Tom Beale : Computing Integrals on Surfaces

Suppose you need to compute an integral over a general surface numerically. How would you do it? You could triangulate the surface, or you might use coordinate charts. Either way is a lot of work, maybe more than you want to do if you have a large number of surfaces. I will describe a fairly simple method, appropriate for smooth, closed surfaces, developed by a former grad student here, Jason Wilson, in his Ph.D. thesis, including proofs that his algorithm works. I will then discuss the extension to integrals for potentials defined by densities on surfaces, such as harmonic functions. In that case the integrand has a singularity; special treatment is needed, and some interesting math comes in. Another of our former Ph.D.'s, Wenjun Ying, has contributed to that work (among many projects of his). Such integrals occur in several scientific contexts; I will especially mention Stokes flow (fluid flow dominated by viscosity), appropriate for modeling some aspects of biology on small scales. For more information, see J. t. Beale, W. Ying, and J. R. Wilson, A simple method for computing singular or nearly singular integrals on closed surfaces at my web site or at http://arxiv.org/abs/1508.00265

• Category: Graduate/Faculty Seminar
• Duration: 01:14:32
• Date:  January 25, 2016 at 11:55 AM
• Views: 139
• Tags:   seminar, Graduate/faculty Seminar