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

该项目的目标是对脆性和延性材料的真实裂纹扩展进行可靠的预测。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处理这个问题。然而，对于这些方法，必须生成网格，在有限元方法中，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网格技术。这些问题可以使用无网格方法绕过，其中节点不通过元素连接，因此不需要复杂的细化算法。然而，对于无网格方法，数学上合理的后验误差估计迄今尚未开发。本课题将针对弹性断裂力学中的线性问题、小应变弹塑性断裂力学和有限弹性力学中的非线性问题，开发误差控制自适应无网格方法。本项目中选择的无网格方法将是再现核粒子方法，因为它基于伽辽金方案，因此允许扩展为有限元方法开发的残差，分层和平均型误差估计技术。首先推导了无网格方法的误差估计量以控制全局能量范数的误差，然后将其推广到断裂力学中面向目标的误差估计量。对于延性材料，必须额外考虑（伪）时间离散化的误差。在使用伽辽金格式的总体概念中，将使用不连续伽辽金方法实现（伪）时间离散化。在这个项目中开发的各种误差估计器将应用于结构力学中感兴趣的技术问题。

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.