This report describes a numerical method for generating 3-D grids for general configurations. The basic method involves the solution of a set of quasi-linear elliptic partial differential equations via pointwise relaxation with a local relaxation factor. It allows specification of the grid spacing off the boundary surfaces and the grid orthogonality at the boundary surfaces. It includes adaptive mechanisms to improve smoothness, orthogonality, and flow resolution in the grid interior

Chen, S. C.

English

NASA Technical Reports Server

.. 
NASA Contractor Report 182192 
Three-Dimensional Adaptive Grid 
Generation for Body-Fitted 
Coordinate System 
s.C. Chen 
Sverdrup Technology, Inc. 
NASA Lewis Research Center Group 
Cleveland, Ohio 
August 1988 
Prepared for 
Lewis Research Center 
Under Contract NAS3-24105 
NI\SI\ 
National Aeronautics and 
Space Administration 
J.AtlG!..EY R;::SE!\RCH CEN.TER 
LIBRARY, NASA 
HAMeTOtJ, VIRG!NIF\ 
111I11111111I~lllIllr 11111 1IIImilmll 1111-
NF00910 
https://ntrs.nasa.gov/search.jsp?R=19880018136 2020-03-20T05:42:22+00:00Z
THREE~DIMENSIONAL ADAPTIVE GRID GENERATION 
FOR BODY-FITTED COORDINATE SYSTEM 
1. SUMMARY 
S. C. Chen· 
Sverdrup Technology, Inc. 
NASA Lewis Research Center Group 
Cleveland, Ohio 44135 
This report describes a numerical method for generating three-
dimensional grids for general configurations. The basic method involves 
the solution of a set of quasi-linear elliptic partial differential equations via 
pointwise relaxation with a local relaxation factor. It allows specification of 
the grid spacing off the boundary surfaces and the grid orthogonality at the 
boundary surfaces. It includes adaptive mechanisms to improve smoothness, 
orthogonality, and flow resolution in the grid interior. 
2. INTRODUCTION 
Three-dimensional computational fluid dynamics codes require compu-
tational grids with suitable resolution, smoothness, and orthogonality. High 
grid resolution allows complex flow physics to be modelled near shocks and 
in shear layers. Smoothness of the metric data prevents the flow solution 
from being dominated by truncation error in the metric coefficients. Grid 
orthogonality at the boundaries simplifies and improves the accuracy of any 
boundary condition involving normal gradients. 
This report describes a numerical method for generating three-
dimensional grids for computational fluid dynamics codes with the potential 
and the ability to satisfy the foregoing requirements. The basic method is 
general and involves the solution of a quasi- linear elliptic partial differ-
ential equation via pointwise successive relaxation with a local relaxation 
factor. The governing equation contains forcing functions that depend upon 
the boundary point distribution and the boundary surface gradient. The 
method allows specification of the grid point distribution on the boundary 
surfaces, the grid spacing off the boundary surfaces, and the grid orthogo-
nality at the boundary surfaces. It includes adaptive mechanisms to improve 
smoothness, orthogonality, and flow resolution in the grid interior. 
• Senior Research Engineer 
The mechanism of adaptation contains two branches: cumulative move-
ment and non-cumulative movement each with its own characteristic. 
In non-cumulative action, the amount of adjustment made on the nodal 
point distribution is governed by a penalty type of behavior, with fixed upper 
and lower limits, on the forcing functions. 
The system appears to be very stable numerically. The overall accuracy 
of the system, in terms of residues of the physical aspects under adaptation, 
is limited by the strength of the penalty function. 
In cumulative action, the adjustments are guided by physical con-
straints. Movement of the grid will not stop unless a homogeneous condition 
is reached. There are no directly predetermined upper or lower limits on 
the forcing functions with this technique. 
The strength of the adaptive mechanism is determined by a scaling 
factor, which in turn controls the magnification of disturbances at each 
iteration. The behavior of this technique resembles an explicit scheme, the 
scaling factor should be small enough to assure stability of the system, and 
yet large enough to achieve better accuracy. 
This paper includes a discussion of the mathematical formulation of the 
method and some representative results. 
3. MATHEMATICAL FORMULATION 
The quasi-linear elliptic governing equation is taken from reference 1 as 
~ ~ . . a2 x ~ kk ax ~~ gO, aE.aE. + ~ 9 PkaE = o. 
i = 1 ,. = 1 "k= 1 k 
(1) 
The metric tensor components gi,. and gkk in equation (1) are defined 
as 
where 
ai = a,. x ak/Jg i,j, k cyclic 
Jg = al • ((12 x (13) 
_ ax 
ai = aEi' 
The Pk in equation (1) are the forcing functions specified by the user. 
They represent one-dimensional stretching in each index coordinate direc-
tion. 
Equation (1) can be rewritten using matrix notation as 
A+BP=O (2) 
where 
2 
(911 a2i aei B= 0 
o 
o 
9
22 32 i 
ae~ 
o 9,,~a'i ) ae; 
Equation (2) can be solved for P k on the boundaries as 
(3) 
where the subscript "0" indicates values on the boundary. Tangential deriva-
tives for terms on the right hand side of equations (3) are determined by 
applying standard difference formulas to the prescribed boundary point dis-
tribution on the surface. Normal derivatives are determined by specifying 
the first normal derivative equal to the desired spacing off the boundary (for 
detail, see reference 2) and using the approximation 
where n indicates the normal direction, the subscript "0" indicates values on 
the boundary, and the subscript "1" indicates values one point away from 
the boundary. 
Once the boundary values are known, the interior values of P k can be 
determined using 
3 3 
Pk (6, 6, ~3) = I: POl • I •• (1 + O'k,I,t}ik,l,l + I: POl •I • 2 (1 + O'k,I,2hk,I,2 
1=1 1=1 
where 
3 
O:k,I,1 = Cal.I • I ((~, - €,,,,.J/(6,,,u - €1",.J)Dl. I • I 
O:k,I,2 = C al .I •2 ((€''''u - €t)/(6", .. - €1",.J)Dl.,.2when C a ~ 0 
O:k,I,1 = tanh(O:k,I,t) 
O:k,I,2 = tanh(O:k,I,2) 
f3k,I,1 = [(€,,,,U - 6",.J/(6 - €1",;J1 C61• 1 
f3k,I,2 = [(6",as - 6_;J/(6",as - 6)C61,2 
3 3 
'Ik,l,l = f3k,l,d (L f3k,I,1 + L f3k,I,2 ) 
m=l m=l 
3 3 
'Ik,I,2 = f3k",2/( L f3k,l,l + L f3k"';')' 
m=l m=l 
when C a < 0 
The value of POl .I . I represents the k-th component of the Po vector on 
the minimum boundary surface in the I-th direction. The value of Pal I I 
represents the k-th component of the Po vector on the maximum bou~d­
ary surface in the I-th direction. The functions 0:, 13, and I have subscript 
notation similar to that of Po. The 0: function represents polynomial ex-
trapolation from a controlled boundary using a constant factor Ca. The I 
function represents the combined effect of the f3 functions, which represent 
power-law factorization with constant exponent Cf3 to control the depth of 
influence away from a controlled boundary. 
The values of the forcing functions can be modified for improved 
smoothness by using 
where 
e = 1 - tanh(Cs l (1 - O"c. 2 )) 
0" = J /(11/ 1 11.11/2 II . 11/3 II)· 
The constants C SI and CS 2 define the rate and order of the adaptation. 
The variable 0" is a measure of the shear of the grid, with J representing the 
Jacobian and 11, 12 , 13 representing the grid cell lengths in each direction. 
A measure of the local orthogonality of the grid can be defined as 
(4) 
The constant C s defines the order of the adaptation. A one-dimensional 
variational form of equation (4) can then be written as 
4 
. t 
If a flow error index E is computed by the flow solver such that E 2: 0, 
a measure of the local flow resolution can be defined as 
,p = J(1 + f(E)). (5) 
where f represents a suitable functional of E. 
A one-dimensional variational form of equation (5) can then be written 
as 
,p' _ Ok(J) _ okf(E) 
k - J 1 + f(E) 
The values of the forcing functions can be modified for improved local 
orthogonality and flow resolution using 
(6) 
where 8)., similar to 8, is the skewness index for the adaptive constributions, 
and 
Ak = C).ltanh(C).2IFklc).~) 
Fk = (W~4>~ + W",,p~)/(W~ + W"') 
with the constants C).l' C).2' and C)., determining the range, rate, and 
order of the adaptation for the non-cumulative mechanism. The variable F 
represents a weighted combination of the skewness and flow error variations. 
When cumulative adjustment is desired, the focus must be on the dis-
turbances of Fk instead of Fk itself. Thus let 
D.Fk = F!n-l) _ F!n) 
Ai. = (D.Fk · F!n) /IF!n)l)(eIFt)IC).~ - 1) 
Ai.· = tanh(C).2 Ai.) 
At = 0 
and then 
when Ai. ~ 0 
when ).k < 0 
(7) 
Equation (7) is used in conjuction with equation (6) for evaluating the 
proper value of the forcing functions at the Nth step. 
4. RESULTS AND DISCUSSION 
Two groups of test cases using a turbine blade geometry are attached. 
Each of these cases represents a particular capabilit.y of the technique. 
5 
Within each group, the nodal point distribution on the boundary lines and 
the basic values of the forcing functions are constructed in exactly the same 
fashion. 
Figure 1 illustrates the effect of adaptation according the the skewness 
of the grid only. Highly sheared grids (around the trailing edge) are suffi-
ciently improved compared to results without this adaptation (not shown). 
Figure 2 shows the effect of the non-cumulative adaptation according 
to the skewness and the flow property with constant error index everywhere 
(equivalent to requiring a zero variation in the Jacobian). The result has 
indicated that the nearly collapsed grid lines (from Figure 1) at the inlet 
corner are greatly improved whereas the grid elsewhere is well preserved. 
Figure 3 illustrates the effect of the non-cumulative adaptation accord-
ing to the skewness and the general orthogonality of the grids. The result 
shows the tendency of the expansion of a good grid and the suppression of 
a sheared grid. _ 
Two representatives of the three-dimensional capability of the current 
technique are illustrated in Figure 4 and 5. Figure 4 shows a C-grid configu-
ration with adaptation according to the skewness of the grid only. Figure 5 
shows an H-grid configuration with the adaptation according to the skewness 
and the non-cumulative orthogonality of the grid. 
Figure 6, i, and 8 illustrate the effects of the solution adaptive mecha-
nisms. The error index used here is a fixed artivicial pattern with 
E = (( J - 1) / J H al J) 4 
= ((Jmax - J)/JHalJ)" 
where J Hall = (Jmax - 1)/2 + 1. 
The error functional I used is 
when J ::; J Hall + 1 
when J > JHal1 + 1 
1 + J(E) = eA .E 
with a constant A. 
Figure 6 represents the optional grid with a skewness adaptation only. 
Figure 7 shows the effect of the non-cumulative solution adaptive mechanism 
with A = 10. Figure 8 illustrates the effect of the cumulative solution adap-
tive technique with A = 2. The results have indicated a positive response on 
the redistribution of the grids. The adjustments achieved by the cumulative 
mechanism are more profound than that indicated by the non-cumulative 
mechanism. 
The stability of the adaptive technique is satisfactory. Failure to reach 
convergence is possible, however, particularly when the spacing requirements 
of the grids are extremely small and when the physical configuration forces 
the appearances of highly sheared grids in the interior. These difficulties 
6 
can be relaxed, only to a degree, by the specification of a suitable strength 
for the adaptive schemes. 
The solution adaptive technique introduced here is still at its early 
developing stage. The adaptation according to a real slow property has not 
yet been tested. Although current test results have shown positive response 
on the grid movements to the adaptive mechanisms, they also appear to 
indicate a major weakness of this technique, that is, the adjustments on the 
grid are ultimately limited by the basic requirements, which are specified 
by the user. The residue of the property under adaptation can only reach 
a relative minimum, not an absolute minimum, in order to maintain the 
numerical stability. 
5. ACKNOWLEDGEMENTS 
The author would like to thank Mr. J.R. Schwab of NASA Lewis for his 
valuable contributions, and the assistance provided by colleagues at Sver-
drup Technology. This work was sponsered by the NASA Lewis Research 
Center under contract NAS3-24105 with J.R. Schwab as monitor. 
6. REFERENCES 
!l] THOMPSON, J. F., WARS!, Z.U.A., and MASTIN, C. W. - Numerical 
Grid Generation: Foundations and Applications. Elsevier Science Pub-
lishing Co. New York, New York. 1985. 
!2] SORENSON, R. L. and STEGER, J. L. - Grid Generation in Three Di-
mensions by Poisson Equations with Control of Cell Size and Skewness 
at Boundary Surfaces. Advance in Grid Generation, FED-Vol 5. 1983. 
7 
00 
1'1,:. I Skl'W11l'lI!l ,ulaptntion only. 
101"1(, "rl,I: •. 
I'll;. 2 Skew11l~HB nnd ZCI'O varl-
allo11 In thc .Jocoblall. 
Nnn-cllmlllntive. 
Fig. J Skewness and orthC'go-
nality control. 
Non-cumulative. 
Magnified views for 
Fig.2 and Fig.J res-
pective.ly. 
o 
Fig. 6 Original grids with the 
skewness control only. 
1'1,:.7 Sollltioll IIllaplatlnn. 
NIII1-cIIIII1I1 /I t I VI'. 
Fi~. 8 Sollltioll nduptnllol1. 
Cumulntivl'. 
MOlln1ficcl vi<'Ws for 
Fig.7 nnel Fig.S rcu-
PC('tiVl')Y· 
Fig.4 Three di=ensional C-grid configuration. 
Fig. 5 Three di:ner.sional H-grid configuration. 
9 
NI\SJ\ Report Documentation Page National Aeronautics and 
Space Administration 
1. Report No. 2. Government Accession No. 3. Recipient's Catalog No. 
NASA CR-182192 
4. Title and Subtitle 5. Report Date 
Three-Dimensional Adaptive Grid Generation for Body-Fitted August 1988 
Coordinate System 6. Performing Organization Code 
7. Author(s) 8. Performing Organization Report No. 
S.C Chen None (E-4331) 
10. Work Unit No. 
582-01-11 
9. Performing Organization Name and Address 
11. Contract or Grant No. 
Sverdrup Technology, Inc. 
NAS3-24105 NASA Lewis Research Center Group 
Cleveland, Ohio 44135 13. Type of Report and Period Covered 
12. Sponsoring Agency Name and Address 
Contractor Report 
Final 
National Aeronautics and Space Administration 14. Sponsoring Agency Code 
Lewis Research Center 
Cleveland, Ohio 44135-3191 
15. Supplementary Notes 
Project Manager, John R. Schwab, Internal Fluid Mechanics Division, NASA Lewis Research Center. Prepared 
for the Second International Conference on Numerical Grid Generation in Computational Fluid Dynamics 
cosponsored by NASA and AFOSR, Miami Beach, Florida, December 5-8, 1988. 
16. Abstract 
This report describes a numerical method for generating three-dimensional grids for general configurations. The 
basic method involves the solution of a set of quasi-linear elliptic partial differential equations via pointwise 
relaxation with a local relaxation factor. It allows specification of the grid spacing off the boundary surfaces and 
the grid orthogonality at the boundary surfaces. It includes adaptive mechanisms to improve smoothness, orthog-
onality, and flow resolution in the grid interior. 
17. Key Words (Suggested by Author(s)) 18. Distribution Statement 
Grid generation Unclassified - Unlimited 
Turbomachinery Subject Category 34 
19. Security Class if. (of this report) 20. Security Classif. (of this page) 21. No of pages 22. Price" 
Unclassified Unclassified 11 A03 
NASA FORM 1626 OCT 86 
"For sale by the National Technical Information Service, Springfield, Virginia 22161 
End of Document 


Three-dimensional adaptive grid generation for body-fitted coordinate system

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server