wit te first-order system least squares approximation for te Neo-Hooke model satisfies divp LS +f α P P LS + u u LS + inf z Z f z

28 Proof. Starting as in te proof of Teorem 3.2, we arrive at div P LS div P LS P, z + f sup + f π f. 70 z Z z Recalling tat P LS, u LS Π V minimizes RP, u 2 and tat RP, u = 0, we ave RP, u RP LS, u LS for all Q, v Π V, were, R P LS, u LS [Q, v ] = 0 71 R P, u[q, v] div Q = A P F u T [QF u T + P v T ] F u v T vf u T div Q =: R. 2P, u[q, v] We replace te auxiliary boundary value problem 23 by te following: Find Ξ H ΓN div, Ω 3 and η H 1 Γ D Ω 3 suc tat div Ξ = z, R 2P, u[ξ, η] = 0 72 olds. For arbitrary Ξ Π wit div Ξ = z and η V, we obtain from 71 tat div P P LS, z = div P P LS, div Ξ = div P P LS, div Ξ + R 2 P, u R 2 P LS = div P P LS, div Ξ div Ξ + R 2 P, u R 2 P LS = R 2 P, u R 2 P LS, u LS, R 2P, u[ξ, η], u LS, u LS olds. Combining tis wit 70 leads to, R 2P, u[ξ, η] R 2P LS, u LS [Ξ, η ], R 2P, u[ξ, η] R 2P LS, u LS [Ξ, η ] div P LS + f R 2 P, u R 2 P LS sup Ξ, u LS R 2P, u[ξ, η] R 2P LS div Ξ, u LS [Ξ, η ]

29 Te second term in 73 can be bounded furter as R 2P, u[ξ, η] R 2P LS, u LS [Ξ, η ] R 2P, u[ξ Ξ, η η ] + R 2P, u R 2P LS, u LS [Ξ, η ]. 74 Te first term in 74 may be bounded using Müller et al., 2014, Lemma 4.3 to get R 2P, u[ξ Ξ, η η ] Ξ Ξ + η η α div η using our regularity assumption. For te second term in 74, R 2P, u R 2P LS P P LS α div η, u LS [Ξ, η ] L Ω + u u LS L Ω Ξ + η can be sown in a similar way using te expression for R NH Tis finises te proof of 69. below 56. Stress reconstruction in te nonlinear case In analogy to 28 one may consider a displacement-pressure finite element approac to te nonlinear variational problem 37 associated wit yperelasticity. In te case of a neo-hookean model wit P u = µ F u F u T + λ det F u 2 1 F u T, 75 2 a pressure-like variable p = λ 2 det F u 2 1 may be introduced leading to te system µ F u F u T, v + p F u T, v = f, v + g, v 0,ΓN 1 det F u 2 1, q 1λ 76 2 p, q = 0 to old for all v and q in suitable test spaces. In order to allow for general u H 1 Γ D Ω 3 and to keep te pressure space as big as possible, one may 29

30 rewrite 75 sligtly as Cof F u P u = µ F u det F u + λ 2 det F u 1 Cof F u det F u 77 and define te pressure variable as p = λ 1 det F u. 2 det F u Tis is motivated by te fact tat, for u H 1 Ω 3 and under te additional assumption tat Cof F u L 2 Ω 3 3, we ave det F u L 1 Ω, see Ciarlet, 1988, Tm Te assumption on te co-factor is not as restrictive as it seems since it is usually fulfilled for te solutions associated wit edge singularities at re-entrant corners. For te same reason, one may also assume tat det F u 1 L Ω wic suggests te well-posedness of te following system: Find u H 1 Γ D Ω 3 and p L 1 Ω suc tat Cof F u µ F u det F u, v 1 2 det F u +p Cof F u, v = f, v + g, v 0,ΓN, q 1 p, q = 0 λ 1 det F u 78 for all v HΓ 1 D Ω 3 and q L Ω. In te small-strain limit, 79 as well as 76 turns into te displacement-pressure formulation of linear elasticity 28. Te discrete problem consists in finding u V and p Q suc tat µ F u Cof F u det F u, v 1 2 det F u +p Cof F u, v = f, v + g, v 0,ΓN, q 1 λ p, q = 0 1 det F u 79 for all v V and q Q. It is natural to use te same combination of spaces V and Q as in te linear case altoug te compatibility is not guaranteed. However, te investigations in Auriccio et al and Auriccio et al indicate tat at least some of tese finite element 30

31 spaces like Taylor-Hood are also safe to use in te yperelastic case for incompressible materials. Te stability and approximation properties of te Fortin-Soulie elements in combination wit piecewise linear pressure for yperelastic models wit incompressible materials needs still to be investigated. But even if te approximation quality is similar to te linear elasticity case, te stress reconstruction procedure is more involved. Te stress P u, p = µ F u F u T + p F u T associated wit a piecewise quadratic u and piecewise linear p is certainly not piecewise linear and terefore does not satisfy te properties of Proposition 4.1. Te element-wise stress reconstruction procedure by Kim 2012 is terefore not immediately applicable in te nonlinear case. Te stress reconstruction can be expected to be more involved for yperelastic material models speaking in favour of our first-order system approac wic produces accurate stress approximations simultaneously. 5.4 Computational results For te confirmation of our teoretical results, we consider some two- and tree-dimensional examples. Altoug te tree-dimensional situation is, of course, more realistic from an application point of view, two-dimensional computations ave te advantage tat te asymptotic beavior is more accessible. Two-dimensional numerical tests We start wit te two-dimensional case and use a plane strain configuration, meaning tat te displacement components u 1 and u 2 of u depend only on x 1 and x 2 and te component u 3 is constant. In tis situation te deformation gradient becomes u 1 2 u 1 0 F = F u = I + u = 1 u u Due to te definition of te Caucy-Green strain tensors B = F F T and C = F T F, tey ave te same structure as F. Moreover, also te stress tensor P as tis structure see 40 and 41. Our numerical simulations in te plane strain situation may terefore be based on a two-dimensional domain wit planar Raviart-Tomas elements for te first two rows of P and piecewise polynomial functions witout any continuity requirement for 31

32 te remaining nonzero entry P 33 of P. For te displacement components u 1 and u 2 we can use continuous piecewise polynomial functions. In fact, we use next-to-lowest-order Raviart-Tomas elements RT 1 T for te plane stress field, piecewise linear discontinuous elements DP 1 T for P 33 and conforming, piecewise quadratic, finite elements P 2 T for eac displacement component. Tus our finite dimensional space is altogeter given by Π V := RT 1 T 2 DP 1 T P 2 T 2. Example 2. In tis example we consider again te polygonal Cook s membrane domain as in Section 4. Wit respect to te vertices 0, 0, 48, 44, 48, 60 and 0, 44, te left part of te boundary is again used as Γ D := {0, x 2 : 0 < x 2 < 44} and te remaining boundary as Γ N. Te volume force is set to f = 0 and for te surface force, g = 0 is prescribed on te upper and lower part of Γ N and g = 0, γ load T wit γ load R on te rigt part Γ R := {48, x 2 : 44 < x 2 < 60} of Γ N. We coose µ = 1 and λ = as Lamé constants, i.e. we simulate a fully incompressible material wic is te numerical most callenging case. We use te formulation in B and as load value γ load = Before we present some numerical results, note tat te exact solution P, u of tis problem is not in Π V, since at te point 0, 44 te boundary condition canges from ard clamped u = 0 to stress-free P n = 0 and te interior angle is larger tan te critical one, cf. Rössle Altoug te regularity assumptions in 63 are not satisfied for te exact solution, we see in Table 3 tat te nonlinear least squares functional F NH works reasonable as a posteriori error estimator. n t dim Π dim V F NH P, u order Table 3. Convergence rates of F NH wit adaptive refinement 2d 32

33 Figure 4. Adaptively refined triangulation for Cook s membrane In te last column of Table 3 te approximation order log F NH P l+1, u l+1 log F NH log log n l t n l+1 t P l, ul of F NH is listed. Here P, u := P l, ul denotes te approximated solution and n t := n l t te number of elements on level l N {0}. One observes tat te optimal convergence rate of 2 is acieved using adaptive 33

34 n t dim Π dim V F NH P, u order Table 4. Convergence rates of F NH wit uniform refinement 2d refinement. Figure 4 sows te mes after four adaptive refinement steps resulting in a triangulation wit 821 triangles. In Table 4 te approximation order using uniform refinement is illustrated. Obviously te optimal convergence rate is not reaced and adaptive refinement is superior. Tis is as expected due to te singularity at 0, 44. adaptive refinement uniform refinement n t divp P 2 order n t divp P 2 order Table 5. Improved convergence rates for balance of momentum 2d In Table 5 we can confirm numerically tat te convergence rate of te term divp P 2 = div P + f 2 is approximately doubled, regardless of using uniform or adaptive refinement. Furtermore te values itself are close to zero wic means tat te approximations satisfy te conservation of linear momentum quite well. Besides te convergence rates we are interested in te quality of te surface traction forces resulting from our stress approximations. Te distribution of te normal component of te traction force acting at te left boundary is sown in Figure 5. For a closer investigation of te accuracy of tese quantities we focus on te integral Γ D P n ds wic constitutes te resultant force acting on te left and boundary segment. Due to f = 0, 34

35 Figure 5. Normal traction at left boundary for Cook s membrane n t γ load = γ load = Table 6. Comparison of Γ D P 21 ds for different load values te divergence teorem implies 0 P n ds = P n ds = g ds = Γ D Γ N Γ R γ load. 80 Γ R Wit te outward normal n = 1, 0 T of Γ D, te load values γ load {0.0005, 0.05} and Γ R = 16 it follows immediately tat { P 21 ds = 16γ load , γ load = = Γ D , γ load = 0.05 olds for te second entry in 80. One observes in Table 6 tat te least squares approac does in bot cases produce quite satisfactory approximations to te resultant force. 35

36 Tree-dimensional numerical tests For fully tree-dimensional examples we use te finite-dimensional spaces Π V := RT 1 T 3 P 2 T 3 on a tetraedral decomposition of te given domain. Example 3. We consider a tree-dimensional Cook membrane problem. For tis purpose we expand te two-dimensional domain of Example 2 in x 3 -direction wit tickness 5. Tus te tree-dimensional polyedral domain is defined troug te vertices 0, 0, 0, 48, 44, 0, 48, 60, 0, 0, 44, 0, 0, 0, 5, 48, 44, 5, 48, 60, 5 and 0, 44, 5. We split te boundary Γ = Ω into te left lateral face Γ D := {0, x 2, x 3 : 0 < x 2 < 44, 0 < x 3 < 5} and Γ N consisting of te remaining five lateral faces. We clamp te body on Γ D and apply a surface force g = 0, γ load, 0 T wit load value γ load R on te rigt part of te boundary Γ R := {48, x 2, x 3 : 44 < x 2 < 60, 0 < x 3 < 5}. On te oter parts of Γ N no surface forces act g = 0. As body force density we use f = 0, coose γ load = 0.05 and Lamé constants µ = 1, λ =, i.e. we consider again a fully incompressible material. Figure 6 sows te mes after tree adaptive refinement steps resulting in a triangulation wit 2892 tetraedra. Te concentration of te refinement in te vicinity of te singularity at te edge at x 1 = 0 and x 2 = 44 is clearly visible. In Tables 7 and 8 te numerically obtained convergence rates corresponding to F NH P, u and divp P 2 using adaptive and uniform refinement, respectively, can be compared. One observes in Table 7 tat we obtain good convergence rates, close to te optimal value 4 3, for te nonlinear functional using adaptive refinement. Moreover we see, similar as in te two-dimensional example, tat te convergence for te balance of momentum is significantly faster tan for te overall functional. Moreover, te value divp P 2 on eac considered level is again close to zero, i.e. linear momentum is conserved quite well. Similar as in te two-dimensional example we consider again te boundary integral values Γ D P n ds. Due to Γ R = 16 5 = 80 te exact values are Val Val 2 := P 0 ds = 80γ load = 4 Val Γ 3 D following te same calculations as in te two-dimensional derivation. We can observe in Table 9 tat our least squares approac yields already on a coarse mes good approximations to te resultant forces and converges to te correct values. 36

37 Figure 6. Adaptively refined triangulation for te 3D Cook s membrane n t dim Π dim V F NH P, u order divp P 2 order Table 7. Convergence rates of F NH refinement 3d and divp P 2 wit adaptive n t dim Π dim V F NH P, u order divp P 2 order Table 8. Convergence rates of F NH refinement 3d and divp P 2 wit uniform Acknowledgement. Te work reported ere was supported by te German Researc Foundation DFG under grant STA 402/11-1. Te autors would also like to tank Jörg Scröder and Alexander Scwarz for many discussions on te 37