Solids in boundary representation ‐ a NURBS based Galerkin formulation

Abstract

AbstractThe present approach deals with a numerical method for the static analysis of solids in boundary representation. A formulation is derived where the geometrical description of the boundary is sufficient to define the elastic boundary value problem for the complete solid. The interior of the domain is described by a radial scaling parameter. Following the idea of the scaled boundary finite element method [1] the scaling of the boundary with respect to a specified scaling center provides a representation for the complete solid. This concept fits perfectly to the boundary representation modeling technique, which is frequently used in CAD to define solids. In CAD the boundary of the solid is described by employing NURBS basis function. In the isogeometric analysis methodology, [2], a trivariate tensor product structure is used to analyze solids. In the present approach, the tensor‐product structure of the solid will be reduced by one dimension to parameterize the physical domain, i. e., the three‐dimensional solid exploits only two‐dimensional NURBS objects, which parameterize the boundary surfaces. Hence, the present approach represents perfectly the isoparametric paradigm, see [3]. However, in [3] the weak form of equilibrium is enforced on the boundary surface. This leads to an ordinary differential equation (ODE) of Euler type, which could be solved by applying a collocation scheme. In the present approach the weak form is employed on the boundary and in scaling direction. The displacement response in the radial scaling direction is approximated by one‐dimensional NURBS. Finally, the Galerkin projection of the weak form yields a linear system of equilibrium equations whose solution gives rise to the displacement response. In conclusion, the presented method is able to analyze solids, which are bounded by an arbitrary number of surfaces. Numerical examples will show the capabilities of the presented method. The accuracy of the method is investigated by comparison to the analytical solution. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)