Lecture 07: Nonlinear Preconditioning Methods and Applications

We consider solving system of nonlinear algebraic equations arising from the discretization of partial differential equations. Inexact Newton is a popular technique for such problems. When the nonlinearities in the system are well-balanced, Newton\u27s method works well, but when a small number of nonlinear functions in the system are much more nonlinear than the others, Newton may converge slowly or even stagnate. In such a situation, we introduce some nonlinear preconditioners to balance the nonlinearities in the system. The preconditioners are often constructed using a combination of some domain decomposition methods and nonlinear elimination methods. For the nonlinearly preconditioned problem, we show that fast convergence can be restored. In this talk we first review the basic algorithms, and then discuss some recent progress in the applications of nonlinear preconditioners for some difficult problems arising in computational mechanics including both fluid dynamics and solid mechanics

Modeling and simulation of arterial walls with focus on damage and residual stresses

Die vorliegende Arbeit behandelt die kontinuumsmechanische Modellierung von Arterienwänden. Ein Schwerpunkt liegt in der Konstruktion von anisotropen Schädigungsmodellen zur Beschreibung von Schädigungseffekten in Arterienwänden, wie sie bei therapeutischen Maßnahmen auftreten. Solche Schädigungseffekte gelten als einer der wesentlichen Faktoren für eine erfolgreiche Behandlung von atherosklerotisch degenerierten Arterien mittels Ballonangioplastie. Ein weiterer Schwerpunkt liegt in der Erarbeitung eines numerischen Modells zur Berücksichtigung von Eigenspannungen in Arterienwänden. Eigenspannungen beeinflussen die Spannungsverteilung in Umfangsrichtung derart, dass sie zu einer Verringerung der Spannungsgradienten in der Arterienwand beitragen. Hierauf aufbauend wird ein neuer Ansatz zur Implementierung von Eigenspannungen vorgeschlagen. Alle Modelle werden an experimentelle Daten angepasst und auf die numerische Simulation von patientenspezifischen Arterienwänden angewendet. Die Quasi-Inkompressibilität des Materials wird zum einen durch die Verwendung einer Penalty-Methode und zum anderen über einen Augmented-Lagrange Ansatz erfüllt. Beide Methoden werden hinsichtlich ihres Einflusses auf die Robustheit numerischer Simulationen untersucht.The present work deals with the continuum-mechanical modeling and analysis of arterial walls. One focus is on the construction of anisotropic damage models that are able to reflect damage effects in arterial tissues under therapeutic loading. Damage effects are assumed to be a main contributor to the success of a balloon angioplasty, which is a method of treatment of atherosclerotic arteries. Another main focus is on the elaboration of a numerical model for the incorporation of residual stresses in arterial walls. Residual stresses influence the circumferential stress distribution in such a way that they prevent large stress gradients in the arterial wall. Thus, a novel approach for the implementation of residual stresses is proposed. All models are adjusted to experimental data and applied to numerical simulations of patient-specific arterial walls. The quasi-incompressibility constraint is ensured by using the Penalty-Method and the Augmented-Lagrange-Method, which are analyzed with respect to their computational robustness

Geometrical Modeling and Numerical Simulation of Heterogeneous Materials

The discretization of the considered body or material by finite elements is a crucial part of the Finite Element Method in addition to the material modeling and the element formulation. Thereby, the treatment of heterogeneous materials is an advanced challenge, because the interfaces between the individual constituents must be taken into account in addition to the free surfaces. In many materials these interfaces exhibit complex geometries, since they are built up in growth and transformation processes. In the present work the numerical analysis of such materials is presented starting from the geometrical construction and ending up with the evaluation of the computational results. The first part is concerned with the simulation of diseased blood vessels and focusses on the reconstruction of patient-specific arterial geometries. The results of two- and three-dimensional finite element simulations show the field of application of the presented method. The consideration of the heterogeneity of modern two-phase steels in the numerical simulation is given in the following part of the present work. Therein the focus is on the application of geometrically simplified structures, which exhibit a similar mechanical response compared to the real microstructure. The applicability of the proposed method is shown in different boundary value problems using a direct micro-macro transition approach.Bei Simulationen unter Verwendung der Finite-Elemente-Methode spielt neben der Materialmodellierung und Elementformulierung die Diskretisierung des zu untersuchenden Körpers oder Materials durch finite Elemente eine große Rolle. Diese Aufgabe wird erschwert, wenn es sich um heterogene Materialien handelt. Bei diesen müssen zusätzlich zu den äußeren freien Oberflächen die inneren Grenzschichten zwischen den jeweiligen Individuen berücksichtigt werden. In vielen Materialien sind diese Grenzflächen durch Wachstums- oder Umwandlungsprozesse entstanden, können somit auch komplexe Strukturen aufweisen und erschweren die geometrische Beschreibung. Die vorliegende Arbeit beschäftigt sich im Wesentlichen mit der numerischen Analyse solcher Materialien ausgehend von der Konstruktion der Geometrien bis hin zur Auswertung der Simulationsergebnisse. Der erste Teil der Arbeit beschäftigt sich mit der Simulation von erkrankten Blutgefäßen und geht dort vor allem auf die Rekonstruktion von patienten-spezifischen Arteriengeometrien ein. Die Ergebnisse von zwei- und dreidimensionalen FE-Berechnung verdeutlichen das Einsatzgebiet der vorgestellten Methodik. Die Berücksichtigung der Heterogenität moderner Zweiphasenstähle in der numerischen Simulation wird im anschließenden Teil der Arbeit vorgestellt. Hierbei liegt der Schwerpunkt auf dem Einsatz von geometrisch vereinfachten Ersatzstrukturen, die ein vergleichbares mechanisches Antwortverhalten zur realen Mikrostruktur liefern. Die Anwendbarkeit dieser Methode wird in verschiedenen Randwertproblemen unter Einsatz eines direkten Mikro-Makro Übergangs gezeigt

Efficient Reduction Techniques for the Simulation and Optimization of Parametrized Systems:Analysis and Applications

This thesis is concerned with the development, analysis and implementation of efficient reduced order models (ROMs) for the simulation and optimization of parametrized partial differential equations (PDEs). Indeed, since the high-fidelity approximation of many complex models easily leads to solve large-scale problems, the need to perform multiple simulations to explore different scenarios, as well as to achieve rapid responses, often requires unaffordable computational resources. Alleviating this extreme computational effort represents the main motivation for developing ROMs, i.e. low-dimensional approximations of the underlying high-fidelity problem. Among a wide range of model order reduction approaches, here we focus on the so-called projection-based methods, in particular Galerkin and Petrov-Galerkin reduced basis methods. In this context, the goal is to generate low cost and fast, but still sufficiently accurate ROMs which characterize the system response for the whole range of input parameters we are interested in. In particular, several challenges have to be faced to ensure reliability and computational efficiency. As regards the former, this thesis presents some heuristic approaches to approximate the stability factor of parameterized nonlinear PDEs, a key ingredient of any a posteriori error estimate. Concerning computational efficiency, we propose different strategies to combine the `Matrix Discrete Empirical Interpolation Method' (MDEIM) with a state approximation resulting either from a proper orthogonal decomposition or a greedy approach. Specifically, we exploit the MDEIM to develop fast and efficient ROMs for nonaffinely parametrized elliptic and parabolic PDEs, as well as for the time-dependent Navier-Stokes equations. The efficacy of the proposed methods is demonstrated on a variety of computationally-intensive applications, such as the shape optimization of an acoustic device, the simulation of blood flow in cerebral aneurysms and the simulation of solute dynamics in blood flow and arterial walls. %and coupled blood flow and mass transport in human arteries. Furthermore, the above-mentioned techniques have been exploited to develop a model order reduction framework for parametrized optimization problems constrained by either linear or nonlinear stationary PDEs. In particular, among this wide class of problems, here we focus on those featuring high-dimensional control variables. To cope with this high dimensionality and complexity, we propose an all-at-once optimize-then-reduce paradigm, where a simultaneous state and control reduction is performed. This methodology is applied first to a data reconstruction problem arising in hemodynamics, and then to several optimal flow control problems