Abstract
In this paper we explore ERBS extraction in application to a heat conduction problem. Two types of basis functions are considered: B-spline and expo-rational B-spline (ERBS) combined with Bernstein polynomials. The first one is a global basis defined on a subdomain whose size is related to the spline degree, while the second one is a strictly local basis under the local geometry. While all the coefficients of the B-spline tensor product surface affect each element of the patch, local surfaces of the blending surface conserve both local manipulation and smoothness of the global surface.We show the conversion between these bases by using an ERBS extraction. The extraction operator allows us to convert the control points of the surface evaluated by B-splines to control points of the ERBS surface and vice versa. This approach is demonstrated on an example of the finite element solution approximation of the heat equation.
Highlights
In the isogeometric framework the solution space for dependent variables is represented in terms of the same functions which represent the geometry [1]
By utilizing the B-spline basis and its modifications, the standard finite element method based on the piecewise linear basis can be modified for the purpose of matching the exact geometry independently of the discretization
Investigations using tensor product NURBS constructions [6], T-spline constructions [7] and their Bézier and Lagrange extraction have shown that the use of a smooth basis in analysis provides computational advantages over standard finite elements in several areas
Summary
In the isogeometric framework the solution space for dependent variables is represented in terms of the same functions which represent the geometry [1]. In the context of isogeometric analysis, structures that precisely approximate the boundary are demanded For this purpose, B-spline based tensor product [3] and triangular surfaces are used [4]. Investigations using tensor product NURBS constructions [6], T-spline constructions [7] and their Bézier and Lagrange extraction have shown that the use of a smooth basis in analysis provides computational advantages over standard finite elements in several areas. ERBS finite elements based on Taylor polynomials as local functions were applied in [11,12,13] to both rectangular and triangular elements. In this paper we propose to extract the linear operator which maps the expo-rational blending basis combined with the Bernstein polynomial basis of the local Bézier surfaces to the global B-spline basis.
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