The Nitsche-based Partition of Unity Method for coupled problems in linear elasticity

Abstract

This thesis addresses the use of Nitsche's method within the Partition of Unity Method (PUM), and in particular within the PUMA software library developed both at Fraunhofer SCAI and at the Institute for Numerical Simulation, which has provided the practical ground for the mathematical developments. While the specific adaptation of Nitsche's method to the PUM has been already published in a paper attached to this thesis, the focus of this work lies on problems arising from linear elasticity, with the aim of handling interface constraints for the coupling of solid and shell models. These include not only single isotropic materials, but also laminated structures consisting of multiple orthotropic elastic materials, which are used to simulate fiber-reinforced composites. This work revealed the lack of efficient and robust smoothers for PUMA's multilevel solver, which should be able to handle anisotropic and higher-order problems. The development of such a smoother, based on the Factorized Sparse Approximate Inverse (FSAI) preconditioner and combined with the Chebyshev iteration, has been the subject of another paper, which is also attached to this thesis. Additionally, the coupling of shell structures parameterized by Non-Uniform Rational B-Splines (NURBS) required the development and implementation of projection algorithms capable of projecting a point into a NURBS curve or surface (also retrieving the corresponding parameters). All in all, this thesis and its attached papers involve the formulation and discretization of PDEs with the Nitsche-based PUM, including both geometrical and analytical complications, its numerical integration (handling all involved operations in an efficient but also rigorous manner), and finally the iterative solution of the resulting linear systems (again with specific emphasis on efficiency), thus encompassing all stages of the numerical solution of linear and elliptic PDEs. The developments are accompanied by several numerical tests, which also showcase the practical work carried out for a significant improvement of the PUMA library.