Tangential-Displacement and Normal-Normal-Stress Continuous Mixed Finite Elements for Elasticity

Transcription

1 angential-displacement and Normal-Normal-Stress Continuous Mixed inite Elements for Elasticity J. Schöberl, A. Sinwel RICAM-Report

2 ANGENIAL-DISPLACEMEN AND NORMAL-NORMAL-SRESS CONINUOUS MIXED INIE ELEMENS OR ELASICIY JOACHIM SCHÖBERL AND ASRID SINWEL Abstract. In this paper we introduce new finite elements to approximate the Hellinger Reissner formulation of elastictiy. he elements are the vector valued tangential continuous Nédélec elements for the displacements, and symmetric, tensor valued, normal-normal continuous elements for the stresses. hese elements do neither suffer from volume locking as the Poisson ratio approaches 1 2, nor suffer from shear locking when anisotropic elements are used for thin structures. We present the analysis of the new elements, discuss their implementation, and give numerical results. 1. Introduction inite element simulation of mechanical problems has a long history, and the theory is well developed. But, as material parameters or some geometric dimensions become bad, the primal method may lead to very poor results. his effect is known as locking. Here, many non-standard methods such as mixed variational principles have been introduced [14]. We consider the Hellinger Reissner formulation, where the displacement vector as well as the stress tensor are approximated as independent unknowns. here are essentially the two possibilities to apply the derivatives either to the displacements, or to the stresses. he first one is easy to discretize, but leads usually to the standard method suffering from locking. he second one needs stress tensor elements with continuous normal vectors, which are more difficult to construct [9, 1, 7, 10]. hey may lead to better approximation properties. An alternative are elements with weak symmetry [6, 26, 27, 12, 8]. In the present paper, we develop a formulation, which is in between these two concepts. he obtained finite elements have good approximation properties and are easy to implement Mathematics Subject Classification. 65N30. Key words and phrases. Mixed inite Elements, Elasticity, Locking. Both authors acknowledges support from the Austrian Science oundation W within project grant Start Y-192, hp-em: ast Solvers and Adaptivity. 1

3 2 JOACHIM SCHÖBERL AND ASRID SINWEL σ nn σ nn u τ igure 1. riangular element u τ igure 2. etrahedral element We propose elements which have continuous tangential components for the displacement vector u, and continuous normal-normal component of the stress tensor σ. he lowest order stable elements for a triangle is drawn in igure 1. he displacement element has one degree of freedom for the tangential component along edges. his is exactly the lowest order Nédélec element. he normal component of the normal stress vector is approximated as linear function along the edges. here are exactly three dofs for each edge which are necessary to fix the rigid body motions in 2D: he displacement is prescribed in tangential direction, and the linear stress can equilibrate the normal force and the angular momentum. he corresponding 3D element is drawn in igure 2. he tangential displacement is prescribed by one value for each edge, and the normal component of the normal stress vector is given as linear function on each face. Again, these six degrees of freedom fix the rigid body motions: Displacements fix the tangential displacement and the torsion, while the stresses compensate the normal force, and the two angular moments. he complete basis functions including the internal bubbles will be given in Section 3 In igure 3, the degrees of freedom of a quadrilateral element are drawn. he goal is to discretize thin domains with anisotropic elements. Standard primal low order methods lead to bad results as the aspect ratio of the element becomes large. In Section 6 we will analyze the robustness of the corresponding Reissner Mindlin plate element. Now, we motivate the anisotropic element by distinguishing stretching and bending degrees of freedom: he mean value of the horizontal displacement prescribed by the bottom and top edge, and the mean values of the stresses at the vertical edges prescribe a one dimensional stretching deformation. he arising method is a mixed method with continuous stresses and non-continuous displacements. It corresponds to the presented elements, since this is a normal-normal continuous

4 MIXED INIE ELEMENS OR ELASICIY 3 σ nn uτ igure 3. Quadrilateral element 1 1 stress tensor, and a displacement vector with non-continuous normal components. A bending deformation is prescribed by the vertical deformation along the short edges, the differences of the horizontal deformations, i.e., the rotations, and the differences of the horizontal stresses, i.e., the bending moments. his leads to a stable element for the imoshenko beam model. A similar dimension reduction works out for thin prismatic and hexahedral elements. 2. Different versions of the Hellinger Reissner formulation he equations of linear elasticity consist of the constitutive relation (1) Aσ = ε(u), where u is the unknown displacement vector, ε(u) := 1[ u + 2 ( u) ] is the strain, σ is the unknown symmetric stress tensor. A is the compliance tensor, which read as Aσ = 1 2µ σd + 1 λ + µ tr(σ)i for an isotropic material with Lamé coefficients µ and λ. Here, tr(τ) denotes the trace of the tensor τ, and τ D is the deviator τ D = τ 1 d tr(τ)i. he stress tensor shall be in equilibrium with the prescribed force vector f, i.e., (2) div σ = f, where the div-operator is applied to each row of the tensor. We assume displacement boundary conditions u = u D at the part Γ D of the boundary, and traction boundary conditions σ n = g

5 4 JOACHIM SCHÖBERL AND ASRID SINWEL at the part Γ N of the boundary. We write σ n for the normal stress vector σn where n is the outer normal vector. he primal mixed formulation is: find σ L SY 2 M := {τ L d d 2 : τ = τ } and u [H 1 ] d such that u = u D on Γ D and (3) Aσ : τ ε(u) : τ = 0 τ L SY M 2, ε(v) : σ = f v + Γ N g v v [H0,D 1 ]d. Since the stresses are L 2 variables, they can be eliminated pointwise from the first equation, and one ends up with the primal form: find u [H 1 ] d such that u = u D and (4) A 1 ε(u) : ε(v) = f v + g v v [H0,D] 1 d. Γ N Alternatively, one may integrate by parts the off-diagonal terms, and obtains the dual mixed formulation: find σ H(div) SY M := {τ L SY 2 M : div τ L d 2} and u [L 2 ] d such that σ n = g on Γ N and Aσ : τ + u div τ = Γ (5) D u D τ n τ H(div) SY M, v div σ = f v v [L2 ] d. he spaces imply the necessary continuity of conforming finite element sub-spaces. he primal mixed formulation requires continuous elements for the displacements, but allows discontinuous elements for the stresses. he dual mixed formulation needs elements with continuous normal-vector for the stresses, but the displacements may be discontinuous. Our goal is to end up with elements, where the tangential component of the displacement and the normal component of the normal stress vector are continuous. he equilibrium equation (2) can be understood in different ways. Posing the equation in L 2 leads to the bilinear-form b(σ, v) = div σ v, and requires v L 2 as well as div σ L 2. esting the equilibrium equation with displacements in H 1, i.e., posing the equation in [H 1 ] = H 1 leads to the form b(σ, v) := div σ, v H 1 H = σ : ε(v). 1 his needs v H 1, and div σ H 1. he later one is clearly satisfied by σ L 2.

6 MIXED INIE ELEMENS OR ELASICIY 5 Our new method tests the equilibrium equation with displacements in H(curl): div σ, v H(curl) H(curl) = f, v H(curl) H(curl) he space is H(curl) = {v [L 2 ] d : curl v L 2 }, where the differential operator curl is defined by and curl v = v for d = 3 curl v = v x y v y for d = 2. x his allows right-hand sides of the form f, v = f 1 v + f 2 curl v, v H(curl) where f 1 is a usual volume force density, and f 2 is a rotational force density. Lemma 1. he dual space of H(curl) is H 1 (div) = {q [H 1 ] d : div q H 1 }. Proof. Any function v H(curl) can be decomposed as v = ϕ + z with ϕ H 1 and z [H 1 ] d and the corresponding bounds in the norms (see [19], heorem 3.4). he dual norm is q H(curl) = sup v H(curl) sup v H(curl) = sup ϕ,z q, v v H(curl) q, v inf v= ϕ+z ϕ H 1 + z H 1 q, ϕ + z ϕ H 1 + z 1 H q, ϕ q, z sup + sup ϕ H 1 ϕ H 1 z [H 1 ] z 1 d H div q, ϕ q, z = sup + sup ϕ H 1 ϕ H 1 z [H 1 ] z 1 d H = div q H 1 + q H 1

7 6 JOACHIM SCHÖBERL AND ASRID SINWEL he proper function space Σ for the stresses is Σ = {σ L SY M 2 : div σ H 1 (div)} he constraint div σ H 1 is no new requirement, the only additional constraint is div(div σ) H 1. hus, the space can be written also as Σ = {σ L SY M 2 : div div σ H 1 } hroughout the following, let h = { } be a shape-regular triangulation of Ω, consisting of triangular or tetrahedral elements in two or three space dimensions, respectively. Let denote the set of all element interfaces (facets), i. e. edges in two or faces in three space dimensions, respectively. In 3d, we also define E as the set of all element edges. he following theorem provides that normal-normal continuous finite elements are proper for the space Σ: heorem 2. Assume that σ is a piece-wise smooth symmetric tensor field such that σ nτ H 1/2 ( ) and n σn is continuous across interfaces. hen there holds div σ H(curl). Proof. Let v be a smooth test function. he divergence operator div : H(div div) H(curl) is then defined as div σ, v = σ : v = { } div σ v σ n v. Ω We further investigate the boundary terms. Using the continuity of σ nn and u τ across element interfaces, we may reorder the surface integrals facet by facet. We can write σ n v = σ nn v n + σ nτ v τ = σ nn [v n ] + σ nτ v τ Using this relation, we obtain div σ, v = { div σ v } σ nτ v τ + [σ nn ] }{{} =0 div σ L2 ( ) v L2 ( ) + σ nτ H 1/2 ( ) v τ H 1/2 ( ) v n C(σ) v H(curl)

8 MIXED INIE ELEMENS OR ELASICIY 7 By density, the continuous functional can be extended to the whole space H(curl): div σ, v = { } div σ v σ nτ v τ Remark 3. he condition σ nτ H 1/2 is a continuity constraint at vertices in 2d and at edges in 3d. If we slightly weaken the regularity, the constraint disappears, and much simpler elements can be used. he analysis in discrete norms in the next section will rectify this violation of conformity. Let us now include boundary conditions into the problem formulation. We assume that u is given on a non-trivial part Γ D Γ of the boundary, and surface tractions σ n are prescribed on Γ N = Γ\Γ D. or simplicity, let all data be homogenous, u = 0 on Γ D, σ n = 0 on Γ N. Boundary conditions for u τ, σ nn turn out to be essential, conditions on u n, σ nτ are natural and can be included into the variational equations. More precisely, we search for (σ, u) Σ 0 V 0 where Σ 0 = {σ H(div div) : σ nn = 0 on Γ N } V 0 = {v H(curl) : v τ = 0 on Γ D } If the solution of the elasticity problem is piecewise smooth, the duality pairing div σ, v can be rewritten by means of domain and boundary integrals. Now, we formulate the problem by means of these integrals. or smooth solutions, this is an equivalent formulation of the elastictiy problem. heorem 4. Let the solutions be piece-wise smooth. hen the elasticity problem (1),(2) is equivalent to the following mixed problem: ind σ Σ 0 and u V 0 such that { Aσ : τ + div τ u τ } nτu τ = 0 τ Σ0 { div σ v σ } nτv τ = f v v V0 Proof. Again, we reorder the surface integrals on element interfaces in a face-wise mode. hen the second line turns out to be the equilibrium equation, plus tangential continuity of the normal stress vector: (div σ + f)v + [σ nτ ]v τ = 0 v

9 8 JOACHIM SCHÖBERL AND ASRID SINWEL Since the space requires continuity of σ nn, the normal stress vector is continuous. Element-wise integration by parts in the first line gives (Aσ ε(u)) : τ + τ nn [u n ] = 0 τ his is the constitutive relation, plus normal-continuity of the displacement. angential continuity of the displacement is implied by the space H(curl). 3. inite element error estimates he displacement space H(curl) and the stress space H(div div) imply the necessary continuity of finite element spaces: to be conforming, the tangential component of the displacement as well as the normalnormal tensor components need to be continuous. or k 1, we define the finite element spaces Σ h := {τ h L SY 2 M : τ h P k, τ nn P max(1,k 1) continuous, τ nn = 0 on Γ N } V h := {v h (L 2 ) d : v h P k, v τ continuous, v τ = 0 on Γ D }. he space V h is the second family of Nédélec. Up to our knowledge, the space Σ h is a new finite element space. In the continuous setting, both v = H(curl) and Σ = H(div div) are equipped with their natural norms. or the finite element analysis, we choose different norms. We define the discrete, mesh dependent norm for the displacements v 2 V h := ε(v) 2 L 2 ( ) + h 1 [v n] 2 L 2 ( ). Note that, using the piecewise strain tensor instead of gradients, we will not require Korn s inequality [18, 22]. his will be useful when treating anisotropic geometries or elements, for which the constant in Korn s inequality deteriorates. or the stress space, we use the L 2 norm σ Σh := σ L2. here the local mesh size h is the average height perpendicular to of the two adjacent elements = { +, }. hen there holds h. We can pose the finite dimensional problem

10 MIXED INIE ELEMENS OR ELASICIY 9 ind σ Σ h, u V h such that a(σ, τ) + b(τ, u) = 0 τ Σ h b(σ, v) = (f, v) v V h. he bilinear forms a : Σ Σ R is defined via (6). In the homogenous, isotropic case, the compliance tensor A is determined by the Lamé constants λ and µ only. In this case a is given by (7). Here tr(τ) denotes the trace of the tensor τ, and τ D is the deviator τ D = τ 1 d tr(τ)i. (6) (7) a(σ, τ) = = Ω Ω Aσ : τ dx 1 2µ σd : τ D + 1 tr(σ) tr(τ) dx. λ + µ he second bilinear form b : Σ V R is generated by the divergence operator. or piecewise smooth functions τ, v it can be evaluated by means of integrals over elements and element boundaries. Using integration by parts, we can find two different formulations defining b: (8) (9) (10) b(τ, v) = div τ, v = div τ v dx τ nτ v τ ds = τ : ε(v) dx + τ nn v n ds. Combining these two forms into one, we also define the compound bilinear form B : (Σ V ) (Σ V ) R by B(σ, u; τ, v) = a(σ, τ) + b(σ, v) + b(τ, u) Discrete stability. In the continuous setting, we have continuity and inf-sup stability of the bilinear form B on Σ 0 V 0. In this subsection, we are concerned with an analogous condition for the discrete problem. It is sufficient to show that [14] (1) Continuity of a, b a(σ, τ) α 2 σ Σh τ Σh σ, τ Σ h b(σ, v) β 2 σ Σh v Vh σ Σ h, v V h (2) Coercivity on the kernel a(σ, σ) α 1 σ 2 Σ h σ Ker B h

11 10 JOACHIM SCHÖBERL AND ASRID SINWEL (3) inf-sup stability inf v V h b(σ, v) sup σ Σ h σ Σh v β 1 Vh We will see that both conditions are satisfied, but the stability constant α 1 depends on material parameters. or isotropic material, the compliance tensor A is determined by the Lamé constants λ and µ. As λ approaches infinity, the material becomes incompressible. hen a is not uniformly coercive on the whole space Σ 0. According to the second condition above, we only need coercivity on the kernel of B. In the continuous case, Ker B is the set of exactly divergence free functions. One can show that, for divergence free functions σ, there holds a(σ, σ) α 1 σ 2 Σ where the constant α 1 > 0 does not depend on λ [14]. he bilinear form a is not uniformly coercive on the discrete kernel Ker B h = {σ Σ h : b(σ, v) = 0 v V h }. o stabilize the problem, we add a further term to the bilinear form a element by element. We define the stabilized form a s (σ, τ) = (Aσ, τ) 0,Ω + h 2 (div σ, div τ) 0,. he solution σ satisfies div σ = f. herefore, to preserve consistency, the term h 2 (f, div τ) 0, has to be added element-wise to the right hand side. We obtain the stabilized problem ind σ Σ h, u V h such that a s (σ, τ) + div τ, u = h2 (f, div τ) 0, τ Σ h div σ, v = (f, v) v V h. We will see that the stabilized form B s (σ, u; τ, v) = a s (σ, τ) + b(σ, v) + b(τ, u) is inf-sup stable by proving an inf-sup condition for b and coercivity of a s on the discrete kernel, where the stability constants do not depend on λ. In the following, we use the H(curl) and the H(div div) conforming transformations from the reference element ˆ to the physical element, which are described more closely in Subsection 4.3. hey are given by v(x) = ˆv(ˆx) σ(x) = 1 ( J 2 ˆσ(ˆx) )

12 MIXED INIE ELEMENS OR ELASICIY 11 Here is the Jacobian of the transformation Φ : ˆ, and J = det denotes the Jacobi determinant. Note that the tangential components of vector fields and the normal-normal components of tensor-valued functions are not changed by these transformations. When treating anisotropic geometries, it will be useful to use different norms in the discrete spaces. or the displacement space, instead of taking the piecewise defined gradient, we use the strain tensor ε(u) on each element. As we will see, all stability estimates are independent of the shape of the physical element. Lemma 5. Assume to be a shape-regular triangularization of the domain Ω. hen the norm for the displacement space Vh is equivalent to the broken H 1 semi norm v 2 V h v Ṽh := ε(v) 2 L 2 ( ) + h 1 P 1 [v n ] 2 L 2 ( ). Moreover, for the discrete stress space we have σ 2 Σ h σ 2 Σh := σ 2 L 2 ( ) + h σ nn 2 L 2 ( ). Proof. We prove the equivalence of the norms Vh and Ṽh first. his equivalence also holds true in the space of H(curl)-conforming functions, which are piecewise smooth: V := {v V 0 : v H 1 ( ) }. Clearly, there holds v Ṽh v Vh We will now prove the opposite bound, namely v Vh v Ṽh. Let Π denote the element-wise projection onto the space of piecewise rigid body motions RM( ) = {a+bx : a R d, b R skw d d }. his projection is defined via v Π v dx = curl(v Π v) dx = 0. As RM( ) is a subspace of P 1 ( ), there holds for any facet (I P 1 )[v] n L 2 ( ) = (I P 1 [v Π (v)] n L 2 ( ) [v Π (v)] n L 2 ( ). ollowing [13], there holds [v Π (v)] n L 2 ( ) h ε(v) 2 L 2 ( ).

13 12 JOACHIM SCHÖBERL AND ASRID SINWEL Combining these estimates, we obtain h 1 [v n] 2 L 2 ( ) = h 1 P 1 [v n ] 2 L 2 ( ) + h 1 (I P 1 )[v] n 2 L 2 ( ) h 1 P 1 [v n ] 2 L 2 ( ) + ε(v) 2 L 2 ( ) = v Ṽ h his relation proves the first equivalence. or the stress space Σ h, we transform σ element-wise to the reference element ˆ. here we use the equivalence of norms on a finitedimensional space, and transform back. his gives σ 2 L 2 (Ω) = = = ˆ σ : σ dx 1 J 4 ˆσ 2 dˆx (ˆn ˆσˆn) 2 dŝ + ˆ (n σn ) 2 ds + σ 2 L 2 (Ω) = σ 2 Σh ˆ 1 J 4 ˆσ 2 dˆx Lemma 6. he bilinear forms a s : Σ h Σ h R, b : Σ h V h R are continuous. Proof. o show continuity of a s in the discrete norms, we need the inverse inequality div σ L 2 ( ) h 1 σ L 2 ( ) σ Σ h. hen we can estimate a s 1 (σ, τ) = 2µ σd : τ D + 1 Ω ( 1 2µ + 1 λ + µ = α 2 σ Σh τ Σh. λ + µ tr(σ) tr(τ) dx + ) σ L 2 τ L 2 + σ L 2 ( ) τ L 2 ( ) h 2 (div σ, div τ) L 2 ( )

14 MIXED INIE ELEMENS OR ELASICIY 13 or the second bilinear form, we use the norm equivalences from above and see b(σ, v) = σ : ε(v) dx + σ nn v n ds σ L 2 ( ) ε(v) L 2 ( ) + σ Σh v Vh β 2 σ Σh v Vh. h 1/2 σ nn L 2 ( )h 1/2 [v n ] L 2 ( ) Lemma 7. he bilinear form b : Σ h V h R is inf-sup stable, there exists a positive constant β 1 > 0, β 1 independent of h such that (11) inf v V h b(σ, v) sup σ Σ h σ Σh v β 1. Vh Proof. he finite element space Σ h can be decomposed into two parts Σ h = Σ f h + Σb h. he first space, Σ f h, is associated to the set of faces. or each face, there exists a family of functions (σj ) 0 j k such that σ j = 0,, : σ f Σ f h } = P k ( ) {σ f nn he second space, Σ b h, consists of element bubble functions, which satisfy σnn b = 0. o construct these spaces, we need tensor-valued basis functions S f i corresponding to the facets of the element. hese basis functions span P 0 on a single element, and the normal-normal component vanishes on all facets but one, where it is constant. We will now define the basis on the reference element ˆ. herefore, let (λ i ) denote the barycentric coordinates of the vertices. In 2d, we propose Ŝ f i = sym[ λ i+1 λ i+2], i = Calculating them explicitely yields ( ) ( Ŝ1 =, 1 0 Ŝ 2 = 1 2 ) ( 0 1, Ŝ 3 = 1 0 ).

15 14 JOACHIM SCHÖBERL AND ASRID SINWEL In 3d, we need four face-based functions Ŝf i and two additional element bubbles Ŝt i to span the six-dimensional space of constant symmetric 3 3 tensor fields. We set them via Ŝ f i = sym[( λ i+1 λ i+2 ) ( λ i+2 λ i+3 )] Ŝ t 1 = sym[( λ 1 λ 2 ) ( λ 3 λ 4 )] Ŝ t 2 = sym[( λ 1 λ 3 ) ( λ 2 λ 4 )]. Explicitely, they read as Ŝ f 1 = , Ŝ f 2 = , Ŝ f 3 = Ŝ f 4 = , Ŝ1 t = , Ŝ2 t = hey can be transformed to the physical element using the transformation given in Subsection 4.3. Using these transformations we can define a piecewise constant tensor field S for any facet such that S nn = δ. Let v V h be given. In order to prove the Lemma, we want to find some σ Σ h satisfying b(σ, v) = div σ v dx σ nτ v τ ds β 1 σ 0 v Vh, σ Σh c v Vh. herefore we decompose the stress function σ = ασ b + βσ f where α, β R are to be specified later. Let σ f be such that σ f nn = h 1 [v n]. his is possible as [v n ] is a polynomial of degree k on each face, and therefore we can simply set. σ f := h 1 [v n] S. Note that, when using Ṽh, we can choose σ f of order one. he approximation properties of Σ h V h are still of optimal order when using facet bubbles of order k 1.

16 MIXED INIE ELEMENS OR ELASICIY 15 Next, we determine the bubble part σ b on each element. In 2d, the space of P 1 bubble functions on the reference element is spanned by ˆΣ b := {Ŝ i λ i, i = 1, 2, 3}. In 3d, we have to add some more functions ˆΣ b {Ŝ i λ i, i = } {Ŝ i P 1 ( ˆ ), i = 1, 2}. Note that the space of constant symmetric tensor fields on the reference element is spanned in 2d by {Ŝ i }, in 3d by {Ŝ i, Ŝ j } respectively. or simplicity, we will do the following calculations for the 2d case. o treat three space dimensions, we would need to add the additional bubble functions. Choose the bubble part σ b such that, on each triangle resp. tetrahedron, ˆσ b = J 2 ) is given by ˆσ b = J 2 (ˆε(ˆv) : Ŝi 1 }{{} ) Ŝi 1 λ i }{{} i P k 1 ( 1 σb P 1. o achieve a bound for the L 2 -norm of σ, we estimate the two parts σ b, σ f separately by v Vh. We will see in both cases, that the constant arising is independent of the shape or the physical element. his will prove useful when treating anisotropic geometries. irst, we treat σ f. Similar to the proof in Lemma 5, we obtain σ f 2 L 2 (Ω) = h σ f nn 2 L 2 ( ) h 1 [v n] 2 L 2 ( ). Here we used that σ f consists of edge bubble functions only, therefore the facet-contributions of σ nn only form a norm on the discrete space. or the bubble part σ b, there holds σ b 2 L 2 (Ω) = α 1 ˆ J 4 ˆσ b 2 J dˆx = ˆ ˆ ˆ i i 1 (ˆε(ˆv) : ( 1 ˆε(ˆv) ˆε(ˆv) 2 J dˆx Ŝi 1 )Ŝ i λ i 2 J dˆx : Ŝ i )Ŝ i λ i 2 J dˆx = ε(v) 2 L 2 ( ).

17 16 JOACHIM SCHÖBERL AND ASRID SINWEL Combining the two inequalities, we obtain σ 2 L 2 (Ω) αc 1 ε(v) 2 L 2 ( ) + βc 2 h 1 [v n] 2 L 2 ( ) We will need the following bound for the bubble part σ b on each element σ b : ε(v) dx c 3 ε(v) 2 L 2 ( ). Again, we prove this by transformation to the reference element ˆ. We obtain σ b : ε(v) dx = = = = = ˆ ˆ ˆ ˆ ˆ 1 J 2 1 J 2 i i 1 ( ˆσ(ˆx) ) : 1 ˆσ : ˆε(ˆv)J dˆx (ˆε(ˆv) : ( 1 ˆε(ˆv) Ŝ i 1 ˆε(ˆv) 2 J dˆx ε(v) 2 dx = ε(v) 2 L 2 ( ) ˆε(ˆv)(ˆx) J dˆx ) : Ŝ i ) 2 λ i J dˆx Ŝi 1 λ i : ˆε(ˆv)J dˆx where we used that the functions (S i ) i form a basis for P 2 2,SY M 0.

18 MIXED INIE ELEMENS OR ELASICIY 17 or σ defined as above we apply Young s inequality and obtain b(σ, v) = ε(v) : σ dx σ nn [v n ] ds = [ ασ b : ε(v) + βσ f : ε(v) ] dx β σnn[v f n ]ds αc 3 ε(v) 2 L 2 ( ) β σf L 2 ( ) ε(v) L 2 ( ) + β αc 3 ε(v) 2 L 2 ( ) βc 2h 1/2 [v n ] L 2 ( ) ε(v) L 2 ( ) + h 1 [v n] 2 L 2 ( ) β h 1 [v n] 2 L 2 ( ) αc 3 ε(v) 2 L 2 ( ) βc 2ε 2 ε(v) 2 L 2 2 ( ) βc 2 h 1 2ε 2 [v n] 2 L 2 ( ) + β h 1 [v n] 2 L 2 ( ) Choosing β = 1, α = (1 + c 2 2)/(2c 3 ), ε 2 = 2c 2 we get b(σ, v) 1 ε(v) 2 L 2 2 ( ) + 1 [v n ] 2 L 2 2 ( ) 1 2c(α, β) v V h σ Σh. o show coercivity of a s, we need the following lemma. We assume that Γ N is a non-trivial part of the boundary Γ = Ω, where Neumann boundary conditions are prescribed. he Dirichlet problem, i. e. Γ N =, is treated separately. Lemma 8. or any z L 2 there exists some p [H 1 ] d, p τ P 1 for all faces, satisfying div p = z in Ω p = 0 on Γ\Γ N. Proof. rom Stokes theory it is well known that there exists a constant γ > 0 such that (div p, z) L2 inf sup γ. z L 2 p H 1 z L 2 p [H 1 ] d

19 18 JOACHIM SCHÖBERL AND ASRID SINWEL his implies the existence of p 1 [H 1 ] d such that div p 1 = z, p = 0 on Γ\Γ N. We are now looking for a polynomial approximation to p 1. herefore, we consider the H 1 conforming finite element space W h proposed by [19]. he degrees of freedom of a function w W h are the vertex values and the mean values of the normal components of w on the element faces. Restricted to a triangle in 2D, w can be written as 3 3 w = w i λ i + w j λ j+1 λ j+2 n j i=1 j=1 whereas on a tetrahedron in 3D w takes the form 4 4 w = w i λ i + w j λ j+1 λ j+2 λ j+3 n j. i=1 j=1 Note that the tangential component of w W h still is in P 1. rom the Stokes problem it is known that there exists a ortin operator Π h : [H 1 ] d W h such that (w Π h w) n dx = 0 Using this operator, we define Π h w 1 c w 1 p 1,h = Π h p 1 which satisfies, due to its construction div p 1,h dx = div p 1 dx = tr(σ) dx. Next, we choose p 2 by solving the following problem locally on each triangle resp. tetrahedron. ind p 2 H 1, p 2 = 0 such that div p 2 = z div p 1,h. hese local solutions exist, as the right hand side z div p 1,h has zero mean value. urther p 2 is in [H 1 ] d globally, as it vanishes on element interfaces. Setting p = p 1,h + p 2, we obtain div p = div p 1,h div p 2 = tr(σ), i. e. the lemma is proven. Using the result above, we can prove stability for the bilinear form a s in the finite element spaces.

20 MIXED INIE ELEMENS OR ELASICIY 19 Lemma 9. he bilinear form a s : Σ h Σ h R is coercive, there exists a constant α 1 independent of the Lamé constant λ such that (12) a s (σ, σ) α 1 σ 2 Σ h σ Ker B h where Ker B h is the discrete kernel (13) Ker B h = {σ Σ h : b(σ, v) = 0 v V h }. Proof. or this proof we assume that the boundary part, where σ n is given, is non-trivial, Γ N 0. or any τ Σ there holds τ 2 L = τ D 2 2 L d tr(τ) 2 L 2 and further τ 2 L 2µa(τ, τ) + 2 tr(τ) 2 L 2. herefore, it is sufficient to show that for σ Ker B h a(σ, σ) c tr(σ) 2 L 2. According to Lemma 8 there exists some p [H 1 ] d such that div p = tr(σ), p = 0 on Γ\Γ N, p 1 c tr(σ) L 2. or this p we may write tr(σ) div(p) dx tr(σ) 2 L 2 = Ω = σ : (div(p)i) dx Ω = d σ : ( p ( p) D) dx Ω [ = d σ : p dx σ D : p ] dx Ω d σ : p dx + d σ D L 2 p H 1 Ω d σ : p dx + d σ D L 2 tr(σ) L 2. Ω

21 20 JOACHIM SCHÖBERL AND ASRID SINWEL or the first term there holds σ : p dx = div σ p dx Ω = div σ p dx = div σ, p = div σ, p I h p (σn) p ds [σ nτ ] p τ ds where we used that σ, v h = 0 for v h V h in the last line. Let I V denote the Nédélec interpolation operator, for which holds (14) q(p I V p) τ dx = 0 q P 1, E E (15) E p I V p L 2 h p H 1. On each face p τ is in P 1. As the interpolation operator I V is polynomial preserving, the face integrals in the estimate above vanish, [σ nτ ](p τ I V p τ ) ds = 0. herefore we may estimate Ω σ : p dx = div σ (p I V p) dx div σ L 2 ( ) p I V p L 2 ( ) c c h div σ L 2 ( ) p 1 h div σ L 2 ( ) tr(σ) L 2. Collecting the results from above, we get tr(σ) 2 L 2 c h 2 div σ 2 L 2 ( ) + d σd 2 L 2. or the bilinear form a s, this implies a s 1 (σ, σ) = Ω 2µ σd λ + µ tr(σ)2 dx + h 2 1 2µ σd 2 L + h 2 div σ 2 2 L 2 ( ) dx c(µ) tr(σ) 2 L 2. div σ 2 dx

22 MIXED INIE ELEMENS OR ELASICIY 21 Remark 10. In the case of Dirichlet boundary conditions, it is not possible to find p H 1 (Ω) d such that div p = tr(σ), p = 0 on Γ p H 1 c tr(σ) L 2, as tr(σ) does not necessarily satisfy the complementarity condition tr(σ) dx = 0. Ω In this case, we choose a different norm for the discrete space Σ h σ 2 Σ h = min ρ R σ ρi 2 L λ ρi 2 L 2. We can find p H 1 (Ω) d 0 such that In this case we can bound div p = tr(σ) tr(σ), p = 0 on Γ p 1 c tr(σ) L 2, tr(σ) tr(σ) L 2 c h div σ L 2 ( ) + d σ D L 2 Setting ρ = 1 tr(σ) we may estimate d σ 2 Σ h σ ρi 2 L λ ρi 2 L 2 c ( σ 1d tr(σ)i 2L2 + 1d ) tr(σ) 2 tr(σ) 2L2 ( ) c σ D 2 L 2 + c(µ)a s (σ, σ). his proves stability for the Dirichlet case. h 2 div σ 2 L 2 ( ) By the stability of the discrete saddle point problem and standard interpolation error estimates we obtain the convergence result heorem 11. he finite element solution (u h, σ h ) satisfies the error estimate u u h Vh + σ σ h Σh ch m 1 u H m

23 22 JOACHIM SCHÖBERL AND ASRID SINWEL for u H m with 1 m k + 1. he constant c is bounded uniformly for λ. Remark 12 (Hybridization). Discretization of the mixed system leads to an indefinite system matrix. o avoid this, the continuity of σ nn can be broken and enforced by inter-element Lagrange multipliers [16, 5, 14]. hen all degrees of freedom for the finite element space Σ h are bound to only one element, and can be eliminated by static condensation. he remaining matrix is symmetric and positive definite. he Lagrange multipliers can be interpreted as normal displacements on the edges of the triangles or faces of the tetrahedra, respectively. 4. inite element basis functions o construct finite elements for the spaces V h H(curl) and Σ h H(div div), we use the 2d-sequence H 1 H 2 ( ) and in 3d H(curl) [ H 1 ( ) ] 2 σ H(div div) div H 1 (div) div H 1 H 1 H 2 ( ) sym curl Q 1 H(curl) [H 1 ( )] 2 H(div div) ε H(curl curl) curl div H 1 (div) div H(div curl) H 1 with the operator Q, its inverse Q 1 and the stress operator σ given by QA = A tr(a)i, Q 1 A = A 1 d 1 tr(a)i, σ = Q 1 ε he additional spaces used in the sequence are H(curl curl) := {ε [L 2 (Ω)] SY M : sym curl Q 1 curl ε [H 1 ] SY M } H(div curl) := {δ [L 2 (Ω)] 3 3 : div sym curl Q 1 δ [H 1 ] 3 } he image of two successive operators is exactly the kernel of the next, in 2d there holds In 3d, we use the relation range(σ( )) = Ker(div) range(div σ( )) = Ker(div) range(sym curl Q 1 curl) = Ker(div)

24 MIXED INIE ELEMENS OR ELASICIY 23 or the construction of shape functions on the reference elements, we use Legendre polynomials (l i ) 0 i k, which are orthogonal polynomials in L 2 ( 1, 1) and span P k [ 1, 1]. he integrated Legendre polynomials (L i ) 2 i k are defined by L i (x) = x 1 l i 1 (ξ) dξ. hey are orthogonal with respect to the H 1 semi norm and vanish on the interval bounds { 1, 1}. We will also need scaled Legendre polynomials (l S i ) 1 i k and scaled integrated Legendre polynomials (L S i ) 2 i k, which are defined for 0 < t 1 and t x t. hey are given by l S i (x, t) = t i l i ( x t ) L S i (x, t) = t i L i ( x t ). hey satisfy L S i ( t, t) = L S i (t, t) = riangular elements. Let the reference triangle be defined by := {(x, y) 0 x, y 1, x + y < 1} with vertices V 1 = (1, 0), V 2 = (0, 1), V 2 = (0, 0). Let λ i, i = 1, 2, 3 denote the barycentric coordinates or i, j 0, let λ 1 = x, λ 2 = y, λ 3 = 1 x y. u i = L S i+2(λ 2 λ 1, λ 2 + λ 1 ) v j = λ 3 l j (2λ 3 1). hen u i is a polynomial of degree i + 2, which vanishes on edges E 1, E 2. he polynomial v j is of degree j + 1 and vanishes on edge E 3. he following H 1 conforming shape functions form a basis for P k ( ) [24, 28] Vertex-based functions: m = 1, 2, 3 φ V m = λ i Edge-based functions: m = 1, 2, 3, E m = [α, β], 0 i k 2 φ Em i = L S i+2(λ α λ β, λ α + λ β ) Cell-based functions: i, j 0, i + j k 3 φ i,j = u i v j = L S i 2(λ 2 λ 1, λ 2 + λ 1 )λ 3 l j (2λ 3 1)

25 24 JOACHIM SCHÖBERL AND ASRID SINWEL We define the local space of order k W k ( ) = span(φ V m) span(φ Em i ) span(φ i,j) = P k ( ). Using this, H(curl) conforming shape functions can be defined as below [24, 28] Edge-based functions: m = 1, 2, 3 Low-order functions: m = 1, 2, 3, E m = [α, β] ϕ N 0 m = λ α λ β λ α λ β High-order functions: 0 i k 1 ϕ Em i = φ Em i Cell-based functions: ype 1, gradient fields: 0 i + j k 2 ϕ,1 i,j ype 2, 0 i + j k 2 = (u i v j ) = u i v j + u i v j ϕ,2 i,j ype 3, 0 j k 2 ϕ,3 j = u i v j u i v j = ( λ 1 λ 2 λ 1 λ 2 )v j he local H(curl) conforming E-space is here holds V k ( ) = span(ϕ N 0 m ) span(ϕ Em i ) span(ϕ i,j) = P k ( ). W k+1 ( ) V k ( ). As a last step, we define H(div div) conforming shape functions on the triangle Edge-based functions, divergence free: m = 1, 2, 3, E m = [α, β], 0 i k ψ Em i Cell-based functions: ype 1, divergence free: i, j 0, 0 i + j k 1 = σ( φ Em i ) ψ,1 i,j = σ( (u i v j )) = 2 (u i v j ) (u i v j )I = ( 2 u i v j + u i v j + v j u i + u i 2 v j ) ( u i v j + 2 u i v j + u i v j )I

26 MIXED INIE ELEMENS OR ELASICIY 25 ype 2, i, j 0, 0 i + j k 1 ψ,2 i,j = ( 2 u i v j u i v j v j u i + u i 2 v j ) ( u i v j 2 u i v j + u i v j )I ype 3, i 0, j 1, 1 i + j k 1 ψ,3 i,j = ( 2 u i v j u i 2 v j ) ( u i v j u i v j )I ype 4, 0 j k 1 ψ,4 j = σ(( λ 1 λ 2 λ 1 λ 2 )v j ) rom these basis functions, we derive the local H(div div) conforming E-space Σ k ( ) = span(ψ Em i ) span(ψi,j) = P k ( ). he way the basis was constructed implies σ(v k+1 ( )) Σ k ( ) etrahedral elements. Let the reference tetrahedron be defined by := {(x, y, z) 0 x, y, z 1, x + y + z < 1} with vertices V 1 = (1, 0, 0), V 2 = (0, 1, 0), V 2 = (0, 0, 1), V 3 = (0, 0, 0). Let λ i, i = denote the barycentric coordinates or i, j 0, let λ 1 = x, λ 2 = y, λ 3 = z, λ 4 = 1 x y z. u i = L S i+2(λ 2 λ 1, λ 2 + λ 1 ) v j = λ 3 l S j (λ 3 λ 2 λ 1, λ 3 + λ 2 + λ 1 ) w l = λ 4 l l (2λ 4 1) he polynomials u i vanish on faces 1, 2, whereas v j = 0 on 3 and w k = 0 on 4. he following functions define a local basis for H 1 ( ) Vertex based functions, m = φ V m = λ m Edge based functions,m = , E m = [α, β], 0 i k 2 φ Em i = L S i+2(λ α λ β, λ α + λ β ) ace based functions, m = , m = [α, β, γ], 0 i + j k 3 u α,β i = L S i+2 (λ α λ β, λ α +λ β ), v γ j = λ γl S (λ γ λ α λ β, λ γ +λ α +λ β ) φ m i,j = L S i+2(λ α λ β, λ α + λ β )λ γ l S j (2λ γ λ, λ)

27 26 JOACHIM SCHÖBERL AND ASRID SINWEL Cell based functions, 0 i + j + l k 4 φ i,j,l = u i v j w l he local space of order k is given by W k ( ) = span(φ V m) span(φ Em i ) span(φ m i,j ) span(φ i,j,l) = P k ( ). Again we use these functions to construct H(curl) conforming shape functions Edge-based functions, m = , E m = [α, β] Low-order functions: ϕ N 0 m = λ α λ β λ α λ β High-order functions: 0 i k 1 ϕ Em i = φ Em i ace-based functions: m = , m = [α, β, γ], u α,β i = L S i+2 (λ α λ β, λ α +λ β ), v γ j = λ γl S (λ γ λ α λ β, λ γ +λ α +λ β ) ype 1, 0 i + j k 2 ϕ m,1 i,j ype 2, 0 i + j k 2 ϕ m,2 i,j = (φ m i,j ) = (uα,β i v γ j ) = u α,β i ype 3, 0 j k 2 ϕ m,3 j v γ j uα,β i v γ j = ( λ α λ β λ α λ β )v γ j Cell-based functions: ype 1, 0 i + j + l k 3 ϕ,1 i,j,l = (φ i,j,l ) = (u iv j w l ) ype 2, 0 i + j + l k 3 ϕ,2a i,j,l = u i v j w l u i v j w l + u i v j w l ϕ,2b i,j,l = u i v j w l + u i v j w l u i v j w l ype 3, 0 j + l k 3 ϕ,3 j,l = ( λ 1 λ 2 λ 1 λ 2 )v j w l We define the local H(curl) conforming E-space V k ( ) = span(ϕ N 0 m ) span(ϕ Em i ) span(ϕ m i,j ) span(ϕ i,j,l) = P k ( ). As in 2D, there holds W k+1 ( ) V k ( ). H(div div) conforming shape functions on the tetrahedron can be constructed as follows

28 MIXED INIE ELEMENS OR ELASICIY 27 ace-based functions, divergence free m = , m = [α, β, γ], u α,β i = L S i+2(λ α λ β, λ α + λ β ), v γ j = λ γl S (λ γ λ α λ β, λ γ + λ α + λ β ) 1 i + j k + 2 ψ m,1 i,j = sym ( ) curl(curl( u α,β i v γ j ) )) Cell-based functions: ype 1, divergence free, 0 i + j + l k ψ,1a i,j,l = sym ( curl(curl(u i v j wl ) )) ) ψ,1b i,j,l = sym ( curl(curl(v j u i w l ) ) ) ψ,1c i,j,l = sym ( curl(curl(w l v j u i ) ) ) ype 2, 0 i + j + l k ψ,2a i,j,l = sym ( v j ( 2 u i ) w l u i curl(curl( v j w l ) ) ), j, l 1 ψ,2b i,j,l = sym ( u i ( 2 v j ) w l v j curl(curl( u i w l ) ) ), l 1 ψ,2c i,j,l = sym ( v j ( 2 w l ) u i w l curl(curl( v j u i ) ) ), j 1 ype 3, 0 i + j k ( ) ψ,3a i,j = sym v i ( φ N 0 E 12 ) w j ψ,3b i,j = sym ( φ N 0 E 12 ( 2 v i ) w j ), i 1 he local H(curl) conforming E-space is defined by Σ k ( ) = span(ψ m i,j ) span(ψ i,j,l) = P k ( ). Similar to the 2D case, we have σ(v k+1 ( )) Σ k ( ) Global inite Element Spaces. he global finite element spaces W h H 1, V h H(curl) and Σ h H(div div) can be derived from the local spaces on the reference elements by element-wise transformation. herefore, let Φ : ˆ be the mapping from the reference element ˆ to the physical element (. By) we denote the Jacobian of Φ,i this transformation, (ˆx) = (ˆx). he Jacobi determinant is ˆx j given by J (ˆx) = det( (ˆx)). Normal and tangential vectors along the element boundary transform as ij τ = ˆτ, n = J ˆn.

29 28 JOACHIM SCHÖBERL AND ASRID SINWEL An H 1 conforming transformation from ˆ to is given by H 1 ( ˆ ) H 1 ( ), ŵ w := ŵ Φ 1. or H(curl), we use the covariant transformation is [21] H(curl, ˆ ) H(curl, ), ˆv v := ˆv Φ 1. he covariant transformation is conforming for H(curl), as the tangential component of a vector field along the boundary of the element is preserved. he following transformation is conforming for H(div div) H(div div, ˆ ) H(div div, ), ˆσ σ := 1 ( J 2 ˆσ ) Φ 1. or this transformation, the normal-normal component of the stress field is preserved, n σn = 1 J 2 n 1 σ n = ˆn ˆσˆn. Using these transformations, and the local finite element spaces W k ( ˆ ), V h ( ˆ ) and Σ k ( ˆ ) defined above, we get finite element spaces W h, V h and Σ h for H 1, H(curl) and H(div div), respectively. 5. Non-conforming elements Until now we have considered finite dimensional spaces Σ h Σ = H(div div) and V h V = H(curl). A piecewise smooth function v is in H(curl), if and only if the tangential component v τ is continuous across element interfaces. In 2D, there are k + 1 degrees of freedom on each edge, which are associated to the tangential displacement along this edge. In this section, we will not assume that the finite element space V h is a subset of H(curl), but that the highest order edge basis function may jump across element edges. hereby we obtain a simplified element. It consists of more internal degrees of freedom, which can be eliminated locally. Let (σ, u) Σ V denote the solution to the continuous problem, and (σ h, u h ) Σ h V h the finite-element solution, where V h is the non-conforming space described above. We will see that the error (σ σ h, u u h ) Σh V h

30 MIXED INIE ELEMENS OR ELASICIY 29 u τ u τ highest order edge dofs u τ lower order edge dofs igure 4. Conforming element igure 5. Non-conforming element can be bounded by the same order of convergence as in the conforming case. he triangle inequality ensures (σ σ h, u u h ) Σh V h (σ I Σ σ, u I V u) Σh V h + (I Σ σ σ h, I V u u h ) Σh V h for suitable interpolation operators I V, I Σ. or the first term we know, provided the solution is sufficiently smooth (σ I Σ σ, u I V u) Σh V h ch k (σ, u) H k H k+1. or the second term, we have that (I Σ σ σ h, I V u u h ) is a function in the finite element space Σ h V h. herefore we can use the discrete inf-sup stability of the bilinear form B (I Σ σ σ h, I V u u h ) Σh V h β B(I Σ σ σ h, I V u u h ; τ h, v h ) sup τ h,v h (τ h, v h ) Σh V h β B(I Σ σ σ, I V u u; τ h, v h ) sup + τ h,v h (τ h, v h ) Σh V h β sup τ h,v h B(σ σ h, u u h ; τ h, v h ) (τ h, v h ) Σh V h Again, the first term can be bounded by the approximation error and is of order h k. or the second term, we use Galerkin orthogonality with respect to σ, as Σ h Σ. B(σ σ h, u u h ; τ h, v h ) B(σ σ h, u u h ; 0, v h ) sup = sup τ h,v h (τ h, v h ) Σh V h v h v h Vh b(σ σ h, v h ) (16) = sup v h v h Vh he edge basis functions associated to the tangential displacement are continuous for order 0 to k 1, only the highest order function jumps.

31 30 JOACHIM SCHÖBERL AND ASRID SINWEL herefore, v h V h can be divided into a conforming part v c, which consists of the lower-order edge basis functions and the internal basis functions, and into the non-conforming part. rom the construction of the basis for V h we know that the restriction of this non-conforming part to a triangle is the gradient of an H 1 edge bubble function of order k + 1. We may write v h = v c + h φ v c H(curl) h φ = 3 r,i φ E i i=1 k+1. We use this and the fact that σ h is a solution to the discrete problem to obtain b(σ σ h, v h ) = b(σ, v h ) (f, v h ) = b(σ, v c ) (f, v c ) +b(σ, }{{} h φ) (f, h φ) =0 = (div σ f) φ dx σ }{{} nτ φ h τ ds =0 = σ nτ φ h τ ds. herefore, we can estimate the right hand side of equation (16) by b(σ σ h, v h ) b(σ σ h, h φ h ) sup sup v h v h Vh φ h h φ h Vh = sup φ h σ nτ φ h τ ( 2 φ h 2 L 2 ( ) + h 1 [ φ h n or an edge bubble function φ h P k+1 ( ) there holds 2 φ h 2 + φ h L 2 ( ) h 1 n 2 L 2 ( ) ds ] 2 ch 2 φ h L 2 ( ). L 2 ( ) ) 1/2 Moreover, φ h restricted to an edge is an integrated Legendre polynomial of degree k + 1. his implies that φ h / τ is orthogonal to polynomials

32 of degree k 1, and sup v h b(σ σ h, v h ) v h Vh c sup MIXED INIE ELEMENS OR ELASICIY 31 φ h c sup φ h c sup φ h c,,,, ch k σ H k (Ω). inf s P k 1 (σ nτ s) φ h τ h 2 φ h L 2 ds σ nτ s L2( ) inf s P k 1 h 2 φ h L 2 φ h τ L 2 ( ) σ nτ s inf L2 ( )h 3/2 φ h L2 ( ) s P k 1 h 2 φ h L 2 h k+1/2 σ nτ H k ( ) his implies an optimal order of convergence of σ in the L 2 norm for the 2D non-conforming elements. 6. Anisotropic Estimates Many technical constructions involve thin structures such as beams, plates or shells. raditionally lower dimensional models are derived and treated by proper finite element methods. Here, the small thickness parameter becomes explicite, and special care must be taken to avoid the so called shear locking. he concept of hierarchical finite element modeling treats the structure directly by anisotropic three dimensional finite elements, and the accuracy of the model corresponds to the polynomial order in thickness direction [11, 3, 17, 20], a posteriori estimates are in [25, 2]. Anisotropic error estimates for scalar equations are developed in [4]. Due to the combination of derivatives of the strain operator, these techniques cannot be applied to conforming finite elements for elasticity problems. he goal of this section is to derive robust anisotropic estimates for the elasticity problem discretized with the proposed new elements. Now, let ω R 2 be a domain identified with the cross-section of a flat three dimensional domain, let I = (0, t), and set Ω = ω I. urthermore, let xy and z be triangulations of ω and I, respectively. hen xy z gives the tensor-product mesh of the three-dimensional domain. We may use just one element in thickness direction, or we may use also a refined mesh to resolve several layers of a laminated plate.

33 32 JOACHIM SCHÖBERL AND ASRID SINWEL We define Lagrangian and Nédélec finite element spaces on the cross section and on the interval as L xy k := {v H 1 (ω) : v P k }, N xy k := {v H(curl, ω) : v [P k ] 2 }, P xy k := {v L 2 (ω) : v P k }, L z k := {v H 1 (I) : v P k }, P z k := N z k := {v L 2 (I) : v P k }. he spaces satisfy L xy k+1 N xy k V h := N xy k and L z k+1 N k z. Note that L z k+1 L xy k+1 N z k gives the Nédélec space on the tensor product domain. Let Σ xy k be the finite element space of symmetric 2 2 tensors with continuous normal-normal components, and set { Σ h := σ = σ xx σ xy σ xz σ xy σ yy σ yz σ xz σ yz σ zz ( ) σxz P xy σ k yz ( ) σxx σ : xy Σ xy σ xy σ k N k z, yy } Nk z, σ zz P xy k L z k+2. By [15] there exist quasi-interpolation operators I xy k such that : L 2 (ω) L xy k u I xy k u l, h m l u m, e 0 l 1, 0 m k + 1, l m, where is the union of the triangle and its neighbours. In [23] quasiinterpolation operators Q xy k : L 2(ω) N xy k for the Nédélec spaces have been introduced. hey commute in the sense of I xy k+1 = Qxy k, and satisfy a corresponding interpolation error estimate u Q xy k u l, h m l u m, e 0 l 1, 0 m k, l m. Note that the estimates for 1, are not contained in [23], but are also easily proven. We define according commuting interpolation operators in thickness direction. On the tensor product domain we define the interpolation operator Q k := Q xy k Iz k+1 I xy k+1 Qz k

34 MIXED INIE ELEMENS OR ELASICIY 33 he in-plane deformation u xy := (u x, u y ) is interpolated by the Nédélec operator in plane, and continuously in the thickness-direction, the transversal displacement u z is interpolated vice versa. Lemma 13. he product space interpolation operator Q k satisfies the anisotropic error estimate (17) ε(u Q k u) L2 ( ) h m xy m xyε(u) L2 ( e ) + hm z m z ε(u) L2 ( e ) Proof. We bound the different components of the strain tensor as follows: ε xy (u xy Ik+1Q z xy k u xy) ε xy (u xy Q xy k u xy) + ε xy (Q xy k (u xy Ik+1u z xy )) h m xy m xyε xy (u xy ) e + ε xy (u xy Ik+1u z xy ) e h m xy m xyε xy (u xy ) e + h m z m z ε xy (u xy ) e he second term ε xy,z involves the coupling of in-plane and transversal deformations and essentially relies on the commuting diagram: 2 ε xy,z (u Q k u) 0, = z (u xy Q xy k Iz k+1u xy ) + xy (u z Q z ki xy k+1 u z) = (I Q xy k Qz k) ( ) z u xy + xy u z h m xy m xyε xy,z (u) e + h m z m z ε xy,z (u) e he last term, ε z (u z Q z k Ixy k+1 u z) is bounded similar to the first one. Anisotropic estimates for the stress tensor are also simply obtained. A careful investigation of the estimates in Section 3 provides that all stability estimates are independent of the aspect ratio of the elements. or this, the weight of the jump-terms must be chosen as the element size norm to the facet. Since the norm equivalence V h Ṽh of Lemma 5 does not hold anymore, σ nn must be chosen at the same order as [u n ]. With the same arguments as in the isotropic case, we obtain the heorem 14 (anisotropic error estimates). Let u h and σ h be the finite element solutions on an anisotropic finite element mesh. hen there holds the anisotropic error estimate (18) u u h Vh + σ σ h ) L2 ch m xy m xyε(u) Ω + ch m z m z ε(u) Ω. he constant c is independent of the aspect ratio of the elements.

35 34 JOACHIM SCHÖBERL AND ASRID SINWEL 1 order 1 order 1, nc order 2 order 2, nc error stress e-04 1e e+06 1e+07 dofs igure 6. Unit square, error σ σ h L 2 (Ω), ν = Numerical Results he first example we consider is the unit square. We suppose the left side is fixed, all other boundaries are free. here is a constant body force acting in vertical direction. We assume that the material is homogenous, isotropic, linear elastic. We calculate stresses σ h and displacements u h using the method and elements described above. We start from a coarse mesh, which is refined adaptively. As an error estimator for the refinement we use a Zienkiewicz-Zhou type estimator. or igure 7, the material parameters are given by Young s modulus E = 1 and the Poisson ratio ν = 0.3. We plot the decrease of the L 2 -error of the stresses, σ σ h L 2 (Ω), as the number of unknowns grows. We use both first and second order elements, and observe an optimal order of convergence. he same behavior can be seen for the non-conforming elements described in Section 5. he elements do not suffer from volume locking, as the material gets nearly incompressible. In igure 7, we plot the L 2 -error of the stresses for a Poisson ratio ν = Again, we see the optimal order convergence. As a second example we consider a U-domain, where the left top is fixed, and a surface load is acting on the right top. he domain is discetized by 26 elements. We use curved elements of order 5 to approximate the circular part of the U-domain. Also the finite element spaces Σ h, V h are 5th order. In igure 7 we plot the xx-component of the stress tensor. he last example is a cylindrical shell, where the radius of the cylinder is R = 0.5. he thickness of the shell is given as t = In igure 7, we used P 1 elements for the stresses σ and P 2 elements for

36 MIXED INIE ELEMENS OR ELASICIY 35 1 order 1, nc order 2, nc error stress e-04 1e e+06 1e+07 dofs igure 7. Unit square, nearly incompressible (ν = ) igure 8. U-domain, σ xx igure 9. Cylindrical shell, σ xy, σ P 1, u P 2 u. We plot the shear stress σ xy. In igure 7, we increase the order by one, i.e. P 2 for σ h and P 3 for u h.

37 36 JOACHIM SCHÖBERL AND ASRID SINWEL igure 10. Cylindrical shell, σ xy, σ P 2, u P 3 References [1] S. Adams and B. Cockburn. A mixed finite element method for elasticity in three dimensions. J. Sci. Comput., pages , [2] M. Ainsworth. A posteriori estimation for fully discrete hierarchical models of elliptic boundary value problems on thin domains. Numer. Math., 80: , [3] S. M. Alessandrini, D. N. Arnold, R. S. alk, and A. L. Madureira. Derivation and justification of plate models by variational methods. In M. ortin, editor, Plates and Shells (Quebec 1996), volume 21 of CRM Proceeding and Lecture Notes, pages AMS, Providence RI. [4]. Apel. Anisotropic finite elements: Local estimates and applications. Advances in Numerical Mathematics. eubner, Stuttgart, [5] D. N. Arnold and. Brezzi. Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO Math. Mod. Anal. Numer., 19:7 32, [6] D. N. Arnold,. Brezzi, and J. Douglas. PEERS: a new finite element for plane elasticity. Jap. J. Appl. Math., 1: , [7] D. N. Arnold, R. S. alk, and R. Winther. inite element exterior calculus, homological techniqeus, and applications. Acta Numer., 15:1 155, [8] D. N. Arnold, R. S. alk, and R. Winther. Mixed finite element methods for linear elasticity with weakly imposed symmetry. Math. Comp, [9] D.N. Arnold. Differential complexes and numerical stability. In Proceedings to the International Congress of Mathematicians (Beijing, 2002), pages , [10] D.N. Arnold, G. Awenou, and R. Winther. inite elements for symmetric tensors in three dimensions (submitted). [11] I. Babuška and L. Li. Hierarchic modeling of plates. Computers and Structures, 40(2): , [12] E. Bertóti. Dual-mixed p and hp finite element methods for elastic membrane problems. Int. J. Num. Meth. Eng., 53:3 29, [13] S. C. Brenner. Korn s inequalities for piecewise Hsp 1 vector fields. Mathematics of Computation, 73: , 2004.