Volumetric Boundary Correspondence for Isogeometric Analysis Based on Unbalanced Optimal Transport

Abstract

Domain parameterization, i.e., constructing a map from a parameter domain to a computational domain, is a key step in isogeometric analysis. Before parameterizing the interior of the computational domain, the boundary correspondence between the parametric domain and the computational domain is required by most domain parameterization methods, and the quality of boundary correspondence has a great effect on the quality of subsequent interior parameterization and analysis. Previous methods manually fulfill this task in general, which is tedious and subject to trial and error. In this paper, we propose an automatic method to compute such a correspondence between the boundary of a unit cube and the boundary of a volumetric computational domain based on the theory of unbalanced optimal transport. Given the boundary of a volumetric computational domain, the main task is to select 8 corner points and 12 curves connecting the 8 corner points on the boundary to divide the boundary into six surface patches (corresponding to the six faces of a unit cube), such that the difference between the Gaussian and mean curvature measures of the input boundary and those of the unit cube is minimized. We formulate this problem as an optimal mass transport problem, which is subject to some restrictions on the areas of the 6 surface patches and the lengths of the 12 boundary curves. To simplify the problem, a spherical intermediate domain is introduced by spherical parameterization of the computational domain in order to reduce the problem to be solved on a sphere. Riemannian L-BFGS method is adopted to solve the optimization efficiently. Experimental examples demonstrate that the proposed approach can produce satisfactory results which are competitive with the manually designed method.