Agglomeration multigrid methods with implicit Runge–Kutta smoothers applied to aerodynamic simulations on unstructured grids
Introduction
For high Reynolds number viscous flows often a severe loss of reliability and efficiency of current solution methods for the compressible Reynolds averaged Navier–Stokes (RANS) equations can be observed. A major factor contributing to the loss in effectiveness of solution methods is that anisotropic meshes are generally used for the economic resolution of the steep gradients occurring in viscous boundary layers. Such meshes have high aspect ratio cells, resulting in a stiffness of the discrete system of governing flow equations. It is this numerical stiffness that creates difficulty in removing certain error modes when computing flow solutions. For analysis of the effects of such stiffness see Pierce and Giles [1].
Even the introduction of multigrid did not overcome the problems inherent to such flows as long as weak smoothers such as explicit Runge–Kutta methods were used. For 2D structured flows it was shown by Rossow, Swanson and Turkel [2], [3], [4] that a significant improvement in both reliability and efficiency can be reached when the explicit smoother is supplemented by an implicit one. In [3] the efficiency of the scheme was demonstrated for a 3D wing flow using the Baldwin–Lomax algebraic turbulence model [5]. In [6] a similar method was introduced into a block-structured code and the superiority of the method when compared with explicit or point-Jacobi preconditioned [7], [8] Runge–Kutta methods was also shown for 3D flows. However, in [6] convergence of the turbulent flow equation given by an SA model was not shown.
Industrial requirements such as automatism as well as the complexity of the considered flow problems in general allow only for a limited number of simulations on structured or block-structured meshes. A more realistic situation is that a computational mesh is generated with an arbitrary mesh generator and the flow solver must find efficiently an accurate solution on the given grid. No further mesh information, in particular no structured mesh information can be assumed to be available. For example, to ensure a robust multigrid algorithm with good damping properties requires the construction of suited coarse grid levels. Within a structured grid environment the construction of coarse grid levels is straightforward, whereas on unstructured grids suited agglomeration techniques are in general not easy to develop. The same is true for prolongation operators. For unstructured grids the development of reliably working interpolation operators is not an easy task, and therefore within this publication we only use pure injection. For structured grids linear and bilinear interpolation operators are straightforward to construct. Concluding, for unstructured mesh algorithms required information to perform reliably and efficiently a computation must be generated by the flow solver itself, since in general necessary grid information is missing.
It is the goal of this article to present an agglomerated FAS multigrid for unstructured meshes to approximate steady state solutions of the RANS equations. The smoother is derived by a general diagonally implicit Runge–Kutta method and based on a first order approximation of the derivative. It was shown [9] that this smoother may be viewed as prototype to derive many well known solution techniques in the CFD literature such as the ones mentioned above [2], [3], [4], [6], line implicit methods [10], [11], [12], [13], [14], point implicit [7], [1], [8] and even explicit Runge–Kutta methods [15].
Hence, the presented method is a generalization of the schemes suggested in [2], [3], [4] and it carries these methods over to unstructured finite volume methods. It should be mentioned that an appropriate set of adjustments to the components of the algorithm is required to achieve an overall robust and efficient solution method for unstructured meshes. Compared to point implicit [16], [17] and line implicit solution algorithms [11], [12], [13] the preconditioner is not restricted to point derivatives and the directions of the strongest coupling. Instead a full Jacobian based on a nearest neighbor discretization is used. The directions of strongest coupling are then integrated into the linear solution method. The use of this additional information in the Jacobian allows in general for much larger CFL numbers when compared with pure point and line implicit methods. Following [16], [17], [11], [12] the directions of strongest coupling are considered in the agglomeration strategy as well. This information is generated by a line search algorithm [12], [13], [14] requiring only a given threshold parameter. Exploiting this methodology is twofold. On the one hand, in our framework it is necessary to make the implicit Runge–Kutta methods applicable on agglomerated meshes (see Section 5). On the other hand, with a flexible algorithm no further mesh information is needed. A classification of surface points as suggested in [18] is not required; and note, such an identification of surface points is in general not available.
Turbulence is modeled using the Spalart–Allmaras one equation model [19], [20]. Since the characterization of the fully coupled discrete system changes, the turbulent flow equation is solved decoupled from the RANS equations. However, the same agglomerated multigrid algorithm is applied to the turbulent flow equation, and both residuals of the mean flow and turbulent flow equations will be reduced to machine accuracy.
The paper is organized as follows: In Section 1 we present the governing equations written in integral form as well as the discretization strategy for both mean and turbulent flow equations. Section 2 is dedicated to the derivation of the implicit Runge–Kutta method which will be used as smoother in the agglomerated FAS multigrid. Considered approximations to the derivative resulting in an efficient preconditioner are also given in this section as well as the iterative linear solution method for the arising linear systems. Details to the multigrid algorithm techniques are given in Section 3. We apply the suggested solution algorithm to several examples, and the results are given in Section 4. Several grids from different mesh generators will be used to show the wide range of applicability. Moreover, if available we investigated the algorithm on sequences of meshes to get an impression of reliability and scalability. In all the calculations presented the residual has been reduced to machine zero and compared to results shown in [16], [17], and in [11], [12] more complex flows are considered. To better understand properties of the developed algorithms theoretical results are presented in Section 5. The GMRES method is exploited to determine approximate spectral data which is applied to polynomial representing the multistage Runge–Kutta methods. A conclusion referring to recent and future work closes the article.
Section snippets
Governing equations and discretization
To describe flow effects we consider for an open domain the three-dimensional Reynolds averaged Navier–Stokes (RANS) equations in conservative variables written in integral form as where denotes the outer unit normal vector, and , and the equation of state
Diagonally implicit Runge–Kutta methods
To derive a solution method to solve the discretized flow equations (12a) and (12b) we consider an s-stage diagonally implicit Runge–Kutta method given by the Butcher scheme (Table 1)
where The stages of this scheme and the discrete evolution are given by Here we skipped the sub-indices “mean” and
Multigrid
An agglomeration multigrid algorithm [29] is used to approximate a steady state solution of the governing equations (1a) and (1b). A multistage preconditioned Runge–Kutta scheme (17) is used as a smoother within the multigrid algorithm. For a detailed description of the FAS multigrid scheme we refer to the textbook [30].
Numerical examples
In the following we will show for several examples the performance of Algorithm (17), where (22) is used as preconditioner. We start with two basic 2D airfoil geometries and then consider 3D examples of increasing complexity. As 2D test cases we have chosen
- (a)
CASE 9, RAE 2822: Re = 6.5e6, M = 0.73, AoA = 2.79° (see [36])
- (b)
MDA30P30N three element high lift airfoil: Re = 9.0e6, M = 0.2, AoA = 16.0° (see [37]).
Computer-aided analysis
Algorithm (17) is motivated by a diagonally implicit Runge–Kutta method. Since several simplifications in the Jacobian are introduced and the solutions of the linear systems (22) are only approximated by a few line Gauss–Seidel sweeps, we cannot expect that the method behaves unconditionally stable ( in general does not work). So, the introduction of the time step ΔT into the preconditioner is a necessary regularizing effect. As a second stabilizing ingredient we suggested a three stage
Conclusion
For an agglomerated FAS multigrid scheme we have presented an implicit smoothing method based on a first order approximation of the derivative of the residual function to approximately solve the RANS equations and the transport-type equation of the SA turbulence model in a loosely coupled manner. The algorithm is realized in a fully unstructured environment. Additional required information given by grid anisotropies identifying the directions of strong coupling is automatically determined by
Acknowledgement
Personal communication and helpful discussions with Dr. Charles Swanson are gratefully acknowledged.
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