Numerical approximation of cardiac electro-fluid-mechanical models

The mathematical modeling of the heart involves several challenges, which are intrinsically related to the complexity of its function. A satisfactory cardiac model must be able to describe a wide range of different processes, such as the evolution of the transmembrane potential in the myocardium, the deformation caused by the muscles contraction, and the dynamics of the blood inside the heart chambers. In this work, we focus on the coupling of the electrophysiology, the active and the passive mechanics, and the fluid dynamics of the blood in the left ventricle (LV) of the human heart. The models describing the previously mentioned processes are called âsingle core modelsâ, and can be regarded as the building blocks of an âintegrated modelâ. In this thesis, we first review the isolated single core mathematical models for the description of the LV function, and discuss their space and time discretizations with particular emphasis on the coupling conditions. We consider both implicit and semi-implicit schemes for the time discretization. The fully discretized single core problems thus obtained are then combined to define integrated electromechanics and electrofluidmechanics problems. We then focus on the numerical coupling strategy for the electromechanics solver in the framework of the active strain formulation. First, we propose a monolithic strategy where the discretized core models are solved simultaneously; then, several novel segregated strategies, where the discretized core models are solved sequentially, are proposed and systematically compared with each other. The segregated strategies are obtained by exploiting a Godunov splitting scheme, which introduces a first order error on the solution. We show that, while the monolithic approach is more accurate and more stable for relatively large timesteps, segregated approaches allow to solve the integrated problem much more efficiently in terms of computational resources. Moreover, with segregated approaches, it is possible to use different timesteps for the different core models in a staggered fashion, thus further improving the computational efficiency of the schemes. The monolithic and the segregated strategies for the electromechanics are used to solve a benchmark problem with idealized geometry: the results are then compared in terms of accuracy and efficiency. We numerically confirm that the segregated strategies are accurate at least of order one. In light of the results obtained, we employ the proposed strategies to simulate the electromechanics of a subject-specific LV for a full heartbeat. We simulate both healthy and pathological scenarios: in the latter case, we account for an ischemic necrosis of the tissue and analyze several clinical indicators such as pressure-volume loops and the end systolic pressure-volume relationship. Finally, we use the proposed strategies to simulate the electrofluidmechanics of a realistic LV during the systolic phase of the heartbeat. When defining the integrated cardiac models, we establish a preprocess pipeline aimed at preparing geometries and data for both idealized and subject-specific simulations. The pipeline is succesfully used for the setting up of large scale simulations in a high performance computing framework, where the (strong and weak) scalability of the proposed coupling strategies is assessed.

Large Eddy Simulation of Particle Dynamics Inside a Differentially Heated Cavity

Christoph Bosshard

In nuclear safety, most severe accident scenarios lead to the presence of fission products in aerosol form in the closed containment atmosphere. It is important to understand the particle depletion process to estimate the risk of a release of radioactivity to the environment should a containment break occur. As a model for the containment, we use the three-dimensional differentially heated cavity (DHC) problem. DHC is a cubical box with a hot wall and a cold wall on vertical opposite sides. On the other walls of the cube we have adiabatic boundary conditions. For the velocity field the no-slip boundary condition is valid. The flow of the air in the cavity is described by the Boussinesq equations. Complex flow patterns develop and the flow characteristics depend on the non-dimensional Rayleigh and Prandtl numbers. The predominant flow type in the DHC is a turbulent natural convection flow. This work aims at reaching Rayleigh numbers and turbulent levels as high as possible given the available computational resources. The method used to simulate the turbulent flow is the large eddy simulation (LES) where the dynamics of the large eddies is resolved by the computational grid and the small eddies are modelled by the introduction of subgrid scale quantities using a filter function. Numerically, the LES equations are discretized by the spectral element method. Particle trajectories are computed using the Lagrangian particle tracking method, including the relevant forces (drag, gravity, thermophoresis). Four different particle sets with each set containing one million particles and diameters of 10 μm, 15 μm, 25 μm and 35 μm are simulated. The complexity and the size of the large three-dimensional problem requires the use of massively parallel supercomputers. Spectral element methods are naturally suitable for parallelisation by distributing the elements among the processors. For the Lagrangian particle tracking we use a method where equal numbers of particles are assigned to every processor. The flow field is broadcast and every particle processor tracks the assigned particles, a procedure which leads to a perfect load balancing. Simulation results for the flow field and particle sizes from 15 μm to 35 μm at a Rayleigh number of 109 are compared to previous results from a direct numerical simulation. First order statistics of the LES flow fields are in very good agreement with the direct numerical simulation while the agreement of second order moments is fair. Also the turbulent structures associated to the maximum of turbulent kinetic energy production are correctly reproduced. Particle statistics in the LES and the direct numerical simulation were similar and the settling rates practically identical. Contrary to previous particle simulations in LES, it was found that no model was necessary for the influence of the unresolved flow scales on the particle motions. This can be explained, because the important settling mechanism is through gravity and particle deposition at the walls by turbophoresis is negligible.

EPFL2012

Isogeometric Analysis for Reduced Fluid-Structure Interaction Models in Haemodynamic Applications

Anna Tagliabue

Isogeometric analysis (IGA) is a computational methodology recently developed to numerically approximate Partial Differential Equation (PDEs). It is based on the isogeometric paradigm, for which the same basis functions used to represent the geometry are then used to approximate the unknown solution of the PDEs. In the case in which Non-Uniform Rational B-Splines (NURBS) are used as basis functions, their mathematical properties lead to appreciable benefits for the numerical approximation of PDEs, especially for high order PDEs in the standard Galerkin formulation. In this framework, we propose an a priori error estimate, extending existing results limited to second order PDEs. The improvements in both accuracy and efficiency of IGA compared to Finite Element Analysis (FEA), encourage the use of this methodology in the haemodynamic applications. In fact, the simulation of blood flow in arteries requires the numerical approximation of Fluid-Structure Interaction (FSI) problems. In order to account for the deformability of the vessel, the Navier-Stokes equations representing the blood flows, are coupled with structural models describing the mechanical response of the arterial wall. However, the FSI models are complex from both the mathematical and the numerical points of view, leading to high computational costs during the simulations. With the aim of reducing the complexity of the problem and the computational costs of the simulations, reduced FSI models can be considered. A first simplification, based on the assumption of a thin arterial wall structure, consists in considering shell models to describe the mechanical properties of the arterial walls. Moreover, by means of the additional kinematic condition (continuity of velocities) and dynamic condition (balance of contact forces), the structural problem can be rewritten as generalized boundary condition for the fluid problem. This results in a generalized Navier-Stokes problem which can be expressed only in terms of the primitive variables of the fluid equations (velocity and pressure) and in a fixed computational domain. As a consequence, the computational costs of the numerical simulations are significantly reduced. On the other side, the generalized boundary conditions associated to the reduced FSI model could involve high order derivatives, which need to be suitably approximated. With this respect, IGA allows an accurate, straightforward and efficient numerical approximation of the generalized Navier-Stokes equations characterizing the reduced FSI problem. In this work we consider the numerical approximation of reduced FSI models by means of IGA, for which we discuss the numerical results obtained in Haemodynamic applications.

2012

Structure-preserving approaches and data-driven closure modeling for model order reduction

Nicolò Ripamonti

In this thesis, we propose model order reduction techniques for high-dimensional PDEs that preserve structures of the original problems and develop a closure modeling framework leveraging the Mori-Zwanzig formalism and recurrent neural networks. Since high-fidelity approximations of PDEs often result in a large number of degrees of freedom, the need for iterative evaluations for numerical optimizations and rapid feedback is computationally challenging.The first part of this thesis is devoted to conserving the high-dimensional equation's invariants, symmetries, and structures during the reduction process. Traditional reduction techniques are not guaranteed to yield stable reduced systems, even if the target problem is stable. In the context of fluid flows, the skew-symmetric structure of the problem entails the preservation of the kinetic energy of the system. By preserving the same structure at the level of the reduced model, we obtain enhanced stability, and accuracy and the reduced model acquires physical significance by preserving a surrogate of the energy of the original problem. Next, we focus on Hamiltonian systems, which, being driven by symmetry, are a source of great interest in the reduction community. It is well known that the breaking of these symmetries in the reduced model is accompanied by a blowup of the system energy and flow volume. In this thesis, geometric reduced models for Hamiltonian systems are further developed and combined with the dynamically orthogonal methods, addressing the poor reducibility in time of advection-dominated problems. The reduced solution is expressed as a linear combination of a finite number of modes and coincides with the symplectic projection of the high-fidelity Hamiltonian problem onto the tangent space of the approximating manifold. An error surrogate is used to monitor the approximation ability of the reduced model and make a change in the rank of the approximating system if necessary. The method is further developed through a combination of DEIM and DMD to reduce non-polynomial nonlinearities while preserving the symplectic structure of the problem and applied to the Vlasov-Poisson system.In the second part of the thesis, we consider several data-driven methods to address the poor accuracy in the under-resolved regime for Galerkin reduced models via a closure term. The closure term is developed systematically from the Mori-Zwanzig formalism by introducing projection operators on the spaces of resolved and unresolved scales, thus resulting in an additional memory integral term. The interaction between different scales turns out to be nonlocal in time and dominated by a high-dimensional orthogonal dynamics equation, which cannot be solved precisely and efficiently. Several classical methods in the field of statistical mechanics are used to approximate the memory term, exploiting the finiteness of the memory kernel support. We conclude this thesis by showing through numerical experiments how long short-term memory networks, i.e., machine learning structures characterized by feedback connections, represent a valid tool for approximating the additional memory term.

EPFL2022