MFEM Workshop 2024 | Robust Containment Queries over Collections of Parametric Curves

The MFEM (Modular Finite Element Methods) project provides high-order mathematical calculations for large-scale scientific simulations. MFEM’s discretization algorithms enable high-performance computing systems to run these simulations more efficiently. The open-source project led by LLNL now has a global user community. Held on October 22-24, 2024, the fourth annual MFEM community workshop brought together users and developers for a review of software features and the development roadmap, a showcase of technical talks and applications, student lightning talks, an interactive Q&A session, and a visualization contest.

Jacob Spainhour of CU Boulder presented “Robust Containment Queries over Collections of Parametric Curves via Generalized Winding Numbers.” The containment query is an important geometric primitive in many multiphysics applications. For example, when initializing multimaterial Arbitrary Lagrangian-Eulerian (ALE) simulations, we often need to determine whether arbitrary quadrature points from the background mesh are inside or outside the regions associated with each material. However, existing methods require expensive refinement to accurately capture curved regions. At the same time, many methods are wholly incompatible with user-defined geometries that contain geometric and numeric gaps and/or self-intersections. In this work, we develop a containment query for 2D regions defined by rational Bezier curves that operates directly on curved objects. Our method relies on the generalized winding number (GWN), a mathematical construction that can be evaluated for each curve independently, making the derived containment query robust to non-watertightness. We use an adaptive algorithm to compute the GWN field exactly, which permits fast evaluation for points considered "distant" to the curve while being numerically stable for points that are arbitrarily close. Overall, this classification scheme greatly expands the types of bounding geometry that can be used directly in shaping applications without the need for otherwise expensive repair techniques. If time permits, we will also discuss our extensions of this idea to 3D shapes defined by parametric surfaces.

Learn more about MFEM at https://mfem.org/. LLNL-PRES-870858