Showing papers in "Mathematical Modelling and Numerical Analysis in 2003"
TL;DR: A quasi-Newton algorithm for solving fluid-structure interaction problems to build an approximate tangent operator which is cost effective and which takes into account the so-called added mass effect.
Abstract: 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 in large displacements.
284 citations
TL;DR: In this paper, a method for handling interfaces between non-matching grids based on an approach suggested by Nitsche (1971) for the approximation of Dirichlet boundary conditions is presented.
Abstract: In this note, we propose and analyse a method for handling interfaces between non-matching grids based on an approach suggested by Nitsche (1971) for the approximation of Dirichlet boundary conditions. The exposition is limited to self-adjoint elliptic problems, using Poisson's equation as a model. A priori and a posteriori error estimates are given. Some numerical results are included.
263 citations
TL;DR: A finite element method for the approximation of second order elliptic problems on composite grids based on continuous piecewise polynomial approximation on each grid and weak enforcement of the proper continuity at an artificial interface defined by edges of one the grids is proposed.
Abstract: In this paper we propose a finite element method for the approximation of second order elliptic problems on composite grids. The method is based on continuous piecewise polynomial approximation on each grid and weak enforcement of the proper continuity at an artificial interface defined by edges (or faces) of one the grids. We prove optimal order a priori and energy type a posteriori error estimates in 2 and 3 space dimensions, and present some numerical examples.
155 citations
TL;DR: A discontinuous Galerkin method for linear elasticity is proposed, based on discontinuous piecewise linear approximation of the displacements, which shows optimal order a priori error estimates, uniform ...
Abstract: We propose a discontinuous Galerkin method for linear elasticity, based on discontinuous piecewise linear approximation of the displacements. We show optimal order a priori error estimates, uniform in the incompressible limit, and thus locking is avoided. The discontinuous Galerkin method is closely related to the non-conforming Crouzeix-Raviart (CR) element, which in fact is obtained when one of the stabilizing parameters tends to infinity. In the case of the elasticity operator, for which the CR element is not stable in that it does not fulfill a discrete Korn's inequality, the discontinuous framework naturally suggests the appearance of (weakly consistent) stabilization terms. Thus, a stabilized version of the CR element, which does not lock, can be used for both compressible and (nearly) incompressible elasticity. Numerical results supporting these assertions are included. The analysis directly extends to higher order elements and three spatial dimensions.
151 citations
TL;DR: In this article, the authors established a priori and a posteriori error estimate for the numerical approximation of some non-linear elliptic problems arising in glaciology, where the stationary motion of a glacier is given by a non-Newtonian fluid flow model which becomes, in first two-dimensional approximation, the so-called infinite parallel sided slab model.
Abstract: The main goal of this article is to establish a priori and a posteriori error estimates for the numerical approximation of some non linear elliptic problems arising in glaciology. The stationary motion of a glacier is given by a non-Newtonian fluid flow model which becomes, in a first two-dimensional approximation, the so-called infinite parallel sided slab model. The approximation of this model is made by a finite element method with piecewise polynomial functions of degree 1. Numerical results show that the theoretical results we have obtained are almost optimal.
144 citations
TL;DR: In this article, an asymptotic representation formula for the steady state voltage perturbation caused by low volume fraction internal conductivity inhomogeneities is established, which generalizes and unifies earlier formulas derived for special geometries and distributions of inhomogenities.
Abstract: We establish an asymptotic representation formula for the steady state voltage perturba- tions caused by low volume fraction internal conductivity inhomogeneities. This formula generalizes and unifies earlier formulas derived for special geometries and distributions of inhomogeneities.
122 citations
TL;DR: In this article, the semi-smooth Newton method is used for a class of variational inequalities in infinite dimensions and it is shown that they are equivalent to certain active set strategies.
Abstract: Semi–smooth Newton methods are analyzed for a class of variational inequalities in infinite dimensions. It is shown that they are equivalent to certain active set strategies. Global and local super-linear convergence are proved. To overcome the phenomenon of finite speed of propagation of discretized problems a penalty version is used as the basis for a continuation procedure to speed up convergence. The choice of the penalty parameter can be made on the basis of an L ∞ estimate for the penalized solutions. Unilateral as well as bilateral problems are considered.
119 citations
TL;DR: In this article, a finite volume coupled scheme with a phase-by-phase upstream weighting scheme is presented, and a subsequence of the sequence of approximate solutions converges to a weak solution of the continuous equations as the size of the discretization tends to zero.
Abstract: Models of two phase flows in porous media, used in petroleum engineering, lead to a system of two coupled equations with elliptic and parabolic degenerate terms, and two unknowns, the saturation and the pressure. For the purpose of their approximation, a coupled scheme, consisting in a finite volume method together with a phase-by-phase upstream weighting scheme, is used in the industrial setting. This paper presents a mathematical analysis of this coupled scheme, first showing that it satisfies some a priori estimates: the saturation is shown to remain in a fixed interval, and a discrete L2 (0,T;H1 (Ω)) estimate is proved for both the pressure and a function of the saturation. Thanks to these properties, a subsequence of the sequence of approximate solutions is shown to converge to a weak solution of the continuous equations as the size of the discretization tends to zero.
116 citations
TL;DR: A modified fixed-point algorithm which combines the standard BGS iterations with a transpiration formulation is proposed which shows the great improvement in computing time with respect to thestandard BGS method.
Abstract: 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 Standard strategies for solving this non-linear problems, are fixed point based methods such as Block-Gauss-Seidel (BGS) iterations Unfortunately, these methods are very CPU time consuming and usually show slow convergence We propose a modified fixed-point algorithm which combines the standard BGS iterations with a transpiration formulation Numerical experiments show the great improvement in computing time with respect to the standard BGS method
105 citations
TL;DR: In this article, a finite volume scheme for multi-dimensional drift-diffusion equations is introduced, which is composed of two continuity equations coupled with a Poisson equation, and the convergence of the scheme and then the existence of solutions to the problem are proved.
Abstract: We introduce a finite volume scheme for multi-dimensional drift-diffusion equations. Such equations arise from the theory of semiconductors and are composed of two continuity equations coupled with a Poisson equation. In the case that the continuity equations are non degenerate, we prove the convergence of the scheme and then the existence of solutions to the problem. The key point of the proof relies on the construction of an approximate gradient of the electric potential which allows us to deal with coupled terms in the continuity equations. Finally, a numerical example is given to show the efficiency of the scheme.
94 citations
TL;DR: In this article, a new formulation of the system of partial differential equations is obtained by the introduction of a new variable (this new variable is similar to the quasi-Fermi level in the framework of semiconductor modelling).
Abstract: We start from a mathematical model which describes the collective motion of bacteria taking into account the underlying biochemistry. This model was first introduced by Keller-Segel [13]. A new formulation of the system of partial differential equations is obtained by the introduction of a new variable (this new variable is similar to the quasi-Fermi level in the framework of semiconductor modelling). This new system of P.D.E. is approximated via a mixed finite element technique. The solution algorithm is then described and finally we give some preliminary numerical results. Especially our method is well adapted to compute the concentration of bacteria.
TL;DR: In this paper, a modification of the Monge-Kantorovitch problem of exponent 2 is introduced to accommodate non-balanced initial and final densities, and the augmented Lagrangian numerical method introduced in [6] is adapted to this "unbalanced" problem.
Abstract: We introduce a modification of the Monge–Kantorovitch problem of exponent 2 which accommodates non balanced initial and final densities. The augmented Lagrangian numerical method introduced in [6] is adapted to this “unbalanced” problem. We illustrate the usability of this method on an idealized error estimation problem in meteorology.
TL;DR: In this paper, a variational inequality weak formulation for the gradient flow was proposed, and convergence to the regularized gradient flow problem was established as h,k,e → 0 in general cases.
Abstract: We study the gradient flow for the total variation functional, which arises in image processing and geometric applications. We propose a variational inequality weak formulation for the gradient flow, and establish well-posedness of the problem by the energy method. The main idea of our approach is to exploit the relationship between the regularized gradient flow (characterized by a small positive parameter e , see (1.7)) and the minimal surface flow [21] and the prescribed mean curvature flow [16]. Since our approach is constructive and variational, finite element methods can be naturally applied to approximate weak solutions of the limiting gradient flow problem. We propose a fully discrete finite element method and establish convergence to the regularized gradient flow problem as h,k → 0, and to the total variation gradient flow problem as h,k,e → 0 in general cases. Provided that the regularized gradient flow problem possesses strong solutions, which is proved possible if the datum functions are regular enough, we establish practical a priori error estimates for the fully discrete finite element solution, in particular, by focusing on the dependence of the error bounds on the regularization parameter e . Optimal order error bounds are derived for the numerical solution under the mesh relation k = O(h2 ) . In particular, it is shown that all error bounds depend on only in some lower polynomial order for small e .
TL;DR: A mathematical model is presented which assumes that cells exert traction forces onto the extracellular matrix, the matrix behaves as a linear viscoelastic material, and the cells move along gradients of exogenously supplied chemical stimuli (chemotaxis) and these stimuli diffuse or are uptaken by the cells.
Abstract: Vasculogenesis and angiogenesis are two different mechanisms for blood vessel formation. Angiogenesis occurs when new vessels sprout from pre-existing vasculature in response to external chemical stimuli. Vasculogenesis occurs via the reorganization of randomly distributed cells into a blood vessel network. Experimental models of vasculogenesis have suggested that the cells exert traction forces onto the extracellular matrix and that these forces may play an important role in the network forming process. In order to study the role of the mechanical and chemical forces in both of these stages of blood vessel formation, we present a mathematical model which assumes that (i) cells exert traction forces onto the extracellular matrix, (ii) the matrix behaves as a linear viscoelastic material, (iii) the cells move along gradients of exogenously supplied chemical stimuli (chemotaxis) and (iv) these stimuli diffuse or are uptaken by the cells. We study the equations numerically, present an appropriate finite difference scheme and simulate the formation of vascular networks in a plane. Our results compare very well with experimental observations and suggest that spontaneous formation of networks can be explained via a purely mechanical interaction between cells and the extracellular matrix. We find that chemotaxis alone is not a sufficient force to stimulate formation of pattern. Moreover, during vessel sprouting, we find that mechanical forces can help in the formation of well defined vascular structures.
TL;DR: In this article, the authors consider the time-harmonic eddy current problem in its electric formulation where the conductor is a polyhedral domain and prove the convergence in energy of the solutions.
Abstract: We consider the time-harmonic eddy current problem in its electric formulation where the conductor is a polyhedral domain. By proving the convergence in energy, we justify in what sense this problem is the limit of a family of Maxwell transmission problems: Rather than a low frequency limit, this limit has to be understood in the sense of Bossavit [11]. We describe the singularities of the solutions. They are related to edge and corner singularities of certain problems for the scalar Laplace operator, namely the interior Neumann problem, the exterior Dirichlet problem, and possibly, an interface problem. These singularities are the limit of the singularities of the related family of Maxwell problems.
TL;DR: In this paper, a general representation formula for the boundary voltage per- turbations caused by internal conductivity inhomogeneities of low volume fraction was proposed. But this formula is not suitable for the case of large volume fractions and cannot be used to obtain accurate estimates for the size of the inhomalities in terms of multiple boundary measure-ments.
Abstract: We recently derived a very general representation formula for the boundary voltage per- turbations caused by internal conductivity inhomogeneities of low volume fraction (cf. Capdeboscq and Vogelius (2003)). In this paper we show how this representation formula may be used to obtain very accurate estimates for the size of the inhomogeneities in terms of multiple boundary measure- ments. As demonstrated by our computational experiments, these estimates are significantly better than previously known (single measurement) estimates, even for moderate volume fractions.
TL;DR: In this paper, a finite volume kinetic method was proposed to compute the transport of a passive pollutant by a flow modeled by the shallow water equations using a new time discretization that allows large time steps for the pollutant computation.
Abstract: The aim of this paper is to present a finite volume kinetic method to compute the transport of a passive pollutant by a flow modeled by the shallow water equations using a new time discretization that allows large time steps for the pollutant computation. For the hydrodynamic part the kinetic solver ensures – even in the case of a non flat bottom – the preservation of the steady state of a lake at rest, the non-negativity of the water height and the existence of an entropy inequality. On an other hand the transport computation ensures the conservation of pollutant mass, a non-negativity property and a maximum principle for the concentration of pollutant and the preservation of discrete steady states associated with the lake at rest equilibrium. The interest of the developed method is to preserve these theoretical properties with a scheme that allows to disconnect the hydrodynamic time step – related to a classical CFL condition – and the transport one – related to a new CFL condition – and further the hydrodynamic calculation and the transport one. The CPU time is very reduced and we can easily solve different transport problems with the same hydrodynamic solution without large storage. Moreover the numerical results exhibit a better accuracy than with a classical method especially when using 1D or 2D regular grids.
TL;DR: In this paper, the authors present a domain decomposition theory on an interface problem for the linear transport equation between a diusive and a non-diusive region, where the diusive region is coupled at the interface at the next order of approximation.
Abstract: We present a domain decomposition theory on an interface problem for the linear transport equation between a diusive and a non-diusive region. To leading order, i.e. up to an error of the order of the mean free path in the diusive region, the solution in the non-diusive region is independent of the density in the diusive region. However, the diusive and the non-diusive regions are coupled at the interface at the next order of approximation. In particular, our algorithm avoids iterating the diusion and transport solutions as is done in most other methods | see for example Bal{Maday (2002). Our analysis is based instead on an accurate description of the boundary layer at the interface matching the phase-space density of particles leaving the non-diusive region to the bulk density that solves the diusion equation.
TL;DR: In this article, two kinds of Zienkiewicz{Zhu (ZZ) type error estimators are derived which originate from dierent backgrounds, and the advantage of keeping two dierent analyses of the estimators is that they allow dierent and partially novel investigations and results.
Abstract: We consider a posteriori error estimators that can be applied to anisotropic tetrahedral nite element meshes, i.e. meshes where the aspect ratio of the elements can be arbitrarily large. Two kinds of Zienkiewicz{Zhu (ZZ) type error estimators are derived which originate from dierent backgrounds. In the course of the analysis, the rst estimator turns out to be a special case of the second one, and both estimators can be expressed using some recovered gradient. The advantage of keeping two dierent analyses of the estimators is that they allow dierent and partially novel investigations and results. Both rigorous analytical approaches yield the equivalence of each ZZ error estimator to a known residual error estimator. Thus reliability and eciency of the ZZ error estimation is obtained. The anisotropic discretizations require analytical tools beyond the standard isotropic methods. Particular attention is paid to the requirements on the anisotropic mesh. The analysis is complemented and conrmed by extensive numerical examples. They show that good results can be obtained for a large class of problems, demonstrated exemplary for the Poisson problem and a singularly perturbed reaction diusion problem.
TL;DR: An implicit finite element method using unstructured grids for an anisotropic bidomain model for electrophysiological waves in the cardiac tissue is proposed.
Abstract: Bidomain models are commonly used for studying and simulating electrophysiological waves in the cardiac tissue. Most of the time, the associated PDEs are solved using explicit finite difference methods on structured grids. We propose an implicit finite element method using unstructured grids for an anisotropic bidomain model. The impact and numerical requirements of unstructured grid methods is investigated using a test case with re-entrant waves.
TL;DR: In this article, a mixed-FEM and BEM coupling method was proposed to solve the electromagnetic field generated in an infinite cylindrical conductor by an alternating current density, and the resulting interface problem between the metal and the dielectric medium was treated by a mixed FEM-BEM coupling.
Abstract: We study in this paper the electromagnetic field generated in an infinite cylindrical conductor by an alternating current density. The resulting interface problem (see [1]) between the metal and the dielectric medium is treated by a mixed-FEM and BEM coupling method. We prove that our BEM-FEM formulation is well posed and leads to a convergent Galerkin method with optimal error estimates. Furthermore, we introduce a fully discrete version of our Galerkin scheme based on simple quadrature formulas. We show that, if the parameter of discretization is sufficiently small, the fully discrete method is well posed and the error estimates remain unaltered.
TL;DR: The Robin penalization is described, and the gap between the penalized and the non-penalized boundary controls for the small penalization parameter is bound.
Abstract: A current procedure that takes into account the Dirichlet boundary condition with non-smooth data is to change it into a Robin type condition by introducing a penalization term; a major effect of this procedure is an easy implementation of the boundary condition. In this work, we deal with an optimal control problem where the control variable is the Dirichlet data. We describe the Robin penalization, and we bound the gap between the penalized and the non-penalized boundary controls for the small penalization parameter. Some numerical results are reported on to highlight the reliability of such an approach.
TL;DR: In this article, the authors derived a constitutive law for the myocardium from the description of both the geometrical arrangement of cardiomyocytes and their individual mechanical behaviour.
Abstract: We derive a constitutive law for the myocardium from the description of both the geometrical arrangement of cardiomyocytes and their individual mechanical behaviour. We model a set of cardiomyocytes by a quasiperiodic discrete lattice of elastic bars interacting by means of moments. We work in a large displacement framework and we use a discrete homogenization technique. The macroscopic constitutive law is obtained through the resolution of a nonlinear self-equilibrum system of the discrete lattice reference cell.
TL;DR: In this paper, a model based on spectral hyperviscosity for the simulation of 3D turbulent incompressible flows is presented, which is reminiscent of Large Eddy Simulation models.
Abstract: This paper presents a model based on spectral hyperviscosity for the simulation of 3D turbulent incompressible flows. One particularity of this model is that the hyperviscosity is active only at the short velocity scales, a feature which is reminiscent of Large Eddy Simulation models. We propose a Fourier{Galerkin approximation of the perturbed Navier{Stokes equations and we show that, as the cuto wavenumber goes to innity, the solution of the model converges (up to subsequences) to a weak solution which is dissipative in the sense dened by Duchon and Robert (2000).
TL;DR: In this article, the authors consider the problem of finding appropriate models for Large Eddy Simulations of turbulent incompressible flows from a mathematical point of view, and propose two variants of the Smagorinsky model: anisotropic turbulent viscosity and a selective model based on vorticity angles.
Abstract: We consider in this paper the problem of finding appropriate models for Large Eddy Simulations of turbulent incompressible flows from a mathematical point of view. The Smagorinsky model is analyzed and the vorticity formulation of the Navier–Stokes equations is used to explore more efficient subgrid-scale models as minimal regularizations of these equations. Two classes of variants of the Smagorinsky model emerge from this approach: a model based on anisotropic turbulent viscosity and a selective model based on vorticity angles. The efficiency of these models is demonstrated by comparisons with reference results on decaying turbulence experiments.
TL;DR: In this article, it was shown that the relaxation frequencies of the relaxed Euler system converges toward a multicomponent Navier-Stokes system with the classical Fick and Newton laws, with a thermal diffusion which can be assimilated to a Soret effect in the case of a fluid mixture.
Abstract: We show that it is possible to construct a class of entropic schemes for the multicomponent Euler system describing a gas or fluid homogeneous mixture at thermodynamic equilibrium by applying a relaxation technique. A first order Chapman–Enskog expansion shows that the relaxed system formally converges when the relaxation frequencies go to the infinity toward a multicomponent Navier–Stokes system with the classical Fick and Newton laws, with a thermal diffusion which can be assimilated to a Soret effect in the case of a fluid mixture, and with also a pressure diffusion or a density diffusion respectively for a gas or fluid mixture. We also discuss on the link between the convexity of the entropies of each species and the existence of the Chapman–Enskog expansion.
TL;DR: The hyperbolic system of governing equations describing one-dimensional blood flow in arterial networks using a discontinuous Galerkin formulation with a spectral/hp element spatial approximation is outlined and the applicability of the methods to the simulation of flows in arterials networks is demonstrated.
Abstract: In this paper we outline the hyperbolic system of governing equations describing one-dimensional blood flow in arterial networks. This system is numerically discretised using a discontinuous Galerkin formulation with a spectral/hp element spatial approximation. We apply the numerical model to arterial networks in the placenta. Starting with a single placenta we investigate the velocity waveform in the umbilical artery and its relationship with the distal bifurcation geometry and the terminal resistance. We then present results for the waveform patterns and the volume fluxes throughout a simplified model of the arterial placental network in a monochorionic twin pregnancy with an arterio-arterial anastomosis and an arterio-venous anastomosis. The effects of varying the time period of the two fetus' heart beats, increasing the input flux of one fetus and the role of terminal resistance in the network are investigated and discussed. The results show that the main features of the in vivo, physiological waves are captured by the computational model and demonstrate the applicability of the methods to the simulation of flows in arterial networks.
TL;DR: A semidiscretization in time of a fourth order nonlinear parabolic system in several space dimensions arising in quantum semiconductor modelling is studied and the rate of convergence proves to be optimal.
Abstract: A semidiscretization in time of a fourth order nonlinear parabolic system in several space dimensions arising in quantum semiconductor modelling is studied. The system is numerically treated by introducing an additional nonlinear potential. Exploiting the stability of the discretization, convergence is shown in the multi-dimensional case. Under some assumptions on the regularity of the solution, the rate of convergence proves to be optimal.
TL;DR: The pulsatile flow of incompressible Newtonian blood is illustrated by numerical simulations carried out in two saccular aneurism types, a side- and a terminal-aneurism.
Abstract: Saccular aneurisms, swelling of a blood vessel, are investigated in order (i) to estimate the development risk of the wall lesion, before and after intravascular treatment, assuming that the pressure is the major factor, and (ii) to better plan medical interventions. Numerical simulations, using the finite element method, are performed in three-dimensional aneurisms. Computational meshes are derived from medical imaging data to take into account both between-subject and within-subject anatomical variability of the diseased vessel segment. The 3D reconstruction is associated with a faceted surface. A geometrical model is then obtained to be finally meshed for a finite element use. The pulsatile flow of incompressible Newtonian blood is illustrated by numerical simulations carried out in two saccular aneurism types, a side- and a terminal-aneurism. High pressure zones are observed in the aneurism cavity, especially in the terminal one.
TL;DR: In this article, the authors used the mathematical model consisting of the diffusion equation with convective terms and of the Navier-Stokes equations with the Korteweg stress and proved the global existence and uniqueness of the solution for the associated initial boundary value problem in a two-dimensional bounded domain.
Abstract: When two miscible fluids, such as glycerol (glycerin) and water, are brought in contact, they immediately diffuse in each other. However if the diffusion is sufficiently slow, large concentration gradients exist during some time. They can lead to the appearance of an “effective interfacial tension”. To study these phenomena we use the mathematical model consisting of the diffusion equation with convective terms and of the Navier-Stokes equations with the Korteweg stress. We prove the global existence and uniqueness of the solution for the associated initial-boundary value problem in a two-dimensional bounded domain. We study the longtime behavior of the solution and show that it converges to the uniform composition distribution with zero velocity field. We also present numerical simulations of miscible drops and show how transient interfacial phenomena can change their shape.