Adaptive Grid Generation Research Articles

Iterative schemes and algorithms for adaptive grid generation in one dimension

Some iterative adaptive grid generators, developed by the author, are numerically explored in detail to assess their relative merits against conventional grid generators, based on a direct method of integration and interpolation. We find that some of these iterative adaptive grid generators are preferable to a direct method of integration and interpolation. In contrast with a direct method, appropriate use of these iterative adaptive grid generators produces adaptive grids with a smooth variation of the grid spacing ratio and resolution. All adaptive grid generators are a subclass of a more general iterative map. General features of this iterated map which are related to the function to be resolved are briefly discussed. The results obtained here supplements recent investigations on these adaptive grid generators by the author.

A new adaptive grid generation by elliptic equations with orthogonality at all of the boundaries

A new adaptive grid generation procedure is proposed, which combines a previously developed method that used Anderson's adaptive grid scheme along with a method for controlling grid spacing and orthogonality on all of the boundaries. The proposed method assigns the desired grid stretching over the smooth region during initial grid system generation and before grid adaptation is performed. After properly interpreting the smoothing term of the weighting function, the desired grid stretching is added to the adaptive grid scheme. Several test cases illustrate the method's feasibility.

A New Theory for One-Dimensional Adaptive Grid Generation and Its Applications

The theory of a new approach to adaptive grid generation in one dimension is developed. The approach is based on approximating either the resolution or the grid spacing ratio on discrete lattice points by continuous variables. The order of accuracy of these approximations in a suitable reference frame characterizes the various methods. Approximations that are first- or second-order accurate in a suitable reference coordinate are derived in this paper. The free parameters associated with these methods provide flexibility in generating a large family of adaptive grids with smooth grid spacing ratio and high resolution. A selected group of this family of adaptive grids may prove very useful in adaptive computations of partial differential equations. The adaptive grids are numerically generated using these approximations. Numerical examples are given that exemplify the usefulness of these adaptive grids.

Adaptive grid generation for the finite element method

Using adaptive FEM it is possible to refine a rudimentary grid iterative with the intention of reaching a favourable relation of accuracy to numerical effort. There is a great number of error indicators or refinement indicators, which serve to decide about the refinement of the meshes. Three of these were tested.

Adaptive grid generation from harmonic maps on Riemannian manifolds

In this paper we describe a new method for generation of solution adaptive grids based on harmonic maps on Riemannian manifolds. The reliability of the method is assured by an existence and uniqueness theorem for one-to-one maps between multidimensional multiconnected domains. We formulate an adaptive Riemannian metric consistent with this theorem. Several examples demonstrating application of the developed procedure are provided.

Hybrid adaptive poisson grid generation and grid smoothness

AbstractA hybrid technique for adaptive grid generation using a variety of Poisson grid generators is described. The technique ensures that, when adaptivity functions are weak, the adapted grid reverts to an arbitrary specified base grid rather than to the grid produced by the homogeneous elliptic generator. The hybrid technique also serves to expose the weakness of the claim that elliptic grid generators produce smooth grids.

An adaptive direct variational grid generation method

Variational grid generation techniques are now used to produce grids suitable for solving numerical partial differential equations in irregular geometries. Variational grid generation methods are very robust but slow. The method considered here is a discrete variational method. This method preserves the robustness of the variational method; also, it is very fast so it can be used in time-dependent problems. Here we discuss geometry and solution adaption for the direct discrete grid generation method, and some examples are presented to demonstrate its capabilities.

Using the grid spacing ratio as a continuous variable in one dimensional adaptive grid generation

A new method for one dimensional adaptive grid generation is introduced based on defining the grid spacing ratio as a continuous variable. In this paper we validate our theoretical results in order to justify their use in numerical construction of adaptive grids in one dimension.

Adaptive grid generation for VSLI device simulation

A grid refinement methodology suitable for arbitrary spatial dimensions and geometries is developed. Using just the structural description of a device-specified by the user or taken directly from the output of a process simulator-strategies for creating an initial grid are proposed, embodying whatever knowledge the simulator can have about how the solution will behave. After solving the PDEs, a grid is adapted by examining the local error involved in the discretization process. A set of procedures is proposed for this purpose and shown to be nearly optimal in terms of cost and efficiency. The key to implementing a robust and efficient scheme lies in the choice of a suitable error indicator; it is shown that some of the more obvious candidates can perform quite poorly. The initial grid construction and error indicator schemes are used to show how the full adaptation procedure works on some practical examples.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Adaptive grid generation from harmonic maps on Riemannian manifolds

Adaptive grid generation from harmonic maps on Riemannian manifolds

Three-dimensional adaptive grid generation on a composite-block grid

The EAGLE three-dimensional composite-block grid code has been extended to an adaptive grid to be coupled with a PDE solver. Both the adaptive control function formulation and the variational formulation were evaluated, and the former was found to be much faster and somewhat more effective. Results for the code coupled with an implicit Euler solver for a three-block configuration on a finite wing are compared with experimental data for transonic flow

Grid cell volume control with an adaptive grid generator

A technique for generating solution adaptive grids controlling cell volume is presented. The grid control functions of the popular Poisson grid generators are defined to control cell size. An analytic derivation showing the relationship of the grid control functions and the cell volume is presented, and several examples are given illustrating the idea. Only one function must be prescribed to control cell size when this technique is implemented in a Poisson grid generator.

Application of variational methods in the fixed and adaptive grid generation

The present paper is devoted to the development of a system of partial differential equations for the purpose of generating fixed and adaptive numerical grids. The resulting equations are applicable to both two- and three-dimensional grid generation problems. Equations have also been presented for both the fixed and adaptive grids in curved surfaces.

Adaptive grid generation

A review of adaptive grid generation is presented with an emphasis on the basic concepts and the interrelationship between the various methods. The concepts are developed in a multifaceted progressive sense with enough detail so as to instill an operative spirit for the methods. The operational capabilities come from an explicit display of the necessary formulas for algorithmic construction. While virtually all adaptive procedures are aimed at problems with rapid solution variations, our main concern is the construction of methods that are not fundamentally restricted by the choice of problem or solution algorithm. Moreover, to maintain a simple treatment for the computational data and to have access to many of the best solution algorithms, we consider coordinate transformations. As a consequence, particular attention is given to grid point motion that occurs in response to the influence from the solution data, regardless of how that data was obtained. After some introductory discussion on the utilization of solution data, the topic of grid point motion is addressed first in one dimension and then in higher dimensions. The basic equidistribution process is first seen from a dozen different viewpoints in one dimension. This is further amplified with the practical notions of precise coefficient specification, the attraction to a given grid, and the action of evolutionary forces. With the definition of the metric, the direct extension into higher dimensions is developed with curve-by-curve methods. This is followed by finite volume methods and variational methods. With the various movement strategies established in a multidimensional context, the next consideration is the temporal coupling of the movement with the solution algorithm. This is undertaken in the discussion of temporal aspects.

Finite elements in CFD: What lies ahead

AbstractThe current state of the art of Finite Element Methods in Computational Fluid Dynamics is reviewed. The aim of this review is to point out what appear currently as the main shortcomings of Finite Element Methods, so as to concentrate the efforts to remove them. The analysis of flows using Finite Elements will only be successful if all steps involved in it are optimized. Therefore, I deem it necessary to describe explicit and implicit flow solvers, unstructured multigrid methods, adaptive refinement schemes, grid generation and graphics in 3‐D, the effective use of supercomputer hardware and memory, as well as the combination of structured and unstructured grids.

A self‐adaptive gridding for inviscid transonic projectile aerodynamics computation

AbstractAn adaptive grid generation technique based on modified variational principles coupled with an exponential clustering has been developed and tested successfully for the computation of steady inviscid transonic projectile aerodynamics. The isoperimetric problem for adaptive gridding is to extremize a grid smoothness functional subject to grid orthogonality and resolution functionals; however, the Lagrange multipliers have been assumed to be variables with zero variation and are properly chosen as functions of local grid size to enhance locally the grid resolution as well as to maintain the weight of three grid characteristics the same over the entire flow field. With computed pressure gradient as the control function for grid adaptation, the resulting Euler equations cannot provide sufficient grid resolution in the boundary layer region of the projectile geometry; hence, a clustering technique is needed to redistribute the points along the normal grid lines. A grid generation code has been developed and coupled to an axisymmetric thin‐layer Navier‐Stokes code for self‐adaptive grid generation. For the three transonic flow cases considered, M∞ = 0.91, 0.96 and 1.10, the distribution of surface pressure calculated from the inviscid option of the Navier‐Stokes code is indeed in excellent agreement with published measured data.

Generation of solution-adaptive computational grids using optimization

A new method for generating solution-adaptive computational grids is presented that builds on the requirement that the adapted grid retains maximum possible smoothness and local orthogonality. The approach taken is one of nonlinear optimization, where an objective function combining measures of grid smoothness, local orthogonality, and cell volume control is minimized using a fast iterative scheme. The method is multidimensional by construction, and accepts any arbitrary grid as input, even a grid that is initially overlapped. Several applications to published test problems allow comparisons with some of the existing adaptive grid generation methods. An example of dynamic adaptation to the solution of finite difference equations is also given that demonstrates the error reduction capabilities of the new adaptive grid generation and optimization method, and suggests further applications to practical engineering problems.

Boundary conformed co‐ordinate systems for selected two‐dimensional fluid flow problems. Part I: Generation of BFGs

AbstractIn this paper the generation of general curvilinear co‐ordinate systems for use in selected two‐dimensional fluid flow problems is presented. The curvilinear co‐ordinate systems are obtained from the numerical solution of a system of Poisson equations. The computational grids obtained by this technique allow for curved grid lines such that the boundary of the solution domain coincides with a grid line. Hence, these meshes are called boundary fitted grids (BFG). The physical solution area is mapped onto a set of connected rectangles in the transformed (computational) plane which form a composite mesh. All numerical calculations are performed in the transformed plane. Since the computational domain is a rectangle and a uniform grid with mesh spacings Δξ = Δη = 1 (in two‐dimensions) is used, the computer programming is substantially facilitated. By means of control functions, which form the r.h.s. of the Poisson equations, the clustering of grid lines or grid points is governed. This allows a very fine resolution at certain specified locations and includes adaptive grid generation. The first two sections outline the general features of BFGs, and in section 3 the general transformation rules along with the necessary concepts of differential geometry are given. In section 4 the transformed grid generation equations are derived and control functions are specified. Expressions for grid adaptation arc also presented. Section 5 briefly discusses the numerical solution of the transformed grid generation equations using sucessive overrelaxation and shows a sample calculation where the FAS (full approximation scheme) multigrid technique was employed. In the companion paper (Part II), the application of the BFG method to selected fluid flow problems is addressed.

Application of a single-equation MG-FAS solver to elliptic grid generation equations (subgrid and supergrid coefficient generation)

A single-equation MG-FAS solver is applied to elliptic grid generation equations obtained from variational principle and to problems of fluid dynamics and electromagnetics on these grids. The problem of automatic generation of subgrid coefficients, given the user defined discretized problem on a defining grid, is addressed in a novel and clear way which has some advantages. The same idea is applied to supergrid generation of the elliptic grid generation equations, which can result in substantial arithmetic savings for solution adaptive grid generation in 3D.

Adaptive Grid Generation by Mean Value Relaxation

A grid movement algorithm has been developed for the purpose of adaptively resolving numerical solutions to physical problems and, in addition, for grid clustering on arbitrary surfaces. Both the solutions and the arbitrary surfaces are represented by grid point data with a continuous definition provided by interpolation between points. Movement is applied relative to this representation. The algorithm comes from a local mean value construction to produce a finite difference molecule for movement. The mean value weights are of a general enough nature to provide for a generous number of clustering possibilities. The movement molecule is executed within an interative cycle in the spirit of point Jacobi or Gauss-Seidel, and as a consequence, corresponds to the solution of some elliptic partial differential equation which satisfies a maximum (minimum) principle due to the mean value construction. From this principle, the movement will always preserve nonsingularity for the continuous transformation. For the discrete representation in the form of a grid, local geometric constraints are established to maintain this preservation.