ISBN 978-3-8439-5021-3

978-3-8439-5021-3, Reihe Mathematik

Thomas Marx
Shape Optimization for Radiative Transfer Models - With Applications in High-Temperature Processes

153 Seiten, Dissertation Technische Universität Kaiserslautern (2022), Softcover, A5

Zusammenfassung / Abstract

In this thesis, we discuss optimal design problems motivated by phosphate production. The foundation of the modeling of high-temperature processes is given by a kinetic integro-differential equation, namely the Radiative Transfer Equation (RTE), which models the propagation of radiation through a medium with absorbing, emitting, and scattering phenomena. This model with volume radiation is placed in a shape optimization context. We model the radiation process, discuss the existence, uniqueness and a-priori boundedness of states, and averaged adjoint states, and prove shape differentiability for the proposed optimal design problem with a material derivative-free approach that bypasses saddle point assumptions. The derivation of the shape derivative is done in its tensor representation. The design problem is treated numerically using a moment discretization and by combining two open-source projects; one for the discretization and the other for the optimization.In the next step, we model the high-temperature process with methods that approximate the RTE due to its inherent numerical complexity in order to solve the optimal design problems more efficiently. Further, we extend the model by coupling the equation with a non-linear heat equation which results in a semi-linear elliptic system. We prove existence, uniqueness, and a priori bounds for the states and averaged adjoint states as well as shape differentiability of the proposed shape cost functional. The numerical investigation is done by a Python-co-developed gradient descent method, which underlines the feasibility of the proposed shape methodology applied to the model of choice. In the last step, we refine the model further by including a combined natural and forced convection field induced by the flame, which is generated by a two-way coupling with the Boussinesq- approximation of the Navier-Stokes Equation, for which we study the optimal design problem numerically.