Volume preserving mesh parameterization based on optimal mass transportation. (January 2017) - Su, Kehua; Chen, Wei; Lei, Na; Zhang, Junwei; Qian, Kun; Gu, Xianfeng
Abstract: In order to convert a finite element mesh model to the spline representation for the purpose of isogeometric analysis, one needs to parameterize the solid. This work introduces a novel volumetric parameterization method, which guarantees to be free of volume distortion. Given a simply connected tetrahedral mesh with a single boundary surface, we first compute a harmonic map from the boundary triangle mesh to the unit sphere by non-linear heat diffusion method; then we use the surface harmonic map as the boundary condition to compute the volumetric harmonic map to parameterize the solid onto the unit solid ball; finally we compute an optimal mass transportation map from the unit solid ball with the push-forward volume element induced by the harmonic map onto itself with the Euclidean volume element. The composition of the volumetric harmonic map and the optimal mass transportation map gives an volume-preserving parameterization. The method has solid theoretic foundation, and is based on conventional algorithms in computational geometry, easy to implement. We have thoroughly tested our algorithm on many solid models in reality. The experimental results demonstrate the efficiency and efficacy of the proposed method. To the best of our knowledge, it is the first work addressing volume-preserving parameterization in the literature. Highlights: The method is based on solid mathematical foundation. The framework can be straightforwardly generalized to arbitrary dimension. The Algorithm is easy to implement. The volume preserving parameterization has broad range of applications such as IGA.