Reliable direct and inverse methods in computational hemodynamics Open Access

Bertagna, Luca (2015)

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In the last 25 years, developments in mathematical models/methods together with the improvements in the data acquisition devices have made possible to use mathematics to study the behavior of the human cardiovascular system. Furthermore, cardiovascular mathematics has not been limited to be used as a descriptive qualitative tool, but instead, has started to be used for quantitative analysis of patients conditions and even treatment design. The robustness of this tool depends on the reliability of the results. Data Assimilation (DA) is a set of techniques that helps to improve the specificity of the model, by incorporating available data into the model and therefore leading to patient specific results. On the other hand, the numerical methods used in the simulations must be accurate enough to guarantee that the computed solution accurately describes the real behavior of the system. This work is divided into two parts. In the first, we focus on the estimation of the compliance of a blood vessel using DA techniques. In particular, we use measurements of the displacement of the vessel wall to estimate its Young's modulus. We adopt the variational approach proposed Perego et al. (2011), and we focus on the issue of the computational costs associated with the solution of the inverse problem. The secon part concerns the accurate simulation of flows at moderately large Reynolds numbers. In particular, we focus on the model proposed in Layton et al. (2012) for the discretization of the Leray system, and we propose a new interpretation of the method as an operator-splitting scheme, for a perturbed version of the Navier-Stokes equations, and we use heuristic arguments to calibrate one of the main parameters of the model. For both these parts we will perform numerical experiments, on 3D geometries, to validate the approaches. In particular, for the first part, we will use synthetic measures to validate our approach, while for the second part, we will test the method on a benchmark proposed by the Food and Drug Administration, comparing out results with experimental data.

Table of Contents

1 Introduction

1.1 Cardiovascular mathematics

1.2 Some of the challenges in cardiovascular mathematics

1.3 Thesis outline

2 Data Assimilation: methods, examples and applications to cardiovascular mathematics

2.1 Introduction

2.2 Stochastic and deterministic approaches

2.3 The Kalman Filter

2.4 Variational method

2.4.1 The method of Lagrange multipliers

2.4.2 Regularization

2.5 DA in hemodynamics

3 Variational estimation of the compliance of a blood vessel

3.1 Motivation 3.2 The forward problem

3.2.1 The fluid-membrane interaction

3.2.2 The discrete problem

3.3 The inverse problem

3.4 Results

3.4.1 Cylinder case

3.4.2 Idealized aortic arch case

4 Reduced Order Modeling for the compliance estimation problem

4.1 Reduced Order Models

4.1.1 Greedy Reduced Basis

4.1.2 Proper Orthogonal Decomposition

4.2 A POD approach for the compliance estimation problem

4.3 Results

4.3.1 Cylinder case

4.3.2 Idealized aortic arch case

5 Deconvolution-based filtering schemes

5.1 Motivation: numerical simulation of turbulent flows

5.2 Non-linear Leray models

5.2.1 The continuous problem

5.2.2 The time-discrete problem

5.3 Indicator functions

5.3.1 Physical phenomenology based indicator functions

5.3.2 Deconvolution based indicator functions

5.4 EFR as an operator-splitting algorithm

5.4.1 The choice of the relaxation parameter

5.4.2 The boundary conditions

5.5 Discretization of the operator-splitting algorithm

5.6 Numerical experiments

5.6.1 Case Re 3500

5.6.2 Case Re 5000

6 Conclusions and future directions

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