Variational Space-Time Elements

在空间中使用有限元离散化被广泛使用的今天。在过去的几十年中，基于合适的空间域分解方法，已经开发出了有效的并行化技术。然而，如果子域变得很小，缩放饱和，特别是对时间相关的问题。为了解决这一问题，大量的项目都在致力于时间并行化，然而，时间步进算法的使用是一个缺点，由于在正时间方向上的信息流的限制，而且，这些经典的时间步进计划否定了有限元方法的最强大的功能之一：非结构化网格。在这个建议中，我们的目标是在一个直接的有限元离散化的空间，时间使用连续的Bubnov-Galerkin方法，适用于大规模并行分析所产生的大规模问题。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Galerkin方法或时间上的测试或试验函数的降阶。此外，这种方法使我们能够处理非光滑系统和守恒性质的Hamiltonian systems.The建设的三维和四维的非线性弹性时空元素，以及新的解决技术的接触约束是在这个建议的主要焦点。新的多场制剂允许了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.