Grid Generation Through Differential Systems

Liao G.G., Cai X., Fleitas D., Luo X., Wang J., Xue J.

A mathematical problem arising from data set alignment is studied. For two given real-valued functions defined on two (2D or 3D) domains, separately, find a transformation between the two domains that optimizes a similarity functional between the functions. Our approach is based on optimal control with PDE (partial differential equation) constraints. Numerical examples are provided to demonstrate the effectiveness of the method.

Greenberg J.B.

Hindman R.G., Kutler P., Anderson D.

A new technique is described for solving supersonic fluid dynamics problems containing multiple regions of continuous flow, each bounded by a permeable or impermeable surface. Region boundaries are, in general, arbitrarily shaped and time dependent. Discretization of such a region for solution by conventional finite difference procedures is accomplished using an elliptic solver which alleviates the dependence on a particular base coordinate system. Multiple regions are coupled together through the boundary conditions. The technique has been applied to a variety of problems including a shock diffraction problem and supersonic flow over a pointed ogive.

Hodge J.K., Leone S.A., McCarty R.L.

Generation de grilles curvilignes generalisees tridimensionnelles pour des corps cartesien, cylindriques et spheriques a l'aide d'equations aux derivees partielles paraboliques. Les equations de grilles parabolisees sont resolues de maniere non iterative en avancant dans deux directions

Visbal M., Knight D.

A NUMERICAL procedure is presented for the generation of boundary-fitted curvilinear coordinates with controllable mesh spacing and orthogonality. The technique is based on a straightforward two-step procedure involving the solution of Poisson's equation. The method is capable of generating orthogonal grids with partial control of the mesh spacing, or nearly orthogonal grids with strict control of the mesh spacing. The inhomogeneous terms in the Poisson equations are automatically adjusted during the grid generation. The technique is applicable to two-dimensional simply-connected regions typical of airfoils, cascades, diffusers, and inlets.

Rai M.M., Anderson D.A.

Coordinate system selection is an important consideration in the asymptotic numerical solution of any fluid-flow or heat transfer problem. This paper uses a new technique that provides a simple way of moving the mesh points in physical space in order to reduce the error in the computed asymptotic solution relative to that obtained using a fixed mesh. Applications to fluid-flow problems are presented, including boundary layer flow and inviscid supersonic flow over cylinders, and wedges with associated detached shocks. The treatment of curved boundaries, stationary and nonstationary boundaries, and systems of PDE's is discussed. Significant error reductions are demonstrated.

Dwyer H.A., Kee R.J., Sanders B.R.

A new method for generating adaptive grids for time-dependent and steady problems in multidimensional fluid mechanics and heat transfer has been developed. The method can be used with many existing grid generation schemes or can be used as an independent grid generation technique. The present adaptive method is based upon the placement of grid points in proportion to the gradients that appear in the dependent variable. The multidimensional results presented in the paper are for the unsteady heat conduction equation and have included steep gradients due to geometry and unsteady boundary conditions. The method has performed in an impressive fashion, although there is a need to control grid skewness better. A study of one-dimensional problems associated with combustion and cell Reynolds number has demonstrated the technique's accuracy and versitility. The paper also discusses the relationship of the method to other grid generation techniques, as well as extensions of the new method.

Thomas P.D., Middlecoff J.F.

The generation of computational grids suitable for obtaining accurate numerical solutions to the three-dimensional Navier-Stokes equations is the subject of intensive research. For a wide class of nozzle configurations, a three-dimensional grid can be constructed by a sequence of two-dimensional grids in successive cross-sectional planes. The present paper is concerned with numerical generation of two-dimensional grids. An effective method of interior grid control is presented based on a modified elliptic system containing free parameters. For a simply connected region, the free parameters are computed from the Dirichlet boundary values. The resulting interior grid point distribution is controlled entirely by a priori selection of the grid point distribution along the boundaries of the section.

Noack R.W., Anderson D.A.

A solution-adaptive parabolic grid generation scheme has been developed. A new upwind finite-volume formulation for space marching solution of the steady Euler equations is also presented. In addition, a solution- adaptive elliptic grid scheme developed previously and applied to unsteady flow solutions is applied in the present steady-flow context. Solutions for hypersonic flow are computed with the present finite-volume formulation coupled to both the parabolic and elliptic adaptive grid schemes. The results show that the solution-adaptive parabolic scheme produces grids that track the flow features of interest as well as the elliptic grid procedure. This is significant because the parabolic procedure is much faster than an elliptic scheme producing a grid in an order of magnitude of less computational time

Jeng Y.N., Shu Y.

Sparis P.D., Karkanis A.

A finite difference approximation of the biharmonic equation is solved using three preconditioned gradient methods for the generation of curvilinear boundary-orthogonal grids in two dimensions. The developed algorithms based on the conjugated gradient, the steepest descent, and the minimal residual method are applied in a number of domains of engineering interest for the purpose of comparison in terms of CPU time and number of iterations to convergence.

Hall D.J., Zingg D.W.

A quantitative study of the numerical error reduction due to adaptive grid redistribution applied to Navier-Stokes computations of steady airfoil flows is presented. A grid redistribution procedure is described and applied to a range of flow cases including attached and separated flows at subsonic and transonic speeds. The error reduction resulting from grid adaptation is sensitive to the adaptive grid generation parameters used, especially those affecting the smoothness of the adapted grid. A suitable set of adaptive grid generation parameters is determined by comparison of adaptive grid solutions with grid-independent solutions. With the given parameters, adaptive grid redistribution is shown to result in an effective reduction in numerical error and improved resolution of flow features.

Slater J.W., Liou M.S., Hindman R.G.

Tai C.H., Chiang D.C., Su Y.P.

Based on upwind differencing of the governing equations, inherent dissipation terms are derived for three-dimensional hyperbolic grid generation. These terms, taking only a negligible amount of mathematic operations, can effectively eliminate grid oscillation that is often encountered in hyperbolic grid generation. In addition to these dissipation terms, a Laplacian-type smoothing is incorporated to increase smoothness. Both implicit and explicit schemes for three-dimensional hyperbolic grid generation are investigated, and a comparison of these two schemes is made. Several different geometries are used to demonstrate the robustness of the new methods. Finally, a wing-body configuration, with severe concave and convex corners, is used to evaluate the versatility of the present methods.

Tu Y., Thompson J.F.

The EAGLE 3D composite-block grid code has been coupled with an implicit Euler flow solver to generate solution-adaptive grids for composite-block configurations with an improved control function formulation. The code has been tested with two complex composite configurations, one of which, an eight-block finned body of revolution at transonic speeds, is described here. The solution-adaptive grids obtained possess continuous slopes across block boundaries and an improved quality of aerodynamics simulation about complex geometries