Adaptive Surface Modeling Using a Quadtree of Quadratic Finite Elements

This article presents special quadrilateral quadratic refinement elements, which provide geometry and field continuity across Tjunctions where two elements are connected to one side of a larger quadrilateral. The main idea in element refinement is to place one or more nodes outside the element area and to modify element shape functions in order to maintain continuity at refinement edges. Special refinement elements allow one to adaptively refine a mesh in such a way that it fits a quadtree data structure. An algorithm of surface modeling starts with a coarse mesh of quadratic quadrilateral elements. Adaptive mesh refinement is done in an iterative manner. At each iteration, the finite element equation system is solved to provide nodal locations with minimization of global approximation error. Elements with excessive local errors are split into four new elements. The mesh refinement iteration process is terminated when no element splits occur. The created mesh of quadratic quadrilaterals can be used directly in finite element analysis.