Approximation of multi-scale elliptic problems using patches of finite elements

In this paper we present a method to solve numerically 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. It resembles the FAC method (see Math. Comp. 46 (174) (1986) 439–456) and its convergence is obtained by a domain decomposition technique (see Math. Comp. 57 (195) (1991) 1–21). However it is of much more flexible use in comparison to the latter. To cite this article: R. Glowinski et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Operator splitting for the numerical solution of free surface flow at low capillary numbers

Free surface flows are pervasive in engineering and biomedical applications. In many interesting cases—particularly when small length scales are involved—surface forces (capillarity) dominate the flow dynamics. In these cases, computing the flow together with the shape of the surfaces, requires specialized solution techniques. This article investigates the capabilities of an operator splitting/finite elements method at handling accurately incompressible viscous flow with free surfaces at low capillary numbers. The test case of flow in the downstream section of a slot coater is used for three reasons: (1) it is an established benchmark; (2) it represents an idealized, yet industrially relevant flow; (3) high-fidelity results obtained with monolithic algorithms are available in literature. The flow and free surface shape attained with the new operator splitting scheme agree very satisfactorily with the results obtained with monolithic solvers. Because of its inherent computational simplicity, the new operator splitting scheme is attractive for large-scale simulations, three-dimensional flows, and flows of complex fluids.     

Direct simulation of the motion of neutrally buoyant balls in a three-dimensional Poiseuille flow

In a previous article the authors introduced a Lagrange multiplier based fictitious domain method. Their goal in the present article is to apply a generalization of the above method to: (i) the numerical simulation of the motion of neutrally buoyant particles in a three-dimensional Poiseuille flow; (ii) study – via direct numerical simulations – the migration of neutrally buoyant balls in the tube Poiseuille flow of an incompressible Newtonian viscous fluid. Simulations made with one and several particles show that, as expected, the Segré–Silberberg effect takes place. To cite this article: T.-W. Pan, R. Glowinski, C. R. Mecanique 333 (2005).     

A kinematically coupled time-splitting scheme for fluid–structure interaction in blood flow

We present a new time-splitting scheme for the numerical simulation of fluid–structure interaction between blood flow and vascular walls. This scheme deals in a successful way with the problem of the added mass effect. The scheme is modular and it embodies the stability properties of implicit schemes at the low computational cost of loosely coupled ones.     

A penalty/Newton/conjugate gradient method for the solution of obstacle problems

Motivated by the search for non-negative solutions of a system of Eikonal equations with Dirichlet boundary conditions, we discuss in this Note a method for the numerical solution of parabolic variational inequality problems for convex sets such as $\text{K={v∣v∈H}{}_0{}^1\text{(Ω)}$, v?ψ a.e. on $\text{Ω}.}$ The numerical methodology combines penalty and Newton's method, the linearized problems being solved by a conjugate gradient algorithm requiring at each iteration the solution of a linear problem for a discrete analogue of the elliptic operator I?μΔ. Numerical experiments show that the resulting method has good convergence properties, even for small values of the penalty parameter. To cite this article: R. Glowinski et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

A parallel finite element sliding mesh technique for the simulation of viscous flows in agitated tanks

A parallel sliding mesh algorithm for the finite element simulation of viscous fluid flows in agitated tanks is presented. Lagrange multipliers are used at the sliding interfaces to enforce the continuity between the fixed and moving subdomains. The novelty of the method consists of the coupled solution of the resulting velocity–pressure‐Lagrange multipliers system of equations by an ILU(0)‐QMR solver. A penalty parameter is introduced for both the interface and the incompressibility constraints to avoid pivoting problems in the ILU(0) algorithm. To handle the convective term, both the Newton–Raphson scheme and the semi‐implicit linearization are tested. A penalty parameter is introduced for both the interface and the incompressibility constraints to avoid the failure of the ILU(0) algorithm due to the lack of pivoting. Furthermore, this approach is versatile enough so that it allows partitioning of sliding and fixed subdomains if parallelization is required. Although the sliding mesh technique is fairly common in CFD, the main advantage of the proposed approach is its low computational cost due to the inexpensive and parallelizable calculations that involve preconditioned sparse iterative solvers. The method is validated for Couette and coaxial stirred tanks. Copyright © 2011 John Wiley & Sons, Ltd.     

A distributed Lagrange multiplier/fictitious domain method for flows around moving rigid bodies: Application to particulate flow

This article discusses the application of a Lagrange multiplier‐based fictitious domain method to the numerical simulation of incompressible viscous flow modeled by the Navier–Stokes equations around moving rigid bodies; the rigid body motions are due to hydrodynamical forces and gravity. The solution method combines finite element approximations, time discretization by operator splitting and conjugate gradient algorithms for the solution of the linearly constrained quadratic minimization problems coming from the splitting method. The study concludes with the presentation of numerical results concerning four test problems, namely the simulation of an incompressible viscous flow around a NACA0012 airfoil with a fixed center but free to rotate, then the sedimentation of 200 and 1008 cylinders in a two‐dimensional channel, and finally the sedimentation of two spherical balls in a rectangular cylinder. Copyright © 1999 John Wiley & Sons, Ltd.     

An efficient preconditioning scheme for iterative numerical solutions of partial differential equations.

Numerical solutions of time dependent and or nonlinear partial differential equations often require several solutions of a sparse linear system. If this system is factorized it may not fit into the computer core; if it is solved by an iterative process like the conjugate gradient algorithm it takes too much computing time. We show that if the small elements of the factorized matrix are deleted then the resulting operator is an excellent preconditioning operator for the conjugate gradient algorithm. Tests on two problems show that 90% of the main storage space can be saved without increasing the computing time as compared with a direct factorization method.     

On a mixed finite element approximation of the Stokes problem (I)

Summary We study in this paper a new mixed finite element approximation of the Stokes' problem in the velocity pressure formulation. This approximation which is based on a new variational principle allows the use of low order Lagrange elements and leads to optimal order of convergence for the velocity and the pressure. Iterative and direct methods for the solution of the approximate problems will be discussed in a forthcoming paper.     

Towards the computation of minimum drag profiles in viscous laminar flow

A numerical method is given for the solution of certain optimum design problems of fluid mechanics. The profile of given area and smallest drag in a uniform laminar flow is computed. This profile is long and slim, its front end is shaped like a wedge of angle 90° and its rear end is shaped like a cusp. Owing to the numerical complexity of the problem the precision of the results is average (around 5%). However, this work is a good illustration of the theoretical method exposed previously and it shows how good precision can be obtained if one is prepared to pay for it. A numerical solution of the adjoint system of the stationary Navier-Stokes equation is also given; this equation will play an important role in optimum design in fluid mechanics.