Error-controlled adaptive meshless methods for elastic and elastoplastic de- formations

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The objective of this project is the reliable prediction of realistic crack growth in brittle and ductile materials. So far, error-controlled finite element and, more recently, extended finite element methods have been used to cope with this problem. For these methods, however, a mesh has to be generated, which even has to align with the crack in the case of the finite element method. Moreover, for adaptive strategies, mesh refinement and possibly remeshing techniques are required. These problems can be bypassed using a meshless method where the nodes are not connected via elements and thus complex refinement algorithms are not required. For meshless methods, however, mathematically sound a posteriori error estimators have not been developed so far.In this project, error-controlled adaptive meshless methods will be developed for linear problems in elastic fracture mechanics and nonlinear problems in small-strain elastoplastic fracture mechanics and finite elasticity. The meshless method chosen in this project will be the reproducing kernel particle method, since it is based on a Galerkin scheme and therefore allows extensions of residual-, hierarchical-, and averaging-type error estimation techniques as developed for the finite element method. The error estimators for meshless methods will first be derived to control the error in the global energy norm and will then be extended to goal-oriented error estimators in fracture mechanics. For ductile materials, the error in the (pseudo) time discretization has to be additionally taken into account. Within the overall concept of using Galerkin schemes, the (pseudo) time discretization will be realized using a discontinuous Galerkin method. The various error estimators developed in this project will be applied to problems of technical interest in structural mechanics.