A mathematical model for breath gas analysis of volatile organic compounds with special emphasis on acetone

Recommended standardized procedures for determining exhaled lower respiratory nitric oxide and nasal nitric oxide have been developed by task forces of the European Respiratory Society and the American Thoracic Society. These recommendations have paved the way for the measurement of nitric oxide to become a diagnostic tool for specific clinical applications. It would be desirable to develop similar guidelines for the sampling of other trace gases in exhaled breath, especially volatile organic compounds (VOCs) which reflect ongoing metabolism. The concentrations of water-soluble, blood-borne substances in exhaled breath are influenced by: (i) breathing patterns affecting gas exchange in the conducting airways; (ii) the concentrations in the tracheo-bronchial lining fluid; (iii) the alveolar and systemic concentrations of the compound. The classical Farhi equation takes only the alveolar concentrations into account. Real-time measurements of acetone in end-tidal breath under an ergometer challenge show characteristics which cannot be explained within the Farhi setting. Here we develop a compartment model that reliably captures these profiles and is capable of relating breath to the systemic concentrations of acetone. By comparison with experimental data it is inferred that the major part of variability in breath acetone concentrations (e.g., in response to moderate exercise or altered breathing patterns) can be attributed to airway gas exchange, with minimal changes of the underlying blood and tissue concentrations. Moreover, it is deduced that measured end-tidal breath concentrations of acetone determined during resting conditions and free breathing will be rather poor indicators for endogenous levels. Particularly, the current formulation includes the classical Farhi and the Scheid series inhomogeneity model as special limiting cases.Comment: 38 page

Adaptive high-resolution finite element schemes

The numerical treatment of flow problems by the finite element method is addressed. An algebraic approach to constructing high-resolution schemes for scalar conservation laws as well as for the compressible Euler equations is pursued. Starting from the standard Galerkin approximation, a diffusive low-order discretization is constructed by performing conservative matrix manipulations. Flux limiting is employed to compute the admissible amount of compensating antidiffusion which is applied in regions, where the solution is sufficiently smooth, to recover the accuracy of the Galerkin finite element scheme to the largest extent without generating non-physical oscillations in the vicinity of steep gradients. A discrete Newton algorithm is proposed for the solution of nonlinear systems of equations and it is compared to the standard fixed-point defect correction approach. The Jacobian operator is approximated by divided differences and an edge-based procedure for matrix assembly is devised exploiting the special structure of the underlying algebraic flux correction (AFC) scheme. Furthermore, a hierarchical mesh adaptation algorithm is designed for the simulation of steady-state and transient flow problems alike. Recovery-based error indicators are used to control local mesh refinement based on the red-green strategy for element subdivision. A vertex locking algorithm is developed which leads to an economical re-coarsening of patches of subdivided cells. Efficient data structures and implementation details are discussed. Numerical examples for scalar conservation laws and the compressible Euler equations in two dimensions are presented to assess the performance of the solution procedure.In dieser Arbeit wird die numerische Simulation von skalaren Erhaltungsgleichungen sowie von kompressiblen Eulergleichungen mit Hilfe der Finite-Elemente Methode behandelt. Dazu werden hochauflösende Diskretisierungsverfahren eingesetzt, welche auf algebraischen Konstruktionsprinzipien basieren. Ausgehend von der Galerkin-Approximation wird eine Methode niedriger Ordnung konstruiert, indem konservative Matrixmodifikationen durchgeführt werden. Anschließend kommt ein sog. Flux-Limiter zum Einsatz, der in Abhängigkeit von der lokalen Glattheit der Lösung den zulässigen Anteil an Antidiffusion bestimmt, die zur Lösung der Methode niedriger Ordnung hinzuaddiert werden kann, ohne dass unphysikalische Oszillationen in der Nähe von steilen Gradienten entstehen. Die resultierenden nichtlinearen Gleichungssysteme können entweder mit Hilfe von Fixpunkt-Defektkorrektur-Techniken oder mittels diskreter Newton-Verfahren gelöst werden. Für letztere wird die Jacobi-Matrix mit dividierten Differenzen approximiert, wobei ein effizienter, kantenbasierter Matrixaufbau aufgrund der speziellen Struktur der zugrunde liegenden Diskretisierung möglich ist. Ferner wird ein hierarchischer Gitteradaptionsalgorithmus vorgestellt, welcher sowohl für die Simulation von stationären als auch zeitabhängigen Strömungen geeignet ist. Die lokale Gitterverfeinerung folgt dem bekannten Rot-Grün Prinzip, wobei rekonstruktionsbasierte Fehlerindikatoren zur Markierung von Elementen zum Einsatz kommen. Ferner erlaubt das sukzessive Sperren von Knoten, die nicht gelöscht werden können, eine kostengünstige Rückvergröberung von zuvor unterteilten Elementen. In der Arbeit wird auf verschiedene Aspekte der Implementierung sowie auf die Wahl von effizienten Datenstrukturen zur Verwaltung der Gitterinformationen eingegangen. Der Nutzen der vorgestellten Simulationswerkzeuge wird anhand von zweidimensionalen Beispielrechnungen für skalare Erhaltungsgleichungen sowie für die kompressiblen Eulergleichungen analysiert

Multiscale Modeling and Simulation of Deformation Accumulation in Fault Networks

Strain accumulation and stress release along multiscale geological fault networks are fundamental mechanisms for earthquake and rupture processes in the lithosphere. Due to long periods of seismic quiescence, the scarcity of large earthquakes and incompleteness of paleoseismic, historical and instrumental record, there is a fundamental lack of insight into the multiscale, spatio-temporal nature of earthquake dynamics in fault networks. This thesis constitutes another step towards reliable earthquake prediction and quantitative hazard analysis. Its focus lies on developing a mathematical model for prototypical, layered fault networks on short time scales as well as their efficient numerical simulation. This exposition begins by establishing a fault system consisting of layered bodies with viscoelastic Kelvin-Voigt rheology and non-intersecting faults featuring rate-and-state friction as proposed by Dieterich and Ruina. The individual bodies are assumed to experience small viscoelastic deformations, but possibly large relative tangential displacements. Thereafter, semi-discretization in time with the classical Newmark scheme of the variational formulation yields a sequence of continuous, nonsmooth, coupled, spatial minimization problems for the velocities and states in each time step, that are decoupled by means of a fixed point iteration. Subsequently, spatial discretization is based on linear and piecewise constant finite elements for the rate and state problems, respectively. A dual mortar discretization of the non-penetration constraints entails a hierarchical decomposition of the discrete solution space, that enables the localization of the non-penetration condition. Exploiting the resulting structure, an algebraic representation of the parametrized rate problem can be solved efficiently using a variant of the Truncated Nonsmooth Newton Multigrid (TNNMG) method. It is globally convergent due to nonlinear, block Gauß–Seidel type smoothing and employs nonsmooth Newton and multigrid ideas to enhance robustness and efficiency of the overall method. A key step in the TNNMG algorithm is the efficient computation of a correction obtained from a linearized, inexact Newton step. The second part addresses the numerical homogenization of elliptic variational problems featuring fractal interface networks, that are structurally similar to the ones arising in the linearized correction step of the TNNMG method. Contrary to the previous setting, this model incorporates the full spatial complexity of geological fault networks in terms of truly multiscale fractal interface geometries. Here, the construction of projections from a fractal function space to finite element spaces with suitable approximation and stability properties constitutes the main contribution of this thesis. The existence of these projections enables the application of well-known approaches to numerical homogenization, such as localized orthogonal decomposition (LOD) for the construction of multiscale discretizations with optimal a priori error estimates or subspace correction methods, that lead to algebraic solvers with mesh- and scale-independent convergence rates. Finally, numerical experiments with a single fault and the layered multiscale fault system illustrate the properties of the mathematical model as well as the efficiency, reliability and scale-independence of the suggested algebraic solver

Incompressible lagrangian fluid flow with thermal coupling

A method is presented for the solution of an incompressible viscous fluid flow with heat transfer and solidification using a fully Lagrangian description of the motion. The originality of this method consists in assembling various concepts and techniques which appear naturally due to the Lagrangian formulation. First of all, the Navier-Stokes equations of motion coupled with the Boussinesq approximation must be reformulated in the Lagrangian framework, whereas they have been mostly derived in an Eulerian context. Secondly, the Lagrangian formulation implies to follow the material particles during their motion, which means to convect the mesh in the case of the Finite Element Method (FEM), the spatial discretisation method chosen in this work. This provokes various difficulties for the mesh generation, mainly in three dimensions, whereas it eliminates the classical numerical difficulty to deal with the convective term, as much in the Navier-Stokes equations as in the energy equation. Even without the discretization of the convective term, an efficient iterative solver, which constitutes the only viable alternative for three dimensional problems, must be designed for the class of Generalized Stokes Problems (GSP), which could be able to behave well independently of the mesh Reynolds number, as it can vary greatly for coupled fluid-thermal analysis. Moreover, it offers a natural framework to treat free-surface problems like wave breaking and rough fluid-structure contact. On one hand, the convection of the mesh during one time step after the resolution of the non-linear system provides explicitly the locus of the domain to be considered. On the other hand, fluid-to-fluid and fluid-to-wall contact, as well as the update of the domain due to the remeshing, must be accurately and efficiently performed. Finally, the solidification of the fluid coupled with its motion through a variable viscosity is considered An efficient overall algorithm must be designed to bring the method effective, particularly in a three dimensional context, which is the ambition of this monograph. Various numerical examples are included to validate and highlight the potential of the method

Incompressible Lagrangian fluid flow with thermal coupling

In this monograph is presented a method for the solution of an incompressible viscous fluid flow with heat transfer and solidification usin a fully Lagrangian description on the motion. The originality of this method consists in assembling various concepts and techniques which appear naturally due to the Lagrangian formulation.Postprint (published version

Space-time residual minimization for parabolic partial differential equations

Many processes in nature and engineering are governed by partial differential equations (PDEs). We focus on parabolic PDEs, that describe time-dependent phenomena like heat conduction, chemical concentration, and fluid flow. Even if we know that a unique solution exists, we can express it in closed form only under very strict circumstances. So, to understand what it looks like, we turn to numerical approximation. Historically, parabolic PDEs are solved using time-stepping. One first discretizes the PDE in space as to obtain a system of coupled ordinary differential equations in time. This system is then solved using the vast theory for ODEs. While efficient in terms of memory and computational cost, time-stepping schemes take global time steps, which are independent of spatial position. As a result, these methods cannot efficiently resolve details in localized regions of space and time. Moreover, being inherently sequential, they have limited possibilities for parallel computation. In this thesis, we take a different approach and reformulate the parabolic evolution equation as an equation posed in space and time simultaneously. Space-time methods mitigate the aforementioned issues, and moreover produce approximations to the unknown solution that are uniformly quasi-optimal. The focal point of this thesis is the space-time minimal residual (MR) method introduced by R. Andreev, that finds the approximation that minimizes both PDE- and initial error. We discuss its theoretical properties, provide numerical algorithms for its computation, and discuss its applicability in data assimilation (the problem of fusing measured data to its underlying PDE)

Numerical Methods for Partial Differential Equations

These lecture notes are devoted to the numerical solution of partial differential equations (PDEs). PDEs arise in many fields and are extremely important in modeling of technical processes with applications in physics, biology, chemisty, economics, mechanical engineering, and so forth. In these notes, not only classical topics for linear PDEs such as finite differences, finite elements, error estimation, and numerical solution schemes are addressed, but also schemes for nonlinear PDEs and coupled problems up to current state-of-the-art techniques are covered. In the Winter 2020/2021 an International Class with additional funding from DAAD (German Academic Exchange Service) and local funding from the Leibniz University Hannover, has led to additional online materials such as links to youtube videos, which complement these lecture notes. This is the updated and extended Version 2. The first version was published under the DOI: https://doi.org/10.15488/9248