Mixed aggregated finite element methods for the unfitted discretization of the Stokes problem

In this work, we consider unfitted finite element methods for the numerical approximation of the Stokes problem. It is well-known that these kinds of methods lead to arbitrarily ill-conditioned systems and poorly approximated fluxes on unfitted interfaces/boundaries. In order to solve these issues, we consider the recently proposed aggregated finite element method, originally motivated for coercive problems. However, the well-posedness of the Stokes problem is far more subtle and relies on a discrete inf-sup condition. We consider mixed finite element methods that satisfy the discrete version of the inf-sup condition for body-fitted meshes and analyze how the discrete inf-sup is affected when considering the unfitted case. We propose different aggregated mixed finite element spaces combined with simple stabilization terms, which can include pressure jumps and/or cell residuals, to fix the potential deficiencies of the aggregated inf-sup. We carry out a complete numerical analysis, which includes stability, optimal a priori error estimates, and condition number bounds that are not affected by the small cut cell problem. For the sake of conciseness, we have restricted the analysis to hexahedral meshes and discontinuous pressure spaces. A thorough numerical experimentation bears out the numerical analysis. The aggregated mixed finite element method is ultimately applied to two problems with nontrivial geometries. No separate or additional fees are collected for access to or distribution of the work

A Quasi-Conforming Embedded Reproducing Kernel Particle Method for Heterogeneous Materials

We present a quasi-conforming embedded reproducing kernel particle method (QCE-RKPM) for modeling heterogeneous materials that makes use of techniques not available to mesh-based methods such as the finite element method (FEM) and avoids many of the drawbacks in current embedded and immersed formulations which are based on meshed methods. The different material domains are discretized independently thus avoiding time-consuming, conformal meshing. In this approach, the superposition of foreground (inclusion) and background (matrix) domain integration smoothing cells are corrected by a quasi-conforming quadtree subdivision on the background integration smoothing cells. Due to the non-conforming nature of the background integration smoothing cells near the material interfaces, a variationally consistent (VC) correction for domain integration is introduced to restore integration constraints and thus optimal convergence rates at a minor computational cost. Additional interface integration smoothing cells with area (volume) correction, while non-conforming, can be easily introduced to further enhance the accuracy and stability of the Galerkin solution using VC integration on non-conforming cells. To properly approximate the weak discontinuity across the material interface by a penalty-free Nitsche's method with enhanced coercivity, the interface nodes on the surface of the foreground discretization are also shared with the background discretization. As such, there are no tunable parameters, such as those involved in the penalty type method, to enforce interface compatibility in this approach. The advantage of this meshfree formulation is that it avoids many of the instabilities in mesh-based immersed and embedded methods. The effectiveness of QCE-RKPM is illustrated with several examples

The diffuse Nitsche method: Dirichlet constraints on phase-field boundaries

We explore diffuse formulations of Nitsche's method for consistently imposing Dirichlet boundary conditions on phase-field approximations of sharp domains. Leveraging the properties of the phase-field gradient, we derive the variational formulation of the diffuse Nitsche method by transferring all integrals associated with the Dirichlet boundary from a geometrically sharp surface format in the standard Nitsche method to a geometrically diffuse volumetric format. We also derive conditions for the stability of the discrete system and formulate a diffuse local eigenvalue problem, from which the stabilization parameter can be estimated automatically in each element. We advertise metastable phase-field solutions of the Allen-Cahn problem for transferring complex imaging data into diffuse geometric models. In particular, we discuss the use of mixed meshes, that is, an adaptively refined mesh for the phase-field in the diffuse boundary region and a uniform mesh for the representation of the physics-based solution fields. We illustrate accuracy and convergence properties of the diffuse Nitsche method and demonstrate its advantages over diffuse penalty-type methods. In the context of imaging based analysis, we show that the diffuse Nitsche method achieves the same accuracy as the standard Nitsche method with sharp surfaces, if the inherent length scales, i.e., the interface width of the phase-field, the voxel spacing and the mesh size, are properly related. We demonstrate the flexibility of the new method by analyzing stresses in a human vertebral body

Modeling isovolumetric phases in cardiac flows by an Augmented Resistive Immersed Implicit Surface method

A major challenge in the computational fluid dynamics modeling of the heart function is the simulation of isovolumetric phases when the hemodynamics problem is driven by a prescribed boundary displacement. During such phases, both atrioventricular and semilunar valves are closed: consequently, the ventricular pressure may not be uniquely defined, and spurious oscillations may arise in numerical simulations. These oscillations can strongly affect valve dynamics models driven by the blood flow, making unlikely to recovering physiological dynamics. Hence, prescribed opening and closing times are usually employed, or the isovolumetric phases are neglected altogether. In this article, we propose a suitable modification of the Resistive Immersed Implicit Surface (RIIS) method (Fedele et al., Biomech Model Mechanobiol 2017, 16, 1779-1803) by introducing a reaction term to correctly capture the pressure transients during isovolumetric phases. The method, that we call Augmented RIIS (ARIIS) method, extends the previously proposed ARIS method (This et al., Int J Numer Methods Biomed Eng 2020, 36, e3223) to the case of a mesh which is not body-fitted to the valves. We test the proposed method on two different benchmark problems, including a new simplified problem that retains all the characteristics of a heart cycle. We apply the ARIIS method to a fluid dynamics simulation of a realistic left heart geometry, and we show that ARIIS allows to correctly simulate isovolumetric phases, differently from standard RIIS method. Finally, we demonstrate that by the new method the cardiac valves can open and close without prescribing any opening/closing times

Explicit Time Stepping for the Wave Equation using CutFEM with Discrete Extension

In this paper we develop a fully explicit cut finite element method for the wave equation. The method is based on using a standard leap frog scheme combined with an extension operator that defines the nodal values outside of the domain in terms of the nodal values inside the domain. We show that the mass matrix associated with the extended finite element space can be lumped leading to a fully explicit scheme. We derive stability estimates for the method and provide optimal order a priori error estimates. Finally, we present some illustrating numerical examples

Modélisation XFEM, Nitsche, Level-set et simulation sous FEniCS de la dynamique de deux fluides non miscibles

À l’heure actuelle, les écoulements à deux fluides non miscibles jouent un rôle très important dans plusieurs domaines, que ça soit en science ou en ingénierie. Leur complexité est tellement élevée que les modèles actuels ne permettent de résoudre que des cas particuliers ou simplifiés avec un degré de précision qui demeurent souvent plutôt modeste. Une nouvelle approche numérique parait être une nécessité pour capturer la complexité physique du phénomène. Pour ce faire nous avons besoin d’outils robustes. Au niveau de l’interface de séparation entre les deux fluides non miscibles, les variables physiques sont discontinues, ce qui pose un défi majeur dans la description des variables et des conditions aux limites à l’interface. Le fait que les densités et les viscosités de chaque fluide soient différentes de part et d’autre de l’interface donne naissance à des défauts et des impuretés dans le champ des vitesses, ce qu’on appelle une discontinuité faible. Pour sa part, l’existence de la force de tension superficielle au niveau de l’interface crée une discontinuité sur le champ de pression, ce qu’on appelle une discontinuité forte. Un autre grand problème se pose au niveau de l’étude numérique du problème, où les méthodes numériques classiques ont une précision assez limitée dans ce genre de situation. L’objectif de ce travail est de fournir une étude complète de la dynamique de l’interface entre deux fluides non miscibles à l’aide d’outils mathématiques, physiques et numériques robustes. D’abord, une étude analytique du problème a été faite où l’équation de Navier-Stokes et les conditions de saut sur les variables physiques au niveau de l’interface de séparation entre les fluides ont été prouvées en détail. Pour traiter les discontinuités, nous avons discrétisé nos variables à l’aide de la méthode XFEM. Dû aux larges distorsions rencontrées dans ce genre d’écoulement, nous avons utilisé l’approche Eulérienne, pour corriger les oscillations des solutions dues aux choix du système de coordonnées nous avons utilisé les techniques de stabilisation SUPG/PSPG. Le traitement de la courbure des interfaces K été fait à l’aide de l’opérateur Laplace Beltrami et le suivi d’interface à l’aide de la méthode ¨Level-set¨. Pour le traitement des conditions de saut au niveau de l’interface la méthode Nitsche est développée dans différents contextes. Après avoir développé un modèle physique et mathématique dans les premières parties de notre travail, nous avons fait une étude numérique à l’aide de la plateforme de calcul FEniCS, qui est une plateforme de développement en langage C++ avec une interface Python. Un code de calcul a été développé dans le cas des écoulements de deux fluides non miscibles avec les modèles physiques et les outils mathématiques développés dans les sections précédentes.The two-phase flow problems have an important role in the multitude of domains in science and engineering. Their complexity is so high that the actual models can solve only particular or simplified cases with a certain degree of precision. A new approach is a necessity to understand the evolution of new ideas and the physical complexity in this kind of flow, to contribute to the study of this field. A good study requires solid and robust tools to have performing results and a maximum of efficacy. At the interface of separation between the two immiscible fluids, the physical parameters are discontinuous, which gives us difficulties for the description of the physical variables at the interface and boundary conditions. The fact that the density and the viscosity are discontinuous at the interface creates kinks in the velocity, which represent a weak discontinuity. The existence of the surface tension at the interface create a discontinuity for the pressure field, it represents a strong discontinuity. The main objective of this work is to make a complete study based on strong and robust physical, mathematical and numerical tools. A strong combination, capable of capturing the physical aspect of the interface between the two fluids with a very good precision. Building such a robust, cost effective and accurate numerical model is challenging and requires lots of efforts and a multidisciplinary knowledge in mathematics, physics and computer science. First, an analytical study was made where the one fluid model of the Navier-Stokes equation was proved from Newton’s laws and jump conditions at the interface was proved and detailed analytically. To treat the problem of discontinuity, we used the XFEM method to discretize our discontinuous variables. Due to the large distortion encountered in this kind of fluid mechanic problems, we are going to use the Eulerian approach, and to correct the oscillation of solutions we will use the SUPG/PSPG stabilization technic. The treatment of the interface curvature k was done with the Laplace Beltrami operator and the interface tracking with the Level-set method. To treat the jump conditions with a very sharp precision we used the Nitsche’s method, developed in different cases. After building a strong mathematical and physical model in the first parts of our work, we did a numerical study using the FEniCS computational platform, which is a platform of computational development based on C++ with a Python interface. A numerical code was developed in this study, in the case of two-phase flow problem, based on the previous mathematical and physical models detailed in previous sections

A unified computational framework for process modeling and performance modeling of multi-constituent materials

This thesis presents new theoretical and computational developments and an integrated approach for interface and interphase mechanics in the process and performance modeling of fibrous composite materials. A new class of stabilized finite element methods is developed for the coupled-field problems that arise due to curing and chemical reactions at the bi-material interfaces at the time of the manufacturing of the fiber-matrix systems. An accurate modeling of the degree of curing, because of its effects on the evolving properties of the interphase material, is critical to determining the coupled chemo-mechanical interphase stresses that influence the structural integrity of the composite and its fatigue life. A thermodynamically consistent theory of mixtures for multi-constituent materials is adopted to model curing and interphase evolution during the processing of the composites. The mixture theory model combines the composite constituent behaviors in an effective medium, thereby reducing the computational cost of modeling chemically reacting multi-constituent mixtures, while retaining information involving the kinematic and kinetic responses of the individual constituents. The effective medium and individual constituent behaviors are each constrained to mutually satisfy the balance principles of mechanics. Even though each constituent is governed by its own balance laws and constitutive equations, interactive forces between constituents that emanate from maximization of entropy production inequality provide the coupling between constituent specific balance laws and constitutive models. The mixture model is cast in a finite strain finite element framework that finds roots in the Variational Multiscale (VMS) method. The deformation of multi-constituent mixtures at the Neumann boundaries requires imposing constraint conditions such that the constituents deform in a self-consistent fashion. A set of boundary conditions is presented that accounts for the non-zero applied tractions, and a variationally consistent method is developed to enforce inter constituent constraints at Neumann boundaries in the finite deformation context. The new method finds roots in a local multiscale decomposition of the deformation map at the Neumann boundary. Locally satisfying the Lagrange multiplier field and subsequent modeling of the fine scales via edge bubble functions results in closed-form expressions for a generalized penalty tensor and a weighted numerical flux that are free from tunable parameters. The key novelty is that the consistently derived constituent coupling parameters evolve with material and geometric nonlinearity, thereby resulting in optimal enforcement of inter-constituent constraints. A class of coupled field problems for process modeling and for performance molding of fibrous composites is presented that provides insight into the theoretical models and multiscale stabilized formulations for computational modeling of multi-constituent materials

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described