ADVANCES N ADAPTVE METHODS N COMPUTATONAL FLUD MECHANCS J. Tinsley Oden Texas nstitute for Computational Mechanics The University of Texas at Austin Austin, Texas 78712 Abstract Recent developments in adaptive methods in computational fluid dynamics (CFD) give hope that one may develop "optimal" schemes for analyzing complex flow; i.e., schemes which deliver the best possible accuracy for a fixed computational effort. This note discusses some of the basic ideas behind adaptive methods and cites examples of recent results obtained using adaptive schemes for compressible flow problems. ADAPTVE FEM'S Suppose that one could estimate reliably the accuracy of a numerical solution, e.g., suppose that it were possible to calculate a collection of numbers Be, each of which was an indication of the actual numerical error in some appropriate norm for cell number e in a finite difference or a finite element mesh over a given flow domain. Then, knowing the computational error (or, at least, knowing a good indication of it), one could legitimately ask the question: how can the structure of the approximation be changed in order to reduce the error below a preassigned limit? Such numbers are called local error indicators. This "preassigned limit" may be determined by many factors, such as the computer budget available to the analyst, the number of man-hours that can be devoted to the task, the precision of the results required in a given calculation, or the capacity of the computer being used. Once this tolerance is assigned, there must follow an adaptive process by which the structure of the approximation is systemically adapted to reduce error, i.e., to improve the local quality of the solution. 1
There are several families of adaptive strategies that can be used to modify the structure of the approximation: a) Mesh Refinement Schemes (h-methods). n these schemes, the mesh is automatically refined when the local error indicator exceeds a preassigned tolerance. Such h-schemes present a very difficult data management problem, since they involve a dynamic regeneration of the mesh, renumbering of grid points, cells or elements, and element connectivities as the mesh is refined. However, the h-methods can be very effective in producing nearoptimal meshes for given error tolerances. The author and his colleagues have developed a very fast code that enables the analyst to use h-methods very efficiently for complex flow geometries. Furthermore, the h-method strategy can also be used to coarsen a mesh (use larger mesh cells and thereby reduce the number of unknowns) when the local error becomes lower than an assigned lower-bound tolerance. A sample calculation obtained with our h-method is shown in Fig. 1. Shown here is a calculation of supersonic flow in a rotor-stator flow interaction problem in which rotor blades are moving relative to the stator in a two-dimensional flow field. The procedure dynamically refines the mesh, assigning large elements where the error is small, small elements where the error tends to be large, and simultaneously models shocks, flow through mesh interfaces and shock interaction. Computed density contours are also given in the figure. b) Moving Mesh Schemes. Moving mesh schemes employ a fixed number of grid points and attempt to dynamically move the grid points to areas of high error in the mesh. Moving mesh schemes can be rather easy to implement, and, therefore, do not share the difficult data management problems of h-methods. However, they suffer from several deficiencies. Without care in their implementation, moving mesh schemes can be unstable and can result in mesh tangling and local degradation of the solution. They can never reduce errors below an arbitrary limit, and these methods often fail when time-dependent boundary conditions are enforced, as they are incapable of handling the migration of regions containing irregularities or singularities in the solution as it evolves in time. Nevertheless, when combined with other types of adaptive strategies, these methods can provide a useful approach toward controlling solution error. c) Subspace Enrichment Methods (p- or spectral Method). The subspace enrichment methods (or spectral-type methods) generally employ a fixed mesh and a fixed number of grid cells and points. Most numerical methods for partial differential equations attempt to approximate the solution in a subclass of discrete functions or by functions in some finite-dimensional subspace of functions in which the actual solution belongs. Thus, subspace enrichment methods attempt to enrich this subclass of functions through the use of higher-order differences, spectral methods, by increasing the local polynomial degree in finite methods, etc. f the error in any cell exceeds a preassigned tolerance, the local order of the approximation is increased to re- 2
duce the error. These methods are very effective in modeling thin boundarylayers around bodies moving in a flowfield, where use of very fine meshes is costly and impractical. The problem of developing the data management scheme required to implement these types of adaptive methods, particularly in cases of complex geometries, is exceedingly difficult and while we have worked on this subject for four years, we have only treated it successfully in recent months. d) Combined Adaptive Methods. The best adaptive schemes for internal flow studies applications involve some combination of the h-methods, moving mesh methods, and subspace--enrichment methods discussed above. n recent months, a new data management scheme has been developed for implementing new spectral methods and combined h-spectral adaptivity on unstructured meshes, and has made some preliminary applications to the Navier-Stokes equations in two dimensions. These techniques are capable of delivering incredible accuracy: exponential convergence of solutions, by a carefully applied recipe of simultaneously refining the mesh and changing the spectral-order. Figures 2-4 show results obtained using our h-p method for the Carter plate problem: compressible viscous flow over a heated flat plate. Computed density contours are shown. The final mesh consisted of 1,102 bilinear elements, 42 biquadratic elements, and 161 bicubic elements and 2,831 degrees of freedom. Standard difference methods for this problem may require an order-of-magnitude more degrees of freedom for comparable results. For additional details, see [2]. Figure 5 shows a representative log-log plot of error versus problem size and emphasizes the fact that h-p techniques provide the best available way to get the most out of one's computational effort. Even more significant is the observation that these special adaptive techniques can produce numerical solutions to problems which are impossible to obtain by conventional finite difference or finite element techniques on the largest existing supercomputers! ndeed, to reproduce the accuracy obtainable by h-p methods on some model elliptic problems, a finite difference mesh consisting of over ten million grid points would be required. Calculation of Error ndicators There are several methods for calculating estimates of the solution error in discrete approximations of boundary- or initial-value problems. n developing error indicators it is always desirable if not theoretically necessary to ensure that the error indicator be bounded above and below by the actual error globally in some appropriate norm, i.e., one attempts to construct a number 8, called the global error indicator, which has the properties (.1) 3
(.2) where C and C 2 are constants independent of the mesh size parameter h, e is the actual approximation error, and. is some norm appropriate for the problem at hand. n (2), e is determined by a collection of local error indicators calculated over each cell K in the mesh, and generally 1 :::;p :::;00. Condition (2) ensures that rate of convergence of the global error indicator is precisely the same as that of the actual error and that by designing an algorithm that systematically reduces e we also reduce lell Except for simple one-dimensional cases, it is generally possible to derive error indicators that satisfy (1) only asymptotically, for sufficiently small h or large p. For example, for the model problem, -V avu + bu. = f in n u = Uo on an au aan = 9 on an2 (n C ll~an = an U an 2 ), one can prove under standard hypotheses that constants C and C 2, independent of h and p, exist such that with e = ~ lflklilk { } 1/2 and. the energy norm, ulw = L BK(u, u) K L: { (alvul 2 + bu. 2 )dx K JK :lulllk = { (avu. Vu + bu 2 )dx K n and CPK is a solution of the local problem (posed over each element K): (.3) 4
Here Bh is a "space of u bubble functions'~containing higher-order polynomials which vanish at the nodes of each element, Uh is the finite element approximations of U on a mesh with mesh size h and polynomial degree p, and rh is the re~idual rh = f + V'. av'uh - buh Thus, every factor on the right-hand-side of (3) is known and (3) represents a well-posed problem for the local error indicators «JK. We have developed and implemented a similar error~timation procedure for non-self-adjoint problems. Ordinarily, one seeks a more easily calculated error indicator than (3) to drive the adaptive process and reserves a scheme such as (3) for the end of a computation to obtain a more precise estimate of the error. For instance, one can show that [2] where hk = dia (K), 11'llo,K denotes the L2- norm over K, ~aauh/ank] is the jump in the computed flux over the boundary and C is a constant. The quantity <PK = c- PK, where PK is the quantity on the right side of the above inequality, represents an easily computed error indicator, which is generally sufficiently accurate to correctly direct the adaptive process through solutions of increasing quality. Space limitations do not permit a discussion of such schemes here, but more details can be found in companion papers [1]. Acknowledgement The support of this work by the Office of Naval Research under Contract N00014-84-K-0409-MOO-P00005 is gratefully acknowledged. References 1. Devloo, P., Oden, J. T., and Pattani, P., "An h-p Adaptive Finite Element Method for the Numerical Simulation of Compressible Flow," Computer MethotU in Applied MechanicJ and Engineering (to appear). 2. Oden, J. T., "Notes on Aposteriori Error Estimates for Finite Element Approximations of Boundary- and nitial-value Problems," TCOM R.ept., No. 88-03, Austin, 1988. 5
Figure 1. Rotor-stator flow interaction: Here one sees a computer-generated mesh around two rotor-blades (on the right) moving with respect to a stator blade (on the left) in a simulation of rotor-stator flow interaction in a turbine engine. The mesh is dynamically rermed. using at a given time only the number of cells needed to deliver a specified level of accuracy. Cells are removed dynamically if they are not needed. Figure (a) also contains an instantaneous plot of pressure contours. Figure (b) shows the solution at a later time; note an entirely new optimal mesh prevails. since the solution has changed. n these calculations. approximately one-third the number of unknowns for a conventional uniform mesh solution are used to obtain equivalent accuracies. 6
Figu~ 1. (b) 7
Supersonic Mach.3 nflow (Dirichlet) Supersonic nflow (Dirichlet) Re.1000 Pr.O.72 Y. 1.4 y outflow 0.75 0.1 L-1.0 Figure 2. Data and geometry for the Carter plate problem. 8
Figure 3. An h-p mesh for the Carter problem consisting of linear. quadratic and cubic elements in the boundary layer. 9
H- f- f- -f- - f- f-... Figure 4. Computed pressure contours. 10
h-method (uniform refinement) h-method (adaptive) LOG11Error11 an adaptive h-p method p-method (uniform refinement) LOG(No. of Unknowns) Figure 5. Plots of convergence rates. t is clear that for very high reesolution of complex flow features, traditional finite difference and finite element methods are grossly inadequate; the use of combined h-spectral methods seems to be the most promising approach for fitting very large problems on today's mainframes, particularly if high accuracy is nequired. 11