Data-driven Stabilization of Nitsche's Method

Abstract

The weak imposition of essential boundary conditions is an integral aspect of unfitted finite element methods, where the physical boundary does not in general coincide with the computational domain. In this regard, the symmetric Nitsche's method is a powerful technique that preserves the symmetry and variational consistency of the unmodified weak formulation. The stabilization parameter in Nitsche's method plays a crucial role in the stability of the resultant formulation, whose estimation is computationally intensive and dependent on the particular cut configuration using the conventional eigenvalue-based approach. In this work, we employ as model problem the finite cell method in which the need for the generation of a boundary-conforming mesh is circumvented by embedding the physical domain in a, typically regular, background mesh. We propose a data-driven estimate based on machine learning methods for the estimation of the stabilization parameter in Nitsche's method that offers an efficient constant-complexity alternative to the eigenvalue-based approach independent of the cut configuration. It is shown, using numerical benchmarks, that the proposed method can estimate the stabilization parameter accurately and is by far more computationally efficient. The data-driven estimate can be integrated into existing numerical codes with minimal modifications and thanks to the wide adoption of accelerators such as GPUs by machine learning frameworks, can be used with virtually no extra implementation cost on GPU devices, further increasing the potential for computational gains over the conventional eigenvalue-based estimate. The proposed model is tested on both Intel CPU as well as NVIDIA GPU hardware, showing that while it is already many times more efficient on the CPU compared to the eigenvalue-based estimate, its efficiency margin is even larger on modern GPU devices.