Adaptive optimization of isogeometric multi-patch discretizations using artificial neural networks

TL;DR

This work presents a novel approach for finding a suitable isogeometric function space for a given PDE without sacrificing the tensor-product structure of the underlying spline space by using the fact that different reparameterizations of the same computational domain lead to different isogeometric function spaces while preserving the geometry.

Abstract

In isogeometric analysis, isogeometric function spaces are employed for accurately representing the solution to a partial differential equation (PDE) on a parameterized domain. They are generated from a tensor-product spline space by composing the basis functions with the inverse of the parameterization. Depending on the geometry of the domain and on the data of the PDE, the solution might not have maximum Sobolev regularity, leading to a reduced convergence rate. In this case it is necessary to reduce the local mesh size close to the singularities. The classical approach is to perform adaptive h-refinement, which either leads to an unnecessarily large number of degrees of freedom or to a spline space that does not possess a tensor-product structure. Based on the concept of r-adaptivity we present a novel approach for finding a suitable isogeometric function space for a given PDE without sacrificing the tensor-product structure of the underlying spline space. In particular, we use the fact that different reparameterizations of the same computational domain lead to different isogeometric function spaces while preserving the geometry. Starting from a multi-patch domain consisting of bilinearly parameterized patches, we aim to find the biquadratic multi-patch parameterization that leads to the isogeometric function space with the smallest best approximation error of the solution. In order to estimate the location of the optimal control points, we employ a trained residual neural network that is applied to the graph surfaces of the approximated solution and its derivatives. In our experimental results, we observe that our new method results in a vast improvement of the approximation error for different PDE problems on multi-patch domains.

Figures (18)

• Figure 1: Stationary solution to the heat equation on an L-shaped domain
• Figure 2: Splitting a quadrilateral domain into four triangles.
• Figure 3: Triangle deformations. The deformed triangular Bézier control nets are shown in black. The green lines visualize the point-wise change of the parameterization from undeformed (red) to deformed (black).
• Figure 4: Comparison of the averaging approaches presented in Section \ref{['biquadratic']}. The deformed tensor-product Bézier control nets are shown in black. The green lines visualize the point-wise change of the parameterization from undeformed (red) to deformed (black).
• Figure 5: $L^2$-projection on a square domain with a corner singularity: Exact solution and convergence of $L^2$-errors.