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A posteriori error analysis of Euler-Galerkin approximations to coupled elliptic-parabolic problems

Alexandre Ern, Sébastien Meunier (2009)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We analyze Euler-Galerkin approximations (conforming finite elements in space and implicit Euler in time) to coupled PDE systems in which one dependent variable, say u , is governed by an elliptic equation and the other, say p , by a parabolic-like equation. The underlying application is the poroelasticity system within the quasi-static assumption. Different polynomial orders are used for the u - and p -components to obtain optimally convergent a priori bounds for all the terms in the error energy norm....

A posteriori error analysis of Euler-Galerkin approximations to coupled elliptic-parabolic problems

Alexandre Ern, Sébastien Meunier (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

We analyze Euler-Galerkin approximations (conforming finite elements in space and implicit Euler in time) to coupled PDE systems in which one dependent variable, say u, is governed by an elliptic equation and the other, say p, by a parabolic-like equation. The underlying application is the poroelasticity system within the quasi-static assumption. Different polynomial orders are used for the u- and p-components to obtain optimally convergent a priori bounds for all the terms in the error energy...

A priori error analysis of a fully-mixed finite element method for a two-dimensional fluid-solid interaction problem

Carolina Domínguez, Gabriel N. Gatica, Salim Meddahi, Ricardo Oyarzúa (2013)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We introduce and analyze a fully-mixed finite element method for a fluid-solid interaction problem in 2D. The model consists of an elastic body which is subject to a given incident wave that travels in the fluid surrounding it. Actually, the fluid is supposed to occupy an annular region, and hence a Robin boundary condition imitating the behavior of the scattered field at infinity is imposed on its exterior boundary, which is located far from the obstacle. The media are governed by the elastodynamic...

A quasi-Newton algorithm based on a reduced model for fluid-structure interaction problems in blood flows

Jean-Frédéric Gerbeau, Marina Vidrascu (2003)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We propose a quasi-Newton algorithm for solving fluid-structure interaction problems. The basic idea of the method is to build an approximate tangent operator which is cost effective and which takes into account the so-called added mass effect. Various test cases show that the method allows a significant reduction of the computational effort compared to relaxed fixed point algorithms. We present 2D and 3D fluid-structure simulations performed either with a simple 1D structure model or with shells...

A Quasi-Newton Algorithm Based on a Reduced Model for Fluid-Structure Interaction Problems in Blood Flows

Jean-Frédéric Gerbeau, Marina Vidrascu (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We propose a quasi-Newton algorithm for solving fluid-structure interaction problems. The basic idea of the method is to build an approximate tangent operator which is cost effective and which takes into account the so-called added mass effect. Various test cases show that the method allows a significant reduction of the computational effort compared to relaxed fixed point algorithms. We present 2D and 3D fluid-structure simulations performed either with a simple 1D structure model or with...

A reduced model for Darcy’s problem in networks of fractures

Luca Formaggia, Alessio Fumagalli, Anna Scotti, Paolo Ruffo (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

Subsurface flows are influenced by the presence of faults and large fractures which act as preferential paths or barriers for the flow. In literature models were proposed to handle fractures in a porous medium as objects of codimension 1. In this work we consider the case of a network of intersecting fractures, with the aim of deriving physically consistent and effective interface conditions to impose at the intersection between fractures. This new model accounts for the angle between fractures...

A short note on incremental thermoelasticity.

Ramón Quintanilla de la Torre (1983)

Stochastica

The equations of classical thermoelasticity have been extensively studied [1], [2], [3], [4], [5]. Only more recently the equations when the initial state is at non-uniform temperature have been established [6], and a well-posedness theorem proved by the author and C. Navarro for these equations [7]. Our goal here is to make a brief comment about dissipation in this last case of an initial state with non-uniform temperature.

Acceleration of a fixed point algorithm for fluid-structure interaction using transpiration conditions

Simone Deparis, Miguel Angel Fernández, Luca Formaggia (2003)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this work, we address the numerical solution of fluid-structure interaction problems. This issue is particularly difficulty to tackle when the fluid and the solid densities are of the same order, for instance as it happens in hemodynamic applications, since fully implicit coupling schemes are required to ensure stability of the resulting method. Thus, at each time step, we have to solve a highly non-linear coupled system, since the fluid domain depends on the unknown displacement of the structure....

Acceleration of a fixed point algorithm for fluid-structure interaction using transpiration conditions

Simone Deparis, Miguel Angel Fernández, Luca Formaggia (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this work, we address the numerical solution of fluid-structure interaction problems. This issue is particularly difficulty to tackle when the fluid and the solid densities are of the same order, for instance as it happens in hemodynamic applications, since fully implicit coupling schemes are required to ensure stability of the resulting method. Thus, at each time step, we have to solve a highly non-linear coupled system, since the fluid domain depends on the unknown displacement of...

Air entrainment in transient flows in closed water pipes : A two-layer approach

C. Bourdarias, M. Ersoy, Stéphane Gerbi (2013)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this paper, we first construct a model for free surface flows that takes into account the air entrainment by a system of four partial differential equations. We derive it by taking averaged values of gas and fluid velocities on the cross surface flow in the Euler equations (incompressible for the fluid and compressible for the gas). The obtained system is conditionally hyperbolic. Then, we propose a mathematical kinetic interpretation of this system to finally construct a two-layer kinetic scheme...

An elastic membrane with an attached non-linear thermoelastic rod

Werner Horn, Jan Sokołowski (2002)

International Journal of Applied Mathematics and Computer Science

We study a thermo-mechanical system consisting of an elastic membrane to which a shape-memory rod is glued. The slow movements of the membrane are controlled by the motions of the attached rods. A quasi-static model is used. We include the elastic feedback of the membrane on the rods. This results in investigating an elliptic boundary value problem in a domain Ω ⊂ R^2 with a cut, coupled with non-linear equations for the vertical motions of the rod and the temperature on the rod. We prove the existence...

An XFEM/DG approach for fluid-structure interaction problems with contact

Luca Formaggia, Federico Gatti, Stefano Zonca (2021)

Applications of Mathematics

In this work, we address the problem of fluid-structure interaction (FSI) with moving structures that may come into contact. We propose a penalization contact algorithm implemented in an unfitted numerical framework designed to treat large displacements. In the proposed method, the fluid mesh is fixed and the structure meshes are superimposed to it without any constraint on the conformity. Thanks to the Extended Finite Element Method (XFEM), we can treat discontinuities of the fluid solution on...

Analysis and Numerical Approximation of an Electro-elastic Frictional Contact Problem

El. Essoufi, El. Benkhira, R. Fakhar (2010)

Mathematical Modelling of Natural Phenomena

We consider the problem of frictional contact between an piezoelectric body and a conductive foundation. The electro-elastic constitutive law is assumed to be nonlinear and the contact is modelled with the Signorini condition, nonlocal Coulomb friction law and a regularized electrical conductivity condition. The existence of a unique weak solution of the model is established. The finite elements approximation for the problem is presented, and error...

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