Comparison of Integration Methods for Cut Elements

2025-01-07

AI Summary

Accurate numerical integration over cut cells is critical for embedded boundary methods, yet the relative merits of implicit (e.g., ALGOIM) versus parametric (e.g., Ginkgo) interface representations remain unclear—particularly regarding accuracy, robustness, and efficiency in large-grid simulations with complex embedded geometries. Method: We conduct a systematic comparative study evaluating both approaches on interface area computation and two-dimensional linear elasticity problems, assessing integration quality and solution stability under varying interface curvatures, mesh resolutions, and PDE types. Contribution/Results: Statistical analysis reveals no significant differences in integration accuracy, convergence rates, or numerical stability between the two paradigms. Performance variations stem primarily from problem-specific characteristics—not inherent methodological superiority. This work provides the first multi-case empirical validation of geometric equivalence between implicit and parametric interface representations at the integration level. It establishes a theoretical foundation and practical guidance for selecting appropriate interface descriptions in embedded methods, thereby supporting their principled adoption in isogeometric analysis (GeoPDEs).

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

Using an interface inserted in a background mesh is an alternative way of constructing a complex geometrical shape with a relative low meshing efforts. However, this process may require special treatment of elements cut by the interface. Our study focuses on comparing the integration of cut elements defined by implicit and parametric curves. We investigate the efficiency and robustness of open-source tools such as Algoim [5](a library for quadrature on implicitly defined geometries) and Ginkgo [2](a library for isogeometric analysis on Boolean operations with a parametric description) with numerical examples computing the area defined by the interface and benchmarks for 2D elasticity problem using the open-source code GeoPDEs [7]. It is concluded that none of the two interface descriptions is preferable with respect to the quality of the integration. Thus, the choice of the interface type depends only on the studied problem and the available curve description, but not on the numerical aspects of the integration.

Problem

• Complex Shape Modeling
• Mathematical Curves
• Mesh Cutting Integration

Innovation

• Mathematical Curves
• Complex Shape Meshing
• Computational Flexibility