Locking effects in the finite element approximation of elasticity problems

Summary 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 asv0, 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.The work of this author was supported in part by the Office of Naval Research under Naval Research Grant N00014-90-J-1030The work of this author was supported in part by the Air Force Office of Scientific Research, Air Force Systems Command, U.S. Air Force, under grant AFOSR 89-0252     

Second-order finite element approximations and a posteriori error estimation for two-dimensional parabolic systems

Summary We discuss an adaptive local refinement finite element method of lines for solving vector systems of parabolic partial differential equations on two-dimensional rectangular regions. The partial differential system is discretized in space using a Galerkin approach with piecewise eight-node serendipity functions. An a posteriori estimate of the spatial discretization error of the finite element solution is obtained using piecewise fifth degree polynomials that vanish on the edges of the rectangular elements of a grid. Ordinary differential equations for the finite element solution and error estimate are integrated in time using software for stiff differential systems. The error estimate is used to control a local spatial mesh refinement procedure in an attempt to keep a global measure of the error within prescribed limits. Examples appraising the accuracy of the solution and error estimate and the computational efficiency of the procedure relative to one using bilinear finite elements are presented.Dedicated to Prof. Ivo Babuka on the occasion of his 60th birthdayThis research was partially supported by the U.S. Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant Number AFOSR 85-0156 and the U.S. Army Research Office under Contract Number DAAL 03-86-K-0112     

On the stability of bilinear-constant velocity-pressure finite elements

Summary An analysis of the Babuka stability of bilinear/constant finite element pairs for viscous flow calculations is given. An unstable mode not of the checkerboard type is given for which the stability constant turns out to beO(h). Thus, the indicated spaces are not stable in general for numerical calculation.Work supported by U.S. Air Force Office of Scientific Research under grant AF-AFOSR-82-0213     

Solution manifolds and submanifolds of parametrized equations and their discretization errors

Summary The paper concerns solution manifolds of nonlinear parameterdependent equations (1)F(u, )=y₀ involving a Fredholm operatorF between (infinite-dimensional) Banach spacesX=Z× andY, and a finitedimensional parameter space . Differntial-geometric ideas are used to discuss the connection between augmented equations and certain onedimensional submanifolds produced by numerical path-tracing procedures. Then, for arbitrary (finite) dimension of , estimates of the error between the solution manifold of (1) and its discretizations are developed. These estimates are shown to be applicable to rather general nonlinear boundaryvalue problems for partial differential equations.This work was in part supported by the U.S. Air Force Office of Scientific Research under Grant 80-0176, the National Science Foundation under Grant MCS-78-05299, and the Office of Naval Research under Contract N-00014-80-C-0455     

Comparisons of regular splittings of matrices

Summary In this article, new comparison theorems for regular splittings of matrices are derived. In so doing, the initial results of Varga in 1960 on regular splittings of matrices, and the subsequent unpublished results of Wonicki in 1973 on regular splittings of matrices, will be seen to be special cases of these new comparison theorems.Dedicated to Fritz Bauer on the occasion of his 60th birthdayResearch supported in part by the Air Force Office of Scientific Research, and by the Department of Energy     

Theh−p version of the finite element method for elliptic equations of order2m

Summary 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)!,kl, l2 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)!,kl=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.Dedicated to Professor Ivo Babuka on the occasion of his 60^th birthdaySupported by the Air Force Office of Science Research under grant No. AFOSR-80-0277 NOETIC TECHNOLOGIES, Inc., St. Louis, MO     

Error analysis of splitting algorithms for polynomials

An error analysis is given for the general splitting algorithm, proposed by Shaw and Traub, for evaluating a polynomial and some of its derivatives. The results show that the usual synthetic division is least likely to be affected by round-off errors if only single-precision arithmetic is available for all the algorithms. However, the new splitting algorithms are better than the synthetic division if extended-precision arithmetic is available for the evaluation of powers ofx.This work supported in part by the United States Air Force under grant AFOSR 76-3020     

Convergence of sequential and asynchronous nonlinear paracontractions

Summary We establish the convergence of sequential and asynchronous iteration schemes for nonlinear paracontracting operators acting in finite dimensional spaces. Applications to the solution of linear systems of equations with convex constraints are outlined. A first generalization of one of our convergence results to an infinite pool of asymptotically paracontracting operators is also presented.Research supported in part by Sonderforschungsbereich 343 Diskrete Strukturen in der MathematikResearch supported in part by NSF Grant DMS-9007030 and by Sonderforschungsbereich 343 Diskrete Strukturen in der Mathematik, Fakultät für Mathematik at the Universität BielefeldResearch supported in part by U.S. Air Force Grant AFOSR-88-0047, by NSF Grants DMS-8901860 and DMS-9007030, and by Sonderforschungsbereich 343 Diskrete Strukturen in der Mathematik, Fakultät für Mathematik at the Universität Bielefeld     

A posteriori error estimation with finite element methods of lines for one-dimensional parabolic systems

Summary Consider the solution of one-dimensional linear initial-boundary value problems by a finite element method of lines using a piecewiseP ^th -degree polynomial basis. A posteriori estimates of the discretization error are obtained as the solutions of either local parabolic or local elliptic finite element problems using piecewise polynomial corrections of degreep+1 that vanish at element ends. Error estimates computed in this manner are shown to converge in energy under mesh refinement to the exact finite element discretization error. Computational results indicate that the error estimates are robust over a wide range of mesh spacings and polynomial degrees and are, furthermore, applicable in situations that are not supported by the analysis.This research was partially supported by the U.S. Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant Number AFOSR 90-0194; by the U.S. Army Research Office under Contract Number DAAL03-91-G-0215; and by the National Science Foundation under Institutional Infrastructure Grant Number CDA-8805910     

Error estimates for the finite element solution of variational inequalities

Summary We study the mixed finite element approximation of variational inequalities, taking as model problems the so called obstacle problem and unilateral problem. Optimal error bounds are obtained in both cases.Supported in part by National Science Foundation grant MCS 75-09457, and by Office of Naval Research grant N00014-76-C-0369     

On the numerical approximation of finite speed diffusion problems

Summary Lagrangian formulations for the Cauchy problems for the generalized-heat and porous-media equations are introduced and equivalence and existence results discussed. Efficient interface tracking finite difference and finite element discretizations of the Lagrangian formulation are discussed. Mixed Euler-Lagrange formulations for mixed problems and the one phase Stefan problem are presented. Numerical experiments are discussed.Dedicated on the occasion of Prof. Ivo Babuka's 60th birthday     

On the finite volume element method

Summary The finite volume element method (FVE) is a discretization technique for partial differential equations. It uses a volume integral formulation of the problem with a finite partitioning set of volumes to discretize the equations, then restricts the admissible functions to a finite element space to discretize the solution. this paper develops discretization error estimates for general selfadjoint elliptic boundary value problems with FVE based on triangulations with linear finite element spaces and a general type of control volume. We establishO(h) estimates of the error in a discreteH ¹ semi-norm. Under an additional assumption of local uniformity of the triangulation the estimate is improved toO(h ²). Results on the effects of numerical integration are also included.This research was sponsored in part by the Air Force Office of Scientific Research under grant number AFOSR-86-0126 and the National Science Foundation under grant number DMS-8704169. This work was performed while the author was at the University of Colorado at Denver     

Nonconforming finite element methods for eigenvalue problems in linear plate theory

The paper deals with nonconforming finite element methods for approximating fourth order eigenvalue problems of type ² w=w. The methods are handled within an abstract Hilbert space framework which is a special case of the discrete approximation schemes introduced by Stummel and Grigorieff. This leads to qualitative spectral convergence under rather weak conditions guaranteeing the basic properties of consistency and discrete compactness for the nonconforming methods. Further asymptotic error estimates for eigenvalues and eigenfunctions are derived in terms of the given orders of approximability and nonconformity. These results can be applied to various nonconforming finite elements used by Adini, Morley, Zienkiewicz, de Veubeke e.a. This is carried out for the simple elements of Adini and Morley and is illustrated by some numerical results at the end.     

On the sharpness of some upper bounds for the spectral radii of S.O.R. iteration matrices

Summary Sharpness is shown for three upper bounds for the spectral radii of point S.O.R. iteration matrices resulting from the splitting (i) of a nonsingularH-matrixA into the usualD–L–U, and (ii) of an hermitian positive definite matrixA intoD–L–U, whereD is hermitian positive definite andL=1/2(A–D+S) withS some skew-hermitian matrix. The first upper bound (which is related to the splitting in (i)) is due to Kahan [6], Apostolatos and Kulisch [1] and Kulisch [7], while the remaining upper bounds (which are related to the splitting in (ii)) are due to Varga [11]. The considerations regarding the first bound yield an answer to a question which, in essence, was recently posed by Professor Ridgway Scott: What is the largest interval in , 0, for which the point S.O.R. iterative method is convergent for all strictly diagonally dominant matrices of arbitrary order? The answer is, precisely, the interval (0, 1].Research supported in part by the Air Force Office of Scientific Research, and the Department of Energy     

On the multi-level splitting of finite element spaces

Summary In this paper we analyze the condition number of the stiffness matrices arising in the discretization of selfadjoint and positive definite plane elliptic boundary value problems of second order by finite element methods when using hierarchical bases of the finite element spaces instead of the usual nodal bases. We show that the condition number of such a stiffness matrix behaves like O((log )²) where is the condition number of the stiffness matrix with respect to a nodal basis. In the case of a triangulation with uniform mesh sizeh this means that the stiffness matrix with respect to a hierarchical basis of the finite element space has a condition number behaving like instead of for a nodal basis. The proofs of our theorems do not need any regularity properties of neither the continuous problem nor its discretization. Especially we do not need the quasiuniformity of the employed triangulations. As the representation of a finite element function with respect to a hierarchical basis can be converted very easily and quickly to its representation with respect to a nodal basis, our results mean that the method of conjugate gradients needs onlyO(log n) steps andO(n log n) computer operations to reduce the energy norm of the error by a given factor if one uses hierarchical bases or related preconditioning procedures. Heren denotes the dimension of the finite element space and of the discrete linear problem to be solved.     

An iterative substructuring algorithm for equilibrium equations

Summary The topic of iterative substructuring methods, and more generally domain decomposition methods, has been extensively studied over the past few years, and the topic is well advanced with respect to first and second order elliptic problems. However, relatively little work has been done on more general constrained least squares problems (or equivalent formulations) involving equilibrium equations such as those arising, for example, in realistic structural analysis applications. The potential is good for effective use of iterative algorithms on these problems, but such methods are still far from being competitive with direct methods in industrial codes. The purpose of this paper is to investigate an order reducing, preconditioned conjugate gradient method proposed by Barlow, Nichols and Plemmons for solving problems of this type. The relationships between this method and nullspace methods, such as the force method for structures and the dual variable method for fluids, are examined. Convergence properties are discussed in relation to recent optimality results for Varga's theory ofp-cyclic SOR. We suggest a mixed approach for solving equilibrium equations, consisting of both direct reduction in the substructures and the conjugate gradient iterative algorithm to complete the computations.Dedicated to R. S. Varga on the occasion of his 60th birthdayResearch completed while pursuing graduate studies sponsored by the Department of Mathematical Sciences, US Air Force Academy, CO, and funded by the Air Force Institute of Technology, WPAFB, OHResearch supported by the Air Force under grant no. AFOSR-88-0285 and by the National Science Foundation under grant no. DMS-89-02121     

Asymptotic expansions andL ∞-error estimates for mixed finite element methods for second order elliptic problems

Summary Asymptotic expansions for mixed finite element approximations of the second order elliptic problem are derived and Richardson extrapolation can be applied to increase the accuracy of the approximations. A new procedure, which is called the error corrected method, is presented as a further application of the asymptotic error expansion for the first order BDM approximation of the scalar field. The key point in deriving the asymptotic expansions for the error is an establishment ofL ¹-error estimates for mixed finite element approximations for the regularized Green's functions. As another application of theL ¹-error estimates for the regularized Green's functions, we shall present maximum norm error estimates for mixed finite element methods for second order elliptic problems.     

On a numerical approach to Stefan-like problems

Summary This paper is devoted to the numerical analysis of a bidimensional two-phase Stefan problem. We approximate the enthalpy formulation byC ⁰ piecewise linear finite elements in space combined with a semi-implicit scheme in time. Under some restrictions related to the finite element mesh and to the timestep, we prove positivity, stability and convergence results. Various numerial tests are presented and discussed in order to show the accuracy of our scheme.This work is supported by the Fonds National Suisse pour la Recherche Scientifique.     

Zur Stabilität von Differenzenverfahren für Systeme linearer gewöhnlicher Randwertaufgaben

Summary In this paper we give a simple stability theory for finite difference approximations to linear ordinary boundary value problems. In particular we consider stability with respect to a maximum norm including all difference quotients up to the order of the differential equation. It is shown that stability in this sense holds if and only if the principal part of the differential equation is discretized in a stable way. This last property is characterized by root conditions which we prove to be satisfied for some classes of finite difference schemes. Our approach simplifies and generalizes some known results of the literature where Sobolev norms or merely the maximum norm are used.

     

The treatment of nonhomogeneous Dirichlet boundary conditions by thep-version of the finite element method

Summary The paper addresses the problem of the implementation of nonhomogeneous essential Dirichlet type boundary conditions in thep-version of the finite element method.Partially supported by the Office of Naval Research under Grant N-00014-85-K-0169Research partially supported by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant Number AFOSR 85-0322