An immersed phase field fracture model for microporomechanics with Darcy–Stokes flow

This paper presents an immersed phase field model designed to predict the fracture-induced flow due to brittle fracture in vuggy porous media. Due to the multiscale nature of pores in the vuggy porous material, crack growth may connect previously isolated pores, which leads to flow conduits. This mechanism has important implications for many applications such as disposal of carbon dioxide and radioactive materials and hydraulic fracture and mining. To understand the detailed microporomechanics that causes the fracture-induced flow, we introduce a new phase field fracture framework where the phase field is not only used as an indicator function for damage of the solid skeleton but also used as an indicator of the pore space. By coupling the Stokes equation that governs the fluid transport in the voids, cavities, and cracks and Darcy’s flow in the deformable porous media, our proposed model enables us to capture the fluid–solid interaction of the pore fluid and solid constituents during crack growth. Numerical experiments are conducted to analyze how the presence of cavities affects the accuracy of predictions based on the homogenized effective medium during crack growth

Parameter-robust discretization and preconditioning of Biot's consolidation model

Biot's consolidation model in poroelasticity has a number of applications in science, medicine, and engineering. The model depends on various parameters, and in practical applications these parameters ranges over several orders of magnitude. A current challenge is to design discretization techniques and solution algorithms that are well behaved with respect to these variations. The purpose of this paper is to study finite element discretizations of this model and construct block diagonal preconditioners for the discrete Biot systems. The approach taken here is to consider the stability of the problem in non-standard or weighted Hilbert spaces and employ the operator preconditioning approach. We derive preconditioners that are robust with respect to both the variations of the parameters and the mesh refinement. The parameters of interest are small time-step sizes, large bulk and shear moduli, and small hydraulic conductivity.Comment: 24 page

DuMux 3 – an open-source simulator for solving flow and transport problems in porous media with a focus on model coupling

Authors: Timo Koch and Dennis Gläser and Kilian Weishaupt and Sina Ackermann and Martin Beck and Beatrix Becker and Samuel Burbulla and Holger Class and Edward Coltman and Simon Emmert and Thomas Fetzer and Christoph Grüninger and Katharina Heck and Johannes Hommel and Theresa Kurz and Melanie Lipp and Farid Mohammadi and Samuel Scherrer and Martin Schneider and Gabriele Seitz and Leopold Stadler and Martin Utz and Felix Weinhardt and Bernd Flemisc

Mécano-biologie des tissus hétérogènes par changement d'échelles : application à l'ostéosarcome

L’ostéosarcome est une tumeur osseuse primitive qui survient principalement chez les adolescents et les jeunes adultes. Le taux de survie à 5 ans est de 70% et chute à 25% pour les patients présentant des métastases ou ne répondant pas aux traitements. De nouveaux développements sont nécessaires pour améliorer la prise en charge spécifique des patients. Ce type de tumeurs présente de fortes hétérogénéités spatiales dans la micro-architecture osseuse, dans la distribution des populations cellulaires mais aussi dans la réponse au traitement. A l’échelle tissulaire et du point de vue de la biophysique, l’ostéosarcome peut être considéré comme un milieu poreux fortement hétérogène et nous supposons que l’ostéosarcome est sensible aux effets mécaniques lors de sa formation et de son évolution. Nous proposons une méthode de changement d’échelles pour caractériser les propriétés mécaniques de tels milieux poreux. La méthode s’appuie sur une approche séquentielle "grid-block" combinée à une méthode "extend-local". La méthodologie est adaptée à des images de grandes tailles et notamment aux images binaires de coupes histologiques d’ostéosarcomes obtenues en routine clinique. Des modèles d’écoulement, de diffusion, d’élasticité et de poroélasticité sont étudiés. Les paramètres équivalents, constants par morceaux, de type perméabilités tissulaires et coefficients de raideurs tissulaires sont déterminés avec fiabilité. Plusieurs résultats méthodologiques ont été obtenus tels que les inégalités portant sur les paramètres équivalents en fonction des conditions aux limites imposées sur les éléments du "grid-block" ou la caractérisation du rôle des méthodes "extendlocal" dans le calcul des paramètres de raideur. Dans une étude clinique préliminaire, des relations entre les propriétés mécaniques tissulaires et les paramètres cellulaires sont données. Une cohorte réduite de patients montre que la réponse au traitement peut être corrélée à l’architecture du micro-environnement et à ses propriétés mécaniques. Ceci pourrait soutenir la recherche de marqueurs mécanobiologiques pour le suivi de la réponse au traitement chez les patients atteints d’ostéosarcome

A finite-element toolbox for the simulation of solid-liquid phase-change systems with natural convection

International audienceWe present and distribute a new numerical system using classical finite elements with mesh adaptivity for computing two-dimensional liquid-solid phase-change systems involving natural convection. The programs are written as a toolbox for FreeFem++ (www.freefem.org), a free finite-element software available for all existing operating systems. The code implements a single domain approach. The same set of equations is solved in both liquid and solid phases: the incompressible Navier-Stokes equations with Boussinesq approximation for thermal effects. This model describes naturally the evolution of the liquid flow which is dominated by convection effects. To make it valid also in the solid phase, a Carman-Kozeny-type penalty term is added to the momentum equations. The penalty term brings progressively (through an artificial mushy region) the velocity to zero into the solid. The energy equation is also modified to be valid in both phases using an enthalpy (temperature-transform) model introducing a regularized latent-heat term. Model equations are discretized using Galerkin triangular finite elements. Piecewise quadratic (P2) finite-elements are used for the velocity and piecewise linear (P1) for the pressure. For the temperature both P2 or P1 discretizations are possible. The coupled system of equations is integrated in time using a second-order Gear scheme. Non-linearities are treated implicitly and the resulting discrete equations are solved using a Newton algorithm. An efficient mesh adaptivity algorithm using metrics control is used to adapt the mesh every time step. This allows us to accurately capture multiple solid-liquid interfaces present in the domain, the boundary-layer structure at the walls and the unsteady convection cells in the liquid. We present several validations of the toolbox, by simulating benchmark cases of increasing difficulty: natural convection of air, natural convection of water, melting of a phase-change material, a melting-solidification cycle, and, finally, a water freezing case. Other similar cases could be easily simulated with this toolbox, since the code structure is extremely versatile and the syntax very close to the mathematical formulation of the model

Estimation d’erreur a posteriori pour l’approximation de problèmes Laplaciens fractionnaires et applications en poro-élasticité

This manuscript is concerned with a posteriori error estimation for the finite element discretization of standard and fractional partial differential equations as well as an application of fractional calculus to the modeling of the human meniscus by poro-elasticity equations. In the introduction, we give an overview of the literature of a posteriori error estimation in finite element methods and of adaptive refine- ment methods. We emphasize the state–of–the–art of the Bank–Weiser a posteriori error estimation method and of the adaptive refinement methods convergence results. Then, we move to fractional partial differential equations. We give some of the most common discretization methods of fractional Laplacian operator based equations. We review some results of a priori error estimation for the finite element discretization of these equations and give the state–of–the–art of a posteriori error estimation. Finally, we review the literature on the use of the Caputo’s fractional derivative in applications, focusing on anomalous diffusion and poro-elasticity applications. The rest of the manuscript is organized as follow. Chapter 1 is concerned with a proof of the reliability of the Bank–Weiser estimator for three–dimensional problems, extending a result from the literature. In Chapter 2 we present a numerical study of the Bank–Weiser estimator, provide a novel implementation of the estimator in the FEniCS finite element software and apply it to a variety of elliptic equations as well as goal-oriented error estimation. In Chapter 3 we derive a novel a posteriori estimator for the L2 error induced by the finite element discretization of fractional Laplacian operator based equations. In Chapter 4 we present new theoretical results on the convergence of a rational approximation method with consequences on the approximation of fractional norms as well as a priori error estimation results for the finite element discretization of fractional equations. Finally, in Chapter 5 we provide an application of fractional calculus to the study of the human meniscus via poro-elasticity equations.Ce manuscrit traite d’estimation d’erreur a posteriori pour la discrétisation d’équations aux dérivées partielles standard et fractionnaires par les méthodes éléments finis ainsi que de l’application de l’analyse fractionnaire à la modélisation du ménisque humain par les équations de poro-élasticité. Dans l’introduction, nous donnons un aperçu de la littérature sur l’estimation d’erreur a posteriori pour les méth- odes éléments finis et des méthodes de raffinement adaptatif. Nous insistons particulièrement sur l’état de l’art de la méthode d’estimation d’erreur a posteriori de Bank-Weiser et sur les résultats de convergence des méthodes adaptatives. Ensuite, nous nous intéressons aux équations aux dérivées partielles fractionnaires. Nous présentons certaines méthodes de discrétisation d’équations basées sur l’opérateur Laplacien fractionnaire et donnons l’état de l’art sur l’estimation d’erreur a posteriori. Finalement, nous donnons un aperçu de la littérature concernant les applications de la dérivée fractionnaire au sens de Caputo en nous concentrant sur le phénomène de diffusion anormale et les applications en poro-élasticité

Computational Multiscale Solvers for Continuum Approaches

Computational multiscale analyses are currently ubiquitous in science and technology. Different problems of interest-e.g., mechanical, fluid, thermal, or electromagnetic-involving a domain with two or more clearly distinguished spatial or temporal scales, are candidates to be solved by using this technique. Moreover, the predictable capability and potential of multiscale analysis may result in an interesting tool for the development of new concept materials, with desired macroscopic or apparent properties through the design of their microstructure, which is now even more possible with the combination of nanotechnology and additive manufacturing. Indeed, the information in terms of field variables at a finer scale is available by solving its associated localization problem. In this work, a review on the algorithmic treatment of multiscale analyses of several problems with a technological interest is presented. The paper collects both classical and modern techniques of multiscale simulation such as those based on the proper generalized decomposition (PGD) approach. Moreover, an overview of available software for the implementation of such numerical schemes is also carried out. The availability and usefulness of this technique in the design of complex microstructural systems are highlighted along the text. In this review, the fine, and hence the coarse scale, are associated with continuum variables so atomistic approaches and coarse-graining transfer techniques are out of the scope of this paper.Abengoa Researc