A physically and geometrically nonlinear formulation for isogeometric analysis of solids in boundary representation

Abstract

AbstractThis contribution concerns a physically and geometrically nonlinear formulation for isogeometric analysis of solids in boundary representation. The parametrization of the proposed approach is inspired by the scaled boundary finite element method, where the geometry of the boundary is sufficient to define the entire solid surface. This suits perfectly the boundary representation modeling technique, which is commonly employed for the design of solids in CAD. In order to provide a boundary‐oriented approach, the solid surface is parametrized by a radial scaling parameter emanating from a scaling center and a parameter in the circumferential direction of the boundary. According to the idea of isogeometric analysis, NURBS basis functions define the geometry of the boundary and approximate the solution on the boundary. We also employ NURBS to approximate the solution in the scaling direction in the interior of the domain. This enables a straightforward treatment of nonlinear problems while preserving the exact geometry of the boundary. The approximation in scaling direction is in principle flexible and requires for coupling of adjacent patches inside the domain. The Galerkin method is employed for the solution in both parametric directions and a linearization is derived which is used within an iterative Newton‐Raphson scheme. We consider nonlinear problems in solid mechanics including elasto‐plastic material behavior and large deformations as well as complex geometries commonly employed in engineering applications. In this paper we study a numerical example of a complex geometry in order to evaluate the performance of the proposed approach.