Fast Isogeometric Boundary Element Method based on Independent Field Approximation

An isogeometric boundary element method for problems in elasticity is presented, which is based on an independent approximation for the geometry, traction and displacement field. This enables a flexible choice of refinement strategies, permits an efficient evaluation of geometry related information, a mixed collocation scheme which deals with discontinuous tractions along non-smooth boundaries and a significant reduction of the right hand side of the system of equations for common boundary conditions. All these benefits are achieved without any loss of accuracy compared to conventional isogeometric formulations. The system matrices are approximated by means of hierarchical matrices to reduce the computational complexity for large scale analysis. For the required geometrical bisection of the domain, a strategy for the evaluation of bounding boxes containing the supports of NURBS basis functions is presented. The versatility and accuracy of the proposed methodology is demonstrated by convergence studies showing optimal rates and real world examples in two and three dimensions.Comment: 32 pages, 27 figure

Optimal-order isogeometric collocation at Galerkin superconvergent points

In this paper we investigate numerically the order of convergence of an isogeometric collocation method that builds upon the least-squares collocation method presented in [1] and the variational collocation method presented in [2]. The focus is on smoothest B-splines/NURBS approximations, i.e, having global $C^{p-1}$ continuity for polynomial degree $p$. Within the framework of [2], we select as collocation points a subset of those considered in [1], which are related to the Galerkin superconvergence theory. With our choice, that features local symmetry of the collocation stencil, we improve the convergence behaviour with respect to [2], achieving optimal $L^2$-convergence for odd degree B-splines/NURBS approximations. The same optimal order of convergence is seen in [1], where, however a least-squares formulation is adopted. Further careful study is needed, since the robustness of the method and its mathematical foundation are still unclear.Comment: 21 pages, 20 figures (35 pdf images

B-Spline Finite Elements and their Efficiency in Solving Relativistic Mean Field Equations

A finite element method using B-splines is presented and compared with a conventional finite element method of Lagrangian type. The efficiency of both methods has been investigated at the example of a coupled non-linear system of Dirac eigenvalue equations and inhomogeneous Klein-Gordon equations which describe a nuclear system in the framework of relativistic mean field theory. Although, FEM has been applied with great success in nuclear RMF recently, a well known problem is the appearance of spurious solutions in the spectra of the Dirac equation. The question, whether B-splines lead to a reduction of spurious solutions is analyzed. Numerical expenses, precision and behavior of convergence are compared for both methods in view of their use in large scale computation on FEM grids with more dimensions. A B-spline version of the object oriented C++ code for spherical nuclei has been used for this investigation.Comment: 27 pages, 30 figure