Comparative study of different quadrature methods for cut elements

2026-02-20

AI Summary

This study addresses the critical challenge of numerical integration over cut cells in unfitted finite element methods, where efficiency, accuracy, and robustness must be simultaneously ensured. Despite the proliferation of quadrature schemes, a systematic evaluation of existing approaches has been lacking. To bridge this gap, this work presents the first unified benchmark of mainstream open-source quadrature methods for cut cells, introducing a comprehensive 2D/3D test suite that encompasses both implicit and explicit boundary representations. The evaluation rigorously assesses each method’s accuracy, computational efficiency, generality, and robustness, while also quantifying the influence of input parameters on integration error. The study provides a reproducible comparison framework, an open dataset, and complete benchmark results, establishing a reliable foundation for future algorithmic development and practical engineering applications.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Uncertainty Representations
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

The quadrature of cut elements is crucial for all Finite Element Methods that do not apply boundary-fitted meshes. It should be efficient, accurate, and robust. Various approaches balancing these requirements have been published, with some available as open-source implementations. This work reviews these open-sources codes and the methods used. Furthermore, benchmarking examples are developed for 2D and 3D geometries. Implicit and explicit boundary descriptions are available for all models. The different examples test the efficiency, accuracy, versatility, and robustness of the codes. Special focus is set on the influence of the input parameter, which controls the desired quadrature order, on the actual integration error. A detailed comparison of the discussed codes is carried out. The benchmarking allows a conclusive comparison and presents a valuable tool for future code development. All tests are published in an accompanying open-source repository.

Problem

• quadrature
• cut elements
• finite element methods
• numerical integration
• benchmarking

Innovation

• cut elements
• quadrature methods
• benchmarking
• finite element methods
• open-source