Non-iterative generation of optimized meshes for finite element simulations with deep learning

Abstract The finite element method is one of the most widely used computational methods in engineering and science. It provides approximate solutions to boundary value problems. The quality of these solutions critically depends on the underlying discretization, the so-called mesh. To optimize the mesh, adaptive refinement methods have been proposed over the last years that can improve mesh quality over a series of iteration steps. Herein, we propose a novel deep learning architecture that can cut short the process of mesh optimization. This architecture exploits fundamental invariance and equivariance properties to keep the amount of training data modest. It can generate high-quality meshes for a given boundary value problem and a desired target approximation error in a direct, non-iterative way. We demonstrate the performance of our method by the application to standard two-dimensional linear-elastic elasticity problems. There, our method generates meshes that reduce the solution error by $$22.6\%$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>22.6</mml:mn> <mml:mo>%</mml:mo> </mml:mrow> </mml:math> (median) compared to uniform meshes with the same computational demand.