c 2004 Society for Industrial and Applied Mathematics

Transcription

1 SIAM J NUMER ANAL Vol 4, No, pp c 004 Society for Industrial and Applied Matematics LEAST-SQUARES METHODS FOR LINEAR ELASTICITY ZHIQIANG CAI AND GERHARD STARKE Abstract Tis paper develops least-squares metods for te solution of linear elastic problems in bot two and tree dimensions Our main approac is defined by simply applying te L norm least-squares principle to a stress-displacement system: te constitutive and te equilibrium equations It is sown tat te omogeneous least-squares functional is elliptic and continuous in te H(div; Ω) d H (Ω) d norm Tis immediately implies optimal error estimates for finite element subspaces of H(div; Ω) d H (Ω) d It admits optimal multigrid solution metods as well if Raviart Tomas finite element spaces are used to approximate te stress tensor Our metod does not degrade wen te material properties approac te incompressible limit Least-squares metods tat impose boundary conditions weakly and use an inverse norm are also considered Numerical results for a bencmark test problem of planar elasticity are included in order to illustrate te robustness of our metod in te incompressible limit Key words least-squares metod, mixed finite element metod, linear elasticity, incompressible limit AMS subject classifications 65M60, 65M5 DOI 037/S Introduction Te primitive pysical equations for linear elastic problems are te constitutive equation, wic expresses a relation between te stress and strain tensors, and te equilibrium equation Tis first-order partial differential system is called te stress-displacement formulation Substituting te stress into te equilibrium equation leads to a second-order elliptic partial differential system called te pure displacement formulation However, te stress-displacement formulation is preferable to te pure displacement formulation for some important practical problems, eg, modeling of nearly incompressible or incompressible materials and modeling of plastic materials were te elimination of te stress tensor is difficult In addition, te stress is usually a pysical quantity of primary interest It can be obtained in te pure displacement metod by differentiating displacement, but tis degrades te order of te approximation A mixed finite element metod is based on te weak form of te stressdisplacement formulation, and it requires a stable combination of finite element spaces to approximate tese variables Unlike mixed metods for second-order scalar elliptic boundary value problems, stress-displacement finite elements are extremely difficult to construct Tis is caused by te symmetry constraint of te stress tensor Recently, Arnold and Winter in [3] constructed a family of stable conforming elements in two dimensions on a triangular tessellation Teir simplest element as stress and 3 displacement degrees of freedom per triangle Te local degrees of freedom are reduced to for te stress and 3 for te displacement for a stable nonconforming element in [4] For previous work on mixed metods for linear elasticity, see [3] and Received by te editors November 0, 00; accepted for publication (in revised form) October 9, 003; publised electronically June 4, 004 ttp://wwwsiamorg/journals/sinum/4-/4835tml Department of Matematics, Purdue University, 50 N University Street, West Lafayette, IN (zcai@matpurdueedu) Tis autor s work was sponsored in part by te National Science Foundation under grants INT and INT and by Korea Researc Foundation under grant KRF C0004 Institut für Angewandte Matematik, Universität Hannover, Welfengarten, 3067 Hannover, Germany (starke@ifamuni-annoverde) 86

2 LEAST-SQUARES METHODS FOR LINEAR ELASTICITY 87 references terein Like scalar elliptic problems, mixed metods lead to saddle-point problems Many solution metods tat work well for symmetric and positive definite problems cannot be applied directly Altoug substantial progress in solution metods for saddle-point problems as been acieved, tese problems may still be difficult and expensive to solve In recent years tere as been increasing interest in te use of least-squares principles for numerical approximations of partial differential equations and systems (see, eg, te survey paper [6], te monograp [8], and references terein) Teir advantages over te usual mixed finite element discretizations include tat te coice of finite element spaces is not subject to te stability condition (see, eg, [9]), tat te resulting algebraic equations can be solved efficiently by standard multigrid metods or preconditioned by well-known tecniques, and tat te value of a least-squares functional provides a free, sarp, and practical a posteriori error indicator wic can be used efficiently in a local refinement process For linear elasticity, in particular, least-squares metods ave an additional edge over mixed metods in tat te known stable mixed elements are very limited and tey ave a large number of degrees of freedom In [], Cai, Manteuffel, McCormick, and Parter proposed a two-stage least-squares approac tat first solves for te displacement gradient and ten solves for te displacement itself (if desired) Pysical quantities suc as te strain, te stress, and te rotation are ten simple linear combinations of te displacement gradient At te first stage, it as four (nine) variables in two (tree) dimensions, compared to five (nine) variables for te stress-displacement formulation One drawback of tis approac is its requirement of sufficient smootness on te original problem if using standard continuous finite element approximations Anoter approac was proposed by engineers in [9] based on a displacement-stress-rotation formulation; it as te same drawback as tat of [] In addition, it introduces extra variables (te rotation): one (tree) variable in two (tree) dimensions For oter least-squares approaces in te engineering literature in solid mecanics, see references in [9] In contrast to tese approaces, our aim is to develop a least-squares approac tat does not ave te above-mentioned drawbacks, and tat computes te stress and te displacement directly Tus it would be easier to extend tis metod to applications suc as nonlinear elasticity, plasticity, etc Te stress components are pysical quantities of primary interest in many practical applications including coupling of elastic deformation wit fluid flow models Te metod to be developed in tis paper is based on te primitive pysical equations of linear elasticity: te stress-displacement formulation, witout introducing any new variables or any new equations Applying te L norm least-squares principle to tis first-order system wit an appropriately scaled constitutive equation, we develop a least-squares formulation for linear elasticity It is sown tat te omogeneous least-squares functional is elliptic and continuous in te H(div; Ω) norm for te stress and in te H norm for te displacement uniformly wit respect to material constants Tis immediately implies optimal error estimates for finite element subspaces of H(div; Ω) d H (Ω) d It also admits optimal multigrid solution metods if Raviart Tomas finite element spaces (see, eg, [9]) are used to approximate te stress tensor Bot discretization accuracy and multigrid convergence rate of te metod do not degrade wen te material properties approac te incompressible limit As usual, te evaluation of te least-squares functional on eac element is a practical and sarp a posteriori error indicator for adaptive mes refinements Te practical performance of te resulting

3 88 ZHIQIANG CAI AND GERHARD STARKE adaptive strategy will be tested numerically for a common bencmark problem of linear elasticity in te final section of tis paper Te metod ere is closely related to our previous work in [, 0] Te main difference is te scale in te constitutive equation Te omogeneous least-squares functionals in [, 0] are equivalent to te H(div; Ω) norm for te stress and te energy norm for te displacement Tis means tat te least-squares variational problems in [, 0] do not apply for incompressible materials and require effective discretizations and efficient solvers for te pure displacement problem wen materials are nearly incompressible Tese tasks remain difficult and expensive altoug some progress as been acieved (see, eg, [8] and references terein) For completeness, we study an inverse norm least-squares functional and sow tat its omogeneous form is equivalent to te L (Ω) d d H (Ω) d norm for te stress and te displacement Tis functional can be used to develop a discrete inverse norm least-squares metod (see, eg, [7]) For some applications, it is convenient to impose boundary conditions weakly by adding boundary functionals Suc a functional is also studied in tis paper See [] for ow to use tese types of functionals to develop a computationally feasible numerical metod An outline of te paper is as follows Te stress-displacement formulation for te linear elastic problem is introduced in section, along wit some notation, te pure displacement formulation, and some regularity estimates Section 3 develops te least-squares functionals based on te stress-displacement formulation and establises teir ellipticity and continuity Section 4 discusses finite element approximations Section 5 studies a least-squares functional wit boundary terms tat enforces boundary conditions weakly Finally, numerical results for a bencmark test problem of linear elasticity are presented in section 6 Notation We use te standard notation and definitions for te Sobolev spaces H s (Ω) d and H s ( Ω) d for s 0; te standard associated inner products are denoted by (, ) s,ω and (, ) s, Ω, and teir respective norms are denoted by s,ω and s, Ω (We suppress te superscript d because teir dependence on dimension will be clear by context We also omit te subscript Ω from te inner product and norm designation wen tere is no risk of confusion) For s =0,H s (Ω) d coincides wit L (Ω) d In tis case, te inner product and norm will be denoted by and (, ), respectively Set HD (Ω) := {q H (Ω) : q = 0 on Γ D } We use H D (Ω) and H ( Ω) to denote te dual of HD (Ω) and H ( Ω) wit norms defined by φ,d = (φ, ψ) (φ, ψ) sup and φ /, Ω = sup 0 ψ HD (Ω) ψ 0 ψ H ψ /, Ω ( Ω) Denote te product space H D (Ω)d = d Set wic is a Hilbert space under te norm i= H D H(div; Ω) = {v L (Ω) : v L (Ω)}, v H(div; Ω) = ( v + v ) (Ω) wit te standard product norm Linear elasticity and preliminaries Let Ω be a bounded, open, connected subset of R d (d = or 3) wit a Lipscitz continuous boundary Ω Denote n = (n,,n d ) t as te outward unit vector normal to te boundary We partition te

4 LEAST-SQUARES METHODS FOR LINEAR ELASTICITY 89 boundary of te domain Ω into two open subsets Γ D and Γ N suc tat Ω = Γ D Γ N and Γ D Γ N = For simplicity, we assume tat Γ D is not empty (ie, mes (Γ D ) 0) Our approaces proposed in tis paper can be easily extended to te pure traction problem (Γ D = ) by excluding te space of infinitesimal rigid motions Let f =(f,,f d ) t be a given body force defined on Ω Te linear elastic problem consists of finding a displacement field u =(u,,u d ) t and a stress tensor σ = ( σ ij tat satisfy te equilibrium equation )d d () and boundary conditions () d j= u = 0 on Γ D and σ ij x j + f i = 0 for i =,,d d σ ij n j = 0 on Γ N for i =,,d j= For simplicity, ere we assume tat te boundary conditions are omogeneous Denote ɛ(u) =(ɛ ij (u)) d d as te linearized strain tensor, were ɛ ij (u) = ( ui + u ) j x j x i Te constitutive law expresses a relation between te stress and te strain tensors: (3) σ = C ɛ(u) or ɛ(u) =A σ, were C and A are te elasticity and te compliance tensors of fourt order, respectively Denote by tr te trace operator tr ( ɛ(u) ) = ɛ (u)+ + ɛ dd (u) = u, were stands for te divergence operator For an isotropic elastic material, te elasticity tensor as te following simple expression: (4) C ɛ(u) =λ tr ( ɛ(u) ) δ +µɛ(u), were δ =(δ ij ) d d is te identity tensor, and positive constants λ and µ are te Lamé constants suc tat µ [µ,µ ] wit 0 <µ <µ and λ (0, ) Materials are said to be nearly incompressible or incompressible wen λ is very large or infinite, respectively Note tat bot te stress and te strain tensors are symmetric Suc symmetry of te stress stems from te conservation of angular momentum For a second-order tensor τ =(τ ij ) d d, define its divergence and normal by τ = τ / x + + τ d / x d τ d / x + + τ dd / x d and n τ = n τ + + n d τ d n τ d + + n d τ dd respectively Tat is, te divergence and normal operators apply to eac row of te tensor Ten te stress-displacement system in (3), (), and () may be rewritten in te compact form { σ Cɛ(u) = 0 in Ω, (5) σ = f in Ω,

5 830 ZHIQIANG CAI AND GERHARD STARKE wit te boundary conditions (6) u = 0 on Γ D and n σ = 0 on Γ N Tere are tree approaces to treating tis first-order partial differential system One is to substitute te stress into te equilibrium equation to get te pure displacement formulation in (7) Numerical metods based on tis formulation are not desirable for accurate approximations of te stress and for some important practical problems suc as te modeling of nearly incompressible or incompressible or plastic materials Anoter approac is to find te unique saddle point (σ, u) HN S (div; Ω)d L (Ω) d of te Hellinger Reissner functional J (τ, v) = (A τ, τ )+( τ + f, v) Here HN S (div; Ω)d denotes te space of square-integrable symmetric tensors wit square-integrable divergence and omogeneous normal on Γ N Equivalently, (σ, u) satisfies te following weak formulation: (A σ, τ )+(u, τ )+( σ, v) =( f, v) (τ, v) H S N (div; Ω) d L (Ω) d Numerical metods based on tis formulation require a stable combination of finite element spaces to approximate te stress and te displacement Known stable mixed elements are very limited and ave a large number of degrees of freedom In addition, te resulting indefinite algebraic system is still difficult and expensive to solve In tis paper, we study te tird approac based on te least-squares principle tat automatically stabilizes te stress-displacement system (see section 3) We complete tis section by deriving te pure displacement formulation and describing some regularity estimates To tis end, eliminating te stress in system (5) (6) yields te pure displacement formulation wic satisfies te following secondorder elliptic partial differential system: µ ɛ(u)+λ ( u) = f in Ω, (7) u = 0 on Γ D, n (µ ɛ(u)+λ ( u) δ) = 0 on Γ N, were stands for te gradient operator Te energy norm associated wit te above problem is defined as follows: (8) v = ( µ ɛ(v) + λ v ) By using Korn s inequality (see [4]), (9) v C ɛ(v) v H D(Ω) d, te energy norm is equivalent to te H norm for a fixed λ In tis paper, we use C wit or witout subscripts to denote a generic positive constant, possibly different at different occurrences, wic is independent of te Lamé constant λ and te mes size introduced in section 4 but may depend on te domain Ω Note tat one could scale te variables and te rigt-and side accordingly so tat µ is equal to one We will frequently use te term uniform in reference to a relation to mean tat it olds independent of λ and

6 LEAST-SQUARES METHODS FOR LINEAR ELASTICITY 83 Te weak form of boundary value problem (7) as a unique solution u HD (Ω)d for any f H D (Ω)d (see [4]) Moreover, te solution satisfies te following H - regularity estimate (see, eg, [8, 4]): (0) u + λ u C f,d Furtermore, if te domain Ω is convex or its boundary is C, and if eiter Γ D or Γ N is empty, ten te H -regularity estimate () u + λ u C f olds Bot te regularity estimates in (0) and () suggest tat te divergence of te displacement as a different scale from te displacement itself for large λ 3 Least-squares variational formulation In tis section, we first discuss an appropriate stress-displacement formulation and ten consider te corresponding least-squares functionals based on suc a first-order system Our primary objective ere is to establis continuity and ellipticity of tese least-squares functionals in te appropriate Hilbert spaces It is convenient to view d d-matrices as d -vectors and vice versa For example, view (σ ij ) d d as (σ,,σ d ) t, were σ j =(σ j,,σ jd )istejt row of (σ ij ) d d for j =,,d Let b = { (, 0, 0, ) t d =, (, 0, 0, 0,, 0, 0, 0, ) t d =3, wic may be viewed as te d d identity matrix I d d or te identity tensor δ Tus, σ d t tr σ =tr(σ ij ) d d = σ ii = b t = b t σ i= σ t d By viewing a d d-matrix as a d -vector, we can ten write te fourt-order elasticity tensor as d d -matrix C = λ bb t +µi It is clear tat C is symmetric and tat C is positive definite for finite λ Te compliance tensor as te form A = ( ) λ (3) I µ dλ +µ bbt Note tat A = C for finite λ Wen te λ approaces, te elasticity tensor blows up and te compliance tensor tends to (3) µ (I d bbt ) µ dev, wic is not invertible For any tensor τ, devτ = τ d (trτ )δ is te deviatoric part of τ Hence, for nearly incompressible or incompressible materials, it is preferable to use te following strain and stress relation: ɛ(u) =A σ = ( ) λ (33) σ (tr σ)δ µ dλ +µ

7 83 ZHIQIANG CAI AND GERHARD STARKE Now, te first-order system for te stress and te displacement of linear elasticity is as follows: { A σ ɛ(u) = 0 in Ω, (34) σ + f = 0 in Ω wit boundary conditions (6), were A is given in (3) It is important to note tat te stress is symmetric; ie, (35) σ σ t = 0, were σ t denotes te transpose of σ as a d d-matrix One can impose tis symmetry condition eiter in te solution space (in a strong sense) or in te equation (in a weak sense) In [], we augment (35) wit te stress-displacement system so tat our least-squares metods ave freedom to treat it eiter strongly or weakly depending on discretization and solution metods In [0], we sow tat te symmetry constraint of te stress is enforced at te continuous level even witout te term σ σ t in te least-squares functionals Tis is because for any τ L (Ω) d d and any v H (Ω) d, we ave (36) τ τ t C A τ ɛ(v) Tus, A τ ɛ(v) = 0 implies tat τ is symmetric Terefore, we will apply te leastsquares principle to first-order system (34) witout augmenting (35) Inequality (36) follows from te symmetry of ɛ(v), (3), and te triangle inequality tat ( ) ( ) t τ τ t =µ µ τ ɛ(v) µ τ ɛ(v) =µ (A τ ɛ(v)) (A τ ɛ(v)) t 4µ A τ ɛ(v) Before defining least-squares functionals, let us first describe solution spaces If Γ N =, ten Ω udx = n u ds = 0, wic implies Ω tr σ dx =0 Ω Terefore, we are at liberty to impose suc a condition for te stress σ Let { H(div; Ω) d if Γ N, X = {τ H(div; Ω) d tr σ dx =0} oterwise, Ω and denote its subspace by Let X N = {τ X : n τ = 0 on Γ N } V B = X N H D(Ω) d For f L (Ω) d, we define te following least-squares functionals: (37) G (σ, u ; f) = A σ ɛ(u) + σ + f,d

8 LEAST-SQUARES METHODS FOR LINEAR ELASTICITY 833 and (38) G(σ, u ; f) = A σ ɛ(u) + σ + f for (σ, u) V B Least-squares variational problems for te stress-displacement of linear elasticity are ten to minimize te above least-squares functionals over V B In tis paper, we concentrate on te least-squares problem based on te L norm functional in (38): find (σ, u) V B suc tat (39) G(σ, u ; f) = inf G(τ, v ; f) (τ, v) V B Note tat te inverse norm functional in (37) can be used to develop a discrete inverse norm least-squares metod (see [7]) as well Remark 3 Since te minimum of te quadratic functional G(σ, u ; f) is zero, by (36) te symmetry of te stress tensor is guaranteed by te first term of te functional, ie, te constitutive equation Remark 3 Te least-squares functionals defined in (37) and (38) differ from tose in [, 0] mainly in te first term wit an extra weigt of C More precisely, te first term of te functionals in [, 0] is C σ C ɛ(u) Note tat C ɛ(u) = λ u +µ ɛ(u) Tis means tat te least-squares variational problems in [, 0] do not apply for incompressible materials and require effective discretizations and efficient solvers for te pure displacement problem wen materials are nearly incompressible Below we establis uniform continuity and ellipticity (ie, equivalence) of te omogeneous functionals G (τ, v; 0) and G(τ, v; 0) in terms of te respective functionals M (τ, v) and M(τ, v) defined on V B by and M (τ, v) = ɛ(v) + τ M(τ, v) = ɛ(v) + τ + τ To do so, we need te following fundamental inequality on te trace of X N : ( ) (30) tr τ C (A τ, τ )+ τ,d τ X N, were C is a positive constant independent of λ Tis inequality sould be a classic result But te only references tat we know for its proof are [] for two dimensions and Diriclet boundary conditions (ie, d = and Γ N = ) and [] for bot two and tree dimensions and general boundary conditions Note tat (A τ, τ )= µ ( τ ) λ tr τ dλ +µ = µ dev τ (3) + d(dλ +µ) tr τ, were dev τ and tr τ are te respective deviatoric and volumetric parts of τ It is ten obvious tat te divergence term in (30) is necessary to bound te L norm of te trace From te definition of te inverse norm and te Caucy Scwarz inequality, we ave tat (3) τ,d τ

9 834 ZHIQIANG CAI AND GERHARD STARKE By (30), (3), and (3) it is easy to see tat τ a ( (A τ, τ )+ τ ),D is equivalent to te L norm; ie, tere exists a positive constant C independent of λ suc tat (33) C τ τ a C τ τ X N Teorem 3 Te omogeneous functionals G (τ, v; 0) and G(τ, v; 0) are uniformly equivalent to te functionals M (τ, v) and M(τ, v), respectively; ie, tere exist positive constants C and C, independent of λ, suc tat (34) M (τ, v) G (τ, v ; 0) C M (τ, v) C and (35) M(τ, v) G(τ, v ; 0) C M(τ, v) C old for all (τ, v) V B Proof It follows from (3) tat ( ) ( A τ = τ λ ( ) ) µ dλ +µ tr τ λ + d tr τ dλ +µ ( ) ( ) ( ) = τ λ (dλ +4µ) tr τ τ µ (dλ +µ) µ Tus, A τ is bounded above by τ in te L norm: (36) A τ τ µ Te upper bounds in bot (34) and (35) follow easily from te triangle inequality, (36), and (3) To sow te validity of te lower bound in (34), we first prove tat τ in te L norm is bounded above by te omogeneous functional: (37) τ CG (τ, v ; 0) (τ, v) V B To tis end, by te triangle inequality and (36) we ave (38) ɛ(v) ɛ(v) Aτ + A τ ɛ(v) Aτ + τ µ Since ɛ(v) = ( v +( v)t ) is symmetric, ten integration by parts; te triangle, Caucy Scwarz, and Korn inequalities; and (36) give ( ( (τ, ɛ(v)) = τ τ t (τ, v), v) = τ τ t ( τ, v)+, v) ( τ,d + τ τ t ) v C ( τ,d + A τ ɛ(v) ) ɛ(v),

10 LEAST-SQUARES METHODS FOR LINEAR ELASTICITY 835 wic, togeter wit (38), implies tat (39) (τ, ɛ(v)) CG (τ, v ; 0)+CG (τ, v ; 0) τ, were G (τ, v ; 0) denotes te square root of G (τ, v ; 0) Now, it follows from te Caucy Scwarz inequality, (39), and (33) tat Hence, (A τ, τ )=(A τ ɛ(v), τ )+(ɛ(v), τ ) Aτ ɛ(v) τ + CG (τ, v ; 0)+CG (τ, v ; 0) τ ( ) CG (τ, v ; 0)+CG (τ, v ; 0) (A τ, τ )+ τ,d CG (τ, v ; 0)+CG (τ, v ; 0) (A τ, τ ) (A τ, τ ) CG (τ, v ; 0), wic, togeter wit (33), implies te validity of (37) Wit (38) and (37), it is ten easy to see tat ɛ(v) is also bounded above by te omogeneous functional G (τ, v ; 0) Tis completes te proof of te lower bound in (34) Since G (τ, v ; 0) G(τ, v ; 0) and τ G(τ, v ; 0), ten te lower bound in (35) follows from (34) terefore completed Te proof of te teorem is 4 Finite element approximation We approximate te minimum of te leastsquares functional G(σ, u; f) in (39) using a Rayleig Ritz type finite element metod For convenience, we use two-dimensional terminology (d = ) Assuming tat te domain Ω is a polygon, let T be a triangulation of Ω wit triangular elements of size O() tat is regular (see [3]) We restrict ourselves to triangular elements for convenience because extension to eiter rectangular or a combination of triangular and rectangular elements is straigtforward Since te omogeneous functional G(σ, u ; 0) is equivalent to te H(div; Ω) norm for te stress and te H norm for te displacement by Teorem 3 and Korn s inequality (9), it is ten natural to approximate te stress (eac row) by te standard H(div; Ω) conforming Raviart Tomas space of order k (see [0]) and te standard (conforming) continuous piecewise polynomials of degree k + for te displacement: (4) (4) Σ k = {τ X N : τ K RT k (K) K T } X N, V k = {v C 0 (Ω) : v K P k (K) K T, v = 0 on Γ D } H D(Ω), were RT k (K) is local Raviart Tomas space of order k defined by ( ) RT k (K) =P k (K) x + P k (K), and P k (K) is te space of polynomials of degree k on triangle K Tese spaces ave te following approximation properties: if k 0 is an integer and l (0, k+ ], ten x (43) inf σ τ H(div; Ω) C l ( σ l + σ l ) τ Σ k

11 836 ZHIQIANG CAI AND GERHARD STARKE for σ H l (Ω) X N wit σ H l (Ω), and (44) inf u V k+ u v C l u l+ for u H l+ (Ω) HD (Ω) Te finite element approximation for minimizing G(σ, u; f) in (39) on V B becomes: find (σ, u ) Σ k V k+ suc tat (45) G(σ, u ; f) = min (τ, v) Σ k k+ V G(τ, v; f) By Teorem 3, (9), and te fact tat Σ k V k+ is a subspace of V B, we conclude tat (45) as a unique solution and is equivalent to te weak form: find (σ, u ) Σ k V k+ suc tat (46) F(σ, u ; τ, v) =( f, τ ) (τ, v) Σ k V k+, were te bilinear form F( ; ) as te form of F(σ, u ; τ, v) =(A σ ɛ(u ), A τ ɛ(v))+( σ, τ ) Moreover, te error (σ σ, u u ) satisfies te ortogonality property (47) F(σ σ, u u ; τ, v) =0 (τ, v) Σ k V k+ Teorem 4 Assume tat te solution, (σ, u), of (39) is in H l (Ω) H l+ (Ω) and tat te divergence of te stress, σ, isinh l (Ω) Letk + be te smallest integer greater tan or equal to l Ten wit (σ, u ) Σ k V k+, te following error estimate olds: (48) σ σ H(div; Ω) + u u C l ( σ l + σ l + u l+ ) Proof Te proof is a simple consequence of te ortogonality property (47) and te approximation properties (43) and (44) of te finite element spaces Σ k V k+ Teorem 3 indicates tat te bilinear form F( ; ) is elliptic and continuous wit respect to te H(div; Ω) norm for te stress and te H norm for te displacement It is ten well known tat multigrid metods applied to te resulting discrete system (46) are optimally convergent (see, eg, [6,, 7, 3]) It is obvious tat te finite element approximation in (45) does not preserve te symmetry of te stress But te finite element approximation of te stress is approximately symmetric Moreover, one can obtain symmetric stress approximation wit te same accuracy as σ by simply computing (49) σ = ( σ + σ t ) It sould also be noted tat many mixed finite element approaces commonly used produce stress approximations wic do not satisfy symmetry exactly (cf [9, sect VII]) Corollary 4 Under te assumptions of Teorem 4, we ave tat (40) σ σ t C l ( σ l + σ l + u l+ )

12 LEAST-SQUARES METHODS FOR LINEAR ELASTICITY 837 and tat (4) σ σ C l ( σ l + σ l + u l+ ) Proof Since te stress σ is symmetric, by te triangle inequality we ave tat σ σ t = (σ σ ) (σ σ ) t σ σ and tat σ σ = (σ σ )+ (σ σ ) t σ σ Now, (40) and (4) follow from te error bound in (48) Neverteless, tere may be a possible reluctance in te engineering community to accept nonsymmetric stress approximation since te symmetry is due to te conservation of angular momentum In order to directly preserve te symmetry of finite element approximations to te stress tensor, one may enforce te symmetry constraint in te finite element approximation space To tis end, let X s N denote te symmetric stress space, X s N = {τ X N τ t = τ in Ω} A simple and obvious coice is to use continuous piecewise polynomials of degree k for eac component of te symmetric stress: (4) Σ k,s = {τ C0 (Ω) X s N : τ K P k (K) K T } X s N Tis space as te following approximation property: if k is an integer and l (0, k], ten (43) inf τ Σ k,s σ τ H(div; Ω) C l σ l+ for σ H l+ (Ω) X N Now, te least-squares finite element approximation problem is to minimize G over Σ k,s V k: find (σ, u ) Σ k,s V k suc tat (44) G(σ, u ; f) = inf (τ, v) Σ k,s V k G(τ, v ; f) It is easy to see tat (44) as a unique solution (σ, u ), tat σ is symmetric, and tat σ as te following error bound: (45) σ σ H(div; Ω) + u u C l ( σ l+ + u l+ ) if te solution, (σ, u), of (39) is in H l+ (Ω) H l+ (Ω) Note tat tis estimate is not optimal in te regularity of te displacement Neverteless, te nodal elements Σ k,s ave many fewer local average degrees of freedom tan te Raviart Tomas elements Σ k Developing a better finite element space of te symmetric stress will be a topic of our furter study

13 838 ZHIQIANG CAI AND GERHARD STARKE 5 Weakly imposed boundary conditions In previous sections, boundary conditions are imposed in te solution space Tis leads to least-squares finite element approximations tat are muc more accurate on te boundary tan in te interior of te domain In te context of te least-squares metod, it is natural to treat boundary conditions weakly troug boundary functionals For many applications, tis is also convenient In tis section, we study a least-squares functional wit boundary terms minimized over a solution space free of boundary conditions We focus on establising continuity and ellipticity of tis functional ere See [] for te development of computable finite element approximations and te corresponding iterative solvers based on tis functional Assume te following nonomogeneous boundary conditions: (5) u = g on Γ D and n σ = on Γ N Let V = X H (Ω) d, and for g H / (Γ D ) and H / (Γ N ) define te least-squares functional as follows: (5) G(σ, u; f, g, ) = A σ ɛ(u) + σ + f + u g, Γ D + n σ, Γ N for (σ, u) V Te least-squares variational problem is ten to minimize G over V: find (σ, u) V suc tat (53) G(σ, u ; f, g, ) = inf G(τ, v ; f, g, ) (τ, v) V To establis te continuity and ellipticity of te omogeneous least-squares functional G(u, σ ; 0, 0, 0) inv, we need te trace inequalities (see [5]) u, Ω u u H (Ω), n v, Ω v H(div; Ω) v H(div; Ω) and te generalized Korn inequality (54) v C ( ɛ(v) + v 0, ΓD ) v H (Ω) d Teorem 5 Te omogeneous functional G(τ, v ; 0, 0, 0) is uniformly equivalent to te functional M(v, τ ); ie, tere exists a positive constant C independent of λ suc tat (55) C M(τ, v) G(τ, v ; 0, 0, 0) CM(τ, v) olds for all (τ, v) V Proof Te upper bound in (55) follows easily from te triangle inequality, (36), and trace inequalities Te proof of te lower bound in (55) is te same as tat for Teorem 3 except te proof on te upper bound of (τ, ɛ(v)) Tis is because v

14 LEAST-SQUARES METHODS FOR LINEAR ELASTICITY 839 and n τ do not satisfy any boundary conditions and our new functional as extra boundary terms Terefore, it suffices to sow tat (56) (τ, ɛ(v)) C G(τ, v ; 0, 0, 0)+C G(τ, v ; 0, 0, 0) τ To tis end, te triangle, Caucy Scwarz, and generalized Korn inequalities give v (n τ ) ds Γ = v (n τ ) ds + v (n τ ) ds D Γ N Ω v, Γ D n τ, Γ D + v, Γ N n τ, Γ N v, Γ τ D H(div; Ω) + v n τ, Γ N Now, it follows from te symmetry of ɛ(v); integration by parts; te triangle, Caucy Scwarz, and generalized Korn inequalities; and (36) tat ( (τ, ɛ(v)) = τ τ t (τ, v), v) = ( τ, v) v (n τ ) ( τ τ t ds +, v) Ω ( τ + τ τ t ) v + v, Γ τ D H(div; Ω) + v n τ, Γ N C ( ) τ + A τ ɛ(v) + n τ, Γ N ( ɛ(v) + τ )+ v, Γ τ, D wic, togeter wit (38), implies (56) and, ence, te teorem 6 Numerical results In tis section, numerical results for a bencmark problem of linear elasticity taken from [] are presented Te problem to be considered is given by a quadratic membrane of elastic isotropic material wit a circular ole in te center Traction forces act on te upper and lower edges of te strip Because of te symmetry of te domain, it suffices to discretize only a fourt of te total geometry Te computational domain is ten given by Ω={x R :0<x < 0, 0 <x < 0,x + x > } (see Figure 6) Te boundary conditions on te top edge of te computational domain (x = 0, 0 <x < 0) are set to σ n =45, te boundary conditions on te bottom (x =0,<x < 0) are set to (σ,σ ) n =0,u = 0 (symmetry condition), and, finally, te boundary conditions on te left (x =0,<x < 0) are given by u =0,(σ,σ ) n = 0 (symmetry condition) Te material parameters are E = for Young s modulus and ν =09 for Poisson s ratio, and teir relation wit te Lamé constants is given by Eν E λ = and µ = ( + ν)( ν) ( + ν) Obviously, te definition of te functional in (38) implies G(σ, u ; f) = ( A σ ɛ(u ) 0,K + σ + f 0,K) =: G K (σ, u ; f) K T K T

15 840 ZHIQIANG CAI AND GERHARD STARKE Fig 6 Computational domain and boundary conditions Table 6 Adaptive finite element approximation (k =, ν =09) # elements dim Σ dim V Functional (σ ) (, 0) l = e l = e- 56 l = e l = e l = e l = e l = e l = e l = e Due to te equivalence (35), te local contributions G K (σ, u ; f) to te leastsquares functional constitute an a posteriori error estimator to be used for adaptive refinement (cf [5]) Te results in Table 6 are computed on a sequence of adaptively refined meses based on tis error estimator In eac refinement step tose triangles wit te largest values of G K (σ, u ; f) (rougly 5 percent) were refined regularly (by dividing eac into four congruent subtriangles) Te Raviart Tomas spaces of order one for te stress approximation are coupled wit standard quadratic conforming elements for te displacement (Σ V in te terminology of section 4) Table 6 provides a strong indication tat te minimum of te functional is inversely proportional to te square of te number of degrees of freedom: F (σ, u ) (dim Σ + dim V ) Tis is te optimal asymptotic convergence rate acievable wit piecewise quadratic finite elements Of particular interest in tis example is te stress component σ at te point (, 0) Te size of tis stress component is responsible for failure of te

16 LEAST-SQUARES METHODS FOR LINEAR ELASTICITY 84 Fig 6 Initial triangulation and result after tree adaptive refinement steps functional ν = ν = # degrees of freedom Fig 63 Adaptive finite element approximation for k =(ν =09, 05) material at tis point For ν =09 te value of σ (, 0)=38873 is given in [] for a reference solution computed by a polynomial approximation of ig degree Te corresponding column in Table 6 sows te convergence of te solutions obtained wit our least-squares approac to tat reference value as te mes is refined Te initial triangulation and te result of tree adaptive refinement steps are sown in Figure 6 Te robustness wit respect to te incompressible limit can be seen in te doubly logaritmic convergence graps in Figure 63 In addition to te numbers of Table 6, te results for te incompressible limit (ν =05) are sown in Figure 63

17 84 ZHIQIANG CAI AND GERHARD STARKE REFERENCES [] D N Arnold, J Douglas, and C P Gupta, A family of iger order mixed finite element metods for plane elasticity, Numer Mat, 45 (984), pp [] D N Arnold, R S Falk, and R Winter, Multigrid in H(div) and H(curl), Numer Mat, 85 (000), pp 97 7 [3] D N Arnold and R Winter, Mixed finite elements for elasticity, Numer Mat, 4 (00), pp [4] D N Arnold and R Winter, Nonconforming mixed elements for elasticity, Mat Models Metods Appl Sci, 3 (003), pp [5] M Berndt, T A Manteuffel, and S F McCormick, Local error estimates and adaptive refinement for first-order system least squares, Electron Trans Numer Anal, 6 (997), pp [6] P B Bocev and M D Gunzburger, Finite element metods of least-squares type, SIAM Rev, 40 (998), pp [7] J H Bramble, R D Lazarov, and J E Pasciak, A least-squares approac based on a discrete minus one inner product for first order systems, Mat Comp, 66 (997), pp [8] S C Brenner and L R Scott, Te Matematical Teory of Finite Element Metods, Springer-Verlag, New York, 994 [9] F Brezzi and M Fortin, Mixed and Hybrid Finite Element Metods, Springer-Verlag, New York, 99 [0] Z Cai, J Korsawe, and G Starke, A least squares mixed finite element metod for te stress-displacement formulation of linear elasticity: Error estimation and adaptive refinement, Numer Metods Partial Differential Equations, to appear [] Z Cai, T A Manteuffel, S F McCormick, and S V Parter, First-order system least squares (FOSLS) for planar linear elasticity: Pure traction problem, SIAM J Numer Anal, 35 (998), pp [] Z Cai and G Starke, First-order system least squares for te stress-displacement formulation: Linear elasticity, SIAM J Numer Anal, 4 (003), pp [3] P G Ciarlet, Te Finite Element Metod for Elliptic Problems, Nort Holland, Amsterdam, 978 [4] P G Ciarlet, Matematical Elasticity Volume I: Tree-Dimensional Elasticity, Nort Holland, Amsterdam, 988 [5] V Girault and P-A Raviart, Finite Element Metods for Navier-Stokes Equations, Springer-Verlag, New York, 986 [6] W Hackbusc, Multi-Grid Metods and Applications, Springer-Verlag, Berlin, 985 [7] R Hiptmair, Multigrid metod for Maxwell s equations, SIAM J Numer Anal, 36 (998), pp 04 5 [8] B Jiang, Te Least-Squares Finite Element Metod, Springer-Verlag, Berlin, 998 [9] B Jiang and J Wu, Te least-squares finite element metod in elasticity Part I: Plane stress or strain wit drilling degrees of freedom, Internat J Numer Metods Engrg, 53 (00), pp [0] P-A Raviart and J-M Tomas, A mixed finite element metod for -nd order elliptic problems, in Matematical Aspects of Finite Element Metods, Lecture Notes in Mat 606, I Galligani and E Magenes, eds, Springer-Verlag, New York, 977, pp 9 35 [] G Starke, Multilevel boundary functionals for least-squares mixed finite element metods, SIAM J Numer Anal, 36 (999), pp [] E Stein, P Wriggers, A Rieger, and M Scmidt, Bencmarks, in Error-Controlled Adaptive Finite Elements in Solid Mecanics, E Stein, ed, Jon Wiley and Sons, New York, 00, pp [3] P S Vassilevski and J Wang, Multilevel iterative metods for mixed finite element discretizations of elliptic problems, Numer Mat, 63 (99), pp