Adaptive multigrid solution for mixed finite elements in 2-dimensional linear elasticity, t

Adaptive multigrid solution for mixedfiniteelements in2-dimensional linear elasticityChristian WienersInstitut f¨u r Computeranwendungen III,Universit¨a t StuttgartPfaffenwaldring27,70569Stuttgart,GermanyAbstractWe explain the implementation of adaptive multigrid methods for mixedfinite element in its hybrid formulation using interelement Lagrange multipliers and staticcondensation.We apply this method to problems in linear elasticity using the stabi-lized BDM elements introduced by S TENBERG.Therefore,a new type of grid transferand a new error indicator is introduced.Numerical examples on unstructured locallyrefined grids are given.AMS Subject Classification:65N30Key words:multigrid methods,mixedfinite elements,Lagange multipliers,linearelasticity1IntroductionIn the last years the application of mixedfinite elements in linear elasticity was discussed intensively:Mechanical conditions for stability and optimal convergence where investigated by S TEIN-R OLFES[14].There,the numerical stability condition for mixed elements is interpreted by mechanical terms and is reduced to an condition which can be verified locally.The implementation of mixed elements is described in detail by K LAAS-S CH¨ODER-M IEHE-S TEIN[12].Here,the stability of the approximation for nearly incom-pressible materials is demonstrated and the performance is compared with other discretizations.An appropriate error indicator is introduced by B RAESS-K LAAS-N IEKAMP-S TEIN-W OBSCHAL[5].The advantage of locally refined grids is shown by several exam-ples.The coupling of mixedfinite elements with boundary elements is considered byB RINK-C ARSTENSEN-S TEIN[10].There,the numerical solving process for the resulting linear systems is not considered. Here,we want to continue the discussion and we introduce an adaptive multigrid solver for mixed elements on locally refined grids.This solver is of optimal complexity:the solution process has complexity,where is the number of degrees of freedom.Thus,our method can be applied effectively to very large problems or to linearized problems in a nonlinear time-dependent process.2 C.Wieners Multigrid methods for mixed and nonconformingfinite elements are considered e.g.by A RBOGAST-C HEN,B RENNER,V ERFUEHRT[1,8,9,15],but the these algorithms has to be adapted for systems in linear elasticity.We explain in detail the modifications for the standard multigrid method which are necessary for the application to mixed elements in elasticity on unstructured locally refined grids.Especially,the construction of the grid transfer is different for mixed elements.Furthermore,we modify the error indicator pro-posed in[5]to make it more efficient.The local components of the indicator build an error bound in the energy norm.The paper is organized as follows.In thefirst section we introduce our notation.Then,we summarize the local multigrid algorithm adapted to unknowns bounded to the edges.In the third section we explain the computation of error bounds for the stabilized BDM dis-cretization.Finally,we give numerical examples in planar linear elasticity.All computa-tions are done with UG,aflexible toolbox for adaptive multigrid methods on unstructured grids[4],using thefinite element library described in[17].2Mixed and hybrid formulations in linear elasticityIn linear elasticity,we want to compute a symmetric stress tensor such that the equilib-rium equationholds in a domain,where describes the body load.Let be a disjunct decomposition of the boundary.We requireonwhere describes the surface traction.Then,the solution will be inonThe stress tensor is coupled with the displacement vector by the constitutive equation where.The idea of interelement Lagrange multipliers is to relax the continuity requirements on .Furthermore,the symmetry and the boundary condition on will be claimed onlyMultigrid solution for mixedfinite elements in linear elasticity3 implicitly.Therefore,let be a polygonal resp.polyhedral domain approximating decomposed into elements and definefor allThen,if is continuous on every edge resp.side,of and is symmetric.Here,denotes the outer unit normal vector and is the approximation on on.The corresponding problem is equivalent tofind such that(3) under the constraints(2),(4)and(5)for all for all.Introducing Lagrange multipliers for the constraints,we get an equivalent saddle point formulation of the problem:find a saddle point of(6)i.e.for all.Since our problem is linear,the saddle point problem is equivalent to the variational equa-tion:find with(7) 3Local multigrid methodsThe standard multigrid method(see e.g.[11])can be applied to unstructured grids and local refinement,but it must be modified in order to avoid a deterioration of the complex-ity of the algorithm.In particular,the smoothing process must be performed only in the refined region.For linear conforming approximations this is investigated using different approaches e.g.by Y SERENTANT[18],B ANK-D UPONT-Y SERENTANT[2],B RAMBLE-P ASCIAK-X U[7],B RAMBLE-P ASCIAK-W ANG-X U[6].These algorithms has to be mod-ified for mixed elements.Here,we explain in detail the concept which was realized by B ASTIAN[3]and which forms the basis of UG.We use a multiplicative multigrid method with smoothing in a slightly enlarged refined region.Up to now,this is the only way of obtaining optimal complexity combined with robustness for a wide range of applications.4 C.Wieners3.1Grids and MultigridsA grid is a closed polygonal or polyhedral set and a decomposition into elements.For an element,the corresponding domain in,is denoted by and the boundary is denoted by.A grid is consistent,if is empty,a common node (corner),a common edge or a common side for all elements.Let be a domain.A multigrid is a sequence of consistent grids approximating.The set of elements is denoted by,. Elements of different grids will be distinguished.For,we assume that there exists a father element for all elements.In general,,but this property will be violated near curved boundaries.For,letbe the set of son elements of.The refinement of an element to can be of type regular,irregular or copy.The application of a refinement rule results in the gener-ation of elements of the corresponding type,e.g.application of an irregular refinement rule to results in being irregular elements.For copy elements we trivially have .Successive regular refinement of an element results in elements which satisfying a minimum angle condition independent of the number of refinements.Succes-sive irregular refinement,however,can decrease the interior angles arbitrarily;hence it is allowed only once.The copy elements are needed in order to cover the domain on all levels.Most of the copy elements can be omitted in thefinal implementation as will be explained below.In the following,we will speak of an element as refined if it is refined either regularly or irregularly(but not copy).Algorithms for refining and derefini ng grids of this type are described in detail in[13].3.2Geometrically based dataWe assume that a linear problem is given on and that we can define afinite dimen-sional solution approximating the continuous solution on for every grid approximat-ing.Therefore,vectors and matrices have to be defined on every grid.Here,we explain the data structure used for interelement Langrange multipliers in two dimension.Then, the unknowns are associated to the edges.The set of edges on level is denoted by. In the UG-context,an edge will be called point or interpolation point.The points on different levels will be distinguished.In general,father points cannot be defined for refined elements,but for all edges on copy elements there exists a well defined father edge.Furthermore,denotes the number of degrees of freedom associated to.resp.denotes the number of degrees of freedom per element resp.in.For the stabilized BDM-elements we consider triangular elements only where and.Let be a multigrid and letbe a vector for the grid on level,the restriction to an element andthe restriction to an interpolation point.On the grid of the top level,the discretized problemMultigrid solution for mixedfinite elements in linear elasticity5 is given by the global stiffness matrix and the global right hand side ,the global solution vector is denoted by.Afinite element discretization is constructed from element stiffness matrices and element right hand sides. The global stiffness matrix and the global right hand side vector are assembled from resp.and a modification due to boundary conditions for somepoints.For the lower levels,we need the stiffness matrix only, because in a multigrid algorithm,auxiliary problems are solved,where the right hand side is replaced by the defect and the solution vector is replaced by the correction vector.Example.The construction of the stiffness matrix for the stabilized BDM elements.The stress tensor on a triangle is approximated inwhere is a bubble function vanishing on,the Lagrange parameters are approximated infor every edgeThis leads to thefinite dimensional problemThe submatrix is a block matrix and can be eliminated in every ele-ment.Then,one has to solve the reduced system for the corresponding Schur complement only,which can be assembled elementwise.3.3Grid transferFor multigrid methods,the grid transfer is essential for the coupling of values on different levels.In general,the interpolation from level to level is given by matrices and can be constructed by local interpolation from a father element to a son element.For mixed elements we have to construct local interpolation matricesThen,the interpolated values on the edges will be different for the elements on the left and on the right side.Therefore,the global interpolation matrix is defined by6 C.Wieners where is a diagonal scaling matrix with. Furthermore,the grid transfer from afine level to a coarse level will be defined by the transposed matrix.Example.The construction of the interpolation a for the stabilized BDM elements. Given the Lagrange parameter on a coarse level,compute for every elementand the father elementandMultigrid solution for mixedfinite elements in linear elasticity7 Let be consistent and the defect.If for is changed,the new defect will change for points with.In order to compute the new defect,one needs for all points with.This motivates the following definition:the local grids are the smallest subsets of the surface grids,such that for all with,i.e.for someOn the base level we have.The local multigrid method will be defined such that the solution vector and the correction are always consistent.If the stiffness matrix and the right hand side are assembled on the local grids,they are not consistent.Thus,and for must not be used on local grids.Nevertheless,the defect on the surface grid can be constructed recursively from the local defects:for set for andfor.We do not need the values for,they will be set to zero in the smoothing step.Therefore,the restriction of must not change for such that .For a simple notation,we set if is a point on a copy element and if for a refined element.Then,the set of points where can be changed get.3.5Smoother and solverWe formulate the multigrid method for the surface grids,but all steps will be defined such that only points on the local grids are used.The basic step on every level is the smoothing.S set forsolve,set for,In the examples considered below,we use an incomplete block-LU decomposition ,where,respect the sparsity pattern of.A multigrid cycle is defined recursively and combines the grid transfer and several smooth-ing steps.8 C.Wieners MCifcall S until(coarse grid solver)elsefor call S(pre-smoothing)(restriction)forfor call MC(coarse grid correction)(prolongation)for call S(post-smoothing)In our examples we use a V-cycle()with one pre-and one post-smoothing step (and).For symmetric problems,the multigrid cycle can used as preconditioner of the conjugated gradient method.CG,,repeatcall MCuntil4Global error bounds and local error indicators for sta-bilized BDM-elementsLet be the exact solution of the problem,an approximation for the solution and resp.an approximation for stress tensor resp.the strain tensor in which satisfy the boundary condition.Then,we have the error bound in the energy normfor some constant(cf.[16]).In our examples,the body load is constant on every element.Thus can be fulfilled exactly.In addition,we assume that the boundary conditions are piecewise linear and that they can be approximate exactly. Using BDM elements,only conforming approximations for stress and stain tensor are available.In order to use this error bound as an error indicator for refinement,we define a conforming interpolation.Therefore,let be the set of all nodes and all midpoints of edges of the grid.Then,will be a piecewise quadraticMultigrid solution for mixedfinite elements in linear elasticity9 function represented by its values at the nodal points.The interpolation points for the Lagange parameter are the two Gaußpoints on every edge.They will be denoted by. The computation of consists of two steps:in thefirst step in every element a quadratic approximation is defined byHere,denotes the cubic bubble function in the component.In the second step,where the element contributionsare used as local indicators:all element whereare marked for refinement.In practice,for gives reasonable results.5Numerical examplesThe efficiency of our algorithm is demonstrated on the problems given in[5].As thefirst example,we consider Cook’s mebrane problem.conv,,,fixed onsurface traction onYoung modulus,Possion ratio10 C.Wienersmixed(Stenberg)nonconforming(Falk)conforming(P2)245122.76 4422722.626 6440222.618 8384922.608 105it1627.402325622.78443409622.61763it1622.1893617422.5375676322.60476154622.60996306222.610115635022.610The second example is a plane strain problem with Young Modulus and the Poisson ratio.We consider a punctured disc with a constant surface load on the top and on the bottom.Due to the symmetry of the problem we compute only a quarter of the domain.Young ModulusPoisson ratiosurface loadstress tensor,full refinement local refinement(Stenberg)P1Falk Stenbergstep it1600643-6.742563-7.9510243-9.5640963-11.15163843-12.35655363-13.0step16-07.4 1464-10.9 35218-13.3 55804-13.94 772322-13.938Multigrid solution for mixedfinite elements in linear elasticity11 References[1]T.A RBOGAST AND Z.C HEN,On the implementation of mixed methods as non-conforming methods for second-order elliptic problems,p.,64(1995), pp.943–972.[2]R.E.B ANK,T.F.D UPONT,AND H.Y SERENTANT,The hierarchical basis multi-grid method,Numer.Math.,52(1988),pp.427–458.[3]P.B ASTIAN,Parallele adaptive Mehrgitterverfahren,Teubner Skripten zur Nu-merik,Teubner-Verlag,1996.[4]P.B ASTIAN,K.B IRKEN,K.J OHANNSEN,S.L ANG,N.N EUSS,H.R ENTZ-R EICHERT,AND C.W IENERS,UG–aflexible software toolbox for solving partial differential equations,Computing and Visualization in Science,1(1997),pp.1–30.[5]D.B RAESS,O.K LAAS,R.N IEKAMP,E.S TEIN,AND F.W OBSCHAL,Error in-dicators for mixedfinite elements in2-dimensional linear elasticity,Comp.Methods Appl.Mech.Engrg.,127(1995),pp.345–356.[6]J.B RAMBLE,J.P ASCIAK,J.W ANG,AND J.X U,Convergence estimates for mult-grid algorithms without regularity assumptions,p.,57(1991),pp.23–45.[7]J.H.B RAMBLE,J.E.P ASCIAK,AND J.X U,Parallel multilevel preconditioners,p.,55(1990),pp.1–22.[8]S.C.B RENNER,A multilevel algorithm for the lowest-order raviart-thomas mixedtriangularfinite element method,SIAM J.Numer.Anal.,29(1992),pp.647–678.[9]12 C.Wieners [15]R.V ERF¨UHRT,A multilevel algorithm for mixed problems,SIAM J.Numer.Anal.,21(1984),pp.264–271.[16]C.W IENERS,Adaptive multigrid solution for mixedfinite elements in2-dimensionallinear elasticity,tech.rep.,Universit¨a t Stuttgart,SFB404Preprint97/13,1997. 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