Object‐oriented hierarchical mesh refinement with CHARMS

Abstract

key words: Finite element, mesh refinement, hierarchical, adaptive approximation, object-oriented design SUMMARY A new approach to the construction of adaptive approximations on finite element meshes and also in more general settings, including approximations on subdivision surfaces, had been recently proposed by Krysl, Grinspun, and Schröder. This paper outlines how the general refinement algorithms may be specialized to some common finite element discretizations in two and three dimensions, discusses the design of nested meshes, and delves into the algorithmic, design, and implementation issues in an object oriented software framework

Adaptive computations using material forces and residual-based error estimators on quadtree meshes

“…This procedure can be performed to construct conforming shape functions for quadtree elements with any number of hanging nodes by using the corresponding polygonal reference element. On using this approach there is no need to restrict the number of hanging nodes to one on each edge (2:1 rule) as is needed in some of the other techniques [2,3,[6][7][8]10]. …”

“…Due to the presence of these hanging nodes, incompatibilities arise in classical finite element approximations. Special techniques have been used to construct conforming approximations over quadtree meshes: constraining hanging nodes to corner nodes [1], adding temporary elements to construct a compatible mesh [2,3], Lagrange multipliers and penalty or Nitsche's method to impose constraints [4,5], using hierarchical enrichment [6,7] or B-splines [8], and natural neighbor basis functions [9,10]. In this paper the method developed in Reference [9] is employed to resolve the problem associated with the presence of hanging nodes.…”

Extended finite element method on polygonal and quadtree meshes

“…As an alternative, directly constructing conforming approximations on a quadtree is appealing-the quadtree data structure is untouched and a standard Galerkin formulation suffices with no changes in the properties of the stiffness matrix. The application of B-spline finite elements [27], hierarchical nodal refinement [28,29], and use of natural neighbor basis functions [18,30] are a few approaches that share this viewpoint.…”

A Dynamic Data Driven Grid System for Intra-operative Image Guided Neurosurgery

“…Our finite element simulation model, currently under development, is based on the conforming hierarchical adaptive refinement method (CHARMS) [16,17]. Inspired by the theory of wavelets, this refinement method produces globally compatible meshes by construction.…”