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Finite element algorithm with adaptive quadtree-octree mesh refinement

Certain difficulties with the use of quadrilateral or hexahedral finite elements are related to mesh refinement and to element compatibility and quality after refinement. In this paper, special refinement elements are presented that make possible connecting two special elements to one edge of an 8 node quadrilateral element (2D). The main idea in refinement elements is to place some midside nodes outside the element area and to modify element shape functions in order to maintain continuity at a refinement edge. Special refinement elements allow to adaptively refine mesh in such a way that it fits the quadtree data structure. In the three-dimensional case, hexahedral elements with 20 nodes are employed. Placement of some nodes outside the element volume allows us to create a compatible octree refinement scheme.