Error estimation by compatibility in patches for plate structures

Error estimation is a key tool in modern finite element technology in order to verify and validate the finite element simulations, as well as to improve results and control the error, when combined with adaptivity. An efficient and practical way to derive a posteriori error estimators is offered by recovery procedures. The error, generally in stress based norms, is estimated by comparing the original finite element solution with the recovered one. The major steps forward in using recovery procedures were taken with the Superconvergent Patch Recovery (SPR) developed by Zienkiewicz and Zhu [1] and the Recovery by Equilibrium in Patches (REP) developed by Boroomand and Zienkiewicz [2]. Both these procedures have been successfully applied to plate problems in [3]. Recently, a new superconvergent procedure called Recovery by Compatibility in Patches (RCP) has been proposed by one of the authors [4] and shown to provide an excellent basis for error estimation in 2D problems [5]. Thin structures like plates and shells constitute an important class among finite element analyses because of their large application fields. With this in mind, the aim of the present paper is to develop an extension of the RCP-based error estimation to Reissner-Mindlin plates finite element analysis. The basic idea of this procedure is the same as the RCP in its original version, that is to recover stress resultants by enforcing compatibility over patches of elements. Displacements computed by the finite element analysis are prescribed on the boundary of the patch, and improved stress resultants are computed by minimizing the complementary energy of such a sub-model. The resulting procedure is simple, efficient, numerically stable and does not need any knowledge of superconvergent points. Some numerical examples involving thin and thick plates under different loading and support conditions are given. References [1] Zienkiewicz OC, Zhu JZ, The superconvergent patch recovery and a posteriori error estimates. Part I: The recovery technique, International Journal for Numerical Methods in Engineering, 33, 1992, 1331–1364. [2] Boroomand B, Zienkiewicz OC, An improved REP recovery and the effectivity robustness test, International Journal for Numerical Methods in Engineering, 40, 1997, 3247–3277. [3] Boroomand B, Ghaffarian M, Zienkiewicz OC, On application of two superconvergent recovery procedures to plate problems, International Journal for Numerical Methods in Engineering, 61, 2004, 1644–1673. [4] Ubertini F, Patch recovery based on complementary energy, International Journal for Numerical Methods in Engineering, 59, 2004, 1501–1538. [5] Benedetti A, de Miranda S, Ubertini F., A posteriori error estimation based on the superconvergent Recovery by Compatibility in Patches, International Journal for Numerical Methods in Engineering, in press.