Thèse Solveurs Hybrides Fluide-Structure pour la Modélisation Tridimensionnelle Personnalisée en Hémodynamique Cardiovasculaire H/F

Université Paris-Saclay GS Sciences de l'ingénierie et des systèmes

Établissement : Université Paris-Saclay GS Sciences de l'ingénierie et des systèmes
École doctorale : Sciences Mécaniques et Energétiques, Matériaux et Géosciences
Laboratoire de recherche : LMPS - Laboratoire de Mécanique Paris-Saclay
Direction de la thèse : Andrea BARBARULO
Début de la thèse : 2026-10-01
Date limite de candidature : 2026-04-21T23:59:59

Les maladies cardiovasculaires constituent l'une des principales causes de morbidité et de mortalité dans le monde, et leur évaluation quantitative repose de plus en plus sur la modélisation computationnelle spécifique au patient. La modélisation de l'écoulement sanguin dans le système circulatoire humain est un problème complexe d'interaction fluide-structure (FSI), dans lequel l'hémodynamique pulsatile interagit avec des parois artérielles compliantes régies par des équations aux dérivées partielles élastiques ou hyperélastiques non linéaires [1-3]. Des simulations de haute fidélité dans des géométries tridimensionnelles (3D) réalistes, reconstruites à partir d'images médicales, sont essentielles pour évaluer les contraintes de cisaillement pariétal, la propagation des ondes de pression et le transfert des charges mécaniques sous des conditions aux limites physiologiques et cliniquement réalistes [3,11]. La méthode de Boltzmann sur réseau (LBM) [5-9] s'est imposée comme une alternative puissante et hautement scalable aux solveurs traditionnels de Navier-Stokes pour les écoulements cardiovasculaires, avec une capacité démontrée pour la modélisation hémodynamique couplée 0D-3D [3] dans le cadre de simulations en boucle fermée du système circulatoire. Néanmoins, le couplage efficace et robuste de la LBM avec des parois vasculaires compliantes demeure un verrou majeur, en particulier lorsque des modèles structuraux d'ordre complet sont utilisés et que de forts effets de masse ajoutée sont présents [6-8,14]. Bien que les cadres classiques d'interaction fluide-structure basés sur la méthode des éléments finis (FEM) soient bien validés [11-13], leur coût computationnel compromet les gains associés à l'utilisation de la LBM et limite davantage l'exploration systématique des paramètres ou les simulations à requêtes répétées.

L'objectif de cette thèse est de développer un cadre réduit d'interaction LBM-FSI mathématiquement et mécaniquement cohérent pour des simulations cardiovasculaires 3D spécifiques au patient. Le premier axe porte sur le développement et la validation d'un solveur haute performance basé sur la LBM pour l'hémodynamique pulsatile, ainsi que sur une stratégie de couplage bidirectionnel stable pour des modèles de paroi réduits (de type membrane/coque) et des modèles hyperélastiques pleinement tridimensionnels [6-8,14]. Le second axe concerne la construction de modèles d'ordre réduit basés sur la projection pour la mécanique des parois artérielles, en explorant des approches telles que la Décomposition Orthogonale Propre (POD) ou la Décomposition Généralisée Propre (PGD) (de type glouton) [15-18], afin d'obtenir des approximations paramétrées des EDP structurelles non linéaires sous-jacentes, en fonction des propriétés matérielles et des descriptions géométriques, tout en préservant la stabilité et la cohérence en présence d'un couplage fluide-structure fort [10,19]. À terme, ce travail vise à faire progresser les méthodologies de Boltzmann sur réseau d'ordre réduit pour l'interaction fluide-structure cardiovasculaire non linéaire, en développant un cadre computationnel capable de conserver la fidélité physique des modèles à haute résolution tout en permettant des simulations rapides, paramétrées et spécifiques au patient, pour des formulations à paroi mince comme à paroi épaisse.

Modeling of blood flow in the human circulatory system is a complex fluid-structure interaction (FSI) problem essential for understanding the underlying mechanisms of cardiovascular conditions/diseases such as aneurysms, vessel wall dissections, and heart failure [1, 2, 3], as well as for evaluating the design and performance of medical devices such as artificial heart valves, stents, and grafts [4]. Pulsatile flow generated in the heart can be modeled as pressure and flow waves entering the aorta and reflecting through vasculature due to vessel tapering, bifurcations, and variations in wall material properties [1]. These multiphysical dynamics can be studied to assess cardiac health and to develop clinically-relevant analysis.
For modeling such flow in three-dimensional (3D) vessels or organs, the lattice Boltzmann method (LBM) [5, 6, 7, 8, 9], originating from classical statistical physics, is a powerful alternative to conventional continuum-based computational fluid dynamics (CFD) methods that employ Navier-Stokes equations. LBM uses simplified kinetic equations combined with a modified molecular-dynamics approach to model both Newtonian and non-Newtonian fluids [9] (as particles that stream and collide over a discrete lattice mesh) in an efficient and parallelizable framework. The accuracy and usefulness of LBM have been demonstrated in a variety of fluid dynamics problems including turbulence and multiphase flow (see [3] and references therein), having also been shown to be particularly suitable for hemodynamics (able to capture many flow features of clinical interest [3, 4]).
However, coupling LBM with a vessel wall governed by elastic, hyperelastic, or viscoelastic partial differential equation (PDE) models remains a significant bottleneck [3, 10]. In patient-specific simulations, realistic geometries reconstructed from medical imaging [11] must be combined with compliant wall models whose material parameters can be uncertain and heterogeneous. Classical cardiovascular FSI formulations typically employ discretizations based on the finite element method (FEM) for the solid domain (using monolithic or strongly-coupled partitioned algorithms [11, 12, 13]). Although well established, such FEM-based approaches can introduce restrictive time-stepping and high computational cost both for thick and anisotropic walls as well as for thin wall/shell formulations. Recent LBM-FEM strategies have improved stability through implicit coupling and stabilized partitionining [6, 7, 8], although application to full-order nonlinear wall models is still limited in terms of practicality and computational efficiency. From a mathematical perspective, the coupled formulation combines an incompressible Navier-Stokes-type system (recovered from the LBM kinetic equations) with a finite-strain solid, yielding a strongly coupled saddle-point problem at the fluid-solid interface [12, 14]. Such challenges motivate the development of reduced-order representations of the solid structure [16, 17, 15] that can preserve both mathematical and mechanical consistency while enabling efficient exploration of thin- and thick-wall models across patient-specific geometries and material parameters [10].
Projection-based reduced-order modeling (ROM) offers an approach to reduce the computational burden associated with physically-realistic (compliant) wall modeling in cardiovascular FSI. Proper Orthogonal Decomposition (POD) and reduced basis methods provide low-dimensional Galerkin approximations of parametrized PDE systems [16], while methods based on the Proper Generalized Decomposition [18, 17] (PGD) enable (possibly greedy) separated representations in material parameters, wall thickness, and loading conditions [17] (such that rapid simulations can be achieved for new, unseen parameters). In the present context, such techniques can be applied to both reduced (membrane/shell-type) wall models and fully 3D hyperelastic (or viscoelastic) solids in order to provide compressed structural representations that maintain physical relevancy. A primary challenge relates to the preservation of stability and consistency in the coupled formulation under strong FSI, particularly in the presence of geometric nonlinearities and nearly-incompressible material behavior. Nevertheless, recent developments in cardiovascular reduced-order modeling indicate that significant gains in computational time/efficiency can be achieved with a reasonable and useful accuracy [10, 15] suitable for clinically-relevant simulation and analysis, supporting the construction of reduced-order parametric wall models for patient-specific geometries and material properties.
Despite significant progress in LBM-based cardiovascular FSI and in projection-based model reduction, these developments have largely evolved separately. Existing LBM-FSI frameworks focus primarily on coupling strategies and wall representations [3, 6, 7, 8], while projection-based reduced-order models for cardiovascular mechanics have largely been developed in FEM-based FSI frameworks rather than within LBM-based solvers [10]. Additionally, many reduced-order approaches in such settings rely on simplified or reduced wall descriptions in lieu of fully 3D nonlinear-elastic or hyperelastic wall models.
Consequently, there is limited mathematical and computational work on integrating projection-based ROM directly within an LBM-based FSI setting in a manner that remains stable across both thin- and thick-wall descriptions and that supports parametric variation in material properties and geometry [16, 17, 19].
Accordingly, the goal of this project is to develop an LBM-FSI solver that employs projection-based reduced-order wall models in order to enable fast patient-specific simulations that are mathematically consistent and mechanically interpretable/faithful (across both thin- and thick-wall solid formulations).

The overarching goals of this project are two-fold:

1) Construction of a robust lattice Boltzmann fluid-structure interaction solver for cardiovascular flows. ***
The objective is to develop, implement in code, and validate a scalable lattice Boltzmann framework for pulsatile hemo-
dynamics in complex 3D geometries, and to propose a mathematically consistent and stable two-way coupling strategy
with compliant solid walls that is suitable for eventual reduced-order approximations. The formulation will support both
physically-reduced (membrane/shell-type) and fully three-dimensional hyperelastic wall models within a unified inter-
face treatment, with particular attention geared towards the mathematical structure of the coupled problem, algorithmic
stability under strong FSI added-mass effects, and accurate transmission of interface data [6, 7, 8, 14].

2) Development of & coupling with projection-based reduced-order models for thin/thick arterial walls. ***
The second objective extends the LBM solver and proposes/constructs mathematically coherent low-dimensional struc-
tural representations (ROMs based on, e.g., POD or the greedy PGD) in order to obtain parametrized approximations of
the underlying nonlinear PDEs governing the solid wall mechanics [16, 17]. Such reduced formulations will be derived
from both membrane/shell-type and volumetric hyperelastic formulations, with emphasis on preserving mechanical con-
sistency, ensuring stability under FSI coupling, and maintaining explicit (parametric) dependence on material properties
and geometric characteristics [10, 19].

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A complementary research direction, contingent on the progress in the above two axes, is on the use of the resulting reduced
LBM-FSI framework and solver towards the treatment of patient-specific cardiovascular configurations that may incorporate
imaging-based geometries and physiological (material wall) boundary conditions. In this context, the reduced structural models
may be applied/leveraged for the efficient exploration of varying material properties, inverse/parameter identification, and/or
uncertainty quantification (all of which require multiquery simulations that can otherwise be computationally prohibitive [3, 10]
and for which parametric reduced-order approaches are well suited).

In short, such work is towards advancing reduced-order LBM-based methodologies for nonlinear FSI in cardiovascular prob-
lems by developing a robust and physically-faithful solver that integrates projection-based wall reduction (for both thin- and
thick-wall formulations) and enables fast, parametrized patient-specific scientific computing.

**** Year 1: literature review and development of a robust LBM-FSI solver. ****
The first phase consists of an in-depth bibliographic study of lattice Boltzmann methods for cardiovascular flows and classical finite element formulations for fluid-structure interaction, with particular attention to stability under strong added-mass effects and to the mathematical structure of the coupled problem. Building on existing LBM implementations, a high-performance solver for pulsatile hemodynamics in realistic three-dimensional vascular geometries will be developed and validated. A stable two-way coupling strategy with compliant arterial walls will be proposed and implemented, supporting both reduced (membrane/shell-type) and fully three-dimensional hyperelastic wall models within a unified interface treatment. Validation will be performed on canonical FSI benchmarks representative of compliant hemodynamics.

**** Year 2: construction of projection-based reduced-order wall models. ****
The second phase focuses on the construction of mathematically and mechanically consistent projection-based reduced-order models for arterial wall mechanics. Starting from full-order membrane-type and volumetric hyperelastic formulations, low-dimensional parametrized approximations of the underlying nonlinear structural PDEs will be derived using Proper Orthogonal Decomposition (POD) and greedy Proper Generalized Decomposition (PGD). Particular emphasis will be placed on preserving stability and consistency under strong fluid-structure interaction, as well as on maintaining explicit parametric dependence on material properties and geometric descriptors. The reduced structural models will be integrated into the LBM-FSI framework and assessed in comparison with corresponding full-order simulations.

**** Year 3: patient-specific simulations and parametric investigations. ****
The final phase focuses on the application of the reduced-order LBM-FSI framework to patient-specific cardiovascular configurations reconstructed from medical imaging and incorporating physiologic boundary conditions. The methodology will be evaluated across thin- and thick-wall regimes, with systematic exploration of material-property variability and preliminary inverse identification studies. Depending on progress, complementary developments may investigate uncertainty quantification or optimization-based calibration strategies enabled by the reduced-order structure, supporting repeated-query simulations in clinically relevant cardiovascular analysis.