Boundary correspondence of planar domains for isogeometric analysis based on optimal mass transport

Abstract

Domain parameterization is the process of setting up a map from a parametric domain to a computational domain. It is a key step in isogeometric analysis and has regained much attention in recent years. A prerequisite for domain parameterization is that a correspondence between the boundary of the parametric domain and that of the computational domain should be established. The boundary correspondence has a significant influence on the subsequent parameterization and numerical simulation. Currently, such correspondence is generally provided manually by users, which is very cumbersome and subjects to trial and error. In this paper, we propose an automatic approach to compute a correspondence between the boundaries of a unit square and a planar domain based on the theory of optimal mass transport (OMT). Given the boundary representation of a planar domain, the problem becomes to select four corner points on the boundary such that the difference between the curvature measure of the boundary of the planar domain and that of the unit square is minimized. We formulate the problem into an optimization problem, the objective function of which includes the transport cost between two curvature measures and the length differences of the opposite edges of the planar domain. Minimizing the objective function is equivalent to maximizing the similarity between the unit square and the computational domain. We develop an efficient algorithm to solve the optimization problem by combining Sinkhorn’s algorithm with the L-BFGS method. Numerous examples show that our approach can produce satisfactory boundary correspondence results which are comparable to manually selected ones.