Face offsetting: A unified approach for explicit moving interfaces

Abstract

Dynamic moving interfaces are central to many scientific, engineering, and graphics applications. In this paper, we introduce a novel method for moving surface meshes, called the face offsetting method, based on a generalized Huygens' principle. Our method operates directly on a Lagrangian surface mesh, without requiring an Eulerian volume mesh. Unlike traditional Lagrangian methods, which move each vertex directly along an approximate normal or user-specified direction, our method propagates faces and then reconstructs vertices through an eigenvalue analysis locally at each vertex to resolve normal and tangential motion of the interface simultaneously. The method also includes techniques for ensuring the integrity of the surface as it evolves. Face offsetting provides a unified framework for various dynamic interface problems and delivers accurate physical solutions even in the presence of singularities and large curvatures. We present the theoretical foundation of our method, and also demonstrate its accuracy, efficiency, and flexibility for a number of benchmark problems and a real-world application.

Woodification: User‐Controlled Cambial Growth Modeling

“…Surface propagation is commonly implemented using the level set method, which operates on a volumetric representation for robustness, but suffer appearance artifacts and additional volumetric work for a surface simulation, which can be reduced by adaptive and compressed level set methods, (e.g., [HNB * 06]). Surface meshes can also be robustly propagated, (e.g., [Jia07]). We utilize the deformable simplicial complex (DSC) [Mis10,MB12] which also maintains an interior and exterior tetrahedral mesh.…”

Anisotropic mesh adaptation for evolving triangulated surfaces

“…In these tests, we propagate each vertex using the fourth-order Runge-Kutta integration scheme and then adapt the surface anisotropically. The time step was controlled using the approach of Jiao [21] to prevent mesh folding. We perform anisotropic mesh smoothing at every time step and invoke the full-edged anisotropic adaptation every few iterations or when the time step becomes too small.…”

Optimizing Surface Triangulation Via Near Isometry with Reference Meshes

“…This equation constrains the search direction within the local tangent space at vertex i without having to project its neighborhood onto a plane. We estimate the tangent space as in [23]. In particular, at each vertex v, suppose v is the origin of a local coordinate frame, and m is the number of the faces incident on v. Let N be an 3 × m matrix whose ith column vector is the unit outward normal to the ith incident face of v, and W be an m × m diagonal matrix with W ii equal to the face area associated with the ith face.…”

“…To address this problem, we introduce an asynchronous step-size control. For each triangle p i p j p k , we solve for the maximum α ≤ 1 such that [23]. We reduce d i at vertex i by a factor equal to the minimum of the αs of its incident faces.…”

“…We integrate the motion of the interface using the face-offsetting method in [23] while redistributing the vertices using the initial mesh as the reference mesh. In this test the angles and areas of the triangles were well preserved even after very large deformation.…”