Alexei Lozinski

A fast solver for Fokker--Planck equation applied to viscoelastic flows calculations: 2D FENE model

A residual current circuit breaker 1 has a partition wall 3 which separates an over-current protection device 4 from a residual current detection circuit 5. A plunger rod 24 extends through a bore 23 in the armature 21 of a coil 20 and is moved independently through the coil 20 to trip the breaker if a residual current is detected by the circuit 5. The plunger rod 24 is moved by a drive rod 31, the operation of which is controlled by a permanent magnet which retains the drive rod 31 retracted and an electromagnet which allows the drive rod 31 to drive forwardly under the action of a spring 32 in the event of a residual current being detected. The plunger rod 30 is reset by a reset lever 40 which is moved when an operating handle 18 of the breaker moves from a non-tripped to a tripped position upon tripping of the breaker.

An energy estimate for the Oldroyd B model: theory and applications

Abstract In this paper, we present energy estimates for the stresses and velocity components in a general setting, for both inertial and inertialess flows of an Oldroyd B fluid. Our results apply to flows in bounded domains in any number of dimensions, subject to Dirichlet and possibly inflow boundary conditions. A novel numerical scheme is introduced and shown to be superior to a conventional Galerkin discretization of the Oldroyd B equations. In particular, the new scheme respects the derived energy estimates and guarantees positive definiteness of the stress tensor τ +((1−β)/We) I at all times, β being a solvent-to-total viscosity ratio and We a Weissenberg number. Numerical results for the planar viscoelastic Poiseuille problem illustrate some differences between the new and conventional schemes and reveal that the conventional scheme may lead to violation of the theoretical energy bounds in certain circumstances.

Finite element approximation of multi-scale elliptic problems using patches of elements

In this paper we present a method for the numerical solution of elliptic problems with multi-scale data using multiple levels of not necessarily nested grids. The method consists in calculating successive corrections to the solution in patches whose discretizations are not necessarily conforming. This paper provides proofs of the results published earlier (see C. R. Acad. Sci. Paris, Ser. I 337 (2003) 679–684), gives a generalization of the latter to more than two domains and contains extensive numerical illustrations. New results including the spectral analysis of the iteration operator and a numerical method to evaluate the constant of the strengthened Cauchy-Buniakowski-Schwarz inequality are presented.

An asymptotic-preserving method for highly anisotropic elliptic equations based on a Micro-Macro decomposition

The concern of the present work is the introduction of a very efficient asymptotic preserving scheme for the resolution of highly anisotropic diffusion equations. The characteristic features of this scheme are the uniform convergence with respect to the anisotropy parameter 0<e?1, the applicability (on cartesian grids) to cases of non-uniform and non-aligned anisotropy fields b and the simple extension to the case of a non-constant anisotropy intensity 1/e. The mathematical approach and the numerical scheme are different from those presented in the previous work P. Degond, F. Deluzet, A. Lozinski, J. Narski, C. Negulescu, Duality-based asymptotic-preserving method for highly anisotropic diffusion equations, Communications in Mathematical Sciences 10 (1) (2012) 1-31] and its considerable advantages are pointed out.

Fokker–Planck simulations of fast flows of melts and concentrated polymer solutions in complex geometries

In 1999, Ottinger introduced a thermodynamically admissible reptation model incorporating chain stretching, anisotropic tube cross sections, double reptation, and the convective constraint release mechanism. In this paper, we describe and use a new high-order Fokker–Planck-based numerical method for the simulation of the Ottinger model in complex geometries. Evidence, in the case of startup homogeneous flows, of the significant CPU time advantage (for comparable levels of accuracy) of our method over a stochastic simulation [Fang et al. (2000)], is presented. For the confined cylinder benchmark problem, differences in the drag behavior observed between the Ottinger model and those of Doi and Edwards (1978a, 1978b, 1978c) and Mead et al. (1998) are explained in terms of double reptation and the differing relaxation spectra.

An Anisotropic Error Estimator for the Crank-Nicolson Method: Application to a Parabolic Problem

In this paper we derive two a posteriori upper bounds for the heat equation. A continuous, piecewise linear finite element discretization in space and the Crank-Nicolson method for the time discretization are used. The error due to the space discretization is derived using anisotropic interpolation estimates and a postprocessing procedure. The error due to the time discretization is obtained using two different continuous, piecewise quadratic time reconstructions. The first reconstruction is developed following G. Akrivis, C. Makridakis, and R. H. Nochetto [Math. Comp., 75 (2006), pp. 511-531], while the second one is new. Moreover, in the case of isotropic meshes only, upper and lower bounds are provided as in [R. Verfurth, Calcolo, 40 (2003), pp. 195-212]. An adaptive algorithm is developed. Numerical studies are reported for several test cases and show that the second error estimator is more efficient than the first one. In particular, the second error indicator is of optimal order with respect to both the mesh size and the time step when using our adaptive algorithm.

AN MSFEM TYPE APPROACH FOR PERFORATED DOMAINS

We follow up on our previous work [C. Le Bris, F. Legoll and A. Lozinski, Chinese Annals of Mathematics 2013] where we have studied a multiscale finite element (MsFEM) type method in the vein of the classical Crouzeix-Raviart finite element method that is specifically adapted for highly oscillatory elliptic problems. We adapt the approach to address here a multiscale problem on a perforated domain. An additional ingredient of our approach is the enrichment of the multiscale finite element space using bubble functions. We first establish a theoretical error estimate. We next show that, on the problem we consider, the approach we propose outperforms all dedicated existing variants of MsFEM we are aware of.

The Langevin and Fokker–Planck Equations in Polymer Rheology

Publisher Summary This chapter discusses the applications of Langevin and Fokker–Planck equations in polymer rheology. It presents the stochastic simulation techniques for solving the Langevin equation. It introduces the stochastic differential equations for dilute polymer solutions modeled by dumbbells. Micro-macro techniques for simulating flows of polymeric fluids are discussed in the chapter. These methods are based on coupling macroscopic techniques for solving the conservation equations with microscopic methods for determining the polymeric stress in the fluid. Some of the early attempts to reduce the statistical error in the stochastic simulations without increasing the number of realizations are described in the chapter. Some of the major advances in the development and implementation of micro-macro techniques presented, such as the method of Brownian configuration fields of Hulsen, van Heel, and van den Brule. The chapter also describes efficient implicit schemes for micro-macro simulations developed by Laso, Ramirez, and Picasso. These schemes give rise to a large nonlinear system of algebraic equations for both the macroscopic and microscopic degrees of freedom at each time step with efficiency being achieved using size reduction techniques. A brief account of the solution of stochastic differential equations for linear polymer melts based on the Doi–Edwards model is discussed in the chapter. The deterministic numerical methods based on the Fokker–Planck equation for several kinetic theory models of polymer fluids are discussed in the chapter.

Duality-based asymptotic-preserving method for highly anisotropic diffusion equations

The present paper introduces an efficient and accurate numerical scheme for the solution of a highly anisotropic elliptic equation, the anisotropy direction being given by a variable vector field. This scheme is based on an asymptotic preserving reformulation of the original system, permitting an accurate resolution independently of the anisotropy strength and without the need of a mesh adapted to this anisotropy. The counterpart of this original procedure is the larger system size, enlarged by adding auxiliary variables and Lagrange multipliers. This Asymptotic-Preserving method generalizes the method investigated in a previous paper [arXiv:0903.4984v2] to the case of an arbitrary anisotropy direction field.

A fictitious domain approach for the Stokes problem based on the extended finite element method

SUMMARY In the present work, we propose to extend to the Stokes problem a fictitious domain approach inspired by extended finite element method and studied for the Poisson problem in a paper of Renard and Haslinger of 2009. The method allows computations in domains whose boundaries do not match. A mixed FEM is used for the fluid flow. The interface between the fluid and the structure is localized by a level-set function. Dirichlet boundary conditions are taken into account using Lagrange multiplier. A stabilization term is introduced to improve the approximation of the normal trace of the Cauchy stress tensor at the interface and avoid the inf-sup condition between the spaces for the velocity and the Lagrange multiplier. Convergence analysis is given, and several numerical tests are performed to illustrate the capabilities of the method. Copyright