No abstract

No abstract

This contribution presents an isogeometric approach in three dimensions for nonlinear problems in solid mechanics and structural dynamics. The proposed approach aligns with the boundary representation modeling technique in CAD.Isogeometric analysis is combined with the parameterization of the scaled boundary finite element method. In this way, the boundary description of 3D solids in CAD can be directly utilized for the analysis in an isogeometric framework. The approximation of the solution on the boundary is based on bi-variate NURBS. We also approximate the interior of the solid with uni-variate B-splines to facilitate the solution of nonlinear problems. The nonlinear case is extended to three dimensions in this present work. The method is further extended to dynamic problems. The mass and damping matrices are derived using the same basis functions. The solution of the entire solid is obtained with the Galerkin method. We study several nonlinear and dynamic problems with simple geometries and arbitrary number of boundaries. Moreover, a fiber reinforced composite serves as demonstration for complex shapes.

The 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.

This contribution is concerned with a coupling approach for nonconforming NURBS patches in the framework of an isogeometric formulation for solids in boundary representation. The boundary representation modeling technique in CAD is the starting point of this approach. We parameterize the solid according to the scaled boundary finite element method and employ NURBS basis functions for the approximation of the solution. Therefore, solid surfaces consist of several sections, which can be regarded as patches and discretized independently. The main objective of this study is to derive an approach for the connection of independent sections in order to allow for local refinement and thus an accurate and efficient discretization of the computational domain. Nonconforming sections are coupled with a mortar approach within a master-slave framework. The coupling of adjacent sections ensures the equality of mutual deformations along the interface in a weak sense and is enforced by constraining the NURBS basis functions on the interface. We apply this approach to nonlinear problems in two dimensions and compare the results with conforming discretizations. K E Y W O R D Sboundary representation, mortar methods, multi-patch coupling, nonconforming NURBS patches, nonlinear isogeometric analysis, solids