Michel Deville

Boundary conditions for incompressible flows

A general framework is presented for the formulation and analysis of rigid no-slip boundary conditions for numerical schemes for the solution of the incompressible Navier-Stokes equations. It is shown that fractional-step (splitting) methods are prone to introduce a spurious numerical boundary layer that induces substantial time differencing errors. High-order extrapolation methods are analyzed to reduce these errors. Both improved pressure boundary condition and velocity boundary condition methods are developed that allow accurate implementation of rigid no-slip boundary conditions.

Finite element simulation of pulsatile flow through arterial stenosis.

The problem of blood flow through a stenosis is solved using the incompressible Navier-Stokes equations in a rigid circular tube presenting a partial occlusion. Calculations are based on a Galerkin finite element method. The time marching scheme employs a predictor-corrector technique using a variable time step. Results are obtained for steady and physiological pulsatile flows. Computational experiments analyse the effect of varying the degree of stenosis, the stricture length, the Reynolds number and Womersley number. The method gives results which agree well with previous computations for steady flows and experimental findings for steady and pulsatile flows.

Chebyshev pseudospectral solution of second-order elliptic equations with finite element preconditioning

Abstract A new Chebyshev pseudospectral algorithm for second-order elliptic equations using finite element preconditioning is proposed and tested on various problems. Bilinear and biquadratic Lagrange elements are considered as well as bicubic Hermite elements. The numerical results show that bilinear elements produce spectral accuracy with the minimum computational work. L -shaped regions are treated by a subdomain approach.

Application of Spectral Elements To Viscoelastic Creeping Flows

The spectral element method is analyzed for the simulation of viscoelastic fluid flows in plane and cylindrical geometries. The implementation of the method and choices of functional spaces is discussed in detail. The method is then applied to the case of the Maxwell-B fluid and compared with the so-called 4 x 4 SUPG finite element method on the perturbed channel flow problem. Analysis of the rate of convergence of the method is provided as well as thorough discussion on the decomposition of the geometry in spectral elements: the choice is between the h- vs. p-refinement. A comparison with other numerical (e.g. pseudo-spectral) methods is performed for the flow of the Maxwell-B fluid in a wavy tube. In each test case, very good agreement is found.

Steady Viscous Flows By Compact Differences in Boundary-fitted Coordinates

Fourth-order compact differencing is applied to the steady solution of two-dimensional viscous incompressible flows at moderate Reynolds number. The physical region where the fluid flow occurs is mapped onto a rectangle by means of the boundary-fitted coordinates transformation method. The design of the general numerical algorithm rests upon discretization of velocity components and pressure field at the same grid points. The validity of this procedure is assessed by the investigation of a Stokes test problem. Fourth-order numerical results are compared to the analytical solution, second-order results and Chebyshev tau approximation. It is shown that fourth-order differences provide good precision, particularly when the ability of generating irregular meshes is fully exploited. Standard problems as Poiseuille and Couette flows, and the square cavity problem are solved. The fourth-order results on coarse mesh compare favourably with other techniques such as finite element method and second-order differences. A global convergence analysis was performed. On a two-dimensional problem with smooth boundary conditions, one observes a rate of convergence of order four for the velocity components and of order three for the pressure. For the square cavity problem with the two corner singularities, the rates of convergence are decreased by almost an order of magnitude. The solution of a plane constricted channel flow enhances the overall improvement in accuracy, gained by the treatment of the geometrically complex region through the mapping technique.

Spectral elements for viscoelastic flows with change of type

The spectral element method is used on periodic flows containing change of type of the vorticity equation. On slightly perturbed viscometric flows in a channel, the method is compared to the so-called EVSS finite element method and to analytical results when they are available. It is found that the method performs very well on all flows examined and that the computational cost is reduced compared to EVSS. The analysis of the uniform perturbed flow of a Maxwell fluid has revealed an unusual condition on the polynomial space containing the extra-stress representations. The technique is also applied to study the increase of the flow resistance in a periodically constricted tube. Results are compared with existing ones produced by means of the mixed pseudospectral-finite difference method (PCFD). Excellent agreement is found when comparing flow resistance. Explanation of the observed numerical difficulties is also investigated by inspection of the vorticity patterns corresponding to several pairs of Weissenberg and Reynolds numbers.

On the spectrum of the iteration operator associated to the finite element preconditioning of Chebyshev collocation calculations

Abstract An analytical investigation is performed of the spectrum of the iteration operator associated to the finite element preconditioning of Chebyshev collocation calculations, on a one-dimensional Dirichlet model problem. Use is made of the techniques developed by Haldenwang et al. for the finite difference preconditioning of the same problem. In the latter case the eigenvalues may be obtained as the diagonal elements of an upper-triangular matrix. This is not possible with finite element preconditioning, where the corresponding operator splits into two parts, one of which only is upper-triangular. Discarding the non-triangular part of the operator (which vanishes asymptotically for large values of N , partition size), this procedure yields an approximate expression of the eigenvalues in good agreement with the properties given earlier by Deville and Mund.

A multidimensional compact higher-order scheme for 3-D Poisson's equation

Abstract A multidimensional compact finite-difference scheme is applied to the solution of a three-dimensional Poissons equation. Excellent precision is obtained by means of a moderate discretization net. The presence of Neumann boundary conditions calls for special attention because these normal conditions affect the global precision. The numerical results for a test case involving five Neumann conditions and one Dirichlet condition on the six faces of a unit cube show good agreement with the analytical solution.

An algorithm for the evaluation of multidimensional (direct and inverse) discrete Chebyshev transforms

Abstract A method is presented for the efficient computation of n-D (n-dimensional) imbricated sums of (direct and inverse) discrete Chebyshev transforms. The algorithm consists essentially in the generalization of well-known onedimensional procedures which are fitted together in order to perform the pre- and post-processing of the Chebyshev coefficients array.

Chebyshev collocation solutions of flow problems

Chebyshev collocation methods are reviewed. For general second-order elliptic equations, the algebraic system obtained through the collocation process is ill-conditioned. Recent work on finite element preconditioning shows that bilinear elements give full satisfaction. For Stokes problems, as interpolants of different degree are used for the velocity and the pressure, the classical nine-node Lagrangian element with biquadratic velocities and bilinear pressures constitutes the best choice. In order to treat the non-linear terms, a Newtons method is designed. Domain decomposition is set up with the jumps of the stress vector across interfaces between adjacent subdomains. Two-dimensional curvy geometries are handled by a coordinate mapping. Problems with singularities are treated by a mixed finite element and spectral approximation. Finally, current developments for Non-Newtonian fluids are evoked.