## 35-XX PARTIAL DIFFERENTIAL EQUATIONS

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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.

Gliomas are primary brain tumors with a high invasive potential and infiltrative spread. Among them, glioblastoma multiforme (GBM) exhibits microvascular hyperplasia and pronounced necrosis triggered by hypoxia. Histological samples showing garland-like hypercellular structures (so-called pseudopalisades) centered around one or several sites of vaso-occlusion are typical for GBM and hint on poor prognosis of patient survival.
This thesis focuses on studying the establishment and maintenance of these histological patterns specific to GBM with the aim of modeling the microlocal tumor environment under the influence of acidity, tissue anisotropy and hypoxia-induced angiogenesis. This aim is reached with two classes of models: multiscale and multiphase. Each of them features a reaction-diffusion equation (RDE) for the acidity acting as a chemorepellent and inhibitor of growth, coupled in a nonlinear way to a reaction-diffusion-taxis equation (RDTE) for glioma dynamics. The numerical simulations of the resulting systems are able to reproduce pseudopalisade-like patterns. The effect of tumor vascularization on these patterns is studied through a flux-limited model belonging to the multiscale class. Thereby, PDEs of reaction-diffusion-taxis type are deduced for glioma and endothelial cell (EC) densities with flux-limited pH-taxis for the tumor and chemotaxis towards vascular endothelial growth factor (VEGF) for ECs. These, in turn, are coupled to RDEs for acidity and VEGF produced by tumor. The numerical simulations of the obtained system show pattern disruption and transient behavior due to hypoxia-induced angiogenesis. Moreover, comparing two upscaling techniques through numerical simulations, we observe that the macroscopic PDEs obtained via parabolic scaling (directed tissue) are able to reproduce glioma patterns, while no such patterns are observed for the PDEs arising by a hyperbolic limit (directed tissue). This suggests that brain tissue might be undirected - at least as far as glioma migration is concerned. We also investigate two different ways of including cell level descriptions of response to hypoxia and the way they are related.

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.

Synapses are connections between different nerve cells that form an essential link in neural signal transmission. It is generally distinguished between electrical and chemical synapses, where chemical synapses are more common in the human brain and are also the type we deal with in this work.
In chemical synapses, small container-like objects called vesicles fill with neurotransmitter and expel them from the cell during synaptic transmission. This process is vital for communication between neurons. However, to the best of our knowledge no mathematical models that take different filling states of the vesicles into account have been developed before this thesis was written.
In this thesis we propose a novel mathematical model for modeling synaptic transmission at chemical synapses which includes the description of vesicles of different filling states. The model consists of a transport equation (for the vesicle growth process) plus three ordinary differential equations (ODEs) and focuses on the presynapse and synaptic cleft.
The well-posedness is proved in detail for this partial differential equation (PDE) system. We also propose a few different variations and related models. In particular, an ODE system is derived and a delay differential equation (DDE) system is formulated. We then use nonlinear optimization methods for data fitting to test some of the models on data made available to us by the Animal Physiology group at TU Kaiserslautern.

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.

Optimal control of partial differential equations is an important task in applied mathematics where it is used in order to optimize, for example, industrial or medical processes. In this thesis we investigate an optimal control problem with tracking type cost functional for the Cattaneo equation with distributed control, that is, \(\tau y_{tt} + y_t - \Delta y = u\). Our focus is on the theoretical and numerical analysis of the limit process \(\tau \to 0\) where we prove the convergence of solutions of the Cattaneo equation to solutions of the heat equation.
We start by deriving both the Cattaneo and the classical heat equation as well as introducing our notation and some functional analytic background. Afterwards, we prove the well-posedness of the Cattaneo equation for homogeneous Dirichlet boundary conditions, that is, we show the existence and uniqueness of a weak solution together with its continuous dependence on the data. We need this in the following, where we investigate the optimal control problem for the Cattaneo equation: We show the existence and uniqueness of a global minimizer for an optimal control problem with tracking type cost functional and the Cattaneo equation as a constraint. Subsequently, we do an asymptotic analysis for \(\tau \to 0\) for both the forward equation and the aforementioned optimal control problem and show that the solutions of these problems for the Cattaneo equation converge strongly to the ones for the heat equation. Finally, we investigate these problems numerically, where we examine the different behaviour of the models and also consider the limit \(\tau \to 0\), suggesting a linear convergence rate.

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.

In this thesis we present a new method for nonlinear frequency response analysis of mechanical vibrations.
For an efficient spatial discretization of nonlinear partial differential equations of continuum mechanics we employ the concept of isogeometric analysis. Isogeometric finite element methods have already been shown to possess advantages over classical finite element discretizations in terms of exact geometry representation and higher accuracy of numerical approximations using spline functions.
For computing nonlinear frequency response to periodic external excitations, we rely on the well-established harmonic balance method. It expands the solution of the nonlinear ordinary differential equation system resulting from spatial discretization as a truncated Fourier series in the frequency domain.
A fundamental aspect for enabling large-scale and industrial application of the method is model order reduction of the spatial discretization of the equation of motion. Therefore we propose the utilization of a modal projection method enhanced with modal derivatives, providing second-order information. We investigate the concept of modal derivatives theoretically and using computational examples we demonstrate the applicability and accuracy of the reduction method for nonlinear static computations and vibration analysis.
Furthermore, we extend nonlinear vibration analysis to incompressible elasticity using isogeometric mixed finite element methods.

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.

The work consists of two parts.
In the first part an optimization problem of structures of linear elastic material with contact modeled by Robin-type boundary conditions is considered. The structures model textile-like materials and possess certain quasiperiodicity properties. The homogenization method is used to represent the structures by homogeneous elastic bodies and is essential for formulations of the effective stress and Poisson's ratio optimization problems. At the micro-level, the classical one-dimensional Euler-Bernoulli beam model extended with jump conditions at contact interfaces is used. The stress optimization problem is of a PDE-constrained optimization type, and the adjoint approach is exploited. Several numerical results are provided.
In the second part a non-linear model for simulation of textiles is proposed. The yarns are modeled by hyperelastic law and have no bending stiffness. The friction is modeled by the Capstan equation. The model is formulated as a problem with the rate-independent dissipation, and the basic continuity and convexity properties are investigated. The part ends with numerical experiments and a comparison of the results to a real measurement.