First-order system least squares finite elements for finite elasto-plasticity

拟议的研究项目的目标是在一阶系统的基础上，在有限弹塑性领域的混合有限元公式的开发，建设和分析。由于目前没有仿真工具能够为这些问题提供可靠的解决方案，因此该项目的目标是在该领域提供一种新的现代离散化技术。最小二乘有限元法（LSFEM），特别是产生精确的近似应力，这往往是主要利益的背景下，固体力学问题。这种方法避免了对应力和位移的有限元空间的相容性（inf-sup）条件的要求，从而导致更一般的组合。此外，一个强大的数值行为，特别是准不可压缩的materials.In大量的调查中，申请人实现了长期的联合经验的基础上，LSFEM和它的应用，固体力学之间的合作支持的数学和力学。在此合作取得的进展的基础上，现在有可能共同解决有限塑性问题。该项目的主要任务可分为三个步骤。首先，该方法的组件将开发的弹性子问题，以便将它们扩展到有限乘性弹塑性模型。之后，将通过优先方案提供的基准问题以及通过计算真实的复杂微观非均匀结构（例如双相钢），对所得公式进行数值验证。为了提高效率和鲁棒性，这些计算是由几个调查有关，例如自适应性，选择的近似空间和适当的weighting.This研究项目的目的是在非传统的离散化方法的结构力学问题的框架内有限弹塑性设置新的质量标准。

The goal of the proposed research project is the development, construction and analysis of mixed finite element formulations in the field of finite elasto-plasticity on the basis of first-order systems. Since there are no simulation tools available at the moment that are capable to provide reliable solutions for such problems, the ambition of this project is to provide a new access with modern discretization techniques in this field. The least squares finite element method (LSFEM), in particular produces accurate approximations of stresses which are often of primary interest in the context of solid mechanical problems. This approach avoids the requirement of a compatibility (inf-sup) condition for the finite element spaces for stresses and displacements leading to more general combinations. Furthermore, a robust numerical behavior particularly for quasi-incompressible materials is obtained.In numerous investigations the applicants achieved a long-term joint experience basis with respect to the LSFEM and its application to solid mechanics supported by a cooperation between mathematics and mechanics. On the basis of the advances obtained within this cooperation it is now possible to jointly address problems of finite plasticity.The main tasks of the project can be divided into three steps. Firstly, the components of the method will be developed for the elastic subproblem in order to extend them to the model of finite multiplicative elasto-plasticity. After that, the resulting formulations will be validated numerically by means of the benchmark problems provided by the Priority Programme as well as by computations of real complex microheterogeneous structures as for instance of dual-phase steels. In order to improve the efficiency and robustness, these computations are guided by several investigations concerning e.g. adaptivity, choice of the approximation spaces and suitable weighting.This research project aims at setting new standards of quality in the field of non-conventional discretization methods for structural mechanical problems in the framework of finite elasto-plasticity.