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

该项目将开发传热和微流问题的数值方法。重点将放在处理移动的几何图形，可能会改变随着时间的推移拓扑结构。 直接受益于拟议工作的应用领域是制造业，例如热成型工艺，凝固和熔化工艺。此外，微流场的数值解在微机电系统器件的设计、微生物游动和沉降过程的模拟中具有重要意义。工程设计往往是一个交互的过程，因此速度和鲁棒性对于底层的数值方法至关重要。 通过让一名研究生参与研究，拟议的活动还将有助于未来劳动力教育的卓越和增长。Galerkin边界积分方法是求解抛物型偏微分方程的一种有效的数值工具。 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算子是稠密矩阵。然而，一个时空版本的快速多极子方法将被用来克服高数值成本与密集矩阵代数。 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。这里将开发几种新技术，这包括处理抛物积分算子的独特奇异性的求积方法和时空边界流形的新离散化方法，这些方法将解决与拓扑变化相关的困难。 所发展的数值方法将应用于瞬态Stokes流问题和Stefan问题。该项目还将研究形状优化技术，以解决固液界面的未知演变。通过最小化能量泛函来解决自由表面问题很好地符合Galerkin离散化的变分公式，并且有望比更传统的时间步进方法更鲁棒和稳定。

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.