Variational Space-Time Elements

利用有限元进行空间离散是当今广泛使用的方法。在过去的几十年里，基于合适的空间区域分解方法，已经发展了高效的并行化技术。然而，如果子域变得很小，伸缩就会饱和，特别是对于时间相关的问题。为了解决这个问题，很多项目都在进行时间并行化，然而，由于时间方向上信息流的限制，时间推进算法的使用是一个缺陷，而且这些经典的时间推进方法否定了有限元方法最强大的特征之一：非结构化网格。在这一建议中，我们的目标是使用连续的Bubnov-Galerkin方法对时空进行直接有限元离散，适合于大规模并行分析所产生的大规模问题。这种方法与经典的时空公式形成了鲜明的对比，后者通常使用不连续的Galerkin方法，或者在时间上对检验函数或试函数进行降阶。此外，这种方法还允许我们处理非光滑系统和哈密顿系统中的守恒性质。非线性弹性力学的三维和四维时空单元的构造以及接触约束的新求解技术是本方案的主要重点。新的多场公式允许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.