Numerical solution of the Euler equations for complex aerodynamic configurations using an edge-based finite element scheme

Abstract

This paper describes the development, validation and application of a new finite element scheme for the solution of the compressible Euler equations on unstructured grids. The implementation of the numerical scheme is based on an edge-based data structure, as opposed to a more traditional element-based data structure. The use of this edge-based data structure not only improves the efficiency of the algorithm, but also enables a straightforward implementation of upwind schemes in the context of finite element methods. The algorithm has been tested and validated on some well documented configurations. A flow solution ahout a complete F-18 fighter is shown to demonstrate the accuracy and robustness of the proposed algorithm. 1. I N T R O D U C T I O N I/ In recent years, significant progress has been made in the development of numerical algorithms for the solution of the compressible Euler and NavierStokes equations. The use of unstructured meshes for computational fluid dynamics problems has become widespread due to their ability to discretize arbitrarily complex geometries and due to the ease of adaption in enhancing the solution accuracy and efficiency through the use of adaptive refinement techniques. However, any unstructured algorithm requires the storage of the mesh connectivity, which implies the increase of computer memory and the use of indirect addressing to retrieve nearest neighbor information. These requirements, in turn, mean that any numerical algorithm will run slower on an unstructured grid than on a structured grid. In order to reduce indirect addressing, new edge-based finite element schemes([l]-141) have been recently introduced. In addition, even more sophisticated data structures such as stars, super edges, and chains were recently developed by Lohner[5]. The use of edge-based data structure has shown to result in remarkable computational savings for three dimensional problems. In the last few years, extensive research has been Copyright 01993 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. W 1 done on upwind type algorithms for the solution of the Euler and Navier-Stokes equations on unstructured meshes([6]-[9]). A significant advantage of upwind discretization is that it is naturally dissipative, in contrast with central-difference discretizations, and consequently does not require any problem-dependent parameters to adjust. So far, all upwind schemes implemented as either node-centered or cell-centered discretizations on unstructured meshes have used the finite volume approach where the control volume must be constructed first. In terms of computational efficiency, node-centered schemes are preferable to their cell-center counterparts. In the node-centered approach([6],[8]), the control volume is typically taken to he part of the neighboring cells that have a vertex at that node. In two dimensions, the part of the cells taken is determined by connecting the centroid of the cell and the midpoints of the two edges that share the node. In 3-D, the part of the cells taken is determined by a surface constructed in a similar way. However, this is somewhat complicated geometrically to do in three dimensions. The switching from element to edge-based data structure renders the implementation of upwind schemes trivial and straightforward in the context of the finite element approach; this is especially attractive for three dimensional application, since there is no need to construct control volumes explicitly and geometrically. The authors have recently developed some high accuracy schemes for the solution of the Euler and Navier-Stokes equations on unstructured grids by using an edge-based data structure[l]. This paper describes the development, validation, and application of an upwind finite element algorithm to the simulation of three dimensional compressible flows around complex aerodynamic configurations. In this scheme, the spatial discretization is accomplished by an edgebased finite element formulation using Roe’s fluxdifference splitting. A MUSCL approach is used to achieve higher-order accuracy. A characteristic analysis based on the introduction of Riemann invariants for one-dimensional flow normal to the boundary is used to treat boundary conditions. Solutions are advanced in time by a multi-stage Runge-Ihtta timestepping scheme. Convergence is accelerated using local time-stepping and implicit residual smoothing. The algorithm has been tested and validated on some well documented configurations. A solution of the flow around a complete F-18 fighter is presented to demonstrate the accuracy and robustness of the proposed algorithm. 2. GOVERNING EQUATIONS The Euler equations governing unsteady compressible inviscid flows can be expressed in the conservative form as au aFj + = o , at a z j where NI is the standard linear finite element shape function associated with node I , UI is the value at node I , and a[ is a constant. The Galerkin finite element approximation is then given by 'd I findUh E &such that for eachNI(1 5 I < n) (3.3) The integrals appearing here are evaluated in the standard finite element form by summing individual element and boundary surface contributions, the compact support of the shape function NI means that the equation can be written as where the summation convention has been employed and Here p, p , and e denote the density, pressure, and specific total energy, respectively, and ui is the velocity of the flow in the coordinate direction zi. This set of equations is completed by the addition of the equation-of-state which is valid for perfect gas, where y is the ratio of the specific heats. In the sequel, we assume that C2 is the flow domain, r its boundary, and nj the unit outward normal vector to the boundary. 3. VARIATIONAL FORMULATION AND FINITE ELEMENT APPROXIMATION Let I be a trial function space and W a weighting function space, both defined to consist of all suitably smooth functions. An equivalent variational formulation of (2.1) is given by find U € 7 such that VWE W I (3.1) Assuming oh is a classical triangulation of with the nodes numbered from 1 to n and P h the boundary of ah, we approximate the trial and weighting spaces 7 and W by their subspaces of finite dimension T h and Wh, which respectively are defined by (3.4) where the summation extends over those elements e and boundary surfaces b that contain node I. Inserting the assumed form for u,, in Eq.(3.4), the left-hand side integral can be evaluated exactly to give where M denotes the finite element consistent mass matrix. For steady state computations, M can be replaced by the lumped (diagonal) mass matrix, denoted by ML. 4. EDG E-BASED UPWIND FINITE ELEMENT SCHEME It is shown in the appendix that for any interior node, Eq. (3.4) can be written as where mI is the number of edges connected to the node I , and in 3D. The coefficient CIJ denotes the weight applied to the average value of the flux on the edge that connects nodes I and J , to obtain the contribution made by the edge to node I , whereas CJI denotes the weight w