Fast Galerkin Methods for Boundary Integral Reformulations of Time Dependent Partial Differential Equations

该项目将发展传热和微流动问题的数值方法。重点将放在处理可能随时间改变拓扑结构的移动几何形状上。应用领域，立即受益于提出的工作是在制造业，例如热成型工艺，凝固和熔化工艺。此外，微流动的数值解在Mems器件的设计、微生物游泳和沉积过程的模拟中也很重要。工程设计通常是一个交互的过程，因此速度和鲁棒性对基础数值方法至关重要。通过让研究生参与研究，拟议的活动也将有助于未来劳动力教育的卓越和发展。伽辽金边界积分法是求解抛物型偏微分方程的有效数值工具。边界积分算子在空间和时间上都是非局部的，这意味着它们的离散对应物是密集矩阵。然而，快速多极方法的时空版本将用于克服与密集矩阵代数相关的高数值成本。这个项目的主要目标是将这种方法扩展到移动几何的问题上。这里将开发几种新技术，包括处理抛物积分算子唯一奇点的正交方法和解决拓扑变化相关困难的时空边界流形的新的离散化方法。所建立的数值方法将应用于瞬态Stokes流动问题和Stefan问题。该项目还将研究形状优化技术，以解决固体-液体界面的未知演变。通过最小化能量泛函来解决自由曲面问题很好地符合伽辽金离散化的变分公式，并且比传统的时间步进方法更具鲁棒性和稳定性。

The project will develop numerical methods for heat transfer and micro flow problems. The focus will be on handling moving geometries that may change the topology over time. Application areas that immediately benefit from the proposed work are in manufacturing, for instance hot forming processes, solidification and melting processes. Moreover, the numerical solution of micro flows is important in the design of Mems devices, the simulation of microbial swimmers and sedimentation processes. Engineering design is often interactive process, and therefore speed and robustness are essential for the underlying numerical methods. By engaging a graduate student in research, the proposed activities will also contribute to excellence and growth in the education of the future workforce.Galerkin boundary integral methods are an effective numerical tool to solve parabolic partial differential equations. Boundary integral operators are non-local in space and time, which implies that their discrete counterparts are dense matrices. However, a space-time version of the fast multipole method will be used to overcome the high numerical cost associated with dense matrix algebra. The main goal of this project is to expand this methodology to problems with moving geometries. Here several new techniques will be developed, this includes quadrature methods to handle the unique singularities of parabolic integral operators and new discretization methods of the space-time boundary manifold which will address the difficulties associated with topology changes. The numerical methods developed will be applied to transient Stokes flow problems and Stefan problems. This project will also investigate shape optimization techniques to solve for the unknown evolution of the solid-liquid interface. Solving free surface problems by minimizing an energy functional fits well within the variational formulation of the Galerkin discretization and promises to be more robust and stable than the more conventional time stepping methods.