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Unstructured Grid Adaptation: Status, Potential Impacts, and Recommended Investments Toward CFD Vision 2030

Unstructured grid adaptation is a powerful tool to control discretization error for Computational Fluid Dynamics (CFD). It has enabled key increases in the accuracy, automation, and capacity of some fluid simulation applications. Slotnick et al. provides a number of case studies in the CFD Vision 2030 Study: A Path to Revolutionary Computational Aerosciences to illustrate the current state of CFD capability and capacity. The authors forecast the potential impact of emerging High Performance Computing (HPC) environments forecast in the year 2030 and identify that mesh generation and adaptivity continue to be significant bottlenecks in the CFD work flow. These bottlenecks may persist because very little government investment has been targeted in these areas. To motivate investment, the impacts of improved grid adaptation technologies are identified. The CFD Vision 2030 Study roadmap and anticipated capabilities in complementary disciplines are quoted to provide context for the progress made in grid adaptation in the past fifteen years, current status, and a forecast for the next fifteen years with recommended investments. These investments are specific to mesh adaptation and impact other aspects of the CFD process. Finally, a strategy is identified to diffuse grid adaptation technology into production CFD work flows.

Park, Michael A.

Adaptive grid relocation algorithms for transonic full potential calculations using one-dimensional or two-dimensional diffusion equation

The effect of solution-adaptive grids using an interpolative grid relocation algorithm on two-dimensional transonic full potential flow calculations is studied with extensions to three dimensions in mind. One-dimensional or two-dimensional diffusion equation is solved to obtain the grid density control function for generating adaptive grids. The results of flow solution for NACA-0012 and Gates-Leajet airfoils show a significant improvement of accuracy in the CP distributions. However, further studies are recommended on the questions (1) what information from the initial flow solution should be passed on to the solution-adaptive grid generation scheme, and (2) how to optimize the accuracy of the solution through the use of adaptive grids.

Nakamura, S.

Grid adaption for hypersonic flow

The methods of grid adaption are reviewed and a method is developed with the capability of adaption to several flow variables. This method is based on a variational approach and is an algebraic method which does not require the solution of partial differential equations. Also the method has been formulated in such a way that there is no need for any matrix inversion. The method is used in conjunction with the calculation of hypersonic flow over a blunt nose body. The equations of motion are the compressible Navier-Stokes equations where all viscous terms are retained. They are solved by the MacCormack time-splitting method. A movie has been produced which shows simultaneously the transient behavior of the solution and the grid adaption.

Abolhassani, Jamshid S.

Transonic airfoil calculations using solution-adaptive grids

A new algorithm for generating solution-adaptive grids (SAG) about airfoil configurations embedded in transonic flow is presented. The present SAG approach uses only the airfoil surface solution to recluster grid points on the airfoil surface, i.e., the reclustering problem is one dimension smaller than the flow-field calculation problem. Special controls automatically built into the elliptic grid generation procedure are then used to obtain grids with suitable interior behavior. This concept of redistributing grid points greatly simplifies the idea of solution-adaptive grids. Numerical results indicate significant improvements in accuracy for SAG grids relative to standard grids using the same number of points.

Holst, T. L.

An adaptive grid algorithm for one-dimensional nonlinear equations

Richards' equation, which models the flow of liquid through unsaturated porous media, is highly nonlinear and difficult to solve. Step gradients in the field variables require the use of fine grids and small time step sizes. The numerical instabilities caused by the nonlinearities often require the use of iterative methods such as Picard or Newton interation. These difficulties result in large CPU requirements in solving Richards equation. With this in mind, adaptive and multigrid methods are investigated for use with nonlinear equations such as Richards' equation. Attention is focused on one-dimensional transient problems. To investigate the use of multigrid and adaptive grid methods, a series of problems are studied. First, a multigrid program is developed and used to solve an ordinary differential equation, demonstrating the efficiency with which low and high frequency errors are smoothed out. The multigrid algorithm and an adaptive grid algorithm is used to solve one-dimensional transient partial differential equations, such as the diffusive and convective-diffusion equations. The performance of these programs are compared to that of the Gauss-Seidel and tridiagonal methods. The adaptive and multigrid schemes outperformed the Gauss-Seidel algorithm, but were not as fast as the tridiagonal method. The adaptive grid scheme solved the problems slightly faster than the multigrid method. To solve nonlinear problems, Picard iterations are introduced into the adaptive grid and tridiagonal methods. Burgers' equation is used as a test problem for the two algorithms. Both methods obtain solutions of comparable accuracy for similar time increments. For the Burgers' equation, the adaptive grid method finds the solution approximately three times faster than the tridiagonal method. Finally, both schemes are used to solve the water content formulation of the Richards' equation. For this problem, the adaptive grid method obtains a more accurate solution in fewer work units and less computation time than required by the tridiagonal method. The performance of the adaptive grid method tends to degrade as the solution process proceeds in time, but still remains faster than the tridiagonal scheme.

Gutierrez, William E.

SAGE: The Self-Adaptive Grid Code

The multi-dimensional self-adaptive grid code, SAGE, is an important tool in the field of computational fluid dynamics (CFD). It provides an efficient method to improve the accuracy of flow solutions while simultaneously reducing computer processing time. Briefly, SAGE enhances an initial computational grid by redistributing the mesh points into more appropriate locations. The movement of these points is driven by an equal-error-distribution algorithm that utilizes the relationship between high flow gradients and excessive solution errors. The method also provides a balance between clustering points in the high gradient regions and maintaining the smoothness and continuity of the adapted grid, The latest version, Version 3, includes the ability to change the boundaries of a given grid to more efficiently enclose flow structures and provides alternative redistribution algorithms.

Davies, Carol B.

A three-dimensional adaptive grid method

A three-dimensional solution-adaptive-grid scheme is described which is suitable for complex fluid flows. This method, using tension and torsion spring analogies, was previously developed and successfully applied for two-dimensional flows. In the present work, a collection of three-dimensional flow fields are used to demonstrate the feasibility and versatility of this concept to include an added dimension. Flow fields considered include: (1) supersonic flow past an aerodynamic afterbody with a propulsive jet at incidence to the free stream, (2) supersonic flow past a blunt fin mounted on a solid wall, and (3) supersonic flow over a bump. In addition to generating three-dimensional solution-adapted grids, the method can also be used effectively as an initial grid generator. The utility of the method lies in: (1) optimum distribution of discrete grid points, (2) improvement of accuracy, (3) improved computational efficiency, (4) minimization of data base sizes, and (5) simplified three-dimensional grid generation.

Nakahashi, K.

Topology and grid adaption for high-speed flow computations

This study investigates the effects of grid topology and grid adaptation on numerical solutions of the Navier-Stokes equations. In the first part of this study, a general procedure is presented for computation of high-speed flow over complex three-dimensional configurations. The flow field is simulated on the surface of a Butler wing in a uniform stream. Results are presented for Mach number 3.5 and a Reynolds number of 2,000,000. The O-type and H-type grids have been used for this study, and the results are compared together and with other theoretical and experimental results. The results demonstrate that while the H-type grid is suitable for the leading and trailing edges, a more accurate solution can be obtained for the middle part of the wing with an O-type grid. In the second part of this study, methods of grid adaption are reviewed and a method is developed with the capability of adapting to several variables. This method is based on a variational approach and is an algebraic method. Also, the method has been formulated in such a way that there is no need for any matrix inversion. This method is used in conjunction with the calculation of hypersonic flow over a blunt-nose body. A movie has been produced which shows simultaneously the transient behavior of the solution and the grid adaption.

Abolhassani, Jamshid S.

The development and application of the self-adaptive grid code, SAGE

The multidimensional self-adaptive grid code, SAGE, has proven to be a flexible and useful tool in the solution of complex flow problems. Both 2- and 3-D examples given in this report show the code to be reliable and to substantially improve flowfield solutions. Since the adaptive procedure is a marching scheme the code is extremely fast and uses insignificant CPU time compared to the corresponding flow solver. The SAGE program is also machine and flow solver independent. Significant effort was made to simplify user interaction, though some parameters still need to be chosen with care. It is also difficult to tell when the adaption process has provided its best possible solution. This is particularly true if no experimental data are available or if there is a lack of theoretical understanding of the flow. Another difficulty occurs if local features are important but missing in the original grid; the adaption to this solution will not result in any improvement, and only grid refinement can result in an improved solution. These are complex issues that need to be explored within the context of each specific problem.

Davies, Carol B.

A simplified self-adaptive grid method, SAGE

The formulation of the Self-Adaptive Grid Evolution (SAGE) code, based on the work of Nakahashi and Deiwert, is described in the first section of this document. The second section is presented in the form of a user guide which explains the input and execution of the code, and provides many examples. Application of the SAGE code, by Ames Research Center and by others, in the solution of various flow problems has been an indication of the code's general utility and success. Although the basic formulation follows the method of Nakahashi and Deiwert, many modifications have been made to facilitate the use of the self-adaptive grid method for single, zonal, and multiple grids. Modifications to the methodology and the simplified input options make this current version a flexible and user-friendly code.

Davies, C.

Multiple-block grid adaption for an airplane geometry

Grid-adaption methods are developed with the capability of moving grid points in accordance with several variables for a three-dimensional multiple-block grid system. These methods are algebraic, and they are implemented for the computation of high-speed flow over an airplane configuration.

Abolhassani, Jamshid Samareh

Grid-Adapted FUN3D Computations for the Second High Lift Prediction Workshop

Contributions of the unstructured Reynolds-averaged Navier-Stokes code FUN3D to the 2nd AIAA CFD High Lift Prediction Workshop are described, and detailed comparisons are made with experimental data. Using workshop-supplied grids, results for the clean wing configuration are compared with results from the structured code CFL3D Using the same turbulence model, both codes compare reasonably well in terms of total forces and moments, and the maximum lift is similarly over-predicted for both codes compared to experiment. By including more representative geometry features such as slat and flap brackets and slat pressure tube bundles, FUN3D captures the general effects of the Reynolds number variation, but under-predicts maximum lift on workshop-supplied grids in comparison with the experimental data, due to excessive separation. However, when output-based, off-body grid adaptation in FUN3D is employed, results improve considerably. In particular, when the geometry includes both brackets and the pressure tube bundles, grid adaptation results in a more accurate prediction of lift near stall in comparison with the wind-tunnel data. Furthermore, a rotation-corrected turbulence model shows improved pressure predictions on the outboard span when using adapted grids.

Lee-Rausch, E. M.

Three-dimensional self-adaptive grid method for complex flows

A self-adaptive grid procedure for efficient computation of three-dimensional complex flow fields is described. The method is based on variational principles to minimize the energy of a spring system analogy which redistributes the grid points. Grid control parameters are determined by specifying maximum and minimum grid spacing. Multidirectional adaptation is achieved by splitting the procedure into a sequence of successive applications of a unidirectional adaptation. One-sided, two-directional constraints for orthogonality and smoothness are used to enhance the efficiency of the method. Feasibility of the scheme is demonstrated by application to a multinozzle, afterbody, plume flow field. Application of the algorithm for initial grid generation is illustrated by constructing a three-dimensional grid about a bump-like geometry.

Djomehri, M. Jahed

Three-dimensional self-adaptive grid method for complex flows

A self-adaptive grid procedure for efficient computation of three-dimensional complex flow fields is described. The method is based on variational principles to minimize the energy of a spring system analogy which redistributes the grid points. Grid control parameters are determined by specifying maximum and minimum grid spacing. Multidirectional adaptation is achieved by splitting the procedure into a sequence of successive applications of a unidirectional adaptation. One-sided, two-directional constraints for orthogonality and smoothness are used to enhance the efficiency of the method. Feasibility of the scheme is demonstrated by application to a multinozzle, afterbody, plume flow field. Application of the algorithm for initial grid generation is illustrated by constructing a three-dimensional grid about a bump-like geometry.

Djomehri, M. Jahed

A solution adaptive grid procedure for an upwind parabolized flow solver

A solution adaptive grid procedure based on an error equi-distribution scheme is developed and applied to a Parabolized Navier-Stokes solver. An improved method for selecting weighting functions is introduced which involves normalizing a combination of flowfield gradients and curvature of a number of dependent variables and then selecting the largest at each point. The scheme re-distributes grid points line-by-line, with grid point motion controlled by forces analogous to tensional and torsional spring forces with the spring constants set equal to the weighting functions. Torsional terms are functions of the grid point positions along neighboring grid lines and provide grid smoothness and stability. A grid-fitting scheme is introduced for external flows in which the number of grid points in the freestream are reduced to a minimum. Results for several problems are presented to demonstrate the improvements obtainable with the solution adaptive grid procedure.

Harvey, Albert D.

The multidimensional self-adaptive grid code, SAGE

This report describes the multidimensional self-adaptive grid code SAGE. A two-dimensional version of this code was described in an earlier report by the authors. The formulation of the multidimensional version is described in the first section of this document. The second section is presented in the form of a user guide that explains the input and execution of the code and provides many examples. Successful application of the SAGE code in both two and three dimensions for the solution of various flow problems has proven the code to be robust, portable, and simple to use. Although the basic formulation follows the method of Nakahashi and Deiwert, many modifications have been made to facilitate the use of the self-adaptive grid method for complex grid structures. Modifications to the method and the simplified input options make this a flexible and user-friendly code. The new SAGE code can accommodate both two-dimensional and three-dimensional flow problems.

Davies, Carol B.

Adaptive grid methods for RLV environment assessment and nozzle analysis

Rapid access to highly accurate data about complex configurations is needed for multi-disciplinary optimization and design. In order to efficiently meet these requirements a closer coupling between the analysis algorithms and the discretization process is needed. In some cases, such as free surface, temporally varying geometries, and fluid structure interaction, the need is unavoidable. In other cases the need is to rapidly generate and modify high quality grids. Techniques such as unstructured and/or solution-adaptive methods can be used to speed the grid generation process and to automatically cluster mesh points in regions of interest. Global features of the flow can be significantly affected by isolated regions of inadequately resolved flow. These regions may not exhibit high gradients and can be difficult to detect. Thus excessive resolution in certain regions does not necessarily increase the accuracy of the overall solution. Several approaches have been employed for both structured and unstructured grid adaption. The most widely used involve grid point redistribution, local grid point enrichment/derefinement or local modification of the actual flow solver. However, the success of any one of these methods ultimately depends on the feature detection algorithm used to determine solution domain regions which require a fine mesh for their accurate representation. Typically, weight functions are constructed to mimic the local truncation error and may require substantial user input. Most problems of engineering interest involve multi-block grids and widely disparate length scales. Hence, it is desirable that the adaptive grid feature detection algorithm be developed to recognize flow structures of different type as well as differing intensity, and adequately address scaling and normalization across blocks. These weight functions can then be used to construct blending functions for algebraic redistribution, interpolation functions for unstructured grid generation, forcing functions to attract/repel points in an elliptic system, or to trigger local refinement, based upon application of an equidistribution principle. The popularity of solution-adaptive techniques is growing in tandem with unstructured methods. The difficultly of precisely controlling mesh densities and orientations with current unstructured grid generation systems has driven the use of solution-adaptive meshing. Use of derivatives of density or pressure are widely used for construction of such weight functions, and have been proven very successful for inviscid flows with shocks. However, less success has been realized for flowfields with viscous layers, vortices or shocks of disparate strength. It is difficult to maintain the appropriate mesh point spacing in the various regions which require a fine spacing for adequate resolution. Mesh points often migrate from important regions due to refinement of dominant features. An example of this is the well know tendency of adaptive methods to increase the resolution of shocks in the flowfield around airfoils, but in the incorrect location due to inadequate resolution of the stagnation region. This problem has been the motivation for this research.

Thornburg, Hugh J.

A time-accurate adaptive grid method and the numerical simulation of a shock-vortex interaction

A time accurate, general purpose, adaptive grid method is developed that is suitable for multidimensional steady and unsteady numerical simulations. The grid point movement is performed in a manner that generates smooth grids which resolve the severe solution gradients and the sharp transitions in the solution gradients. The temporal coupling of the adaptive grid and the PDE solver is performed with a grid prediction correction method that is simple to implement and ensures the time accuracy of the grid. Time accurate solutions of the 2-D Euler equations for an unsteady shock vortex interaction demonstrate the ability of the adaptive method to accurately adapt the grid to multiple solution features.

Bockelie, Michael J.