Scalable Parallel Computational Methods for Partial Differential Equations with Moving and Varying Boundaries

该项目将研究新的并行数值方法，依赖于一种特殊类型的有限元网格，这里称为lrlfh网格（局部精炼局部拟合分层网格），以及先进的多网格方法与域嵌入/虚拟域和域分解技术的耦合，用于求解具有移动和变化边界的线性和非线性偏微分方程。这些问题的有效解决在各种科学和工业应用中非常重要，例如，颗粒流动，运动部件系统的最佳控制，最佳形状设计。在具有不同几何形状的问题中，需要解决与不同网格相关的大量有限元/有限体积方程组，在精度和算法效率之间找到适当的平衡是非常困难的，特别是在并行平台上。最近的研究结果表明，将多网格预处理与域分解和虚拟/嵌入域技术相结合，可以设计出针对上述问题的高效并行方法。该项目的主要目标是：开发和研究用于复杂二维和三维几何的局部精炼局部拟合分层网格的并行生成器。开发和研究基于前置条件的可扩展并行迭代求解方法，该方法基于高级多网格算法与虚拟（嵌入）域和域分解技术的耦合。将网格生成器和迭代求解方法应用于具有复杂移动边界的流动问题的数值模拟，例如，由Navier-Stokes方程控制的颗粒流动，以及并行平台（工作站集群和IBM SP/2）上的最佳形状设计问题。这项工作将对化学、机械和航空航天工程中遇到的各种现象的模拟产生影响。

This project will investigate novel parallel numerical methods, relying on a special type of finite element meshes, called here LRLFH-meshes (Locally Refined Locally Fitted Hierarchical meshes), and on coupling of advanced multigrid methods with domain embedding/fictitious domains and domain decomposition techniques, for solving linear and nonlinear Partial Differential Equations with moving and varying boundaries. The efficient solution of such problems is very important in various scientific and industrial applications, e.g., particulate flows, optimal control of systems with moving parts, optimal shape design. In the case of problems with varying geometry, requiring the solution of a large number of finite element/finite volume systems of equations associated with varying meshes, finding the right balance between accuracy and algorithmic efficiency is very difficult, particularly on parallel platforms. Recent results indicate that highly efficient parallel methods can be designed for the above problems by coupling multigrid preconditioning with domain decomposition and fictitious/embedding domain techniques on meshes which are block structured or almost structured, respectively. The main objectives of this project are:To develop and investigate parallel generators for Locally Refined Locally Fitted Hierarchical meshes for complex two and three dimensional geometries.To develop and investigate scalable parallel iterative solution methods with preconditioning based on coupling advanced multigrid algorithms with fictitious (embedding) domain and domain decomposition techniques.To apply the mesh generators and iterative solution methods, to be developed, to the numerical simulation of flow problems with complex moving boundaries, e. g., particulate flows governed by the Navier-Stokes equations, and to optimal shape design problems on parallel platforms (clusters of workstations and IBM SP/2).The work will have impact on the simulation of various phenomena encountered in chemical, mechanical and aerospace engineering.