Numerical modelling of fluid-structure interaction with application in hemodynamics

The thesis deals with numerical algorithms for fluid-structure interaction problems with application in blood flow modelling. It starts with a short introduction on the mathematical description of incompressible viscous flow with non-Newtonian viscosity and a moving linear viscoelastic structure. The mathematical model consists of the generalized Navier-Stokes equation used for the description of fluid flow and the generalized string model for structure movement. The arbitrary Lagrangian-Eulerian approach is used in order to take into account moving computational domain. A part of the thesis is devoted to the discussion on the non-Newtonian behaviour of shear-thinning fluids, which is in our case blood, and derivation of two non-Newtonian models frequently used in the blood flow modelling. Further we give a brief overview on recent fluid-structure interaction schemes with discussion about the difficulties arising in numerical modelling of blood flow. Our main contribution lies in numerical and experimental study of a new loosely-coupled partitioned scheme called the kinematic splitting fluid-structure interaction algorithm. We present stability analysis for a coupled problem of non-Newtonian shear-dependent fluids in moving domains with viscoelastic boundaries. Here, we assume both, the nonlinearity in convective as well is diffusive term. We analyse the convergence of proposed numerical scheme for a simplified fluid model of the Oseen type. Moreover, we present series of experiments including numerical error analysis, comparison of hemodynamic parameters for the Newtonian and non-Newtonian fluids and comparison of several physiologically relevant computational geometries in terms of wall displacement and wall shear stress. Numerical analysis and extensive experimental study for several standard geometries confirm reliability and accuracy of the proposed kinematic splitting scheme in order to approximate fluid-structure interaction problems.

Numerical analysis of the relation between interactions and structure in a molecular fluid

Coarse graining is a popular technique used in physics to speed up the computer simulation of molecular fluids. An essential part of this technique is a method that solves the inverse problem of determining the interaction potential or its parameters from the given structural data. Due to discrepancies between model and reality, the potential is not unique, such that stability of such method and its convergence to a meaningful solution are issues.rnrnIn this work, we investigate empirically whether coarse graining can be improved by applying the theory of inverse problems from applied mathematics. In particular, we use the singular value analysis to reveal the weak interaction parameters, that have a negligible influence on the structure of the fluid and which cause non-uniqueness of the solution. Further, we apply a regularizing Levenberg-Marquardt method, which is stable against the mentioned discrepancies. Then, we compare it to the existing physical methods - the Iterative Boltzmann Inversion and the Inverse Monte Carlo method, which are fast and well adapted to the problem, but sometimes have convergence problems.rnrnFrom analysis of the Iterative Boltzmann Inversion, we elaborate a meaningful approximation of the structure and use it to derive a modification of the Levenberg-Marquardt method. We engage the latter for reconstruction of the interaction parameters from experimental data for liquid argon and nitrogen. We show that the modified method is stable, convergent and fast. Further, the singular value analysis of the structure and its approximation allows to determine the crucial interaction parameters, that is, to simplify the modeling of interactions. Therefore, our results build a rigorous bridge between the inverse problem from physics and the powerful solution tools from mathematics. rn