Continuous space-time multi-level hp Galerkin-Petrov finite elements for the direct numerical simulation of laser power bed fusion processes

本项目旨在开发一种新的离散化技术来分析激光粉床熔化过程。LPBF是制造复杂形状承载零件最重要的添加剂制造技术之一。它在空间和时间上具有多尺度行为的特征。虽然这一过程显然也表现出多物理性质，但尤其是对热历史的预测对这一过程至关重要。然而，由于所需的空间分辨率和所涉及的大量时间步长，对热历史的准确预测使当前的数值技术受到限制。本项目中提出的方法是在惠普环境下的连续Galerkin-Petrov有限元格式，从而导致最优收敛速度。一个特别的重点在于算法效率，为此，将本着多级惠普方法的精神，在分层网格上进行离散化。此外，拟议的提法将允许在时间上实现并行，考虑到这一进程涉及的时间尺度，这被认为是至关重要的。由于在四维空间中进行边界协调离散是不可行的，因此发展的层次化网格上的连续Galerkin-Petrov有限元格式将与有限单元法相结合。FCM是一种嵌入式区域格式，它允许在简单网格上进行非边界协调离散，而不会损失精度，即使模拟的物理伪影具有复杂的形状和拓扑。所开发的方法将在许多基准中得到验证，包括对其算法复杂性的分析。此外，还计划与美国NIST国家科学和技术研究所一起进行彻底的验证。在第二阶段，计划在高性能计算机上实现时间并行化的好处，包括开发合适的并行求解器。然后，该技术将扩展到对其他物理现象和耦合多物理问题的模拟。

This project aims at developing a new discretizational technology for the analysis of laser powder bed fusion processes (LPBF). LPBF is one of the most important additive manufacturing technologies to produce complex shaped load bearing parts. It is characterized by a multi-scale behavior in space and time. While the process clearly exhibits a multi-physical nature as well, it is especially the prediction of the thermal history which is crucial to this process. However, due to the required spatial resolution as well as the numerous time steps involved, the accurate prediction of the thermal history brings current numerical techniques to their limit.The proposed methodology to be developed in this project is a continuous Galerkin-Petrov finite element formulation in an hp setting which leads to optimal convergence rates. A special focus lies on algorithmic efficiency, for which the discretization will be formulated on hierarchical grids in the spirit of the multi-level hp-method. Further, the proposed formulation will allow for a parallelization in time, which is considered crucial given the time scales involved in the process. Since a boundary-conforming discretization in 4D is not feasible, the developed continuous Galerkin-Petrov finite element formulation on hierarchical grids will be combined with the Finite Cell Method (FCM). The FCM is an embedded domain formulation which allows for a non-boundary conforming discretization on simple grids without loss of accuracy even if the simulated physical artifact possesses a complex shape and topology. The developed methodology will be verified at numerous benchmarks including an analysis of its algorithmic complexity. Additionally, a thorough validation is planned together with the National Institute of Science and Technology, NIST, USA. In a second phase, it is planned to harvest the benefits of time-parallelization on high-performance computers, including a development of suitable parallel solvers. The technology will then be extended for the simulation of other physical phenomena and coupled multi-physical problems.