Application of a Generalized Quadrature Free Discontinuous Galerkin Method in Aeroacoustics

Abstract

A quadrature free discontinuous Galerkin method using a general polynomial basis set is developed. It is shown that this method allows for the use of orthogonal and near orthogonal polynomials, whose properties not only lead to sparse elemental matrices, but also reduce the computational effort required in the evaluation of elemental boundary integrals. Expansion of the nonlinear flux expressions is accomplished easily using nodal polynomials such as Lagrange polynomials. Since most of the work involved in a discontinuous Galerkin solver is concentrated in the calculation of the boundary integrals involving coordinate transformations and nonlinear flux expansions, the present approach proves to be an efficient one. The method is applied to solve benchmark problems in aeroacoustics, involving uniform and non-uniform mean flows in two dimensions. Basis sets consisting of polynomials of up to 5th degree are used on an unstructured grid, and a study of both h and p refinement is presented. The performance of the method is compared to the quadrature free method using moment polynomials. Introduction The discontinuous Galerkin finite element method (DG method) has been used in computational aeroacoustics extensively over the past decade. This is due to the method’s high order of accuracy under a finite element framework. In this method, each element in the computational domain is treated as a separate entity. The governing equations are solved locally, and the global solution is discontinuous across the element boundaries. Because of this local formulation, the method is well suited for time accurate aeroacoustic simulations on unstructured grids using parallel computing. Traditionally, the DG method has been regarded as expensive in terms of both computational storage and time considerations, due to the requirement of numerical ∗Graduate Research Assistant, Student Member, AIAA †Boeing/A. D. Welliver Professor, Fellow AIAA Copyright c © 2003 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. quadrature rules in the evaluation of the integrals in the element matrices. The quadrature free DG method1 tries to resolve this problem by using a simple polynomial basis consisting of moment polynomials, the products of which can be integrated without numerical quadrature. However, with the use of moment polynomials in the basis set, calculation of elemental boundary integrals and the expansion of nonlinear flux terms is cumbersome. The calculation of the flux terms on the element interfaces requires a transformation of the coordinates. The transformation matrices associated with this coordinate transformation are fairly dense, and the elemental boundary calculations take up a major portion of the computation time, particularly when basis sets of higher degree polynomials are used.2 The flux terms in computational aeroacoustics involve the products of the components of a nonuniform mean flow and acoustic perturbations. The expansion of such flux expressions involving spatially varying components requires the truncation of a polynomial. The calculation of the coefficients of the polynomial is required at each time step for each element. Moreover, the coefficients of the polynomial will involve products of higher order derivatives of the spatially varying mean flow variables, which may not be readily available. In the present work, a generalized quadrature free discontinuous Galerkin method is developed, that addresses these issues. Also the results from its application to aeroacoustics problems are presented. The generalized method allows for the use a basis set made up of any polynomial function, without the use of numerical quadrature to evaluate the integrals of the discontinuous Galerkin formulation. This allows for the use of polynomials such as Lagrange, Chebyshev or similar orthogonal or near orthogonal polynomials. The shape of such polynomials along the elemental boundaries is the same, regardless of the element coordinates. The use of this property considerably reduces the computation time required for boundary integrals. With the use of a Lagrange polynomial basis set, the nonlinear flux terms are expanded easily, without the need for higher order derivatives of the flow variables. 1 OF 7 AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS 9th AIAA/CEAS Aeroacoustics Conference and Exhibit 12-14 May 2003, Hilton Head, South Carolina AIAA 2003-3120 Copyright © 2003 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. The method is then applied to solve benchmark problems in aeroacoustics, involving uniform and nonuniform mean flows in two spatial dimensions. The test cases considered are the propagation of an initial Gaussian pulse near a wall in a uniform subsonic mean flow, and the excitation of shear layer instabilities by an acoustic source in a parallel supersonic jet. Basis sets consisting of polynomials of up to 5th degree are used, on an unstructured grid consisting of uniform triangles. The inflow and outflow boundary conditions are implemented using a combination of a buffer layer3 and the stabilized perfectly matched layer (PML) technique as developed by Hu.4 Results from numerical experiments for h and p refinements using a Lagrange basis set are presented. A numerical study of the dissipation parameter in the flux formulation for the case of nonuniform mean flow is presented. The performance of the method is compared to the original quadrature free method using moment polynomials. The DG method The Runge-Kutta DG method, which was developed initially by Cockburn and Shu5 can be applied to discretize a system of first order conservation equations, such as the linearized Euler equations in aeroacoustics. Here, for the sake of simplicity, the governing equation is chosen to be the conservation equation of a scalar variable: ut +∇ ·~F = 0 (1) Let the domain be divided into smaller non-overlapping elements Ω(x,y,z) that span the domain. Let the solution in each element be approximated using an expansion of a basis set B, given by uΩ = N+1 ∑ j=1 b ju j B ⇒{bk,1 ≤ k ≤ N +1} (2) The number of basis functions N +1 depends on the order of expansion and the number of dimensions. The formulation of the discontinuous Galerkin method is obtained by the application of the traditional Galerkin method to equation (1) using equation(2), and integration by parts. It is convenient to carry out the integrations in the formulation in a transformed space (denoted by ∆). Let the coordinates in the transformed space be ξ, η and ζ. Then, in each element,