Farzin Shakib

A new finite element formulation for computational fluid dynamics: X. The compressible Euler and Navier-Stokes equations

A space-time element method is presented for solving the compressible Euler and Navier-Stokes equations. The proposed formulation includes the variational equation, predictor multi-corrector algorithms and boundary conditions. The variational equation is based on the time-discontinuous Galerkin method, in which the physical entropy variables are employed. A least-squares operator and a discontinuity-capturing operator are added, resulting in a high-order accurate and unconditionally stable method. Implicit/explicit predictor multi-corrector algorithms, applicable to steady as well as unsteady problems, are presented; techniques are developed to enhance their efficiency. Implementation of boundary conditions is addressed; in particular, a technique is introduced to satisfy nonlinear essential boundary conditions, and a consistent method is presented to calculate boundary fluxes. Numerical results are presented to demonstrate the performance of the method.

A multi-element group preconditioned GMRES algorithm for nonsymmetric systems arising in finite element analysis

Abstract A multi-element group, domain decomposition algorithm is presented for solving linear nonsymmetric systems arising in finite element analysis. The iterative strategy employed is based on the generalized minimum residual (GMRES) procedure originally proposed by Saad and Shultz. Two levels of preconditioning are investigated. Coding is presented which fully exploits vector-architectured computers. Applications to problems of high-speed compressible flow illustrate the effectiveness of the scheme.

A new finite element formulation for computational fluid dynamics: IX. Fourier analysis of space-time Galerkin/least-squares algorithms

Abstract A Fourier stability and accuracy analysis of the space-time Galerkin/least squares method as applied to a time-dependent advective-diffusive model problem is presented. Two time discretizations are studied: a constant-in-time approximation and a linear-in-time approximation. Corresponding space-time predictor multi-corrector algorithms are also derived and studied. The behavior of the space-time algorithms is compared to algorithms based on semidiscrete formulations.

Symmetrization of conservation laws with entropy for high-temperature hypersonic computations

Abstract Results of Hughes, Franca, and Mallet are generalized to conservation law systems taking into account high-temperature effects. Symmetric forms of different equation sets are derived in terms of entropy variables. First, the case of a general divariant gas is studied; it can be specialized to the usual Navier-Stokes equations, as well as to situations where the gas is vibrationally excited, and undergoes equilibrium chemical reactions. The case of a gas in thermochemical nonequilibrium is considered next. Transport phenomena, and in particular mass diffusion, are examined in the framework of symmetric advective-diffusive systems. Suitably defined finite element methods are shown to satisfy automatically the second law of thermodynamics, which guarantees a priori the stability of the discrete solution.

A comparison of internal energy calculation methods for diatomic molecules

Various methods of calculating the internal energy of diatomic molecules are studied. An accurate and efficient method for computing the eigenvalues of the vibrational Schrodinger equation for an arbitrary potential is developed. The method is based on a finite‐element discretization using the cubic Lobatto element. A combination of spectrum slicing and the Laguerre algorithm is used to solve for the eigenvalues. A simple method to compute the quasibound states is presented. For N2 molecules, all vibrational–rotational states of 11 available electronic potentials are computed and summed to obtain the exact internal energy function with temperature. The total computation required 314 sec of CPU time on NASA’s Cray 2 computer. Various approximate models are discussed and compared with exact calculations. It is shown that the splitting of the macroscopic internal energy into separate electronic, rotational, and vibrational energies is not justified at high temperatures.

A new finite element formulation for computational fluid dynamics: development of an hourglass control operator for multidimensional advective-diffusive systems

Abstract An hourglass correction term is developed for multidimensional advective-diffusive systems. This term allows the use of lower order quadrature thereby making the method computationally more efficient. Numerical examples demonstrate the quality of the Galerkin least-squares method with the correction term applied to the solution of the two- and three-dimensional Navier-Stokes equations.

Application of the Galerkin/least-squares formulation to the analysis of hypersonic flows. II - Flow past a double ellipse

A finite element method for the compressible Navier-Stokes equations is introduced. The discretization is based on entropy variables. The methodology is developed within the framework of a Galerkin/least-squares formulation to which a discontinuity-capturing operator is added. Results for four test cases selected among those of the Workshop on Hypersonic Flows for Reentry Problems are presented.

Calculation of Two-Dimensional Compressible Euler Flows with a New Petrov-Galerkin Finite Element Method

We present an overview of a new finite element method for the compressible Euler equations. Our discretization is based on entropy variables. The method is developed within the framework of a Petrov-Galerkin formulation. Two perturbations are added to the weighting function; one is a generalization of the SUPG operator and the other is designed to enhance shock capturing capability. We present results of tests selected among the GAMM Workshop problems. The calculations were performed collaboratively by Stanford University and the Avions Marcel Dassault-Breguet Aviation company.