Formoptimierung für Fluid-Struktur Interaktionsprobleme

In this thesis, shape optimization for unsteady fluid-structure interaction problems via the method of mappings is investigated theoretically and numerically. New existence and regularity results are proven. A framework for deriving differentiability for the solution of nonlinear, unsteady, parameter-dependent partial differential equations is developed and applied to show differentiability of the states with respect to domain variations.In dieser Arbeit wird Formoptimierung für instationäre Fluid-Struktur Interaktionsprobleme mittels der Method of Mappings theoretisch und numerisch untersucht. Es werden neue Existenz- und Regularitätsaussagen bewiesen. Des Weiteren wird ein theoretischer Rahmen entwickelt, womit Differenzierbarkeit für die Lösung von nichtlinearen, instationären und parameterabhängigen Differentialgleichungen gezeigt werden kann, und angewandt, um Differenzierbarkeit des Zustandes bezüglich Gebietsvariationen zu zeigen

Shape Optimization of hemolysis for shear thinning flows in moving domains

We consider the $3$D problem of hemolysis minimization in blood flows, namely the minimization of red blood cells damage, through the shape optimization of moving domains. Such a geometry is adopted to take into account the modeling of rotating systems and blood pumps. The blood flow is described by generalized Navier-Stokes equations, in the particular case of shear thinning flows. The velocity and stress fields are then used as data for a transport equation governing the hemolysis index, aimed to measure the red blood cells damage rate. For a sequence of converging moving domains, we show that a sequence of associated solutions to blood equations converges to a solution of the problem written on the limit moving domain. Thus, we extended the result given in (Soko\l{}owski, Stebel, 2014, in \textit{Evol. Eq. Control Theory}) for $q \geq 11/5$, to the range $6/5< q < 11/5$, where $q$ is the exponent of the rheological law. We then show that the sequence of hemolysis index solutions also converges to the limit solution. This shape continuity properties allows us to show the existence of minimal shapes for a class of functionals depending on the hemolysis index

Adaptive Finite Element Simulation of Fluid-Structure Interaction with Application to Heart-Valve Dynamics

The goal of this work is the development of concepts for the efficient numerical solution of fluid-structure interaction (FSI) problems with applications to heart-valve dynamics. The main motivation for further development in this field is an increasing demand from the medical community for scientifically rigorous investigations of cardiovascular diseases, which are responsible for the major fraction of mortalities in industrialized countries. In this work, the ALE (arbitrary Lagrangian Eulerian) description of fluid equations is utilized for the numerical modeling and simulation of fluid-structure interactions. Using this approach, the fluid equations can easily be coupled with structural deformations. The focal goal is the modeling, numerical analysis, and simulation of prototypical heart-valve dynamics, which requires the investigation of the following issues: the analysis of various fluid-mesh motion techniques, a comparison of different second-order time-stepping schemes, and the prescription of specific boundary conditions on the artificial outflow boundary. To control computational costs, we apply a simplified version of an a posteriori error estimation using the dual weighted residual (DWR) method. This method is used for mesh adaption during the computation. The last, novel aspect comprises a discussion of optimal control problems for wall stress minimization, in which the state is determined by a fluid-structure interaction system. The concepts developed in this work are demonstrated with several numerical tests in two and three dimensions. The programming code is validated by computing several FSI benchmark tests. The focal computation is related to a prototypical two-dimensional aortic heart-valve simulation. The concepts illustrated by this example were developed in cooperation with a cardiologist

Reduced Models for Optimal Control, Shape Optimization and Inverse Problems in Haemodynamics

The objective of this thesis is to develop reduced models for the numerical solution of optimal control, shape optimization and inverse problems. In all these cases suitable functionals of state variables have to be minimized. State variables are solutions of a partial differential equation (PDE), representing a constraint for the minimization problem. The solution of these problems induce large computational costs due to the numerical discretization of PDEs and to iterative procedures usually required by numerical optimization (many-query context). In order to reduce the computational complexity, we take advantage of the reduced basis (RB) approximation for parametrized PDEs, once the state problem has been reformulated in parametrized form. This method enables a rapid and reliable approximation of parametrized PDEs by constructing low-dimensional, problem-specific approximation spaces. In case of PDEs defined over domains of variable shapes (e.g. in shape optimization problems) we need to introduce suitable, low-dimensional shape parametrization techniques in order to tackle the geometrical complexity. Free-Form Deformations and Radial-Basis Functions techniques have been analyzed and successfully applied with this aim. We analyze the reduced framework built by coupling these tools and apply it to the solution of optimal control and shape optimization problems. Robust optimization problems under uncertain conditions are also taken into consideration. Moreover, both deterministic and Bayesian frameworks are set in order to tackle inverse identification problems. As state equations, we consider steady viscous flow problems described by Stokes or Navier-Stokes equations, for which we provide a detailed analysis and construction of RB approximation and a posteriori error estimation. Several numerical test cases are also illustrated to show efficacy and reliability of RB approximations. We exploit this general reduced framework to solve some optimization and inverse problems arising in haemodynamics. More specifically, we focus on the optimal design of cardiovascular prostheses, such as bypass grafts, and on inverse identification of pathological conditions or flow/shape features in realistic parametrized geometries, such as carotid artery bifurcations

Geometry identification and data enhancement for distributed flow measurements

The measurement of fluid motion is an important tool for researchers in fluiddynamics. Measurements with increasing precision did expedite the development of fluid-dynamic models and their theoretical understanding. Several well-established experimental techniques provide point-wise information on the flow field. In recent years novel measurement modalities have been investigated which deliver spatially resolved three-dimensional velocity measurements. Note that for methods such as particle tracking and tomographic particle imaging optical access to the flow domain is necessary. For other methods like magnetic resonance velocimetry, CT-angiography, or x-ray velocimetry this is, however, not the case. Such a property and also the fact that those methods are able to provide three-dimensional velocity fields in a rather short acquisition time makes them in particular suited for in-vivo applications. Our work is motivated by such non-invasive velocity measurement techniques for which no optical access to the interior of the geometry is needed and also not available in many cases. Here, an additional difficulty is that the exact flow geometry is in general not known a priori. The measurement techniques we are interested in, are extensions of already available medical imaging modalities. As a prototypical example, we consider magnetic resonance velocimetry, which is also suited for the measurement of turbulent fluid motion. We will also discuss computational examples using such measurement data. General purpose. Our main goal is a suitable post-processing of the available velocity data and also to obtain additional information. The measurements available from magnetic resonance velocimetry consist of several components given on a fixed field of view. The magnitude of the MRT signal corresponds to a proton density and thus e.g. the density of water molecules. Those data typically give a clear indication of the position and size of the flow geometry. The velocity data, on the other hand, are substantially perturbed outside the flow domain. This is a typical feature of measurements stemming from magnetic resonance velocimetry. Note that the surrounding noise usually has a notably higher magnitude than the actual measurements. Thus, a first necessary step will be to somehow separate the domain containing valuable velocity data from the noise surrounding it. For this reason, we apply some kind of image segmentation where we make use of the given den-sity image. Since the velocity values are given on the same field of view the segmentation directly transfers to those data. Due to the measurement procedure also the segmented velocity data are contaminated by measurement errors. Therefore, besides segmentation, additional post-processing is necessary in order to make the flow measurements available for further usage. In a second step, we propose a problem adapted data enhancement method which is able to provide a smoothed velocity field on the one hand, and also provides additional information on the other hand, like for instance the pressure drop or an estimate for the wall shear stress. The two main steps will therefore be: (i) The identification of the flow geometry, where we make use of the available density measurements. (ii) The denoising and improvement of the segmented velocity data, by using a suitable fluid-dynamical model. Outline. In part I of this thesis, we introduce our basic approach to the geom- etry identification and velocity enhancement problems described above. Both problems are formulated as optimal control problems governed by a partial differential equation and we shortly discuss some general aspects of the analysis and the solution of such problems in section 4. In part II, we thoroughly discuss and analyze the geometry identification problem introduced in section 2. The procedure is formulated as an inverse ill-posed problem and we propose a Tikhonov regularization for its stable solution. We show that the resulting optimal control problem has a solution and discuss its numerical treatment with iterative methods. Finally, a systematic discretization can be realized using finite elements which is also demonstrated by numerical tests. The velocity enhancement problem is introduced in part III. We propose a linearized flow-model which directly incorporates the available measurements. The resulting modeling error can be quantified in terms of the data error. The reconstruction method is then formulated as an optimal control problem subject to the linearized equations. We show the existence of a unique solution and derive estimates for the reconstruction error. Additionally, a reconstruction for the pressure is obtained for which we derive similar error estimates. We discuss the systematic discretization using finite elements and show preliminary computational examples for the verification of the derived estimates. In order to verify the applicability of the proposed methods to realistic data, we consider an application using experimental data in part IV. We use measurements of a human blood vessel stemming from magnetic resonance velocimetry obtained at the University Medical Center in Freiburg. After a suitable pre-processing of the available data, we apply the geometry identification method in order to obtain a discretization of the blood vessel. Using the generated mesh, we reconstruct an enhanced velocity field and the pressure from the available velocity data

Variational Methods for the Estimation of Transport Fields with Application to the Recovery of Physics-Based Optical Flows Across Boundaries

In this thesis we develop a method for the estimation of the flow behaviour of an incom- pressible fluid based on observations of the brightness intensity of a transported visible substance which does not influence the flow. The observations are given in a subregion of the flow as a sequence of discrete images with in- and outflow across the image boundaries. The resulting mathematical problem is ill-posed and has to be regularised with information of the underlying fluid flow model. We consider a constrained optimisation problem, namely the minimisation of a tracking type data term for the brightness distribution and a regularisation term subject to a system of weakly coupled partial differential equations. The system consists of the time- dependent incompressible Navier-Stokes equations coupled by the velocity vector field to a convection-diffusion equation, which describes the transport of brightness patterns in the image sequence. Due to the flow across the boundaries of the computational domain we solve a boundary identification problem. The usage of (strong) Dirichlet boundary controls for this purpose leads to theoretical and numerical complications, so that we will instead use Robin-type controls, which allow for a more convenient theoretical and numerical framework. We will prove well-posedness and investigate the functionality of the proposed approach by means of numerical examples. Furthermore, we discuss the connection to Dirichlet-control problems, e. g. the approximation of Dirichlet-controls by the so-called penalised Neumann method, which is based on the Robin-type controls for a varying penalty parameter. We will show via numerical tests that Robin-type controls are suitable for the identifi- cation of the correct fluid flow. Moreover, the examples indicate that the underlying physical model used for the regularisation influences the flow reconstruction process. Thus appropriate knowledge of the model is essential, e. g. the viscosity parameter. For a time- independent example we will present a heuristic, which, beside the boundary identification, automatically evaluates the viscosity in case the parameter is unknown. The developed physics-based optical flow estimation approach is finally used for the data set of a prototypical application. The background of the application is the approximation of horizontal wind fields in sparsely populated areas like desert regions. A sequence of satellite images documenting the brightness intensity of an observable substance distributed by the wind (e. g. dust plumes) is thereby assumed to be the only available data. Wind field information is for example needed to simulate the distribution of other, not directly observ- able, substances in the lower atmosphere. For the prototypical example we compute a high quality reconstruction of the underlying fluid flow by a (discrete) sequence of consecutive spatially distributed brightness intensities. Thereby, we compare three different models (heat equation, Stokes system and the original fluid flow model) in the reconstruction process and show that using as much model knowledge as possible is essential for a good reconstruction result

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