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

该项目将开发用于传热学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的移动几何图形上。直接受益于拟议工作的应用领域是制造业，例如热成形工艺、凝固和熔化工艺。此外，微流场的数值解对于MEMS器件的设计、微生物游泳者的模拟和沉积过程的模拟都具有重要的意义。工程设计往往是一个交互的过程，因此速度和稳健性对于基本的数值方法是至关重要的。通过吸引研究生参与研究，建议的活动也将有助于卓越和未来劳动力的教育成长。Galerkin边界积分方法是求解抛物型偏微分方程组的有效数值工具。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稠密矩阵。然而，将使用快速多极子方法的时空版本来克服与稠密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具有移动几何的问题。这里将发展几种新的技术，包括处理抛物积分算子的唯一奇性的求积方法和解决与拓扑变化相关的困难的时空边界流形的新的离散化方法。所发展的数值方法将应用于瞬变的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-液界面未知的演化问题。通过最小化能量泛函来求解自由面问题与Galerkin离散化的变分形式很好地吻合，并且有望比更传统的时间推进方法更稳健和稳定。

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.