A significant step to engineering design is to take into account uncertainties and to develop optimal designs that are robust with respect to perturbations. Furthermore, it is often of interest to optimize for different conflicting objective functions describing the quality of a design, leading to a multi-objective optimization problem. In this context, generating methods for solving multi-objective optimization problems seek to find a representative set of solutions fulfilling the concept of Pareto optimality. When multiple uncertain objective functions are involved, it is essential to define suitable measures for robustness that account for a combined effect of uncertainties in objective space. Many tasks in engineering design include the solution of an underlying partial differential equation that can be computationally expensive. Thus, it is of interest to use efficient strategies for finding optimal designs. This research aims to present suitable measures for robustness in a multi-objective context, as well as optimization strategies for multi- objective robust design. This work introduces new ideas for robustness measures in the context of multi- objective robust design. Losses and expected losses based on distances in objective space are used to describe robustness. A direct formulation and a two-phase formulation based on expected losses are proposed for finding a set of robust optimal solutions. Furthermore, suitable optimization strategies for solving the resulting multi-objective robust design problem are formulated and analyzed. The multi-objective optimization problem is solved with a constraint-based approach that is based on solving several constrained single-objective optimization problems with a hybrid optimization strategy. The hybrid method combines a global search method on a surrogate model with adjoint- based optimization methods. In the context of optimization with an underlying partial differential equation, a one-shot approach is extended to handle additional constraints. The developed concepts for multi-objective robust design and the proposed optimiza- tion strategies are applied to an aerodynamic shape optimization problem. The drag coefficient and the lift coefficient are optimized under the consideration of uncertain- ties in the operational conditions and geometrical uncertainties. The uncertainties are propagated with the help of a non-intrusive polynomial chaos approach. For increasing the efficiency when considering a higher-dimensional random space, it is made use of a Karhunen-Loève expansion and a dimension-adaptive sparse grid quadrature.

In this thesis, we present the basic concepts of isogeometric analysis (IGA) and we consider Poisson's equation as model problem. Since in IGA the physical domain is parametrized via a geometry function that goes from a parameter domain, e.g. the unit square or unit cube, to the physical one, we present a class of parametrizations that can be viewed as a generalization of polar coordinates, known as the scaled boundary parametrizations (SB-parametrizations). These are easy to construct and are particularly attractive when only the boundary of a domain is available. We then present an IGA approach based on these parametrizations, that we call scaled boundary isogeometric analysis (SB-IGA). The SB-IGA derives the weak form of partial differential equations in a different way from the standard IGA. For the discretization projection on a finite-dimensional space, we choose in both cases Galerkin's method. Thanks to this technique, we state an equivalence theorem for linear elliptic boundary value problems between the standard IGA, when it makes use of an SB-parametrization, and the SB-IGA. We solve Poisson's equation with Dirichlet boundary conditions on different geometries and with different SB-parametrizations.

This research explores the development of web based reference software for characterisation of surface roughness for two-dimensional surface data. The reference software used for verification of surface characteristics makes the evaluation methods easier for clients. The algorithms used in this software are based on International ISO standards. Most software used in industrial measuring instruments may give variations in the parameters calculated due to numerical changes in calculation. Such variations can be verified using the proposed reference software. The evaluation of surface roughness is carried out in four major steps: data capture, data align, data filtering and parameter calculation. This work walks through each of these steps explaining how surface profiles are evaluated by pre-processing steps called fitting and filtering. The analysis process is then followed by parameter evaluation according to DIN EN ISO 4287 and DIN EN ISO 13565-2 standards to extract important information from the profile to characterise surface roughness.

In this thesis we develop a shape optimization framework for isogeometric analysis in the optimize first–discretize then setting. For the discretization we use isogeometric analysis (iga) to solve the state equation, and search optimal designs in a space of admissible b-spline or nurbs combinations. Thus a quite general class of functions for representing optimal shapes is available. For the gradient-descent method, the shape derivatives indicate both stopping criteria and search directions and are determined isogeometrically. The numerical treatment requires solvers for partial differential equations and optimization methods, which introduces numerical errors. The tight connection between iga and geometry representation offers new ways of refining the geometry and analysis discretization by the same means. Therefore, our main concern is to develop the optimize first framework for isogeometric shape optimization as ground work for both implementation and an error analysis. Numerical examples show that this ansatz is practical and case studies indicate that it allows local refinement.

This work aims at including nonlinear elastic shell models in a multibody framework. We focus our attention to Kirchhoff-Love shells and explore the benefits of an isogeometric approach, the latest development in finite element methods, within a multibody system. Isogeometric analysis extends isoparametric finite elements to more general functions such as B-Splines and Non-Uniform Rational B-Splines (NURBS) and works on exact geometry representations even at the coarsest level of discretizations. Using NURBS as basis functions, high regularity requirements of the shell model, which are difficult to achieve with standard finite elements, are easily fulfilled. A particular advantage is the promise of simplifying the mesh generation step, and mesh refinement is easily performed by eliminating the need for communication with the geometry representation in a Computer-Aided Design (CAD) tool. Quite often the domain consists of several patches where each patch is parametrized by means of NURBS, and these patches are then glued together by means of continuity conditions. Although the techniques known from domain decomposition can be carried over to this situation, the analysis of shell structures is substantially more involved as additional angle preservation constraints between the patches might arise. In this work, we address this issue in the stationary and transient case and make use of the analogy to constrained mechanical systems with joints and springs as interconnection elements. Starting point of our work is the bending strip method which is a penalty approach that adds extra stiffness to the interface between adjacent patches and which is found to lead to a so-called stiff mechanical system that might suffer from ill-conditioning and severe stepsize restrictions during time integration. As a remedy, an alternative formulation is developed that improves the condition number of the system and removes the penalty parameter dependence. Moreover, we study another alternative formulation with continuity constraints applied to triples of control points at the interface. The approach presented here to tackle stiff systems is quite general and can be applied to all penalty problems fulfilling some regularity requirements. The numerical examples demonstrate an impressive convergence behavior of the isogeometric approach even for a coarse mesh, while offering substantial savings with respect to the number of degrees of freedom. We show a comparison between the different multipatch approaches and observe that the alternative formulations are well conditioned, independent of any penalty parameter and give the correct results. We also present a technique to couple the isogeometric shells with multibody systems using a pointwise interaction.