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

该项目的目标是在脆性和韧性材料的现实裂纹扩展的可靠预测。到目前为止，误差控制有限元和最近的扩展有限元方法已被用来科普这个问题。然而，对于这些方法，必须生成网格，在有限元方法的情况下，网格甚至必须与裂纹对齐。此外，对于自适应策略，需要网格细化和可能的重新网格化技术。使用无网格方法可以绕过这些问题，其中节点不通过元素连接，因此不需要复杂的细化算法。对于无网格方法，然而，数学上健全的后验误差估计尚未开发至今，在这个项目中，误差控制的自适应无网格方法将被开发用于弹性断裂力学中的线性问题和小应变弹塑性断裂力学和有限弹性中的非线性问题。在这个项目中选择的无网格方法将是再生核粒子法，因为它是基于Galerkin计划，因此允许扩展的剩余，分层，和平均型误差估计技术开发的有限元法。无网格方法的误差估计将首先导出控制误差的整体能量范数，然后将扩展到目标导向的误差估计在断裂力学。对于韧性材料，（伪）时间离散化的误差必须另外考虑。在使用Galerkin格式的总体概念内，（伪）时间离散化将使用间断Galerkin方法来实现。在这个项目中开发的各种误差估计将被应用到结构力学的技术兴趣的问题。

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.