SPP 1748: Reliable Simulation Techniques in Solid Mechanics - Development of Non-Standard Discretisation Methods, Mechanical and Mathematical Analysis

该优先计划的主要目标是开发现代非传统离散方法，例如基于混合（Galerkin或最小二乘）有限元或不连续Galerkin公式，包括几何和物理非线性问题的数学分析，例如不可压缩性，各向异性和不连续性（裂缝，接触）。其目的是汇集德国的力学和数学专业知识，并建立新的和加强现有的网络。在这种合作的框架内，不同工作组之间应交流经验，以产生协同作用，节省时间和费用，提高效率。此外，该项目还旨在引领该研究联盟在非常规离散技术领域取得国际领先地位。具体而言，优先项目将推动非常规有限元公式的以下研究方向：·对有限变形的可靠非协调有限元法（FEM）方法的结构要求的深刻数学理解，·数学上合理的变分公式，·（准）不可压缩的有限变形下的鲁棒和无刚度离散，各向同性和各向异性材料行为以及具有振荡系数的域，·在后面提到的极端情况下所有过程变量的精确近似，·关于显著网格变形的不敏感行为，·自适应网格细化和不连续性的收敛：·创建变分基以及用于不连续性的合适离散化技术：收敛性，稳定性和近似特性，·基于等几何公式的不连续性解决方案，·新的裂纹扩展和裂纹分支模型，·基于非传统离散技术的接触公式超过了Mortar方法。

The main objective of this Priority Programme is the development of modern non-conventional discretisation methods, based on e.g. mixed (Galerkin or least-squares) finite element or discontinuous Galerkin formulations, including the mathematical analysis for geometrically as well as physically non-linear problems in the fields of e.g. incompressibility, anisotropies and discontinuities (cracks, contact). It is the aim to pool the expertise of mechanics and mathematics in Germany and to create new and strengthen existing networks. In the framework of this cooperation the experiences should be exchanged in between the different working groups to create synergies, save time and costs and raise the efficiency. Furthermore, it is intended to lead this research union to international excellence in the field of non-conventional discretisation techniques.In detail the Priority Programme will drive research towards the following directions concerning non-conventional finite element formulations:· deep mathematical understanding of the structural requirements of reliable non-conforming finite element method (FEM) approaches for finite deformations,· mathematically sound variational formulations,· robust and stiffening-free discretisations at finite deformations for (quasi-)incompressible, isotropic and anisotropic material behaviour as well as for domains with oscillating coefficients,· accurate approximation of all process variables in the latter mentioned extremal cases,· insensitive behaviour concerning significant mesh deformation, · convergence of adaptive mesh refinement,and discontinuities:· creation of a variational basis as well as suitable discretisation techniques for discontinuities: convergence, stability and approximation properties,· resolution of discontinuities based on isogeometric formulations,· novel crack growth and crack branching models,· contact formulations based on non-conventional discretisation techniques exceeding Mortar-methods.