Numerical approximation of the Poisson problem with small holes, using augmented finite elements and defective boundary conditions.

We consider the Poisson problem in a domain with small holes, as a template for developing efficient and accurate numerical approximation schemes for partial differential equations defined on domains with low-dimensional inclusions, such as embedded fibers. We propose a reduced model based on the projection of Dirichlet boundary constraints on a finite dimensional approximation space, obtaining in this way a Poisson problem with defective interface conditions. We analyze the existence of the solution of the reduced problem and for arbitrarily small holes we prove its convergence towards the original problem, the rate of which depends on the size of the inclusion and on the number of modes of the finite dimensional space. The numerical discretization of the reduced problem is addressed by the finite element method, using a computational mesh that does not fit to the holes in the framework of a fictitious domain approach. We propose a stable, optimally convergent and robust formulation with respect to the hole size that exploits an augmented Galerkin formulation based on the addition of suitable non-polynomial functions to the finite element approximation space. The properties of the discretization method are supported by numerical experiments

Une nouvelle méthode numérique pour l'intéraction fluide-structure de corps minces dans des écoulements tridimensionels

This PhD dissertation aims to develop a new modelling and computational approach for the simulation of slender bodies immersed in three dimensional flows (3D). Thanks to the special geometric configuration of the slender structures, we can model this problem by mixed-dimensional coupled equations in which the solid balance equations are formulated in a one-dimensional (1D) domain. Addressing these types of problems presents several challenges. From a mathematical perspective, the main two difficulties involve defining well-posed trace operators of co-dimension two (from the 3D to the 1D domain) and ensuring the accuracy of solutions obtained with the mixed-dimensional formulation when compared to the fully 3D one. From a computational point of view, the non-standard mathematical formulation of the coupled problem makes it difficult to guarantee the convergence of the discrete solutions with standard numerical approaches. The main advantages of the approach we present in this manuscript lies in its strong mathematical basis. Indeed, while many standard mixed-dimensional formulations yield solutions with poor regularity due to ill-posed trace operators, our reduced order method generates solutions within standard Hilbert spaces. This facilitates the application of Galerkin projection-based approximation methods such as the finite element method (FEM). In the second chapter, we establish the continuous formulation of the 3D fluid-structure interaction coupled problem using incompressible Navier-Stokes equations for the description of the fluid dynamics and a linear Timoshenko beam model for modeling the response of the slender structure. These two models are coupled with a mixed-dimensional version of fluid-structure interface conditions, combining the fictitious domain (FD) approach with the projection of kinematic coupling conditions onto a finite-dimensional Fourier space via Lagrange multipliers. We then develop a discrete formulation based on the finite element method and a semi-implicit treatment of the Dirichlet-Neumann coupling conditions, employing a partitioned procedure for the resolution of the fluid-structure interaction problem. We establish the energy stability of the scheme and provide extensive numerical evidence of the accuracy and robustness of the discrete formulation, notably with respect to a full order model with standard coupling conditions. In the third and fourth chapter we conduct a mathematical analysis on the approximation error of our reduced order coupled method, examining both the modeling and numerical approximation errors resulting from the mixed-dimensional formulation and the fictitious domain finite element method, respectively. We explore these aspects in two simplified frameworks. We first consider a two-dimensional Poisson problem (2D) with a fixed immersed boundary and non-homogeneous Dirichlet boundary conditions. We then extend this analysis to the 2D stationary Stokes problem with rigid-body Dirichlet boundary conditions on the immersed interface. In both cases, after proving the existence of solutions for the reduced order problem, we prove its convergence, when the size of the obstacle is small, to the full order problem with standard Dirichlet boundary conditions. In particular, our estimates highlight the need to consider enough Fourier modes to achieve convergence on the Lagrange multipliers, which is an essential aspect in addressing the fluid-structure interaction coupled problem. Subsequently, the numerical discretization of the reduced order problem is analyzed. As standard for this family of methods, the convergence obtained with the fictitious domain finite element method is sub-optimal, due to the discontinuity of the solution at the interface. Furthermore, the stability and accuracy of the scheme depend on the ratio between the mesh size and the obstacle size, which can be restrictive for very small obstacles. To address the limitations of the fictitious domain approach, we propose and analyze two modified finite element method, one stabilized and one augmented. Finally, we develop a 2D fluid-structure interaction formulation where small particles are immersed in a Stokesian flow, applying reduced order interface coupling conditions. The properties of the reduced order model and the corresponding numerical methods are illustrated by some numerical experiments. Using a semi-implicit scheme for the resolution of the 3D fluid-structure interaction problem requires to iterate over the fluid and solid solvers multiple times, which can be computationally expensive. The most efficient approach for the time discretization of the fluid-structure interaction problem would be to adopt an explicit coupling scheme, solving this way the fluid and structure sub-problems only once per time step. However, for standard (Dirichlet-Neumann) explicit coupling schemes, a large fluid/solid density ratio combined with a slender and lengthy geometry gives rise to unconditional numerical instability. Subsequently, in the last chapter, we introduce a Robin-based loosely coupled scheme specifically designed for 3D mixed-dimensional formulation and prove its unconditional stability. We also provide numerical evidence of the accuracy of the explicit scheme through several test cases.Ce projet de doctorat a pour objectif de développer une nouvelle approche computationnelle pour la simulation de corps élancés immergés dans un écoulement tridimensionnel (3D). Grâce à la configuration géométrique particulière des structures élancées, nous pouvons modéliser ce problème par des équations couplées en dimensions mixtes pour lesquelles les équations d'équilibre du solide sont formulées dans un domaine unidimensionnel (1D). Ce type de problèmes pose plusieurs difficultés à surmonter. D'un point de vue mathématique, ils impliquent de définir des opérateurs de trace bien posés de codimension deux (du domaine 3D au domaine 1D) mais aussi de garantir que les solutions obtenues avec la formulation mixte sont proches de celles obtenues avec une formulation complètement 3D. D'un point de vue computationnel, la formulation mathématique non classique du problème couplé rend également difficile de garantir la convergence des solutions discrètes avec des approches numériques classiques. Les principaux avantages de l'approche que nous présentons dans ce manuscrit résident dans sa solide base mathématique. En effet, tandis que de nombreuses formulations mixtes donnent des solutions avec une faible régularité en raison d'opérateurs de trace mal posés, notre méthode réduite génère des solutions dans des espaces de Hilbert classiques. Cela facilite l'application de méthodes d'approximation basées sur la projection de Galerkin telles que la méthode des éléments finis (MEF).Dans le deuxième chapitre, nous établissons la formulation continue du problème couplé 3D d'interaction fluide-structure en considérant les équations de Navier-Stokes incompressibles pour la description de la dynamique du fluide et un modèle de poutre linéaire de Timoshenko pour la modélisation de la réponse de la structure élancée. Ces modèles sont couplés avec une version en dimensions mixtes des conditions d'interface fluide-structure, associant l'approche de domaine fictif (DF) avec la projection des conditions de couplage cinématique sur un espace de Fourier de dimension finie via des multiplicateurs de Lagrange. Nous développons ensuite une formulation discrète basée sur la méthode des éléments finis et un traitement semi-implicite des conditions de couplage Dirichlet-Neumann, en utilisant une procédure partitionnée pour la résolution du problème d'interaction fluide-structure. Nous établissons la stabilité énergétique du schéma et fournissons des preuves numériques détaillées sur la précision et la robustesse de la formulation discrète, notamment par rapport à un modèle complet avec des conditions de couplage fluide-structure classiques.Dans les troisième et quatrième chapitres, nous effectuons une analyse mathématique sur l'erreur d'approximation de notre méthode réduite couplée, en examinant les erreurs de modélisation et d'approximation numériques résultant respectivement de la formulation en dimensions mixtes et de la méthode des éléments finis avec domaine fictif. Nous explorons ces aspects dans deux cadres simplifiés. Nous considérons d'abord un problème de Poisson 2D avec une frontière immergée statique et des conditions aux bords de Dirichlet non homogènes. Nous étendons ensuite cette analyse au problème de Stokes bidimensionnel stationnaire avec des conditions aux bords de type solides rigides sur l'interface immergée. Dans les deux cas, après avoir prouvé l'existence de solutions pour le problème réduit, nous prouvons sa convergence, lorsque la taille de l'obstacle tend vers zéro, vers le problème complet avec des conditions aux bords de Dirichlet classiques. En particulier, nos estimations mettent en évidence la nécessité de considérer suffisamment de modes de Fourier pour obtenir une convergence sur les multiplicateurs de Lagrange, ce qui est un aspect essentiel pour l'analyse du problème couplé d'interaction fluide-structure. Ensuite, nous analysons la discrétisation numérique du problème réduit. De façon assez classique pour cette famille de méthodes, la convergence obtenue avec la méthode des éléments finis avec domaine fictif est sous-optimale, en raison de la discontinuité de la solution à l'interface. De plus, la stabilité et la convergence du schéma dépend du rapport entre la taille du maillage et la taille de l'obstacle, ce qui peut être contraignant pour des obstacles de très petite taille. Pour pallier les limites de l'approche domaine fictif, nous proposons et analysons deux méthodes éléments finis alternatives, une méthode stabilisée et une méthode enrichie. Enfin, nous développons une formulation d'interaction fluide-structure 2D où de petites particules sont immergées dans un écoulement de Stokes, en appliquant des conditions de couplage d'interface réduites. Les propriétés du modèle réduit et des méthodes numériques correspondantes sont illustrées par des exemples numériques.L'utilisation d'un schéma semi-implicite pour la résolution du problème d'interaction fluide-structure 3D exige d'itérer de nombreuses fois sur les solveurs fluide et solide, ce qui peut être coûteux en termes de temps de calcul. L'approche la plus efficace pour la discrétisation temporelle du problème d'interaction fluide-structure consisterait à adopter un schéma de couplage explicite, permettant ainsi de résoudre les sous-problèmes fluide et solide une seule fois par pas de temps. Cependant, pour les schémas de couplage explicite classiques (Dirichlet-Neumann), un rapport élevé entre la densité du fluide et la densité du solide associé à une géométrie mince amène souvent à de l'instabilité numérique. Par conséquent, dans le dernier chapitre, nous introduisons un schéma faiblement couplé qui repose sur des conditions d'interface de Robin spécifiquement conçu pour une formulation 3D en dimensions mixtes et prouvons sa stabilité inconditionnelle. Nous fournissons également des preuves numériques de la précision du schéma explicite par plusieurs cas test.Questa tesi di dottorato ha come obiettivo lo sviluppo di un nuovo approccio computazionale per la simulazione di corpi slanciati immersi in un flusso tridimensionale (3D). Sfruttando la particolare configurazione geometrica delle strutture slanciate, è possibile modellizzare il problema tramite equazioni accoppiate con dimensioni miste, in cui le equazioni di bilancio del solido sono formulate in un dominio unidimensionale (1D). Questo tipo di problemi presenta diverse difficoltà. Da un punto di vista matematico, si tratta di definire degli operatori di traccia ben definiti di codimensione due (dal dominio 3D al dominio 1D) oltre al garantire che le soluzioni ottenute con la formulazione in dimensioni miste siano vicine a quelle ottenute con una formulazione completamente 3D. Dal punto di vista computazionale, la formulazione matematica non convenzionale del problema accoppiato rende difficile garantire la convergenza delle soluzioni discrete con approcci numerici classici. In effeti, molte formulazioni miste producono soluzioni con scarsa regolarità a causa di operatori di tracia mal definiti, il nostro metodo ridotto genera soluzioni in spazi di Hilbert classici. Questo facilita l'applicazione di metodi di approssimazione basati sulla proiezione di Galerkin come il metodo degli elementi finiti (MEF)Nel secondo capitolo, presentiamo la formulazione continua del problema accoppiato 3D di interazione fluido-struttura, considerando le equazioni di Navier-Stokes incomprimibili per la descrizione della dinamica del fluido e un modello di trave lineare di Timoshenko per la modellizazione della risposta dinamica della struttura sottile. Questi modelli sono accoppiati con una versione delle condizioni di interfaccia fluido-struttura in dimensioni miste, che associa l'approccio del dominio fittizio (FD) con la proiezione delle condizioni di accoppiamento cinematico su uno spazio di Fourier a dimensioni finite tramite moltiplicatori di Lagrange. Successivamente, sviluppiamo una formulazione discreta basata sul metodo degli elementi finiti e un trattamento semi-implicito delle condizioni di accoppiamento Dirichlet-Neumann, utilizzando una procedura partizionata per la risoluzione del problema di interazione fluido-struttura. Stabilizziamo il regime di energia dello schema e forniamo numerose prove numeriche dell'accuratezza e della robustezza della formulazione discreta, in particolare rispetto a un modello classico (ALE) con condizioni di accoppiamento fluido-struttura convenzionali.Nei capitoli terzo e quarto, presentiamo l'analisi matematica dell'errore di approssimazione del metodo accoppiato e ridotto, esaminando sia gli errori di modellizazione che di approssimazione numerica derivanti rispettivamente dalla formulazione a dimensioni miste e dal metodo degli elementi finiti con dominio fittizio (FEM). Esploriamo questi aspetti in due contesti semplificati. Iniziamo col considerare un problema di Poisson 2D con una frontiera immersa fissa e condizioni al bordo di Dirichlet non omogenee. Estendiamo poi questa analisi al problema di Stokes 2D stazionario con condizioni al bordo di tipo solido rigido sull'interfaccia immersa. In entrambi i casi, dopo aver dimostrato l'esistenza della soluzione per il problema ridotto, ne proviamo la convergenza, quando le dimensioni dell'ostacolo tendono a zero, al problema completamente risolto con condizioni al bordo di Dirichlet standard. In particolare, le stime ricavate evidenziano la necessità di considerare abbastanza modi di Fourier per ottenere una convergenza sui moltiplicatori di Lagrange, che è un aspetto fondamentale per l'analisi del problema accoppiato di interazione fluido-struttura. Successivamente, analizziamo la discretizzazione numerica del problema ridotto. Come è consuetudine per questa famiglia di metodi, la convergenza ottenuta con il metodo degli elementi finiti di dominio fittizio è subottimale, a causa della discontinuità della soluzione all'interfaccia. Inoltre, la stabilità e l'accuratezza dello schema dipendono dal rapporto tra la dimensione caratteristica della griglia del fluido e la dimensione dell'ostacolo, il che può essere limitante per ostacoli molto piccoli. Per superare le limitazioni dell'approccio del dominio fittizio, proponiamo e analizziamo due metodi degli elementi finiti alternativi, un metodo stabilizzato e un metodo arricchito. Infine, sviluppiamo una formulazione dell'interazione fluido-struttura 2D in cui piccole particelle sono immerse in un flusso di Stokes, applicando condizioni di accoppiamento ridotte all'interfaccia. Le proprietà del modello ridotto e dei metodi numerici corrispondenti sono illustrate da alcuni esempi numerici.L'uso di uno schema semi-implicito per la risoluzione del problema di interazione fluido-struttura 3D richiede di iterare più volte sui solvers fluido e solido, il che può essere costoso in termini di tempo di calcolo. L'approccio più efficiente per la discretizzazione temporale del problema di interazione fluido-struttura è lo schema di accoppiamento esplicito, che consente di risolvere i sotto-problemi fluido e solido solo una volta per time step. Tuttavia, per gli schemi di accoppiamento esplicito standard (Dirichlet-Neumann), l'alto rapporto fra la densità del fluido e del solido insieme alla geometria slanciata causa spesso instabilità numerica. Nell'ultimo capitolo, introduciamo uno schema debolmente accoppiato basato su condizioni di interfaccia di Robin specificamente progettato per una formulazione 3D a dimensioni miste e ne dimostriamo la stabilità incondizionata. Forniamo inoltre prove numeriche della precisione dello schema esplicito attraverso diversi esempi

High order accurate schemes for Euler and Navier–Stokes equations on staggered Cartesian grids

International audienceWe present here a new class of staggered schemes for solving the compressible 1D Euler equations in internal energy formulation on uniform grids. The schemes are applicable to arbitrary equation of states and can be extended to high order of accuracy in both time and space on smooth flows. High order accuracy in time is reached thanks to Cauchy–Kovalevskaya procedure. Modifications on the initial schemes are performed to give sufficient conditions for stability on 1D wave equations. Results obtained for wave equations are extended to 1D Euler equations and then to 2D compressible Navier–Stokes equations using directional splitting methods. Results on the conservation of total energy are given, proper shock capturing is observed experimentally. Numerical results are provided up to 4th-order accuracy in 1D and 2D

Numerical approximation of the Poisson problem with small holes, using augmented finite elements and defective boundary conditions.

We consider the Poisson problem in a domain with small holes, as a template for developing efficient and accurate numerical approximation schemes for partial differential equations defined on domains with low-dimensional inclusions, such as embedded fibers. We propose a reduced model based on the projection of Dirichlet boundary constraints on a finite dimensional approximation space, obtaining in this way a Poisson problem with defective interface conditions. We analyze the existence of the solution of the reduced problem and for arbitrarily small holes we prove its convergence towards the original problem, the rate of which depends on the size of the inclusion and on the number of modes of the finite dimensional space. The numerical discretization of the reduced problem is addressed by the finite element method, using a computational mesh that does not fit to the holes in the framework of a fictitious domain approach. We propose a stable, optimally convergent and robust formulation with respect to the hole size that exploits an augmented Galerkin formulation based on the addition of suitable non-polynomial functions to the finite element approximation space. The properties of the discretization method are supported by numerical experiments

A mixed-dimensional formulation for the simulation of slender structures immersed in an incompressible flow

We consider the simulation of slender structures immersed in a three-dimensional (3D) flow. By exploiting the special geometric configuration of the slender structures, this particular problem can be modeled by mixed-dimensional coupled equations (3D for the fluid and 1D for the solid). Several challenges must be faced when dealing with this type of problems. From a mathematical point of view, these include defining wellposed trace operators of codimension two. On the computational standpoint, the nonstandard mathematical formulation makes it difficult to ensure the accuracy of the solutions obtained with the mixed-dimensional discrete formulation as compared to a fully resolved one. We establish the continuous formulation using the Navier-Stokes equations for the fluid and a Timoshenko beam model for the structure. We complement these models with a mixed-dimensional version of the fluid-structure interface conditions, based on the projection of kinematic coupling conditions on a finite-dimensional Fourier space. Furthermore, we develop a discrete formulation within the framework of the finite element method, establish the energy stability of the scheme, provide extensive numerical evidence of the accuracy of the discrete formulation, notably with respect to a fully resolved (ALE based) model and a standard reduced modeling approach