An Integral equation-based solver for the Laplace-Beltrami operator on triangulated surfaces
Abstract
Project Summary The goal of my project An integral equation-based solver for the Laplace-Beltrami operator on triangulated surfaces is to develop a fast, high-order accurate solver for inverting the surface Laplacian on boundaries in three dimensions which have been discretized as piecewise curvilinear triangles. I intend to design both iterative (based on the fast multipole method) and direct (based on fast hierarchical linear algebra methods) solvers which will be based on an integral equation formulation for the Laplace-Beltrami problem. The Laplace-Beltrami operator (also known as the surface Laplacian) arises in several applications in computational electromagnetics - namely in electro/magnetostatics in multiply connected geometries and alternative representations of exterior time-harmonic electromagnetic fields (e.g. the generalized Debye source representation). On globally defined analytic surfaces, the surface Laplacian can usually be inverted using spectral methods or other high-order schemes in the parameterization domain. However, many applications arising in real-world engineering problems require the use of Computer Aided Drawing (CAD) or Engineering (CAE) software, which generally only output triangulated surfaces. In this case, alternative methods for inverting the surface Laplacian (and various other PDEs and integral equations) are needed which do not rely on analytic parameterizations. The integral equation method I intend to build for inverting the surface Laplacian will require development in three areas: (1) generating curvilinear triangles from CAD software, (2) extending a quadrature method for weakly singular layer potentials known as Quadrature by Expansion to three dimensions, and (3) deriving an integral equation formulation of the Laplace-Beltrami problem. Constructing a fast solver for the surface Laplacian is only one piece of a larger effort to design high-order integral equation methods for computational electromagnetics problems in complicated geometry. When carefully formulated, integral equations have many advantages over standard PDEbased methods: matrix-conditioning which is indicative of the physical problem, compatibility with complex geometries and highly-adaptive discretizations, and efficient computation in unbounded domains, just to name a few. This project is most relevant to ONR’s Science and Technology Organization, Code 31: Command, Control, Communications, Computers, Intelligence, Surveillance and Reconnaissance. In particular, the work is most directly related to the effort of Division 311: the Applied and Computational Analysis Program. 1 of 1
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Aug 12, 2016
- Source ID
- N000141512669
Entities
People
- Michael P. O Neil
Organizations
- New York University
- Office of Naval Research
- United States Navy