Analysis of Systems of Hyperbolic Partial Differential Equations Coupled to Switched Differential Algebraic Equations

  • Simplified ODE models describing blood flow rate are governed by the pressure gradient. However, assuming the orientation of the blood flow in a human body correlates to a positive direction, a negative pressure gradient forces the valve to shut, which stops the flow through the valve, hence, the flow rate is zero, whereas the pressure rate is formulated by an ODE. Presence of ODEs together with algebraic constraints and sudden changes of system characterizations yield systems of switched differential-algebraic equations (swDAEs). Alternating dynamics of the heart can be well modelled by means of swDAEs. Moreover, to study pulse wave propagation in arteries and veins, PDE models have been developed. Connection between the heart and vessels leads to coupling PDEs and swDAEs. This model motivates to study PDEs coupled with swDAEs, for which the information exchange happens at PDE boundaries, where swDAE provides boundary conditions to the PDE and PDE outputs serve as inputs to swDAE. Such coupled systems occur, e.g. while modelling power grids using telegrapher’s equations with switches, water flow networks with valves and district heating networks with rapid consumption changes. Solutions of swDAEs might include jumps, Dirac impulses and their derivatives of arbitrary high orders. As outputs of swDAE read as boundary conditions of PDE, a rigorous solution framework for PDE must be developed so that jumps, Dirac impulses and their derivatives are allowed at PDE boundaries and in PDE solutions. This is a wider solution class than solutions of small bounded variation (BV), for instance, used in where nonlinear hyperbolic PDEs are coupled with ODEs. Similarly, in, the solutions to switched linear PDEs with source terms are restricted to the class of BV. However, in the presence of Dirac impulses and their derivatives, BV functions cannot handle the coupled systems including DAEs with index greater than one. Therefore, hyperbolic PDEs coupled with swDAEs with index one will be studied in the BV setting and with swDAEs whose index is greater than one will be investigated in the distributional sense. To this end, the 1D space of piecewise-smooth distributions is extended to a 2D piecewise-smooth distributional solution framework. 2D space of piecewise-smooth distributions allows trace evaluations at boundaries of the PDE. Moreover, a relationship between solutions to coupled system and switched delay DAEs is established. The coupling structure in this thesis forms a rather general framework. In fact, any arbitrary network, where PDEs are represented by edges and (switched) DAEs by nodes, is covered via this structure. Given a network, by rescaling spatial domains which modifies the coefficient matrices by a constant, each PDE can be defined on the same interval which leads to a formulation of a single PDE whose unknown is made up of the unknowns of each PDE that are stacked over each other with a block diagonal coefficient matrix. Likewise, every swDAE is reformulated such that the unknowns are collected above each other and coefficient matrices compose a block diagonal coefficient matrix so that each node in the network is expressed as a single swDAE. The results are illustrated by numerical simulations of the power grid and simplified circulatory system examples. Numerical results for the power grid display the evolution of jumps and Dirac impulses caused by initial and boundary conditions as a result of instant switches. On the other hand, the analysis and numerical results for the simplified circulatory system do not entail a Dirac impulse, for otherwise such an entity would destroy the entire system. Yet jumps in the flow rate in the numerical results can come about due to opening and closure of valves, which suits clinical and physiological findings. Regarding physiological parameters, numerical results obtained in this thesis for the simplified circulatory system agree well with medical data and findings from literature when compared for the validation

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Author:Damla Kocoglu
URN (permanent link):urn:nbn:de:hbz:386-kluedo-62059
Advisor:Stephan Trenn
Document Type:Doctoral Thesis
Language of publication:English
Publication Date:2021/01/06
Year of Publication:2021
Publishing Institute:Technische Universität Kaiserslautern
Granting Institute:Technische Universität Kaiserslautern
Acceptance Date of the Thesis:2020/12/12
Date of the Publication (Server):2021/01/07
Number of page:IX, 140
Faculties / Organisational entities:Fachbereich Mathematik
DDC-Cassification:5 Naturwissenschaften und Mathematik / 510 Mathematik
Licence (German):Creative Commons 4.0 - Namensnennung, nicht kommerziell, keine Bearbeitung (CC BY-NC-ND 4.0)