Mechanically based models: Adaptive refinement for B‐spline finite element

Abstract

SUMMARYThis article presents two new methods for adaptive reÿnement of a B-spline ÿnite element solution within an integrated mechanically based computer aided engineering system.The proposed techniques for adaptively reÿning a B-spline ÿnite element solution are a local variant of np-reÿnement and a local variant of h-reÿnement. The key component in the np-reÿnement is the linear co-ordinate transformation introduced into the reÿned element. The transformation is constructed in such a way that the transformed nodal conÿguration of the reÿned element is identical to the nodal conÿguration of the neighbour elements. Therefore, the assembly proceeds as with classic ÿnite elements, while the solution approximation conforms exactly along the inter-element boundaries. For the h-reÿnement, this transformation is introduced into a construction that merges the super element from the ÿnite element world with the hierarchical B-spline representation from the computational geometry. In the scope of developing sculptured surfaces, the proposed approach supports C 0 as well as the Hermite B-spline C 1 continuous shapes. For sculptured solids, C 0 continuity only is considered in this article.The feasibility of the proposed methods in the scope of the geometric design is demonstrated by several examples of creating sculptured surfaces and volumetric solids. Numerical performance of the methods is demonstrated for a test case of the two-dimensional Poisson equation.

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.…”

B spline-based method for 2-D large deformation analysis

“…In fact, a considerable body of literatures now exists on the application of uniform and non-uniform B spline techniques to the solution of partial differential equations (PDEs) and mechanics problems. The recent studies of B spline method can be found in some articles [17][18][19][20][21][22][23]. The B spline basis functions have compact support and lead to banded stiffness matrices.…”