Boundary Correspondence for Iso-Geometric Analysis Based on Deep Learning

One of the key problems in isogeometric analysis(IGA) is domain parameterization, i.e., constructing a map between a parametric domain and a computational domain. As a preliminary step of domain parameterization, the mapping between the boundaries of the parametric domain and the computational domain should be established. The boundary correspondence strongly affects the quality of domain parameterization and thus subsequent numerical analysis. Currently, boundary correspondence is generally determined manually and only one approach based on optimal mass transport discusses automatic generation of boundary correspondence. In this article, we propose a deep neural network based approach to generate boundary correspondence for 2D simply connected computational domains. Given the boundary polygon of a planar computational domain, the main problem is to pick four corner vertices on the input boundary in order to subdivide the boundary into four segments which correspond to the four sides of the parametric domain. We synthesize a dataset with corner correspondence and train a fully convolutional network to predict the likelihood of each boundary vertex to be one of the corner vertices, and thus to locate four corner vertices with locally maximum likelihood. We evaluate our method on two types of datasets: MPEG-7 dataset and CAD model dataset. The experiment results demonstrate that our algorithm is faster by several orders of magnitude, and at the same time achieves smaller average angular distortion, more uniform area distortion and higher success rate, compared to the traditional optimization-based method. Furthermore, our neural network exhibits good generalization ability on new datasets.