A posteriori error estimates and adaptivity for finite element solutions in finite elasticity

Methods for a posteriori error estimation for finite element solutions are well established and widely used in engineering practice for linear boundary value problems. In contrast here we are concerned with finite elasticity and error estimation and adaptivity in this context. In the paper a brief outline of continuum theory of finite elasticity is first given. Using the residuals in the equilibrium conditions the discretization error of the finite element solution is estimated both locally and globally. The proposed error estimator is physically interpreted in the energy sense. We then present and discuss the convergence behaviour of the discretization error in uniformly and adaptively refined finite element sequences.

[1]  J. Z. Zhu,et al.  Superconvergence recovery technique and a posteriori error estimators , 1990 .

[2]  K. Mattiasson,et al.  On the Accuracy and Efficiency of Numerical Algorithms for Geometrically Nonlinear Structural Analysis , 1986 .

[3]  K. Washizu.  Variational Methods in Elasticity and Plasticity , 1982 .

[4]  Ivo Babuška,et al.  On the Rates of Convergence of the Finite Element Method , 1982 .

[5]  F. Hartmann.  The Mathematical Foundation of Structural Mechanics , 1985 .

[6]  Leszek Demkowicz,et al.  Toward a universal h-p adaptive finite element strategy , 1989 .

[7]  Ernst Rank,et al.  An expert system for the optimal mesh design in the hp‐version of the finite element method , 1987 .

[8]  Ivo Babuška,et al.  Computational error estimates and adaptive processes for some nonlinear structural problems , 1982 .

[9]  John R. Whiteman,et al.  Pointwise superconvergence of recovered gradients for piecewise linear finite element approximations to problems of planar linear elasticity , 1990 .

[10]  Ivo Babuška,et al.  A posteriori error analysis and adaptive processes in the finite element method: Part I—error analysis , 1983 .

[11]  O. Zienkiewicz,et al.  Analysis of the Zienkiewicz–Zhu a‐posteriori error estimator in the finite element method , 1989 .

[12]  J. Z. Zhu,et al.  The superconvergent patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity , 1992 .

[13]  I. Babuska,et al.  A feedback element method with a posteriori error estimation: Part I. The finite element method and some basic properties of the a posteriori error estimator , 1987 .

[14]  D. Kelly,et al.  The self‐equilibration of residuals and complementary a posteriori error estimates in the finite element method , 1984 .

[15]  T. Strouboulis,et al.  Recent experiences with error estimation and adaptivity, part II: Error estimation for h -adaptive approximations on grids of triangles and quadrilaterals , 1992 .

[16]  Cv Clemens Verhoosel,et al.  Non-Linear Finite Element Analysis of Solids and Structures , 1991 .

[17]  J. Oden,et al.  Toward a universal h - p adaptive finite element strategy: Part 2 , 1989 .

[18]  G. Strang,et al.  Geometric nonlinearity: potential energy, complementary energy, and the gap function , 1989 .

[19]  J. Oden,et al.  Variational Methods in Theoretical Mechanics , 1976 .

[20]  T. Strouboulis,et al.  Recent experiences with error estimation and adaptivity. Part I: Review of error estimators for scalar elliptic problems , 1992 .

[21]  W. Rheinboldt,et al.  On the Discretization Error of Parametrized Nonlinear Equations , 1983 .

[22]  Leszek Demkowicz,et al.  Toward a universal adaptive finite element strategy part 3. design of meshes , 1989 .

[23]  J. Z. Zhu,et al.  The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique , 1992 .

[24]  Ivo Babuška,et al.  Basic principles of feedback and adaptive approaches in the finite element method , 1986 .

[25]  O. C. Zienkiewicz,et al.  A simple error estimator and adaptive procedure for practical engineerng analysis , 1987 .

[26]  W. Rheinboldt,et al.  LOCAL ERROR ESTIMATES FOR PARAMETRIZED NONLINEAR EQUATIONS , 1985 .

[27]  Ivo Babuška,et al.  A posteriori error analysis and adaptive processes in the finite element method: Part II—adaptive mesh refinement , 1983 .