Data-driven Stabilization of Nitsche's Method

TL;DR

The paper addresses the costly per-cell estimation of the stabilization parameter $\lambda$ in Nitsche's method for unfitted finite element methods by introducing a data-driven surrogate trained on local cut configurations. Representing the cut by a line and distances to predefined feature points, the authors train a neural network to predict $\lambda$ with $O(1)$ per-cell cost, significantly reducing computation compared to the generalized eigenvalue approach. Across 2D Poisson-based finite cell benchmarks, the data-driven estimator achieves relative errors below about $5\%$ while delivering large speedups, especially on GPUs, and integrates into existing codes with minimal modifications. The method offers a practical, scalable path to efficient stabilization in large-scale unfitted simulations using 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.

Figures (13)

• Figure 1: The representation of the cut configuration and its normalization. (a), (c) two cut configurations before normalization with the same cut line and flipped starting and end points. (b), (d) the normalized form of the cut configurations in (a) and (c), respectively. Note that the normalized representation can distinguish between the cut configurations in (a) and (c)
• Figure 2: The rectangular cut configuration, where the cut sliver is defined using the width of the rectangle $d$
• Figure 3: The stabilization parameter from the rectangular cut configuration test, where the width of the rectangular cut sliver $d$, see Figure \ref{['fig:rectangle_cut_config']}, is progressively reduced in (a) linear scale and (b) logarithmic scale. The data is presented in order of the values of the stabilization parameter $\lambda$
• Figure 4: (a) Logarithmic spacing of feature points on the edges of the cell, where the start points are placed on the top edge and the end points are placed on the other edges according to the normalization technique in Section \ref{['sec:cut_config']} and (b) the distribution of the stabilization parameter $\lambda$ with linear and logarithmic spacing of start and end points
• Figure 5: The distribution of the stabilization parameter $\lambda$ in the training dataset, where 399 feature points with logarithmic spacing are placed on each edge of the cell, resulting in a total of $477,603$ data points