Integral parameterization of volumetric domains via deep neural networks

Abstract

Isogeometric Analysis (IGA) is a promising technique that integrates geometric modeling with numerical analysis. An essential step in IGA is domain parameterization, which aims to establish a parametric representation for a given computational domain. Specifically, it involves defining a spline-based mapping from the standard parametric domain to the computational domain. Typically, domain parameterization is performed in two stages: identifying an appropriate boundary correspondence and then parameterizing the interior region. However, this separation of the parameterization process often leads to a degradation in the quality of the parameterization. To attain high-quality parameterization, it is essential to optimize both the boundary correspondence and the interior mapping simultaneously. This approach is referred to as integral parameterization. Previous research has introduced integral parameterization methods for planar domains using neural networks. The goal of the current paper is to extend the method to handle integral parameterization of volumetric domains. We utilize Multi-Layer Perceptrons (MLPs) to represent the inverse parameterization mappings, incorporating efficient distortion measures into the loss function. To ensure stable training and achieve accurate results, we employ several techniques, including a four-stage training procedure and the smooth cuboid approach. The performance of our method is evaluated on multiple volumetric domains, and experimental results demonstrate its superiority over existing state-of-the-art techniques.