Scaled boundary isogeometric analysis of large deformations in solids

Abstract

AbstractThe so‐called scaled boundary isogeometric analysis combines the advantages of the isogeometric analysis and the scaled boundary finite element method. Here, the parameterization of the solid follows the idea of the scaled boundary finite element method (SB‐FEM), where the boundary of the domain is scaled in respect to a specified scaling center inside the domain. In the framework of isogeometric analysis (IGA), the NURBS functions that describe the geometry also interpolate the unknown displacement field. Such a combined approach is advantageous for three‐dimensional solids, as a radial scaling parameter describes the interior of the solid and only the geometry of the boundary is required for the analysis. The motivation is therefore to fit the idea of the boundary representation modeling technique, which is the way solids are designed in CAD –namely only by their boundary surfaces. Our contribution introduces a formulation for geometrical nonlinear 2D problems. The derived formulation addresses hyperelastic material behavior for large deformations and plane strain conditions for the two‐dimensional domain. We solve the boundary value problem by applying the weak form of equilibrium in both radial scaling and circumferential direction of the boundary. Thereafter, the nonlinear deformation behavior requires a linearization and the application of the Newton‐Raphson iterative scheme. Finally, we study the performance of this approach on a numerical problem by comparison to the standard finite element method.