Efficient isogeometric finite element formulations for coupled systems on deforming surfaces
Zimmermann, Christopher; Sauer, Roger Andrew (Thesis advisor); Elgeti, Stefanie Nicole (Thesis advisor)
Aachen (2019) [Dissertation / PhD Thesis]
Page(s): 1 Online-Ressource (v, 106, xxx Seiten) : Illustrationen, Diagramme
This thesis presents various isogeometric finite element formulations for the efficient computation of coupled systems on deforming surfaces. The surface deformation is described by the Kirchhoff-Love thin shell equation, which is a fourth-order nonlinear partial differential equation (PDE). A curvilinear surface parameterization is used throughout the proposed formulations to allow for general surface shapes and deformations. In the first part of this thesis, a novel adaptive local surface refinement technique based on locally refinable non-uniform rational B-splines (LR NURBS) is presented. For a convenient embedding into general finite element codes, the Bézier extraction operator for LR NURBS is formulated. An automatic remeshing technique is presented that allows adaptive local refinement and coarsening of LR NURBS. This technique is applied to frictionless and frictional sliding contact of deforming thin shells. Several numerical examples demonstrate the benefit of LR NURBS: Compared to uniform refinement, LR NURBS can achieve high accuracy at lower computational cost. In the second part of this thesis, a general theory and isogeometric finite element formulation for studying mass conserving phase transitions on deforming thin shells is presented. The mathematical problem is governed by the coupling of the Kirchhoff-Love shell PDE with the Cahn-Hilliard PDE for phase transitions, which can be derived from surface mass balance in the framework of irreversible thermodynamics. The coupling leads to a system of two fourth-order nonlinear PDEs, that live on an evolving two-dimensional manifold. Structured NURBS and unstructured spline spaces are utilized for these interpolations. In order to use multi-patch NURBS discretizations, an approach for continuity constraints at patch interfaces is proposed. The resulting finite element formulation is discretized in time by the generalized-α scheme. An adaptive time stepping approach is presented and the coupled system is fully linearized within a monolithic Newton-Raphson approach. The behavior of the coupled system is illustrated by several numerical examples exhibiting phase transitions on deforming tori and spheres. Within the last part of this thesis, an adaptive phase field formulation for dynamic fracture of brittle shells is presented. The fracture evolution equation employs a fourth-order phase field formulation and bases on Griffith’s theory of brittle fracture. This fourth-order PDE is coupled with the Kirchhoff-Love shell PDE. LR NURBS are used for the surface discretization. The temporal discretization is based on the generalized-α scheme and the discretized coupled system is solved within a monolithic Newton-Raphson scheme. The proposed formulation is fully adaptive in space and time, by using an adaptive local refinement approach and an adaptive time stepping scheme. The interaction between surface deformation and crack evolution is demonstrated by several numerical examples.