Error-controlled adaptive meshless methods for elastic and elastoplastic de- formations
Final Report Abstract
The main outcome of this research project is the derivation of new goal-oriented a posteriori error estimators of residual type and adaptivity for meshless methods, more precisely for the reproducing kernel particle method (RKPM), thereby also taking the error from numerical quadrature into account, and for related Galerkin methods such as the eXtended Finite Element Method (XFEM) which, similar to meshless methods, also relies on shape functions that are not tied to a mesh. All error estimators derived are based on their finite element counterparts that are well established. Since both RKPM and FEM are Galerkin methods, an extension of available error estimators from FEM to RKPM is generally possible. However, the error estimators need to be adjusted to the properties of meshless methods. First and foremost, the different treatment of Dirichlet boundary conditions needs to be taken into account into the error estimator, because this yields an additional discretization error. This error can be related to the error in the energy norm by an inverse estimate, which was successfully applied in this project to derive the error estimator for RKPM. In XFEM, the Dirichlet boundary conditions are fulfilled exactly, however, owing to the enrichment functions as used to model the crack behavior, additional residual terms arise in the error estimator. The error estimators mentioned above were extended to goal-oriented error estimators (i.e. error estimators that control the error of a given quantity of interest). In RKPM, the dual problem, that is required for this type of error estimators, can be solved approximately using a different discretization than the one for the primal problem. This can be easily carried out using RKPM with no additional costs, whereas a mesh-based method, such as XFEM, would require several transfers of the solutions from one mesh to the other accompanied by a loss of accuracy from those transfers and complicated transfer algorithms. This adds versatility to the RKPM error estimator and was successfully applied to various quantities of interest, e.g. to the J-integral as a crack propagation criterion in fracture mechanics. However, the J-integral is nonlinear and therefore needs to be linearized to be used in the dual problem for both RKPM and XFEM error estimators. In meshless methods, the integration error is equally important as the discretization error. This is mainly owing to the fact that the shape function supports overlap. A methodology to reduce or even correct the integration error has been developed in this project by means of correcting the residual that is obtained by violating the integration constraints. As it turns out, a crucial point is the Galerkin orthogonality which is violated by the meshless solution owing to the quadrature error. The recovery of the Galerkin orthogonality is therefore an important step in obtaining numerical results without quadrature error and leading to a variationally consistent approach with optimal convergence rates. Elastoplastic deformations are treated in this project by means of a blind prediction of a crack initiation and propagation experiment of ductile materials as carried out by Sandia National Laboratories. With the help of the applicant, a numerical simulation based on enriched meshless methods of this blind prediction could be successfully realized.
Publications
- Corrected stabilized non-conforming nodal integration in meshfree methods. In M. Griebel and M. A. Schweitzer, editors, Meshfree Methods for Partial Differential Equations VI, pages 75–92. Springer, Berlin, 2012
M. Rüter, M. Hillman, and J. S. Chen
- An arbitrary order variationally consistent integration for Galerkin meshfree methods. Int. J. Numer. Meth. Engng, 95:387–418, 2013
J. S. Chen, M. Hillman, and M. Rüter
- Goal-oriented explicit residual-type error estimates in XFEM. Comput Mech, 52:361–376, 2013
M. Rüter, T. Gerasimov, and E. Stein
(See online at https://doi.org/10.1007/s00466-012-0816-5)
