SFU Applied & Computational Math Seminar Series: Nathan King

Many geometry processing tasks can be performed by solving partial differential equations (PDEs) on surfaces. These PDEs usually involve boundary conditions (e.g., Dirichlet or Neumann) defined anywhere on the surface, not just on the physical (exterior) boundary of an open surface. This talk discusses how to handle BCs on the interior of a surface while solving PDEs with the closest point method (CPM). The CPM is an embedding method, i.e., it solves the surface PDE by solving a PDE defined in a space surrounding the surface. The PDE is commonly solved using standard Cartesian numerical methods (e.g., finite-differences and Lagrange interpolation). Complex surfaces with high-curvatures and/or thin regions impose restrictions on the size of the embedding space. Therefore, for complex surfaces, fine resolution grids must be used to fit within the embedding space. We develop a matrix-free solver that can scale to millions of degrees of freedom to allow for PDEs to be solved on complex shapes. Our use of a closest point surface representation provides a general framework to handle any surface that allows closest point computation, e.g., parametrizations, point clouds, level-sets, neural implicits, etc. The surface can be open or closed, orientable or not, of any codimension, and even mixed-codimension. Therefore, the approach presented provides a general framework for geometry processing on complex surfaces given by general surface representations.