Grid Generation Problem Research Articles

Computational morphogenesis of free‐form grid structures with Voronoi diagram

AbstractFree‐form grid structures have developed increasingly with the development of computer‐aided design and advanced construction technology. However, the grid generation is still a difficulty for designing free‐form grid structures. The computational morphogenesis of free‐form grid structures with the Voronoi diagram is proposed to solve this difficulty. Voronoi diagram is introduced to form the irregular grid of structures. Morphologies of plane and curved surface Voronoi grid structure are formulated to realize parametric modeling of the structural grid. A comprehensive structural index is proposed for evaluating free‐form grid structures quantitatively from the aspects of both architectural and structural behaviors. Taking the seeds as optimization variables and the comprehensive structural index as optimization objective, the grid generation problem is transformed into an optimization problem. Numerical examples show that the proposed morphogenesis is feasible and effective for the grid generation of free‐form structures. The free‐form grid structures generated by the proposed method have not only appealing geometry but also good mechanical behavior. The consideration of the symmetry can accelerate the grid generation process. Different grid distributions can be realized with different supports.

On Indecomposable Polyhedra and the Number of Steiner Points

The existence of indecomposable polyhedra, that is, the interior of every such polyhedron cannot be decomposed into a set of tetrahedra whose vertices are all of the given polyhedron, is well-known. However, the geometry and combinatorial structure of such polyhedra are much less studied. In this article, we investigate the structure of some well-known examples, the so-called Schönhardt polyhedron [10] and the Bagemihl's generalization of it [1], which will be called Bagemihl's polyhedra. We provide a construction of an additional point, so-called Steiner point, which can be used to decompose the Schönhardt and the Bagemihl's polyhedra. We then provide a construction of a larger class of three-dimensional indecomposable polyhedra which often appear in grid generation problems. We show that such polyhedra have the same combinatorial structure as the Schönhardt's and Bagemihl's polyhedra, but they may need more than one Steiner point to be decomposed. Given such a polyhedron with n ≥ 6 vertices, we show that it can be decomposed by adding at most ▪ interior Steiner points. We also show that this number is optimal in theworst case.

Parallel stochastic methods for PDE based grid generation

The efficient generation of meshes is an important step in the numerical solution of various problems in physics and engineering. We are interested in situations where global mesh quality and tight coupling to the physical solution is important. We consider elliptic PDE based mesh generation and present a method for the construction of adaptive meshes in two spatial dimensions using domain decomposition that is suitable for an implementation on parallel computing architectures. The method uses the stochastic representation of the exact solution of a linear mesh generator of Winslow type to find the points of the adaptive mesh along the subdomain interfaces. The meshes over the single subdomains can then be obtained completely independently of each other using the probabilistically computed solutions along the interfaces as boundary conditions for the linear mesh generator. Further to the previously acknowledged performance characteristics, we demonstrate how the stochastic domain decomposition approach is particularly suited to the problem of grid generation — generating quality meshes efficiently. In addition we show further improvements are possible using interpolation of the subdomain interfaces and smoothing of mesh candidates. An optimal placement strategy is introduced to automatically choose the number and placement of points along the interface using the mesh density function. Various examples of meshes constructed using this stochastic–deterministic domain decomposition technique are shown and compared to the respective single domain solutions using a representative mesh quality measure. A brief performance study is included to show the viability of the stochastic domain decomposition approach and to illustrate the effect of algorithmic choices on the solver’s efficiency.

Orthogonal grid generation of an irregular region using a local polynomial collocation method

In this study, a 2-D orthogonal grid generation model is developed by solving the governing equations of coordinate transformation with a local polynomial collocation method accompanied with the moving least squares (MLS) approach. This method was developed in a way that on the boundaries both the governing equation and boundary condition are satisfied, so it is more robust and accurate than conventional collocation methods. Though the method used to solve the coordinate transforming equations is meshless, it does not deteriorate the value of present work, because most numerical models in modern use are grid-dependent, and grid generation of service to these models is still strongly desired, particularly for finite difference models in irregular domains. Before applying to grid generation problems, the performance of present method is tested by a bench mark potential flow problem. Additional to two basic grid generation problems, a bottleneck problem of previous works, which contains zero-degree corners in the domain, is carried out. Finally, the model is applied to the orthogonal grid generation in a multi-connected domain. The correctness is testified by checking the orthogonality of the generated results.

Aerodynamic shape optimization of hovering rotor blades using a coupled free wake–CFD and adjoint method

This paper concerns a numerical optimization method for designing helicopter rotor blades in hover based on adjoint method and by using a coupled free wake–CFD approach for proper calculation of aerodynamic loads. The Euler flow solver and adjoint equations are formulated in a rotating coordinate system. The rotor wake effects are modeled by a free wake method and included into the CFD calculation. The approach is applied for the redesign of a transonic Caradonna–Tung two-bladed rotor through minor surface changes. The inviscid torque was reduced about 15% from that of original rotor due the reduction of wave drag. Results show that this technique is capable of converging to the optimum aerodynamic shape of rotor blades with minimum inviscid rotor torque coefficient under fixed thrust constraint. The advantage of this method is that satisfactory results can be achieved by minimum computational efforts, particularly for the grid generation problem and the required CPU time.

On 2D structured mesh generation by using mappings

A grid generation problem in two-dimensional domains is considered by using a quasi-conformal mapping of the parametric domain with a given square mesh onto the physical domain where the grid is required. To this end, a harmonic mapping is first applied, which, by the Radó theorem, is a diffeomorphism subject to some known conditions. However, the discrete harmonic mapping may produce folded meshes on a nonconvex domain with a strongly bent boundary. We demonstrate that it is caused by the truncation error. With the aim of controlling grid node location, an additional mapping is used. The Dirichlet problem for the universal elliptic partial differential equations is solved to construct the mapping. This allows to produce unfolded grids with a prescribed cell shape. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1072–1091, 2011

Smoothness and Convex Area Functionals—Revisited

In this paper, we introduce two new functionals within the context of the variational grid generation problem: an area functional and a smoothness functional. Both of them are based on an improved adaptive algorithm which focuses on the most folded grid cells. It is shown that when they are minimized in order to generate smooth and convex grids on irregular planar regions the optimization process converges notably faster than it does with other functionals previously reported.

Generating elliptic grids in three dimensions by a method of false transients

A finite difference method based scheme incorporating a method of false transients and an approximate factorisation technique is presented for solution of a system of Poisson's equations used for grid generation. A time step cycling process with repeated endpoints is incorporated into the scheme to enhance the convergence rate. The scheme required much less computational effort than all other numerical schemes compared in this article, to obtain a high quality grid over a body (converged solution) in three dimensions. Although, the superiority of the scheme has been demonstrated for a grid generation problem, it may be applied for other problems requiring the numerical solution of a set of similar partial differential equations. References Ly, E., and Gear, J. A., Time-Linearized Transonic Computations Including Shock Wave Motion Effects, Journal of Aircraft , 39 , 6, Nov-Dec 2002, pp. 964--972. Ly, E., and Nakamichi, J., Time-Linearised Transonic Computations Including Entropy, Vorticity and Shock Wave Motion Effects, The Aeronautical Journal , Nov. 2003, pp. 687--695. Ly, E., and Norrison, D., Automatic Elliptic Grid Generation by an Approximate Factorisation Algorithm, ANZIAM Journal , 48 (CTAC2006), pp. C188--C202, July 2007. Mathur, J. S., and Chakrabartty, S. K., An Approximate Factorization Scheme for Elliptic Grid Generation with Control Functions, Numerical Methods for Partial Differential Equations , 10 , 6, 1994, pp. 703--713. Thompson, J. F., Thames, F. C., and Mastin, C. W., Boundary-Fitted Curvilinear Coordinate Systems for Solution of Partial Differential Equations on Fields Containing any Number of Arbitrary Two-Dimensional Bodies , NASA Contractor Report CR-2729, Washington DC, USA, July 1977, 253 pages. Warming, R. F., and Beam, R. M., On the Construction and Application of Implicit Factored Schemes for Conservation Laws, in SIAM-AMS Proceedings , 11 , USA, 1978, pp. 85--129. Catherall, D., Optimum Approximate-Factorization Schemes for Two-Dimensional Steady Potential Flows, AIAA Journal , 20 , 8, 1982, pp. 1057--1063. Gear, J. A., Time Marching Approximate Factorization Algorithm for the Modified Transonic Small Disturbance Equation, Research Report Number 6, Department of Mathematics, RMIT University, Melbourne, Australia, Feb. 1996, 11 pages. Hoffmann, K. A., Computational Fluid Dynamics for Engineers , Engineering Educational System, Texas, USA, 1989. Ly, E., Improved Approximate Factorisation Algorithm for the Steady Subsonic and Transonic Flow over an Aircraft Wing, in Proceedings of the 21st Congress of the International Council of the Aeronautical Sciences (ICAS98) , AIAA and ICAS, Melbourne, Australia, Sep. 1998, Paper A98-31699.

2차원 지면 효과에 대한 난류 유동장 해석 Part I. 중첩 격자 기법 적용에 대한 연구

Study on the feasibility assessment for applying the overlap grid method to numerical calculations on the ground effect has been performed. The objective of the present study is to settle the problem in the grid generation process. A low Mach number preconditioned turbulent flow solver using the overlap grid and the multi-block grid methods has been developed and applied to the ground effect simulation around the RAE 101 airfoil. It has been verified that the overlap grid method not only can provide sufficiently accurate solutions but also work out the grid generation problem in the ground effect simulations.伀ȀȀƱᅸறᅸȀᦱᅸༀ⦱ᅸȀȀȀȀȀ㢱ᅸࠀ冱ᅸԀ垱ᅸ܀徱ᅸࠀȀ榱ᅸጀȀȀȀȀȀ莱ᅸĀȀ

Automatic elliptic grid generation by an approximate factorisation

A procedure for automatic numerical generation of a structured grid system with coordinate lines coinciding with all boundaries of a general two-dimensional region containing a body of arbitrary shape is presented. The solution procedure incorporated the method of false transients and the approximate factorisation algorithm, where a sequence of time steps is cycled in a geometric fashion with repeated endpoints, and has a capability for clustering grid lines close to the body. The procedure requires significantly much less computational effort to obtain a converged solution than a point or line successive over-relaxation iterative scheme. Although, the superiority of the presented algorithm has been demonstrated for the grid generation problem, it can be utilised for other problems requiring the solution of a set of elliptic partial differential equations of similar nature.

First-order system least-squares (FOSLS) for modeling blood flow

The modeling of blood flow through a compliant vessel requires solving a system of coupled nonlinear partial differential equations (PDEs). Traditional methods for solving the system of PDEs do not scale optimally, i.e., doubling the discrete problem size results in a computational time increase of more than a factor of 2. However, the development of multigrid algorithms and, more recently, the first-order system least-squares (FOSLS) finite-element formulation has enabled optimal computational scalability for an ever increasing set of problems. Previous work has demonstrated, and in some cases proved, optimal computational scalability in solving Stokes, Navier–Stokes, elasticity, and elliptic grid generation problems separately. Additionally, coupled fluid–elastic systems have been solved in an optimal manner in 2D for some geometries. This paper presents a FOSLS approach for solving a 3D model of blood flow in a compliant vessel. Blood is modeled as a Newtonian fluid, and the vessel wall is modeled as a linear elastic material of finite thickness. The approach is demonstrated on three different geometries, and optimal scalability is shown to occur over a range of problem sizes. The FOSLS formulation has other benefits, including that the functional is a sharp, a posteriori error measure.

Unstructured triangular surface grid generation and intrinsic grid analysis using rational triangular patches

PurposeTo focus on grid generation which is an essential part of any analytical tool for effective discretization.Design/methodology/approachThis paper explores the application of the possibility of unstructured triangular grid generation that deals with derivationally continuous, smooth, and fair triangular elements using piecewise polynomial parametric surfaces which interpolate prescribed R3 scattered data using spaces of parametric splines defined on R2 triangulations in the case of surfaces in engineering sciences. The method is based upon minimizing a physics‐based certain natural energy expression over the parametric surface. The geometry is defined as a set of stitched triangles prior to the grid generation. As for derivational continuities between the two triangular patches C0 and C1 continuity or both, as per the requirements, has been imposed. With the addition of a penalty term, C2 (approximate) continuity can also be achieved. Since, in this work physics‐based approach has been used, the grid is analyzed using intersection curves with three‐dimensional planes, and intrinsic geometric properties (i.e. directional derivatives), for derivational continuity and smoothness.FindingsThe triangular grid generation that deals with derivationally continuous, smooth, and fair triangular elements has been implemented in this paper for surfaces in engineering sciences.Practical implicationsThis paper deals with the important problem of grid generation which is an essential part of any analytical tool for effective discretization. And, the examples to demonstrate the theoretical model of this paper have been chosen from different branches of engineering sciences. Hence, the results of this paper are of practical importance for grid generation in engineering sciences.Originality/valueThe paper is theoretical with worked examples chosen from engineering sciences.

A linearly constrained optimization problem for planar grid generation

We consider the problem of planar grid generation. This problem can be easily reformulated as an unconstrained optimization problem with the usual variational method. This is a very simple technique to produce grids on planar domains, but it cannot be used in automatic grid generation codes. As a matter of fact, its formulation depends on some parameters, whose right value can disagree considerably among different domains. A simple modification of the variational method is proposed. This modification is mainly based on a linearly constrained optimization problem and it gives a significant increase of the robustness of the method.

A class of network optimization methods for planar grid generation

The grid generation problem considers the question of the computation of a grid Q over a given domain Ω from the knowledge of Q on the boundary ∂ Ω. It is usually given in terms of the computation of the grid vertices belonging to the interior of Ω, that in the direct optimization formulation are computed as the minimizer of a particular minimization problem. In this paper the grid generation problem is reformulated as a network optimization problem on a particular graph, that is a nonlinear minimum cost flow problem, where in place of the standard Euclidean vector norm is proposed the use of the well-known vector p-norm, where p ⩾ 1 . Some results coming from our numerical experience on the examples proposed in the Rogue's Gallery of Grids are reported.

Non‐self‐overlapping structured grid generation on an n‐sided surface

AbstractMost existing meshing algorithms for a 2D or shell figure requires the figure to have exactly four sides. Generating structured grids in the n‐sided parametric region of a trimmed surface thus usually requires to first partition the region into four‐sided sub‐regions. We address the automatic structured grid generation problem in an n‐sided region by fitting a planar Gregory patch so that the partition requirement is naturally avoided. However, self‐overlapping may occur in some portions of the algebraically generated grid; this severely limits its usage in most of engineering and scientific applications where a grid system with no self‐intersecting is strictly required. To solve the problem, we use a functional optimization approach to move grid nodes in the u−v domain of the trimmed surface to eliminate the self‐overlapping. The derivatives of a Gregory patch, which are extremely difficult to compute analytically, are not required in our method. Thus, our optimization algorithm compares favourably at least in terms of speed with some other mesh optimization algorithms, such as the elliptic PDE method. In addition, to overcome the difficulty of guessing a good initial position of every grid node for the conjugate gradient method, a progressive optimization algorithm is incorporated in our optimization. Experiment results are given to illustrate the usefulness and effectiveness of the presented method. Copyright © 2004 John Wiley & Sons, Ltd.

New boundary constraints for elliptic systems used in grid generation problems

New boundary constraints for elliptic partial differential equations as used in grid generation problems in generalized curvilinear coordinate systems are proposed in this paper. These constraints, based on the principle of local conservation of thermal energy in the vicinity of the boundaries, are derived using the Green’s Theorem. They uniquely determine the so called decay parameters in the inhomogeneous terms of these elliptic systems. These constraints 1 Invention under US Patent application process. 1 are designed for boundary clustered grids where large gradients in physical quantities need to be resolved adequately. It is observed that the present formulation also works satisfactorily for mild clustering. Therefore, a closure for the decay parameter specification for elliptic grid generation problems has been provided resulting in a fully automated elliptic grid generation technique. Thus, there is no need for a parametric study of these decay parameters since the new constraints fix them uniquely. It is also shown that for Neumann type boundary conditions, these boundary constraints uniquely determine the solution to the internal elliptic problem thus eliminating the nonuniqueness of the solution of an internal boundary value grid generation problem with Neumann boundary conditions.

Three-dimensional Bézier interpolation in solid modeling and grid generation

Bézier and B-spline patches are popular tools in surface modeling. With these methods, a surface is represented by the product of univariate approximations. The extension of this concept to three-dimensions is straightforward and can be applied to the problem of grid generation. This report will demonstrate how rational Bézier basis functions can be used to generate three-dimensional solids that can be used in solid modeling and in the generation of grids. Examples will be given demonstrating the ability to generated three-dimensional grids directly from a set of data points without having to first set up the boundary surfaces. Many geometric grid properties can be imposed by the proper choice of the control net, the weights, and the twist models.

Boundary-fitted parametrization of unstructured grids on arbitrary surfaces

This paper presents a numerical algorithm for handling arbitrary surfaces which are described through an unstructured set of grid points. Surface unstructured grids can be produced using any CAD software and the present method allows its transformation into high-quality structured grids which are appropriate for flow and heat-transfer calculations, by means of finite-difference techniques. The connectivity and the coordinates of the initial surface nodes are the data for the calculation. The structured grid is defined by introducing a convenient set of two parametric coordinates that satisfy the surface Laplace equation. These equations are solved in the unstructured surface grid using the finite element technique, and provide the parametric values over the primitive unstructured grid nodes. The curvilinear coordinate lines, which coincide with the structured grid lines, are defined by searching for the solution contours inside the surface patch. The quality of the so-calculated grid is further improved via a structured-to-structured grid transformation method which improves grid characteristics, such as orthogonality, smoothness and user-defined grid-lines clustering. The proposed method has been successfully applied in several surface grid generation problems of industrial interest, some of them being illustrated herein.

On the invertibility of mappings arising in 2D grid generation problems

In adapting a grid for a Computational Fluid Dynamics problem one uses a mapping from the unit square onto itself that is the solution of an elliptic partial differential equation with rapidly varying coefficients. For a regular discretization this mapping has to be invertible. We will show that such result holds for general elliptic operators (in two dimensions). The Carleman-Hartman-Wintner Theorem will be fundamental in our proof. We will also explain why such a general result cannot be expected to hold for the (three-dimensional) cube.

A direct method in dacorogna-moser's approach of grid generation problems

The deformation method of grid generation is further developed for general domains. By an initial mapping from the general domain to a sector of an annulus in Rn, the grid generation problem is solved explicitly. The resulting grid has prescribed cell sizes. The construction is valid for any dimension.