Fast, High-Order Accurate Integral Equation Methods and Application to PDE-Constrained Optimization

Abstract

Over the last several decades, the development of fast, high-order accurate, and robust integral equation methods for computational physics has gained increasing attention. Using integral equation formulation as a global statement in contrast to a local partial differential equation (PDE) formulation offers several unique advantages. For homogeneous PDEs, the boundary integral equation (BIE) formulation allows accurate handling of complex and moving geometries, and it only requires a mesh on the boundary, which is much easier to generate as a result of the dimension reduction. With the acceleration of fast algorithms like the Fast Multipole Method (FMM), the computational complexity can be reduced to O(N), where N is the number of degrees of freedom on the boundary. Using standard potential theory decomposition, inhomogeneous PDEs can be solved by evaluating a volume potential over the inhomogeneous source domain, followed by a solution of the homogeneous part.

Despite the advantages of BIE methods in easy meshing, near-optimal efficiency, and well conditioning, the accurate evaluation of nearly singular integrals is a classical problem that needs to be addressed to enable simulations for practical applications. In the first half of this thesis, we develop a series of product integration schemes to solve this close evaluation problem. The use of differential forms provides a dimensional-agnostic way of integrating the nearly singular kernels against polynomial basis functions analytically. So the problem of singular integration gets reduced to a matter of source function approximation. In 2D, this procedure has been traditionally portrayed by building a connection to complex Cauchy integral, then supplemented by a complex monomial approximation. In $3$D, the closed differential form requirement leads to the design of a new function approximation scheme based on harmonic polynomials and quaternion algebra. Under a similar framework, we develop a high-order accurate product integration scheme for evaluating singular and nearly singular volume integral equations (VIE) in complex domains using regular Cartesian grids discretization. A high-order accurate source term approximation scheme matching smooth volume integrals on irregular cut cells is developed, which requires no function extension.

BIE methods have been widely used for studying Stokes flows, incompressible flows at low Reynolds' number, in both biological systems and microfluidics. In the second half of this thesis, we employ the BIE methods to simulate and optimize Stokes fluid-structure interactions. In 2D, a hybrid computational method is presented for simulating cilia-generated fluid mixing as well as the cilia-particle hydrodynamics. The method is based on a BIE formulation for confining geometries and rigid particles, and the method of regularized Stokeslets for the cilia. In 3D, we use the time-independent envelop model for arbitrary axisymmetric microswimmers to minimize the power loss while maintaining a target swimming speed. This is a quadratic optimization problem in terms of the slip velocity due to the linearity of Stokes flow. Under specified reduced volume constraint, we find prolate spheroids to be the most efficient micro-swimmer among various families of shapes we considered. We then derive an adjoint-based formulation for computing power loss sensitivities in terms of a time-dependent slip profile by introducing an auxiliary time-periodic function, and find that the optimal swimmer displays one or multiple traveling waves, reminiscent of the typical metachronal waves observed in ciliated microswimmers.