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

本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（局部精化局部拟合分层网格），并耦合先进的多重网格方法与域嵌入/虚拟域和区域分解技术，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方案在各种科学和工业应用中是非常重要的，颗粒流，运动部件系统的最优控制，最优形状设计。在具有变化的几何形状的问题的情况下，需要与变化的网格相关联的方程的大量有限元/有限体积系统的解决方案，找到准确性和算法效率之间的正确平衡是非常困难的，特别是在并行平台上。最近的研究结果表明，高效的并行方法可以设计上述问题的耦合多重网格预处理与区域分解和虚拟/嵌入域技术的网格块结构或几乎结构，分别。本项目的主要目标是：开发和研究二维和三维复杂几何体的局部精化局部拟合层次网格的并行生成器;开发和研究基于虚拟（嵌入）区域和区域分解技术的耦合先进多重网格算法的可扩展并行迭代求解方法;将开发的网格生成器和迭代求解方法应用于复杂移动边界流动问题的数值模拟，例如，流动问题的数值模拟。例如，在一个实施例中，粒子流的Navier-Stokes方程，并在并行平台（集群工作站和IBM SP/2）的最佳形状设计问题。工作将在化学，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.