## 510 Mathematik

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This thesis is concerned with the efficient and accurate simulation and optimization of linear Timoshenko beam networks subjected to external loads.
For this we develop a solution scheme, which is based on so-called analytic ansatz-functions known to provide analytic solutions
for the unknowns of a single beam subjected to given boundary data. In the present work we prove that this concept can be extended to a network of beams
which enables the analytic calculation of the unique deformation of the whole network.
Moreover, we show that this approach is equivalent to a finite element method where only one finite element with a particular shape function per
beam is required. With this we can provide the analytic solution on each node of the network, from which the state of the whole network can be uniquely determined - no further spatial discretization is needed.
Based on this analytic solution scheme for the simulation problem, we investigate a number of problems from structural optimization, more precisely
topology optimization, material and multi-material optimization, cross section optimization, geometry optimization and simultaneous versions of the before mentioned.
Hereby we provide the problem formulation, analytic formulae for first order derivatives, an implementation and numerical examples for each type of optimization model.
In the next part of this thesis we develop a novel approach for geometry, cross section and material optimization for a system of high-pressure steam pipes in a power plant.
This industrial application is efficiently modeled by including a special parametrization for the geometry of pipes.
Then we formulate
an optimization problem with an objective function considering monetary costs and a large number of constraints.
Moreover, we derive analytic expressions for the first order derivatives of the problem and provide an implementation. We demonstrate the
capabilities of our program with numerical tests for a real-world instance.

This paper presents a new approach for the determination of the wind speed distribution based on wind speed data. This approach is based on the fact that, in general, wind speed distributions restricted to seasons of year or months are different. Therefore, instead of one Weibull density function, a convex combination of Weibull density functions is considered for a calendar year. This model improves the maximum likelihood of the estimated wind speed distribution. Numerical results including a Kolmogorov–Smirnov test are given for a site at Jamaica. Numerical comparisons are carried out for different sites and various known methods for the estimation of the wind speed distribution.

In an example for the determination of publicly registered land values in urban building conflict situations an interval value duality and regression are applied. The result is incumbent upon a new interval value optimum – beginning with non-ordered point sets of publicly registered German land values for residential land. These two mathematical approaches with set-valued order relations describe the relation to the socio-economy in an urban building conflict situation.

This thesis deals with properties of singular orbits in multi-body-systems.
Here, systems are examined, in which the dynamics defining potential solely depends on the relative positions of the particles. Celestial mechanics, which is the motion of any number of particles due to Newton's law of universal gravitation, can be regarded as a standard example. With this in mind, the class of so-called long-ranged and moderated potentials are introduced, which among others also includes the Coulomb field of electrostatics. In these systems, the flow which is induced by the set of differential equations of motion, is generally not complete, and the non-global solutions are called singular. The goal is to show their improbability in due situations, that is to show that the set of initial conditions leading to non-global solution is a null-set in the sense of Lebesgue.
An important distinction of singular orbits are collision and non-collision orbits: the former are defined by the property that all particles have definitive limit points as time approaches the singularity. Previous results in the field of celestial mechanics show the improbability of collisions in general, and the improbability of non-collision singularities in the four-body-problem. Some generalizations to a wider class of potentials is possible with the techniques imposed there, however, certain restrictions like on the homogeneity of the force field cannot be dropped, and even within these restrictions optimal bounds cannot be shown.
Hence, the provision of a new technique seems to be necessary, in order to generalize the results to a wider class of systems, which is done within this work...

In this paper we present a chain of mathematical models that enables the numerical simulation of the airlay process and the investigation of the resulting nonwoven material by means of virtual tensile strength tests. The models range from a highly turbulent dilute fiber suspension flow to stochastic surrogates for fiber lay-down and web formation and further to Cosserat networks with effective material laws. Crucial is the consistent mathematical mapping between the parameters of the process and the material. We illustrate the applicability of the model chain for an industrial scenario, regarding data from computer tomography and experiments. By this proof of concept we show the feasibility of future simulation-based process design and material optimization which are long-term objectives in the technical textile industry.

Mathematical modeling of biochemical pathways is an important resource in Synthetic Biology, as the predictive power of simulating synthetic pathways represents an important step in the design of synthetic metabolons. In this paper, we are concerned with the mathematical modeling, simulation, and optimization of metabolic processes in biochemical microreactors able to carry out enzymatic reactions and to exchange metabolites with their surrounding medium. The results of the reported modeling approach are incorporated in the design of the first microreactor prototypes that are under construction. These microreactors consist of compartments separated by membranes carrying specific transporters for the input of substrates and export of products. Inside the compartments of the reactor multienzyme complexes assembled on nano-beads by peptide adapters are used to carry out metabolic reactions. The spatially resolved mathematical model describing the ongoing processes consists of a system of diffusion equations together with boundary and initial conditions. The boundary conditions model the exchange of metabolites with the neighboring compartments and the reactions at the surface of the nano-beads carrying the multienzyme complexes. Efficient and accurate approaches for numerical simulation of the mathematical model and for optimal design of the microreactor are developed. As a proof-of-concept scenario, a synthetic pathway for the conversion of sucrose to glucose-6-phosphate (G6P) was chosen. In this context, the mathematical model is employed to compute the spatio-temporal distributions of the metabolite concentrations, as well as application relevant quantities like the outflow rate of G6P. These computations are performed for different scenarios, where the number of beads as well as their loading capacity are varied. The computed metabolite distributions show spatial patterns, which differ for different experimental arrangements. Furthermore, the total output of G6P increases for scenarios where microcompartimentation of enzymes occurs. These results show that spatially resolved models are needed in the description of the conversion processes. Finally, the enzyme stoichiometry on the nano-beads is determined, which maximizes the production of glucose-6-phosphate.

We use the example of hot rolling to develop a comprehensive optimization approach that is based on a mathematical model of the underlying manufacturing process. More precisely, we study an optimal control problem that is designed to minimize cutting scrap while taking industrial specifications and technical limitations into account.
It is well known that the associated control-to-observation map is non-differentiable due to changes of state resulting from elasto-viscoplastic material behavior and frictional contact. However, we still want to apply gradient-based methods to solve the optimal control problem and therefore have to compute derivatives of cost functional and constraints. To resolve this issue, we first regularize all non-differentiabilities before computing sensitivity information via direct differentiation.
We moreover present solution techniques for the regularized problem and discuss numerical results for real-world examples.

Let K→X be a smooth Lie algebra bundle over a σ-compact manifold X whose typical fiber is the compact Lie algebra k. We give a complete description of the irreducible bounded (i.e., norm continuous) unitary representations of the Fréchet–Lie algebra Γ(K) of all smooth sections of K, and of the LF-Lie algebra Γc(K) of compactly supported smooth sections. For Γ(K), irreducible bounded unitary representations are finite tensor products of so-called evaluation representations, hence in particular finite dimensional. For Γc(K), bounded unitary irreducible (factor) representations are possibly infinite tensor products of evaluation representations, which reduces the classification problem to results of Glimm and Powers on irreducible (factor) representations of UHF C∗-algebras. The key part in our proof is the result that every irreducible bounded unitary representation of a Lie algebra of the form k⊗RAR, where AR is a unital real complete continuous inverse algebra, is a finite product of evaluation representations. On the group level, our results cover in particular the bounded unitary representations of the identity component Gau(P)0 of the group of smooth gauge transformations of a principal fiber bundle P→X with compact base and structure group, and the groups SUn(A)0 with A a complete involutive commutative continuous inverse algebra.