Aggrégation de bactéries. Simulations numériques de modèles de réaction-diffusion à l'aide des éléments finis mixtes

Nous partons d'un modèle mathématique assez général donné dans le livre de J.D. Murray (Mathematical Biology) constitué d'un système de trois équations aux dérivées partielles. Les fonctions inconnues sont la densité de bactéries, la concentration de chemoattractants et la concentration de stimulants (nourriture). On considère ensuite un modèle où l'on prend en compte des bactéries actives et des bactéries inactives entrant cependant toutes deux dans la formation de motifs (bactéries du type Bacillus subtilis). Les schémas d'approximation et de résolution sont des extensions naturelles de ceux décrits dans le rapport de recherche Inria n° 4667 (2002

Aggrégation de bactéries. Simulations numériques de modèles de réaction-diffusion à l'aide des éléments finis mixtes

Nous partons d'un modèle mathématique assez général donné dans le livre de J.D. Murray (Mathematical Biology) constitué d'un système de trois équations aux dérivées partielles. Les fonctions inconnues sont la densité de bactéries, la concentration de chemoattractants et la concentration de stimulants (nourriture). On considère ensuite un modèle où l'on prend en compte des bactéries actives et des bactéries inactives entrant cependant toutes deux dans la formation de motifs (bactéries du type Bacillus subtilis). Les schémas d'approximation et de résolution sont des extensions naturelles de ceux décrits dans le rapport de recherche Inria n° 4667 (2002

Simulations numeriques dans la fabrication des circuits a semiconducteurs (Process modelling)

Résumé disponible dans les fichiers attaché

2D simulation of chemotactic bacteria aggregation

Projet M3NWe 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 . 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 approximat- ed 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

Arclength continuation methods and applications to 2D drift-diffusion semiconductor equations

Projet MENUSINIn this paper, the homotopy deformation method to solve the nonlinear stationary semiconductor equations with Fermi-Dirac statistic is used. This method introduces an artificial transient problem. The time discretization is based on the nonlinear implicit scheme with local time steps. In order to have an automatic adaptation of local time step parameters, we introduce arclength predictor-c- orrector continuation methods. The fondamental goal of these methods is to overcome the unstabilities or the failure of the classical Newton-Raphson's schemes which appear when the nonlinearity is Strong or near Limit or Bifurcation points. The approximate procedure of our system using a Galerkin method that makes use of a mixed finite element approach is used. A peculiar feature of this mixed formulation is that the electric displacement $D$ and the current densities $j_n$ and $j_p$ for electrons and holes, are taken as unknowns, together with the potential $\phi$ and quasi_Fermi levels $\phi_n$ and $\phi_p$. This allows $D$, $j_n$ and $j_p$ to be determined directly and accurately. The above algorithms appear to be efficient, robust and to give satisfactory results. Numerical results are presented, in one and two dimension, for some realistic devices : an Heterojunction Diode (quasi-1D problem) and an Heterojunction Bipolar Transistor (HBT) working in amplifier mode

Numerical Simulation of the 2D Non-Parabolic MEP Energy-Transport Model with a Mixed Finite Elements Scheme

In the Mixed Finite Element approximation scheme has been used to simulate a consistent hydrodynamical model for electron transport in semiconductors, free of any fitting parameters , based on the Maximum Entropy Principle (MEP) and assuming the Parabolic Band approximation. In this paper we describe an application of the above numerical method in the case of the two dimensional Non-Parabolic MEP energy-transport model. We can consider this paper as a generalization of what has been done in . As done in results of the simulation of 2D-MESFET and 2D-MOSFET Silicon devices are presented

Numerical simulation of 2D Silicon MESFET and MOSFET described by the MEP based energy-transport model with a mixed finite elements scheme

The Mixed Finite Element approximation scheme presented in is used to simulate a consistent hydrodynamical model for electron transport in semiconductors, free of any fitting parameters, formulated on the basis of the maximum entropy principle (MEP) in \cite{AnRo,Ro1,Ro2}.. 2D-MESFET and 2D-MOSFET Silicon devices are simulated in the parabolic band approximation. Comparison with the results obtained by the Stratton model are presented for completeness

Numerical simulation of 2D Silicon MESFET and MOSFET described by the MEP based energy-transport model with a mixed finite elements scheme

The Mixed Finite Element approximation scheme presented in is used to simulate a consistent hydrodynamical model for electron transport in semiconductors, free of any fitting parameters, formulated on the basis of the maximum entropy principle (MEP) in \cite{AnRo,Ro1,Ro2}.. 2D-MESFET and 2D-MOSFET Silicon devices are simulated in the parabolic band approximation. Comparison with the results obtained by the Stratton model are presented for completeness

Conservative cross diffusions and pattern formation through relaxation

Analyse mathématique et numérique de sytèmes "cross-diffusion"This paper is aimed at studying the formation of patches in a cross-diffusion system without reaction terms when the diffusion matrix can be negative but with positive self-diffusion. We prove existence results for small data and global a priori bounds in space-time Lebesgue spaces for a large class of 'diffusion' matrices. This result indicates that blow-up should occur on the gradient. One can tackle this issue using a relaxation system with global solutions and prove uniform a priori estimates. Our proofs are based on a duality argument à la M. Pierre which we extend to treat degeneracy and growth of the diffusion matrix. We also analyze the linearized instability of the relaxation system and a Turing type mechanism can occur. This gives the range of parameters and data for which instability can occur. Numerical simulations show that patterns arise indeed inthis range and the solutions tend to exhibit patches with stiff gradients on bounded solutions, in accordance with the theory

Conservative cross diffusions and pattern formation through relaxation

Analyse mathématique et numérique de sytèmes "cross-diffusion"This paper is aimed at studying the formation of patches in a cross-diffusion system without reaction terms when the diffusion matrix can be negative but with positive self-diffusion. We prove existence results for small data and global a priori bounds in space-time Lebesgue spaces for a large class of 'diffusion' matrices. This result indicates that blow-up should occur on the gradient. One can tackle this issue using a relaxation system with global solutions and prove uniform a priori estimates. Our proofs are based on a duality argument à la M. Pierre which we extend to treat degeneracy and growth of the diffusion matrix. We also analyze the linearized instability of the relaxation system and a Turing type mechanism can occur. This gives the range of parameters and data for which instability can occur. Numerical simulations show that patterns arise indeed inthis range and the solutions tend to exhibit patches with stiff gradients on bounded solutions, in accordance with the theory