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

This article presents two new methods for adaptive re nement of a B-spline nite element solution within an integrated mechanically based computer aided engineering system. The proposed techniques for adaptively re ning a B-spline nite element solution are a local variant of np-re nement and a local variant of h-re nement. The key component in the np-re nement is the linear co-ordinate transformation introduced into the re ned element. The transformation is constructed in such a way that the transformed nodal con guration of the re ned element is identical to the nodal con guration of the neighbour elements. Therefore, the assembly proceeds as with classic nite elements, while the solution approximation conforms exactly along the inter-element boundaries. For the h-re nement, this transformation is introduced into a construction that merges the super element from the nite element world with the hierarchical B-spline representation from the computational geometry. In the scope of developing sculptured surfaces, the proposed approach supports C0 as well as the Hermite B-spline C1 continuous shapes. For sculptured solids, C0 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