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

Abstract

Quadtree is a hierarchical data structure that is well-suited for h-adaptive mesh refinement. Due to the presence of hanging nodes, classical shape functions are non-conforming on quadtree meshes. In this paper, we use natural neighbor basis functions to construct conforming interpolants on quadtree meshes. To this end, the recently proposed construction of polygonal basis functions is adapted to quadtree elements. A fast technique for calculating stiffness matrix on quadtree meshes is introduced. Residual-based error estimators and material force technique are used to estimate the error on quadtree meshes. The performance of the adaptive technique is demonstrated through the solution of linear and nonlinear boundary-value problems.

Extended finite element method on polygonal and quadtree meshes

“…The convergence of conforming polygonal finite elements is established in Reference [1] and use of quadtree finite elements to solve linear and nonlinear boundary-value problems is presented in References [18,19]. In this section, four numerical examples are presented to illustrate the performance of the X-FEM on polygonal and quadtree meshes.…”

“…The crack paths are presented in Figures 18c and 18d, and final trajectory is in agreement with theory. In addition to enabling modeling of microcracks on quadtree meshes, there exists the possibility of developing a posteriori error estimators on such meshes [19], which can provide improved accuracy at modest increase in computational costs.…”

“…Polygonal meshes are generated by using the Voronoi tessellation of the domain [19]. To this end, first a set of random seed points are inserted in the domain, and then the Voronoi diagram of this set of generators is constructed.…”

“…A public-domain package is used to generate the CVT mesh [25]. Nonuniform discretizations with local refinement are obtained if a nonuniform density function is used in the CVT algorithm; a few such examples are presented in Reference [19].…”

“…Many new contributions on barycentric polygonal interpolation have been realized in geometry modeling and graphics, and in finite element methods [1,[9][10][11][12][13][14][15][16]. In Reference [1], natural neighbor based (Laplace) interpolants [17] are used to construct shape functions on irregular polygons (n-gons), which is adapted to quadtree meshes (resolves the issue of hanging nodes) in References [18,19]. Barycentric coordinates are non-negative, form a partition of unity, have linear precision, and are linear on element edges.…”