Jacques Periaux

Numerical simulation of compressible Euler flows

Keywords: Euler : equation d ; ecoulement : compressible ; elements : finis ; simulation Reference Record created on 2005-11-18, modified on 2016-08-08

Presentation of Problems and Discussion of Results

The workshop was organized with the specific purpose to bring together a small number of scientists highly concerned with the numerical solution of the compressible Navier-Stokes equations.

Adaptive finite element methods for three dimensional compressible viscous flow simulation in aerospace engineering

We have briefly discussed here the numerical solution of the compressible Navier-Stokes equations written in non conservative form. We have shown by these first results that approximations satisfying some “Inf-sup” condition lead, for moderate Reynolds and Mach numbers, to accurate results without upwinding or artificial viscosity. The solution techniques which have been discussed are extended presently to conservative formulations in view of genuine hypersonic simulations including real gas effects. We can hope from the first results that, for higher Mach and Reynolds numbers, we will obtain accurate solutions while introducing less dissipation in the schemes; the accuracy, particularly of pressure, being a critical point in view of further calculations in hypersonic design.

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

Abstract 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.

Numerical vs. non-numerical robust optimisers for aerodynamic design using transonic finite-element solvers

HOW to solve shape optimization with un1 Approximate-gradient structured meshes ?We consider exact gradients, descent or one-shot algorithms and hierarchical parametrization. approach A~~l i ca t ionsa re transonic flows governed by equaLet 7 be a set of control parameters for the shape of an tions around an airfoil or in a 3D nozzle. airfoil, and W(y) the corresponding flow variables, implicitly defined from the discretized steady Euler equation written as follows Introduction Q(r , W(7)) = 0 , (1) The optimization of complex systems as those arising from aerodynamics is a challenging field from both intuition standpoint and implementation efficiency. The complexity of the multi-point turbulent 3D aerodynamics is difficult to master intuitively and optimization is no more a convex problem. The analysis of such systems stands at (or yet beyond) the limits of the possibilities of existing methods and computers. In order to answer the first point, powerful non-convex optimization algorithms have to be devised and a part of this paper will concentrate on non-numerical Genetic Algorithms. For addressing the efficiency problems, sensitivity analyses are attempted. The variable-domain question will be considered from two standpoints: in 2D, a variablemesh method is applied and an approximate sensitivity analysis is proposed and assessed; in 3D, a transpiration approach is chosen and combined with a hierarchical gradient method. The plan is the following: Section 1 : Approximate-gradient approach [2]), Section 2 : Towards 3D unstructured multilevel for shapes (see [9]). Section 3 : Genetic algorithms for solving aerodynamic optimum shape design problems. if the mesh is of fixed topology, with a deformation parametrized by 7, then the discrete flow variables W are a smooth function of y. Introducing a discrete cost function : the gradient is given by: where II is an adjoint state, solution of the linear system: and the cost functional to be minimized is given as follows : j(7) = function of q ( y ) (5) where Pl(y) is the pressure variable from W(y), and where I belongs to a set of points on the airfoil wall. In these conditions, the whole chain can be exactly differentiated, the gradient of j is expressed with an adjoint-state and a conjugate gradient algorithm can be applied to the minimization of j. In the study of this section, the Euler system (I) , is discretized on the triangulations by means of unstructured, MUSCL or centraldifferenced, finite-volume methods ([4]). The applicaINRIA, B.P. 93,06902 Sophia Antipolis Cedex, France tion of the ODYSSEE automated differentiator in order ~DASSAULT Aviation, 78, Quai Marcel Dassault, 92214 Saintto derive the exact adjoint is studied in ([lll); Cloud, France in the present study, the linearized adjoint Euler system ~ N A I , PR of China and Dassault (1) is based on a first-order Van Leer Flux Vector split§Laforia, Tour 46/00, 2e elage, University of Paris V1, BP 169, ting (i41) to allow an easier by-hand differentiation. 75252 Paris Cedex, France Om1995 bv Dervieux. Published bv the American Institute of The airfoil shape is defined from 20 to 80 spline control Aeronautics and Astronautics, Inc, with permission ordinates. The mesh is deformed by an elasticity sytem RAE Zep MdJ.73 d=2 #kg (woigb 0 1 I) Cp initid Cpopt ----Figure 1: Successive airfoil shapes during one optimizaFigure 2: Shock drag reduction: initial and final shapes.

Recent improvements in Galerkin and upwind Euler solvers and application to 3-D transonic flow in aircraft design

Abstract In this paper, recent studies in the implementation of Euler solvers relying on the Galerkin approximation or the upwind one are presented. The control of non-linear instabilities is achieved in the context of strongly non-regular 3-D tetrahedrizations. For this stabilization, monotone first-order accurate schemes are introduced for both approaches.

Acceleration Procedures for The Numerical Simulation of Compressible and Incompressible Viscous Flows

Applying operator splitting methods to the numerical simulation of compressible or incompressible viscous flows leads to the solution of Stokes type linear problems and of nonlinear elliptic systems. Once discretized, these problems involve a large number of variables and therefore require efficient solution methods.

Finite Element Analysis of Laminar Viscous Flow Over a Step by Nonlinear Least Squares and Alternating Direction Methods

We use in this paper a method described in DINH-GLOWINSKIMANTEL-PERIAUX [1], for which we refer for more details for solving the time dependent Navier-Stokes equations for incompressible viscous fluids. This method combine finite elements for the space discretization and alternating directions for the time discretization. The use of the splitting associated to the alternating direction method decouples the two main difficulties of the original problem, namely non linearity and incompressibility. However this method is a natural extension of those described in [2] [3] since least squares and conjugate gradient algorithms are still the main ingredients used to treat the nonlinearity. Results of numerical experiments using this technique for the numerical simulation of the Navier-Stokes flow in a prescribed for analysis channel with a step are presented.

Use of optimal control theory for the numerical simulation of transonic flow by the method of finite elements

It is shown that the transonic equation for compressible potential flow is equivalent to an optimal control problem of a linear distributed parameter system. This problem can be discretized by the finite element method and solved by a conjugate gradient algorithm. Thus a new class of methods for solving the transonic equation is obtained. Il is particularly well adapted to problems with complicate two or three dimensional geometries and shocks.

EUROPT — A European Initiative on Optimum Design Methods in Aerodynamics

This volume contains a set of methodologies and solutions for a number of selected optimum design problems in aerospace engineering. The methodologies for the solution of these problems cover optimization and inverse problems, external and internal flows, subsonic and transsonic regimes, different flow solvers and different discretizations schemes. These were presented at the EUROPT BRITE/EURAM project. This book should be of interest to engineers and scientists working in computing, optimization and control theory.