Higher order Unfitted Finite Element Methods for moving domain problems

移动域问题的高阶不拟合有限元方法

基本信息

批准号：
319609890
负责人：
Professor Dr. Christoph Lehrenfeld
金额：
--
依托单位：
Institut für Numerische und Angewandte Mathematik (NAM)
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2016
资助国家：
德国
起止时间：
2015-12-31 至 2020-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/319609890?language=en
关键词：
Higher order Unfitted Finite Element

项目摘要

In physics, biology, chemistry and engineering many applications of simulation science involve complex and evolving geometrical shapes. In many important problems these geometrical shapes exhibit topology changes or strong deformation which makes the numerical treatment very challenging. The highly accurate and efficient numerical solution of PDEs on these evolving domains is a challenging task. In the past decade research on this topic has been started. Especially methods using a geometry description which is separated from the computational mesh and in turn provide a more flexible handling of the geometry compared to traditional conforming mesh descriptions have become very popular in recent years. Although big progress has been made, there are important problems which require further research. Especially numerical methods which are flexible with respect to the geometrical configuration, robust and high order accurate at the same time are missing. The goal of this project is to fill this gap. Within this project finite element methods which are suitable for an integration with a separated geometry description and have provable higher order error bounds will be developed and analysed. This entails several issues. First, a higher order representation of the geometry is necessary. For that most often an implicit description with an indicator function is used. While the geometry description typically works well, the practical usability of the description poses a problem. For suitable finite element formulations integrals on cells which are cut are required. Due to the fact that these domains are only described implicitly, an accurate numerical integration is difficult. While for a second order accuracy robust and established methods exist, the extension to higher order accuracy requires new approaches in the discretization. Another important problem consists of the handling of the geometry evolution. With an implicit geometry description the evolution of the geometry can be handled without adapting the computational mesh, but new challenges arise with respect to the time discretization for problems on evolving domains. Within this project discretization methods will be developed which solve these problems. An important part of the related research consists of a rigorous error analysis which provides optimal order error bounds for the discretization methods. The methods considered in the project shall be implemented and investigated numerically based on the open source C++ software toolbox DUNE.

在物理，生物学，化学和工程学中，模拟科学的许多应用都涉及复杂而不断发展的几何形状。在许多重要问题中，这些几何形状表现出拓扑变化或较强的变形，这使得数值治疗非常具有挑战性。 PDE在这些不断发展的域上的高度准确和有效的数值解决方案是一项具有挑战性的任务。在过去的十年中，有关该主题的研究已经开始。尤其是使用与计算网格分离的几何描述的方法，并且与传统的符合网格描述相比，几何描述又提供了更灵活的几何处理方法。尽管取得了很大的进步，但仍有重要的问题需要进一步研究。尤其是缺少相对于几何配置，稳健和高阶精度的数值方法。该项目的目的是填补这一空白。在此项目有限元方法中，将开发和分析具有分离的几何描述并具有可证明的高阶误差范围的集成。这需要几个问题。首先，需要几何形状的高阶表示。因为通常使用指示功能的隐式描述。尽管几何描述通常效果很好，但描述的实际可用性却带来了问题。对于需要切割的细胞上的合适的有限元配方积分。由于这些域仅被隐式描述，因此很难进行准确的数值集成。尽管存在二阶精度，并且存在鲁棒和已建立的方法，但更高订单精度的扩展需要离散化的新方法。另一个重要的问题包括处理几何发展。通过隐式几何描述，可以在不适应计算网格的情况下处理几何形状的演变，但是对于不断发展的域上问题的时间离散化而产生了新的挑战。在此项目中，将开发解决这些问题的离散方法。相关研究的重要部分包括严格的错误分析，该分析为离散方法提供了最佳的订单误差界限。项目中考虑的方法应根据开源C ++软件工具箱沙丘进行数值实施和研究。