Variational Space-Time Elements

目前，空间有限元离散已得到广泛应用。在过去的几十年里，基于合适的空间域分解方法，高效的并行化技术得到了发展。然而，如果子域变得很小，尺度饱和，特别是对于时间相关的问题。为了解决这个问题，很多项目都在研究时间并行化；然而，时间步进算法的使用是一个缺点，由于限制信息流在正时间方向。此外，这些经典的时间步进方案否定了有限元法最强大的特征之一：非结构化网格。在这个提议中，我们的目标是使用连续布布诺夫-伽辽金方法对时空进行直接的有限元离散化，适用于出现的大规模问题的大规模并行分析。这种方法与经典时空公式形成鲜明对比，经典时空公式通常使用不连续伽辽金方法或在时间上降低测试或试验函数的阶数。此外，这种方法允许我们处理非光滑系统和哈密顿系统中的守恒性质。本文主要研究了三维和四维空间弹性元的构造以及新的接触约束求解技术。新颖的多场公式可以理解弹性系统的微分几何结构，并设计合适的时空元素。

Discretisation in space using finite elements are widely used today. Efficient parallelization techniques have been developed throughout the past decades, based on suitable domain decomposition methods in space. Nevertheless, if subdomains becomes to small, scaling saturates especially for time dependent problems. To deal with this issue, a wide range of projects are working on temporal parallelization; however, the usage of time stepping algorithms is a drawback due to the restriction of the flow of informations in positive temporal direction.Moreover, these classical time stepping scheme negates one of the most powerful features of the finite element method: unstructured meshes. In this proposal, we aim at a direct finite element discretisation of the space-time using a continuous Bubnov-Galerkin approach, suitable for massive parallel analysis of the arising large-scale problem. This approach is in stark contrast to classical space-time formulation, which often use discontinuous Galerkin methods or a reduced order of the test or trial functions in time. Moreover, this approach allows us to deal with non-smooth systems and conservation properties in Hamiltonian systems.The construction of three- and fourdimensional space-time elements for non-linear elasticity as well as new solution techniques for contact constrains are in the main focus of this proposal. Novel multifield formulations allows to understand the differential-geometrical structure of the elastic system and to design suitable space-time elements.