Parameterization of Contractible Domains Using Sequences of Harmonic Maps

Abstract

Abstract. In this paper, we propose a new method for parameterizing a contractible domain (called the computational domain) which is defined by its boundary. Using a sequence of harmonic maps, we first build a mapping from the computational domain to the parameter domain, i.e., the unit square or unit cube. Then we parameterize the original domain by spline approximation of the inverse mapping. Numerical simulations of our method were performed with several shapes in 2D and 3D to demonstrate that our method is suitable for various shapes. The method is particular useful for isogeometric analysis because it provides an extension from a boundary representation of a model to a volume representation.

Planar domain parameterization with THB-splines

“…In the first case we plan to use results on harmonic mappings between multiply connected domains from Duren and Hengartner (1997). In the second case, since the theoretical results for harmonic mappings between three-dimensional domains are less strong than in the bivariate case, we may also try to use other mappings, similar to the approach used in Nguyen and Jüttler (2012).…”

“…A related approach has been used in Nguyen and Jüttler (2012). In that paper, the harmonic mapping was computed by a web-spline-based approach (Höllig, 2003), which requires a careful choice of the parameter that controls the influence of the boundary conditions.…”

Two-dimensional domain decomposition based on skeleton computation for parameterization and isogeometric analysis

“…The domain can then be parameterized using any of the methods proposed in [20,16,37,38]. The criteria for choosing the optimal knot vector are described in [39][40][41].…”

Spline Parameterization of Complex Planar Domains for Isogeometric Analysis

“…Kreisselmeier-Steinhauser function value is used for assigning a constraint to each sub-problem. Harmonic functions are used by Jüttler et al [18] for parameterization of the computational domain. Parameterization of the domain is obtained by establishing bijective mapping, between original domain and the parameter domain (unit square or unit cube).…”