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Efficient Smoothing and Deformation of Structured

Volume Grids

William T. Jones

Computer Sciences Corporation

NASA Langley Research Center

Hampton, VA 23666 USA

email: w.t.jones@larc.nasa.gov

Abstract

Methods for the efficient smoothing and reuse of structured volume grids

required for numerical analysis are presented. These techniques increase the

grid quality while reducing the time required for its generation. The goal is to

reduce the need for intervention during the grid generation process. The

smoothing algorithm employs a multigrid acceleration technique in the solution

of the 3-Dimensional Poisson equations. A robust blending function is included

which is used to calculate hybrid source terms for grid control. The deformation

technique presented is purely algebraic and provides for the propagation of

geometric changes into an existing volume grid while preserving the original

grid character. Results of the application of both the smoothing and deformation

algorithms are contained and accompanied by comparisons with traditional

techniques.

Introduction

In recent years, Computational Fluid Dynamics (CFD) has emerged from a field

of concentrated research to become an effective tool used in the design and

analysis of a variety of common engineering problems. Overall, CFD remains a

complex process though it is often numerical grid generation that becomes the

major bottleneck for complicated geometry. The high level of confidence in

structured numerical analysis techniques fuels a continuing need for

improvement in the mature, but restrictive field of structured mesh generation.

Modern structured grid generation tools [1,2] have focused on interactivity and

as such have increased the burden on the user. Complex geometry often includes

a number of special needs that require specific expertise and experience on thepart of the user. Often, producing a structured grid about a complex geometry

proves so formidable that a volume grid is used simply because it has positive

Jacobians. This, however, should seldom be considered acceptable practice.

It is possible to produce such a grid using algebraic means. However, this

practice is time consuming and seldom results in a grid of high quality. One

such method, for example, requires extracting intermediate computational hard

planes from a bad block. These 2-Dimensional planes would be corrected

independently with standard surface grid generation techniques. The volume

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mesh can then be generated piecewise in regions bounded by the inserted hard

planes. Inserting the corrected planes back into the volume essentially subdivides

the grid, thereby giving increased control over the developing interior of the

volume mesh. While this technique may be successful in yielding a grid with all

positive Jacobeans, it is hardly efficient. First, some knowledge, experience, and

luck are required to determine which of the computational planes will have the

greatest impact on improving the interior volume mesh when corrected. Also,

there can be complicated logistics involved in the independent correction of hard

planes that may in fact be coupled. We must also consider the lack of grid

continuity in the piecewise-generated volume mesh as well as the time and

manpower involved in the whole operation.

Another possibility is to use elliptic grid generation. Elliptic grid generation

techniques focus on the solution of a system of 2nd

order non-linear partial

differential equations resulting from the metric transformation of the Poisson

equations [3]. As such, relaxation methods are typically employed. For

complicated 3-dimensional geometry, these methods can be computationally

expensive. However, the manpower requirement is far less than that of the

previous method. The computational cost of a solution to the system can be

greatly reduced with the aid of a multigrid convergence acceleration algorithm

[4-6]. The algorithm consists of using successively coarser mesh levels as a

means to generate solution corrections at each relaxation sweep. The coarse grid

corrections have the obvious benefit of requiring fewer operations, but

additionally improve iterative convergence rates by efficiently eliminating low

frequency error components.

Elliptic grid generation also requires formulation of the source terms defined by

the Poisson system. The specification of source terms, sometimes termed

forcing functions, is used to control grid quality. Several formulations are

available each having their own specific advantages and disadvantages. The

correct choice therefore becomes highly case specific and somewhat of an art

form. An alternate approach can be used which applies exponential blending to

smoothly combine different source term formulations within a given volume.

This hybrid formulation provides for the use of the appropriate forcing function

in areas for which it yields the most benefit [7].

In a design environment, it is often necessary to analyze a number of

perturbations to a given baseline configuration. For example, it may be

necessary to consider a matrix of design parameter variations such as body

length, thickness, and wing span, or flap deflection angles. It is inefficient at

best to require the complete generation of a numerical grid for each perturbation

of the geometry. Therefore, it is desirable to define a method of propagating

these geometric perturbations into the existing baseline mesh. The method must

be robust such that it can handle modifications to a variety of geometry rather

than tailored to a specific configuration. An algebraic algorithm for 3-

dimensional grid deformation is presented which provides such capability.

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Grid Smoothing

Structured volume grids that result from algebraic initialization are seldom

suitable for numerical analysis of complex geometry. Often the initialized grid is

excessively skewed or even folded, resulting in negative Jacobians. Systems of

partial differential equations (PDEs) can be established and used to improve grid

quality. For domains where all physical boundaries are prescribed, an elliptic

system of equations in the form of the Poisson equations can be solved for the

grid points in the physical domain [3]. Consider the Poisson system given as,

( )

( )( )

,,R

,,Q

,,P

2

2

2

==

=

Here P,Q, and R are source terms analogous to those defined in the steady state

heat equation. After metric transformation, a system of 2nd

order non-linear

PDEs results,

( )

uRuQuPJ

uuu2uuu

2

132312332211rrr

rrrrrr

++=

+++++

where [ ]r

u x y zT= , , J = Jacobian of transformation

kjkiij = i,j=1,3 ij is the ijth

cofactor of the Jacobian Matrix [J]

[ ]

=

zzz

yyy

xxx

J

The discrete equations are formed using central differences for the 1st

and 2nd

derivative terms. One-sided differences are used for the mixed derivative terms.

An upwinding scheme is used on the 1st

derivative terms based on the magnitude

of the associated source term [8].

Grid Control

Grid control is specified through appropriate choice of the source terms defined

by the Poisson system. A variety of different formulations are available [9-11].The formulations can be grouped according to their region of influence. Some

source term formulations provide favorable control on the interior of the mesh,

such as that of Thomas and Middlecoff, but do not provide adequate control of

the gridline intersection angle at the boundaries. Typically numerical analysis

tools common to the field of CFD require some degree of grid cell orthogonality

at boundaries defined by solid surfaces. Other formulations, such as GRAPE or

Hilgenstock-White, provide reasonable control at the boundaries, but are not

suited for use on the interior of the mesh. These latter methods are formulated

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based on boundary constraints and therefore the strength of the source term must

decay away from the boundaries.

a) Algebraic grid b) Laplace (no control)

c) Thomas & Middlecoff d) GRAPE

e) Hilgenstock-White f) Hybrid GRAPE/Thomas&Middlecoff

g) Hybrid Hilgenstock-White/Thomas & Middlecoff

Figure 1 Comparison of various source term formulations

The simple 2-dimensional grid shown in figure 1a is used to demonstrate theinfluence of various source term formulations. Unfortunately, these formulations

produce less than satisfactory results when used independently as shown in

figures 1b-1e. Of particular interest here is the streamwise clustering in the

center of the sample such as might be required for improved resolution of a

shock. Figure 1c shows the result of using the Thomas & Middlecoff source

terms. This formulation maintains the clustering on the interior of the mesh

while improving the overall smoothness. However, there is no control over the

gridline intersection angle at the boundaries. The formulations used in figures

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1d and 1e provide for improved control of boundary intersection angle, but do

not maintain streamwise clustering on the interior. Often the decay rate for these

boundary source terms is chosen such that they reduce to zero in the interior of

the mesh. The result is that the grid approaches that of figure 1b on the interior

with a loss of the streamwise clustering. Figures 1f and 1g represent the result of

using a hybrid form. Here the boundary values decay not to zero, but to values

of Thomas and Middlecoff source terms in the interior. The result is control

over the boundary intersection angle with the interior clustering preserved.

In the hybrid formulation, source terms are combined to yield an acceptable

global result. We start by calculating the source terms on the boundary, B, and

the interior source terms, I. We then define a new term on all boundaries of the

region as,

boundaryboundaryboundary IBf =

Note that if the interior source term is chosen as zero, the original formulation of

figures 1d and 1e is recovered. Values of f are blended from the boundary into

the interior with

( )

),,(fc),,(fc

),,(fc),,(fc

),,(fc),,(fc,,F

max6min5

max4min3

max2min1

+++++=

and the hybrid sorce term, H, is now defined simply as,

( ) ( ) ( ) ,,F,,I,,H +=

Exponential blending is used to quickly decay the values of F. This is analogous

to decaying boundary source terms to that of the interior values. Assuming

Dirichlet boundaries, it has been suggested to use blending coefficients of the

form [10]

)(a6

a5

)(a4

a3

)(a2

a1

max

max

max

ecec

ecec

ecec

==

==

==

This however, can hinder the robustness of the solution when competing

constraints result in largely varying values of a given source term along adjacentedges. It is therefore important not only to decay the boundary source terms

from the edges, but also that the sum of the coefficients of the blending function

sum to one everywhere including the edges and corners. An alternate form of

exponentially based blending is presented here which satisfies this requirement.

( )[ ] ( )[ ]( ) kkkkkkk eeeeeeec ++++++= 1113

1

6

1

6

1

3

11

( )[ ] ) ( )[ ]( ) kkkkkkk eeeeeeec ++++++= 1113

1

6

1

6

1

3

12

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-110

-110

-110

:where

npermutatiocyclicbyformedcccc 6543 ,,,

This form of exponential blending has proved to be highly robust yielding well

behaved hybrid source terms in regions of strongly competing constraints such as

those arising in viscous CFD grids.

Multigrid Acceleration

The solution of the non-linear system representing the elliptic grid generation

equations can be greatly enhanced with the use of a Full Approximation Storage

(FAS) multigrid convergence acceleration algorithm. The algorithm is

summarized here for completeness.

Assume a sequence of successively coarser grids Gk: k=0,K with K being the

finest grid. We assume standard coarsening such that, for any grid Kk

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500 1000 1500 2000

Work Units (WU)

-2

-1.75

-1.5

-1.25

-1

-0.75

-0.5

-0.25

0

ln(R/Ro)

4 Level MG (2954 sec., 269.5 WU)

Single Grid (23969 sec., 2396 WU)

SmoothedOriginal

Figure 2 Results of Multigrid Elliptic Smoothing

Original Grid

Perturbation IIPerturbation I

Figure 3 Results of Algebraic Grid Deformation

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Conclusion

Both multigrid acceleration, as applied to the solution to the 3-dimensional grid

generation equations, and grid deformation are techniques needed for reducing

the overall time required in production mode generation of structured grids for

numerical analysis. The algorithms described are applicable to a variety of

engineering problems and aid in improving overall grid quality, reducing user

intervention, and shortening the overall turnaround time of the structured grid

generation process.

The methods described above have been implemented in the Coordinate and

Sensitivity Calculator for Multidisciplinary Design Optimization (CSCMDO)developed in the GEOmetry LABoratory (GEOLAB) at the NASA Langley

Research Center (LaRC) [12,14]. Results shown in the figures were generated

with CSCMDO. Additionally, the CSCMDO software is to be used as an

integral part of the NASA LaRC Framework for Interdisciplinary Design and

Optimization (FIDO) environment.

Acknowledgment

The author would like to thank the members of the NASA Langley Research

Center GEOLAB for their helpful discussions concerning the implementation of

these methods in the CSCMDO package.

Reference[1] Steinbrenner, J. P., Chawner, J. R., GRIDGENs Synergistic

Implementation of CAD and Grid Geometry Modeling, Proceedings of the

5th

International Conference on Numerical Grid Generation in

Computational Field Simulations, pp. 363-372, 1996.

[2] Bertin, D., Casties, C., Lordon, J., A New Automatic Grid Generation

Environment for CFD Applications, 4th

International Conference on

Numerical Grid Generation in CFD and related Fields, pp. 391-402, 1994.

[3] Thompson, J. F., Thames, F. C., Mastin, C. W., Boundary-Fitted

Curvilinear Coordinate Systems for Solution of Partial Differential

Equations on Fields Containing any number of Two-Dimensional Bodies,

NASA CF-2729, 1977.

[4] Brandt, A., Multi-Level Adaptive Solutions to Boundary-Value Problems,Math. Comp., 31, p 333-390. ICASE Report 76-27 (1977).

[5] Brandt, A., Guide to Multigrid Development, Lecture Notes in

Mathematics (Springer-Verlag), Vol. 960, 1981.

[6] Wesseling, P., An Introduction to Multigrid Methods, John Wiley and

Sons Publishing, 1992.

[7] Steinbrenner, J. P., Chawner, J. R., The GRIDGEN Version 9 Multiple

Block Grid Generation Software, MDA Engineering report 94-01, 1994.

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10/10

[8] K. Miki and K. Tago, Three-dimensional composite grid generation by

domain decomposition and overlapping technique, pp. 549-558, Numerical

Grid Generation in Computational Fluid Mechanics, 1988. Pineridge Press

Limited.

[9] Thomas, P. D., Middlecoff, J. F., Direct Control of the Grid Point

Distribution in Meshes Generated by Elliptic Equations, AIAA Journal,

Vol. 18, pp. 652-656, 1979.

[10]Sorenson, Reese L., The 3DGRAPE Book: Theory, Users Manual,

Examples, NASA TM 102224, July, 1989.

[11] White, J. A., Elliptic Grid Generation with Orthogonality and SpacingControl on an Arbitrary Number of Boundaries, AIAA 90-1568, 1990.

[12]Jones, W. T., Samareh-Abolhassani, J., A Grid Generation System for

Multi-disciplinary Design Optimization, AIAA-95-1689, 1995.

[13] Soni, B. K., Two- and Three-Dimensional Grid Generation for Internal

Flow Applications of Computational Fluid Dynamics, AIAA-85-1526,

1985

[14]Bischof, C. H., Jones, W. T., Mauer A., and Samareh, J. A., Experiences

with the Application of the ADIC automatic differentiation tool to the

CSCMDO 3-D Volume Grid Generation Code, AIAA Paper 96-0716,

1996.