a locking free fem for cavitation computation in nearly incompressible nonlinear elasticity abstract a reduced selective integration finite element method coupled with a gradient flow method is introduced for the cavitation computation in nearly incompressible nonlinear elasticity numerical experiments show that for the iso parametric biquadratic finite element approximation of the displacement field the method is locking free and efficient for the nearly incompressible cavitation computation and has the same optimal convergence rates as in the compressible case for the elastic energy as well as the deformation and stress field if and only if the reduced integration is applied to the bulk energy term using no more than 2 gaussian quadrature nodes in each dimension

stagnation analysis of dgmres we study problems for which the iterative method dgmres for solving drazin inverse solution of singular linear systems of equations which makes no progress in its initial iterations particularly our tool for analysis is nonlinear system of equations the stagnation system that characterize this behavior for problems of dimension two and index one we can solve this system explicitly we partially extend this result to higher dimensions for a class of eigenvector matrices called extreme we also study the real matrices

locking free nonconforming finite elements for three dimensional elasticity problem abstract two locking free nonconforming finite elements are presented for three dimensional elasticity problem with pure displacement boundary condition convergence rate of the elements are uniformly optimal with respect to the energy norm and l 2 norm errors are o h 2 and o h 3 respectively lastly a numerical experiment is carried out which coincides with the theoretical analysis

overlapping schwarz methods for mixed linear elasticity and stokes problems overlapping schwarz preconditioners are introduced and studied for saddle point problems with a penalty term such as stokes equations and mixed formulations of linear elasticity these preconditioners are based on the solution of local saddle point problems on overlapping subdomains and the solution of a coarse saddle point problem numerical experiments show that these are parallel and scalable preconditioners since the rate of convergence of the preconditioned operator is independent of the mesh size h the number of subdomains n and the penalty parameter

theh p version of the finite element method for elliptic equations of order2m the problems of elliptic partial differential equations stemming from engineering problems are usually characterized by piecewise analytic data it has been shown in 3 4 5 that the solutions of the second order and fourth order equations belong to the spacesb 1 where the weighted sobolev norms of thek th derivatives are bounded bycd k l k l k l l 2 wherec andd are constants independent ofk in this case theh p version of the finite element method leads to an exponential rate of convergence measured in the energy norm 6 12 13 theh p version was implemented in the code probe1 18 and has been very successfully used in the industry we will discuss in this paper the generalization of these results for problems of order2m we will show also that the exponential rate can be achieved if the exact solution belongs to the spacesb 1 where the weighted sobolev norm of thek th derivatives is bounded bycd k l k l k l m 1 c andd are independent ofk in addition if the data is piecewise analytic then in fact the exact solution belongs to the spacesb m 1 problems of this type are related obviously to many engineering problems such as problems of plates and shells and are also important in connection with well known locking problems

the rate of convergence of gmres on a tridiagonal toeplitz linear system the generalized minimal residual method gmres is often used to solve a nonsymmetric linear system ax b but its convergence analysis is a rather difficult task in general a commonly used approach is to diagonalize a x x 1 and then separate the study of gmres convergence behavior into optimizing the condition number of x and a polynomial minimization problem over a s spectrum this artificial separation could greatly overestimate gmres residuals and likely yields error bounds that are too far from the actual ones on the other hand considering the effects of both a s spectrum and the conditioning of x at the same time poses a difficult challenge perhaps impossible to deal with in general but only possible for certain particular linear systems this paper will do so for a nonsymmetric tridiagonal toeplitz system sharp error bounds on and sometimes exact expressions for residuals are obtained these expressions and or bounds are in terms of the three parameters that define a and chebyshev polynomials of the first kind

the tortoise and the hare restart gmres when solving large nonsymmetric systems of linear equations with the restarted gmres algorithm one is inclined to select a relatively large restart parameter in the hope of mimicking the full gmres process surprisingly cases exist where small values of the restart parameter yield convergence in fewer iterations than larger values here two simple examples are presented where gmres 1 converges exactly in three iterations while gmres 2 stagnates one of these examples reveals that gmres 1 convergence can be extremely sensitive to small changes in the initial residual

a right inverse for the divergence operator in spaces of piecewise polynomials in the first part of this paper we study in detail the properties of the divergence operator acting on continuous piecewise polynomials on some fixed triangulation more specifically we characterize the range and prove the existence of a maximal right inverse whose norm grows at most algebraically with the degree of the piecewise polynomials in the last part of this paper we apply these results to thep version of the finite element method for a nearly incompressible material with homogeneous dirichlet boundary conditions we show that thep version maintains optimal convergence rates in the limit as the poisson ratio approaches 1 2 this fact eliminates the need for any reduced integration such as customarily used in connection with the more standardh version of the finite element method

enhanced assumed strain elements and locking in membrane problems abstract the assumptions which guarantee convergence of eas elements first introduced by simo and rifai can be reduced to two postulates specifically the stability condition refers to a relaxed stability of an associated hu washizu principle we investigate the problem of volume locking of membrane problems estimates of a certain type which for instance imply volume locking of the q1 element are fortunately not present with eas elements finally we verify an equivalence of the eas concept with a hellinger reissner method which was observed in numerical computations by andelfinger and ramm

local characterizations of block covers and their applications let g always be a finite group and h g and let f be a held of characteristic p not necessarily a splitting field for g where p is a fixed prime and p 1 igi let b e fg be an fg block with block idempotent e and let b e fh be an fh block with block idempotent e we can regard b as an f g x g module we say that b covers b or e covers

new conditions for non stagnation of minimal residual methods in the solution of large linear systems a condition guaranteeing that a minimal residual krylov subspace method makes some progress i e that it does not stagnate is that the symmetric part of the coefficient matrix be positive definite this condition results in a well established worst case bound for the convergence rate of the iterative method due to elman this bound has been extensively used e g when the linear system comes from discretized partial differential equations to show that the convergence of gmres is independent of the underlying mesh size in this paper we introduce more general non stagnation conditions which do not require the symmetric part of the coefficient matrix to be positive definite and that guarantee for example the non stagnation of restarted gmres for certain values of the restarting parameter

the optimal convergence rate of the p version of the finite element method optimal error estimates for the p version of the finite element method in two dimensions are proven for the case when u in h k omega or u has singularities induced by the corners of the domain the case of nonhomogenous essential boundary conditions is also analyzed

extensions of nilpotent blocks over arbitrary fields in this paper with a suitable condition we describe the algebraic structure of block extensions of nilpotent blocks over arbitrary fields thus generalize the main result of b kulshammer and l puig on block extensions of nilpotent blocks over algebraically closed fields

on investigating gmres convergence using unitary matrices for a given matrix a and right hand side b this paper investigates unitary matrices generating with some right hand sides c the same gmres residual norms as the pair a b we give characterizations of this class of unitary matrices and point out the relationship with krylov subspaces and krylov residual subspaces for the pair a b we investigate the eigenvalues of these unitary matrices in relation to the convergence behavior of gmres for the pair a b and describe the indispensable role of the eigenvector information we conclude with a formula for the gmres residual norms generated by a normal matrix b in terms of its eigenvalues and components of the right hand side c in the eigenvector basis

truncation strategies for optimal krylov subspace methods optimal krylov subspace methods like gmres and gcr have to compute an orthogonal basis for the entire krylov subspace to compute the minimal residual approximation to the solution therefore when the number of iterations becomes large the amount of work and the storage requirements become excessive in practice one has to limit the resources the most obvious ways to do this are to restart gmres after some number of iterations and to keep only some number of the most recent vectors in gcr this may lead to very poor convergence and even stagnation therefore we will describe a method that reveals which subspaces of the krylov space were important for convergence thus far and exactly how important they are this information is then used to select which subspace to keep for orthogonalizing future search directions numerical results indicate this to be a very effective strategy

an extension of the feti domain decomposition method for incompressible and nearly incompressible problems incompressible and nearly incompressible problems are treated herein with a mixed finite element formulation in order to avoid ill conditioning that prevents accuracy in pressure estimation and can lead to a lack of convergence for iterative solution algorithms a multilevel dual domain decomposition method is then chosen as an iterative algorithm the original feti and feti dp methods are extended to deal with such problems when the discretization of the pressure field is discontinuous throughout the elements a dedicated augmentation of the algorithms is proposed and the different methods are compared with several preconditioners for bidimensional test cases the resulting approaches are both optimal and numerically scalable and their costs are estimated with a complexity analysis

relations between blocks of a finite group and its subgroup let g be a group of finite order g and p be a prime number let k be the field which is generated by the gth roots of unity over the p adic number field r be the ring of local integers in k and p be the maximal ideal in r we denote by o the ring r or the residue class field f of r modulo p we view the group algebra og as a right o g x g module in the usual way so that y x y x yy y e og and x y e g an indecomposable direct summand of the o g x g module og is the same as a block ideal of og hence if 1 e e 0 e

the cycle convergence of restarted gmres for normal matrices is sublinear we prove that the cycle convergence of the restarted gmres applied to a system of linear equations with a normal coefficient matrix is sublinear

spline approximation of thin shell dynamics abstract a spline based method for approximating thin shell dynamics is presented here while the method is developed in the context of the donnell mushtari thin shell equations it can be easily extended to the byrne flugge lur ye equations or other models for shells of revolution as warranted by applications the primary requirements for the method include accuracy flexibility and efficiency in smart material applications to accomplish this the method was designed to be flexible with regard to boundary conditions material nonhomogeneities due to sensors and actuators and inputs from smart material actuators such as piezoceramic patches the accuracy of the method was also of primary concern both to guarantee full resolution of structural dynamics and to facilitate the development of pde based controllers which ultimately require real time implementation several numerical examples provide initial evidence demonstrating the efficacy of the method

the green correspondence and normal subgroups in a recent paper harris and knorr 6 have discovered a connection between the brauer correspondence and the covering relation between blocks of a finite group and blocks of its normal subgroups green s module point of view of blocks 4 suggests that this result has a moduletheoretic proof and that there is a more general result on indecomposable modules we shall show that this is the case by giving a short moduletheoretic proof of the harris kniirr theorem and then use the ideas of the proof to formulate and establish a result relating the green correspondence is with normal subgroups let us establish some notation let g be a finite group and h a normal subgroup let k be a field of prime characteristic p and b a block of h so b is an indecomposable algebra summand of kh we shall regard b as a k h x hi module in the usual way let d be a defect group of b k n d and b equal the block of k with bf b that is the brauer correspondent of b finally set l n d so k is a normal subgroup of l

approximation by smooth multivariate splines abstract one of the important properties of univariate splines is that in most senses smooth splines approximate just as well as do piecewise polynomials on the same mesh this report shows this to be untrue in the multivariate setting in particular it details the cost in approximating power one may have to pay for the luxury of a smooth piecewise polynomial approximant in an extreme case piecewise polynomials of total degree r on a rectangular grid with all derivatives of order or rho continuous will fail to approximate certain smooth functions at all as the grid goes to zero unless rho is kept below r 3 2 during the analysis of approximation on a certain regular triangular grid a novel kind of bivariate b spline is introduced this b spline in contrast to the established multivariate b spline derived from a simplex can be made to have all its breaklines in a given regular grid this makes it a prime candidate for use in the construction of smooth multivariate piecewise polynomial approximation and its properties will be explored further

locking effects in the finite element approximation of elasticity problems we consider the finite element approximation of the 2d elasticity problem when the poisson ratiov is close to 0 5 it is well known that the performance of certain commonly used finite elements deteriorates asv 0 a phenomenon calledlocking we analyze this phenomenon and characterize the strength of the locking androbustness of varioush version schemes using triangular and rectangular elements we prove that thep andh p versions are free of locking with respect to the error in the energy norm a generalization of our theory to the 3d problem is also discussed

a preconditioned and shifted gmres algorithm for the pagerank problem with multiple damping factors google has become one of the most popular and successful search engines in recent years google s success can be attributed to its simple and elegant algorithm pagerank in practice one often needs to solve the pagerank problem with multiple damping factors or with multiple damping factors and multiple personalization vectors the conventional pagerank algorithm has to solve these problems one by one the shifted gmres m algorithm can be used to solve them in the same search subspace however there are two disadvantages to this algorithm the first is near singularity and the second is stagnation in this paper we first present a modified and shifted gmres m algorithm to deal with the problem of near singularity in order to overcome the drawback of stagnation and to improve convergence we propose a polynomial preconditioner for the modified algorithm we show that the resulting algorithm can circumvent the drawbacks of near singularity and stagnation that occur in its original counterpart

on the robustness of the h and p versions of the finite element method abstract the classical h version of the finite element method achieves accuracy by refining the mesh and keeping p the degree of polynomials used at a fixed low level we compare the robustness of such algorithms to that of the more recent p version which uses a fixed mesh and increases p we consider two physical parameter dependent problems that can lead to poor approximations near a limiting value of the parameter locking our results show that p type algorithms are generally more robust with respect to the parameter values than h type ones

nonlinear krylov acceleration applied to a discrete ordinates formulation of the k eigenvalue problem we compare a variant of anderson mixing with the jacobian free newton krylov and broyden methods applied to an instance of the k eigenvalue formulation of the linear boltzmann transport equation we present evidence that one variant of anderson mixing finds solutions in the fewest number of iterations we examine and strengthen theoretical results of anderson mixing applied to linear problems

coefficient parameter polynomial continuation we establish the theory for a very general approach to constructing homotopies for computing all geometrically isolated solutions to polynomial systems we assume that the coefficients for a polynomial system are given by parameters in engineering applications these would be derived from physical parameters since the coefficients themselves can be the parameters this is completely general we show that by considering the space of parameters instead of the space of coefficients we can exploit any special structure the coefficient parameter formulas impose on the solution set in particular we prove that we can ignore generic positive dimensional solution sets we also show that any side conditions additional equations can be used to reduce the number of paths these results unify and generalize all current approaches to polynimial continuation the effect of these new results is to greatly increase the efficiency of polynomial continuation in several engineering areas for example we establish a 36 path homotopy for a electrocardiographic application that needed 193 paths under the previous most appropriate theory also we show that a 16 path homotopy suffices for the inverse kinematics problem for 6 revolute joint manipulators a problem previously requiring 32 paths

a locking free finite difference method on staggered grids for linear elasticity problems abstract a finite difference method on staggered grids is constructed on general nonuniform rectangular partition for linear elasticity problems stability optimal order error estimates in discrete h 1 norms on general nonuniform grids and second order superconvergence on almost uniform grids have been obtained these theoretical results are uniform about the lame constant 0 so the finite difference method is locking free the method and theoretical results can be extended to three dimensional problems numerical experiments using the method show agreement of the numerical results with theoretical analysis

a multiscale method for linear elasticity reducing poisson locking we propose a generalized finite element method for linear elasticity equations with highly varying and oscillating coefficients the method is formulated in the framework of localized orthogonal decomposition techniques introduced by malqvist and peterseim 2014 assuming only l infinity coefficients we prove linear convergence in the h 1 norm also for materials with large lame parameter a the theoretical a priori error estimate is confirmed by numerical examples

the rate of convergence of gmres on a tridiagonal toeplitz linear system ii abstract this paper continues the recent work of the authors r c li w zhang the rate of convergence of gmres on a tridiagonal toeplitz linear system numer math 112 2009 267 293 electronically published on 19 december 2008 on the rate of convergence of gmres for a tridiagonal toeplitz linear system ax b much simpler formulas than the earlier ones for gmres residuals when b is the first or the last column of the identity matrix are established and these formulas allow us to confirm the rate of convergence that was conjectured but only partially proven earlier simpler and sharper bounds than earlier ones when all b s entries except its first and last ones are zeros are also obtained

the analysis of restart dgmres for solving singular linear systems abstract a surprising phenomenon concerning with restart dgmres a sidi dgmres a gmres type algorithm for drazin inverse solution of singular nonsymmetric linear systems linear algebra appl 335 2001 189 204 is presented that small values of the restart parameter may converge faster than larger values we take three examples where dgmres 2 converge while dgmres 3 stagnates to interpret the phenomenon two of these examples reveals that dgmres convergence can be extremely sensitive to small changes in the initial residual

complete stagnation of gmres for normal matrices in this paper we study the problem of complete stagnation of the generalized minimum residual gmres method for normal matrices we first characterize all n i n nonsingular normal matrices a such that gmres a b stagnates completely for some vector b also we give necessary and sufficient conditions for the non existence of a real stagnation vector for real normal matrices the number of real stagnation vectors for normal matrices is studied moreover we characterize all the eigenvalues of nonsingular normal matrices a m 3 c such that gmres a b stagnates completely for some b c 3 using the results derived by a greenbaum v ptak and z strakos in 1996 we consider the complete stagnation of unitary matrices and derive another characterization for all nonsingular normal matrices a m 3 r such that gmres a b stagnates completely for some vector b r 3

complete stagnation of gmres we study problems for which the iterative method gmres for solving linear systems of equations makes no progress in its initial iterations our tool for analysis is a nonlinear system of equations the stagnation system that characterizes this behavior we focus on complete stagnation for which there is no progress until the last iteration we give necessary and sufficient conditions for complete stagnation of systems involving unitary matrices and show that if a normal matrix completely stagnates then so does an entire family of nonnormal matrices with the same eigenvalues finally we show that there are real matrices for which complete stagnation occurs for certain complex right hand sides but not for real ones 2003 elsevier science inc all rights reserved

a locking free nonconforming triangular element for planar elasticity with pure traction boundary condition a new nonconforming triangular element for the equations of planar linear elasticity with pure traction boundary conditions is considered by virtue of construction of the element the discrete version of korn s second inequality is directly proved to be valid convergence rate of the finite element methods is uniformly optimal with respect to l error estimates in the energy norm and l 2 norm are o h 2 and o h 3 respectively

well posedness of the fundamental boundary value problems for constrained anisotropic elastic materials we consider the equations of linear homogeneous anisotropic elasticity admitting the possibility that the material is internally constrained and formulate a simple necessary and sufficient condition for the fundamental boundary value problems to be well posed for materials fulfilling the condition we establish continuous dependence of the displacement and stress on the elastic moduli and ellipticity of the elasticity system as an application we determine the orthotropic materials for which the fundamental problems are well posed in terms of their young s moduli shear moduli and poisson ratios finally we derive a reformulation of the elasticity system that is valid for both constrained and unconstrained materials and involves only one scalar unknown in addition to the displacements for a two dimensional constrained material a further reduction to a single scalar equation is outlined

locking and robustness in the finite element method for circular arch problem in this paper we discuss locking and robustness of the finite element method for a model circular arch problem it is shown that in the primal variable i e the standard displacement formulation the p version is free from locking and uniformly robust with order p k and hence exhibits optimal rate of convergence on the other hand the h version shows locking of order h 2 and is uniformly robust with order h p 2 for p 2 which explains the fact that the quadratic element for some circular arch problems suffers from locking for thin arches in computational experience if mixed method is used both the h version and the p version are free from locking furthermore the mixed method even converges uniformly with an optimal rate for the stress

on locking and robustness in the finite element method a numerical scheme for the approximation of a parameter dependent problem is said to exhibit locking if the accuracy of the approximations deteriorates as the parameter tends to a limiting value a robust numerical scheme for the problem is one that is essentially uniformly convergent for all values of the parameter precise mathematical definitions for these terms are developed their quantitative characterization is given and some general theorems involving locking and robustness are proven a model problem involving heat transfer is analyzed in detail using this mathematical framework and various related computational results are described applications to some different problems involving locking are presented

an analysis of thep version of the finite element method for nearly incompressible materials in this paper we analyze the behavior of the so calledp version of the finite element method when applied to the equations of plane strain linear elasticity we establish optimal rate error estimates that are uniformly valid independent of the value of the poisson ratio v in the interval 0 1 2 this shows that thep versiondoes not exhibit the degeneracy phenomenon which has led to the use of various only partially justified techniques of reduced integration or mixed formulations for more standard finite element schemes and the case of a nearly incompressible material

locking free mixed hp finite element methods for curvilinear domains mixed hp finite element methods fems are robust with respect to poisson ratio locking when straight sided quadrilateral elements are used for the p version which uses fixed and generally large elements curved elements are necessary to represent curved domains accurately we show that mixed hp curved quadrilateral elements retain the robustness of their straight sided counterparts provided degenerate elements are avoided we also show that hp version triangles are robust we use these concepts to develop a locking free fem for geometrically nonlinear elasticity