A NURBS based Galerkin approach for the analysis of solids in boundary representation

The paper is concerned with a new numerical method to solve the elasticity problem of solids in boundary representation. A formulation is derived where the geometrical description of the boundary is sufficient for defining the equations of elasticity of the complete solid. While the interior of the domain is described by a radial scaling parameter, the scaling of the boundary with respect to the specified scaling center leads to the complete solid. This idea fits perfectly to the boundary representation modeling technique commonly employed in CAD. In the present approach the tensor-product structure of the solid will be reduced by one dimension to parametrize the physical domain, i.e., the three-dimensional solid exploits only two-dimensional NURBS objects, which parametrize the boundary surfaces. For the analysis, the weak form of the equilibrium equations is enforced for the entire solid. In particular the weak form is employed in the scaling direction and the circumferential direction. Applying the isogeometric paradigm, the NURBS functions that describe the boundary of the geometry form also the basis for the approximation of the displacement at the boundary. The displacement response in the radial scaling direction, on the other hand, is approximated by a one-dimensional NURBS. Overall, the Galerkin projection of the weak form yields a linear system of equilibrium equations whose solution gives rise to the displacement response. The accuracy of the method is validated by means of analytical reference data. In conclusion, the proposed method allows the analysis of solids which are bounded by an arbitrary number of contour boundaries.