# Nonlinear formulations and coupling of patches for isogeometric analysis of solids in boundary representation

• Nichtlineare Formulierungen und Kopplung von Patches für die isogeometrische Analyse von Festkörpern in Oberflächendarstellung

Aachen : Rheinisch-Westfälische Technische Hochschule Aachen, Fakultät für Bauingenieurwesen, Lehrstuhl für Baustatik und Baudynamik (2020)
Book, Dissertation / PhD Thesis

In: Schriftenreihe des Lehrstuhls für Baustatik und Baudynamik der RWTH Aachen University 10 (2020)
Page(s)/Article-Nr.: 1 Online-Ressource (XII, 175 Seiten) : Illustrationen, Diagramme

Dissertation, RWTH Aachen University, 2020

Abstract

This thesis is concerned with the development of a nonlinear isogeometric approach that enables the analysis of solids designed with the boundary representation modeling technique in Computer-Aided Design (CAD). The main goal of isogeometric analysis (IGA) is to unify design and analysis by employing the same model description in the entire analysis process. Thus, the geometrical approximation error of the finite element method (FEM) can be circumvented. Instead of a volumetric description of the solid, the geometry in CAD is, however, described only by the boundary surfaces. These are typically defined by non-uniform rational B-splines (NURBS). Thus, IGA requires a tri-variate tensor-product structure, which is not available from CAD. Apart from simple geometric cases, such as extruded models, the creation of volumetric discretizations constitutes a challenging task in IGA. This motivates the development of a boundary-oriented approach, where the boundary is sufficient to analyze the entire solid in an isogeometric framework. The scaled boundary finite element method (SBFEM) constitutes a boundary-oriented method, where only the boundary of the domain is discretized reducing thus the dimension of the problem by one. This suits perfectly the way solids are designed in CAD. However, SBFEM is in its original form a semi-analytical method for linear elasticity. Further measures are required to allow the treatment of nonlinear problems. The main objective of this work is the derivation of isogeometric formulations that exploit the geometry of the boundary and are suitable for a wide class of nonlinear problems. To this end, IGA is combined with the parameterization of SBFEM. The solid is partitioned into sections in relation to its boundary surfaces. The parameterization distinguishes between circumferential parameters on the boundary and a radial scaling parameter in the interior of the domain. The derivation is presented for the two-dimensional as well as the three-dimensional case. The exact geometry of the boundary is exploited in order to align with IGA. The interior of the domain is described by uni-variate B-splines in order to allow the analysis of nonlinear problems. A further objective of this work is to enable local refinement of sections for an efficient discretization. Local refinement implies non-conforming discretizations between adjacent sections. For their coupling a mortar formulation is derived, which is directly applicable to a wide class of discretizations and nonlinear problems. It is based on a master-slave relation. The equality of mutual deformations is fulfilled in a weak sense by constraining the NURBS basis functions on the non-conforming interface. The numerical studies of this work demonstrate the capabilities of the developed approach. The results show that the accuracy and efficiency are comparable to other established numerical methods. The main advantage of the proposed approach is that it enables an isogeometric analysis framework in alignment with the boundary representation modeling technique in CAD. Therefore, it can be viewed as a competitive alternative to commonly used numerical methods.