Averaging techniques for reliable a posteriori FE-error control in elastoplasticity with hardening

Averaging techniques are popular tools in adaptive finite element methods for numerical simulation in continuum mechanics since they provide efficient a posteriori error control. In this paper, the reliability of any averaging estimator is shown for low order finite element methods in one time-step of elastoplasticity with hardening. The constants and higher-order terms are effected by the hardening and the smoothness of given right-hand sides, but are independent of the structure of a shape-regular mesh. Since it involves a different functional analytical framework, the case of perfect plasticity is excluded from this paper.

[1]  Carsten Carstensen,et al.  Coupling of FEM and BEM for interface problems in viscoplasticity and plasticity with hardening , 1996 .

[2]  Carsten Carstensen,et al.  Averaging technique for FE – a posteriori error control in elasticity. Part II: λ-independent estimates , 2001 .

[3]  王东东,et al.  Computer Methods in Applied Mechanics and Engineering , 2004 .

[4]  E. Zeidler.  Nonlinear functional analysis and its applications , 1988 .

[5]  Roger Temam,et al.  Mathematical Problems in Plasticity , 1985 .

[6]  Carsten Carstensen,et al.  Averaging technique for FE – a posteriori error control in elasticity. Part I: Conforming FEM , 2001 .

[7]  Claes Johnson,et al.  A Mixed Finite Element Method for Plasticity Problems with Hardening , 1977 .

[8]  Grigorii Aleksandrovich Seregin,et al.  Regularity for minimizers of some variational problems in plasticity theory , 1993 .

[9]  Rolf Rannacher,et al.  A posteriori error control in finite element methods via duality techniques: Application to perfect plasticity , 1998 .

[10]  H. Weinert.  Ekeland, I. / Temam, R., Convex Analysis and Variational Problems. Amsterdam‐Oxford. North‐Holland Publ. Company. 1976. IX, 402 S., Dfl. 85.00. US $ 29.50 (SMAA 1) , 1979 .

[11]  Weimin Han,et al.  Computational plasticity : the variational basis and numerical analysis , 1995 .

[12]  R. Verfürth,et al.  Edge Residuals Dominate A Posteriori Error Estimates for Low Order Finite Element Methods , 1999 .

[13]  David R. Owen,et al.  On error estimates and adaptivity in elastoplastic solids: Applications to the numerical simulation of strain localization in classical and Cosserat continua , 1994 .

[14]  Carsten Carstensen,et al.  All first-order averaging techniques for a posteriori finite element error control on unstructured grids are efficient and reliable , 2003, Math. Comput..

[15]  C. Carstensen.  QUASI-INTERPOLATION AND A POSTERIORI ERROR ANALYSIS IN FINITE ELEMENT METHODS , 1999 .

[16]  I. Ekeland,et al.  Convex analysis and variational problems , 1976 .

[17]  An Adaptive Finite Element Method for Problems in Perfect Plasticity , 1999 .

[18]  Rüdiger Verfürth,et al.  A posteriori error estimation and adaptive mesh-refinement techniques , 1994 .

[19]  G Crosta.  Mathematical Review: MR1202731 Serëgin, G. A. On the regularity of minimizers of some variational problems in the theory of plasticity. (Russian) Algebra i Analiz 4 (1992), no. 5, 181--218; translation in St. Petersburg Math. J. 4 (1993), no. 5, 989–1020 , 1994 .

[20]  Rolf Rannacher,et al.  A posteriori error estimation and mesh adaptation for finite element models in elasto-plasticity , 1999 .

[21]  F. Jones.  Lebesgue Integration on Euclidean Space , 1993 .

[22]  Carsten Carstensen,et al.  Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part II: Higher order FEM , 2002, Math. Comput..

[23]  Carsten Carstensen,et al.  Domain decomposition for a non-smooth convex minimization problem and its application to plasticity , 1997, Numer. Linear Algebra Appl..

[24]  Claes Johnson,et al.  On plasticity with hardening , 1978 .

[25]  Carsten Carstensen,et al.  Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part I: Low order conforming, nonconforming, and mixed FEM , 2002, Math. Comput..

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

[27]  Carsten Carstensen,et al.  Numerical Analysis of Time-Depending PrimalElastoplasticity with Hardening , 2000, SIAM J. Numer. Anal..

[28]  Claes Johnson,et al.  Adaptive finite element methods in computational mechanics , 1992 .

[29]  David R. Owen,et al.  Transfer operators for evolving meshes in small strain elasto-placticity , 1996 .

[30]  Carsten Carstensen,et al.  Numerical analysis of the primal problem of elastoplasticity with hardening , 1999, Numerische Mathematik.

[31]  P. Suquet,et al.  Discontinuities and Plasticity , 1988 .