Azzeddine Soulaïmani

A finite element method for magnetohydrodynamics

Abstract This paper presents a finite element method for the solution of 3D incompressible magnetohydrodynamic (MHD) flows. Two important issues are thoroughly addressed. First, appropriate formulations for the magnetic governing equations and the corresponding weak variational forms are discussed. The selected ( B ,q) formulation is conservative in the sense that the local divergence-free condition of the magnetic field is accounted for in the variational sense. A Galerkin-least-squares variational formulation is used allowing equal-order approximations for all unknowns. In the second issue, a solution algorithm is developed for the solution of the coupled problem which is valid for both high and low magnetic Reynolds numbers. Several numerical benchmark tests are carried out to assess the stability and accuracy of the finite element method and to test the behavior of the solution algorithm.

Finite-element solution of compressible viscous flows using conservative variables

Abstract A finite element method for solving the compressible Navier-Stokes equations is presented. These equations are solved in conservation form and using conservative variables. Appropriate finite element approximations are discussed when a Galerkin variational formulation is used. For high-speed flows, the formulation is stabilized using the stream line upwind Petrov-Galerkin method for which we propose an analytical expression for the matrix τ. A new discontinuity-capturing operator is also proposed to accurately solve flow problems exhibiting sharp gradients. The non-linear systems of equations arising from the discretization are solved using an iterative strategy based on the generalized minimal residual (GMRES) algorithm. Numerical results for transonic and supersonic flows are presented that demonstrate the effectiveness of the proposed finite element method.

An arbitrary Lagrangian-Eulerian finite element method for solving three-dimensional free surface flows

Abstract This paper discusses numerical solution of unsteady three-dimensional free surface flows. The governing equilibrium equations are written in the framework of the Arbitrary Lagrangian-Eulerian kinematic description. The corresponding variational formulation is established afterwards. Since the variational problems are nonlinear with respect to the moving coordinates, a second-order approximate variational problem is derived after a consistent linearization of the referential motion. Stability of the discrete formulations is ensured with the help of a new stabilization method. A robust preconditioned GMRES algorithm is then used to solve the resulting set of nonlinear equations. Finally, the computational algorithms are assessed through numerical studies of various problems: a large sloshing flow in a three-dimensional reservoir, a discharge flow from a reservoir, simulation of a liquid vortex produced inside a cylindrical container with a disk rotating at the bottom and a three-dimensional practical hydraulic problem.

Preconditioning techniques for the solution of the Helmholtz equation by the finite element method

This paper discusses 2D solutions of the Helmholtz equation by finite elements. It begins with a short survey of the absorbing and transparent boundary conditions associated with the DtN technique. The solution of the discretized system by means of a standard Galerkin or Galerkin Least-Squares (GLS) scheme is obtained by a preconditioned Krylov subspace technique, specifically a preconditioned GMRES iteration. The stabilization paremeter associated to GLS is computed using a new formula. Two types of preconditioners, ILUT and ILU0, are tested to enhance convergence.

Numerical Tracking of Shallow Water Waves by the Unstructured Finite Volume WAF Approximation

The depth averaged shallow water equations (s.w.e.) are useful and reliable for dam break flow and flood modeling. For their approximation, the finite volume method (FVM) in conjunction with Riemann solvers permits shock capturing. This work provides an overview of the FV weighted average flux (WAF) method applied to s.w.e., and its implementation on unstructured triangular or quadrilateral meshes. The inviscid fluxes are given by the HLLC solver, and stabilization is ensured by the proper WAF limiters inherited from the total variation diminishing (T.V.D.) theory. Additional numerical improvements are incorporated to the model, such as enhancing the calculation of bed slopes, using an optional semi-implicit discretization of the friction source term, and affecting a depth tolerance to dry areas. The model performance is displayed through the simulation of well known synthetic and experimental examples including CADAM test 1 and test 2, which all show that the predictions are accurate and that the triangular mesh seems more efficient than quadrilateral. The applicability to real cases is assessed by simulating the flooding flow in a breakwater on “rivière des Prairies” having an irregular bathymetry.

A conservative stabilized finite element method for the magneto-hydrodynamic equations

We present a finite element solution of the 3D magneto-hydrodynamics equations. The formulation takes explicitly into account the local conservation of the magnetic field, giving rise to a conservative formulation and introducing a new scalar variable. A stabilization technique is used in order to allow equal linear interpolation on tetrahedral elements of all the variables. Numerical tests are performed in order to assess the stability and the accuracy of the resulting methods. The convergence rates are calculated for different stabilization parameters. Well-known MHD benchmark tests are calculated

Enhanced GMRES Acceleration Techniques for some CFD Problems

Large linear systems arising in CFD problems are often solved using iterative methods which typically combine an accelerator such as GMRES and a preconditioner. This paper presents results using several preconditioning techniques, which are not standard in CFD. Numerical tests are carried out for solving three-dimensional incompressible, compressible and magnetohydrodynamic (MHD) problems. A selection of numerical results is presented showing in particular that the flexible GMRES algorithm preconditioned with ILUT factorization provides a fairly robust iterative solver.

Nonlinear Aeroelasticity Modeling Using a Reduced Order Model Based on Proper Orthogonal Decomposition

Investigations of nonlinear aeroelasticity of flexible struc- tures subjected to unsteady transonic flows were carried out by means of an aeroelasticity model coupled with a reduced order CFD model based on POD ( proper orthogonal decomposition) method. The reduced order model is a three-dimensional with moving fluid boundaries. The CFD model order was reduced from more than 150000 of the full order model to 200 of the re- duced order model and Limit Oscillation Cycle (LCO) was ob- served. The dynamic responses of the system were simulated with the coupled model. Qualitatively, the numerical simulations on AGARD 445.6 from the nonlinear aeroelasticity model coupled with the reduced order CFD model agree with those from the model coupled with the full order CFD model.

Investigations of Nonlinear Aeroelasticity Using A Reduced Order Fluid Model Based on POD method

Investigations of nonlinear aeroelasticity of flexible structures subjected to unsteady transonic flows were carried out using a three-dimensional aeroelasticity reduced-order fluid model which is based on the Proper Orthogonal Decomposition (POD). The number of degrees of freedom was reduced from more than 150000 in the full-order model to 200 unknown variables for the reduced-order model (ROM). The dynamic responses of the aeroelastic system, particularly, the wing Limit Cycle Oscillation (LCO) in the transonic regime was simulated. Numerical simulations on the AGARD 445.6 wing show good agreement between the ROM and the original full-order fluid model.

A finite element formulation of compressible flows using various sets of independent variables

Abstract This paper presents a finite element method for the simulation of compressible flows. The Navier–Stokes and Euler equations are solved in the conservation form using various sets of independent variables. A variational formulation is developed based upon a variant of the Petrov–Galerkin method, and uses a shock-capturing operator. An adaptive algorithm based on a particular residual norm is proposed. Several numerical examples are presented to demonstrate the performances of each set of variables in solving compressible high-speed flows.