A direct procedure is presented for locally bicubic interpolation on a structured, curvilinear, two-dimensional grid. The physical (Cartesian) space is transformed to a computational space in which the grid is uniform and rectangular by a generalized curvilinear coordinate transformation. Required partial derivative information is obtained by finite differences in the computational space. The partial derivatives in physical space are determined by repeated application of the chain rule for partial differentiation. A bilinear transformation is used to analytically transform the individual quadrilateral cells in physical space into unit squares. The interpolation is performed within each unit square using a piecewise bicubic spline

Yarrow, Maurice

Zingg, David W.

English

NASA Technical Reports Server

~ 
NASA Technical Memorandum 10221 3 
A Direct Procedure for 
Interpolation on a Structured 
Curvilinear Two-Dimensional 
Grid 
David W. Zingg, University of Toronto, Downsview, Ontario, Canada 
Maurice Yarrow, Sterling Software, Arnes Research Center, Moffett Field, California 
July 1989 
. 
National Aeronautics and 
Space Administration 
Ames Research Center 
Moffett Field, California 94035 
https://ntrs.nasa.gov/search.jsp?R=19890018062 2020-03-20T01:57:28+00:00Z
Summary 
A direct procedure is presented for locally bicubic interpolation on a structured, curvi- 
linear, two-dimensional grid. The physical (Cartesian) space is transformed to a compu- 
t ational space in which the grid is uniform and rectangular by a generalized curvilinear 
coordinate transformation. Required partial derivative information is obtained by finite 
differences in the computational space. The partial derivatives in physical space are de- 
termined by repeated application of the chain rule for partial differentiation. A bilinear 
transformation is used to analytically transform the individual quadrilateral cells in phys- 
ical space into unit squares. The interpolation is performed within each unit square using 
a piecewise bicubic spline. 
1. Introduction 
Problems in computational fluid dynamics (CFD) are often solved using a numeri- 
cally generated grid which is structured but nonuniform, consisting of quadrilat'erals in 
two dimensions and hexahedra in three dimensions. The grid is structured in that each 
cell contains an implied set of coordinate directions. A uniform rectangular grid in com- 
putational space is obtained froin the physical space grid by a generalized curvilinear 
coordinate transformation (ref. 1). This facilitates the application of finite differences and 
finite volume methods. 
In many applications of CFD, such as multiblock algorithms using overlapping grids, 
surface definition, and graphical representation of flow fields, data which are known at grid 
points must be interpolated at intermediate points. In the overlapping grid technique de- 
scribed by Benek et al. (ref. 2), interpolation is accomplished using an iterative technique. 
The purpose of the present paper is to present an alternative approach which involves no 
iteration. 
The interpolation procedure presented uses a piecewise bicubic interpolant rather 
than the simpler bilinear interpolant. The bilinear interpolant has the often desirable 
property that it introduces no new extrema. However, the bicubic interpolant is inore 
accurate and produces a smooth interpolant. Furthermore, bicubic interpolation is capable 
of preserving derivative information generated by a finite-difference flow solver. Iniportant, 
considerations regarding monotonicity and conservative properties of the interpolant are 
not addressed in this paper. Examples of alternative techniques for bivariate interpolation 
on a rectangular grid are given by Carlson and Fritsch (ref. 3) and Schiess (ref. 4). 
The generalized curvilinear coordinate transformation is given in Sect,ion 2, and the 
iterative interpolation technique of Benek et al. is reviewed in Section 3. The direct, 
interpolation procedure is presented in Section 4. The Appendix contains the formulas for 
bicubic spline interpolation on a unit square. 
1 
I 
I 2. Generalized Curvilinear Coordinate Transformation 
The generalized curvilinear coordinate transformation maps a nonuniform, nonrect- 
angular grid in physical (Cartesian) space to a uniform, rectangular computational grid. 
The Cartesian coordinates (x, y)  are mapped to the curvilinear coordinates ( t , ~ )  by the 
following general transformation: 
where we consider two space dimensions only and ignore time dependence for simplicity. 
rectangular, and of unit length, i.e., 
, The mapping is chosen such that the resulting grid in computational space is uniform, 
Hence, finite-difference representations of tlt and a, are easily formulated. The mapping 
is one-to-one except at topologic,al singularities or cuts, where one physical point may be 
mapped to many computational points. 
The chain rule for partial differentiation gives, in matrix form, 
, The coordinate transformation is generally not known analytically and hence must be 
determined numerically. When the roles of the coordinate systems in equation (3) are 
reversed, t8he chain rule gives 
I 
Therefore, we must have 
= J [  ys 
- 2 9  
All of the t,erins involving tlt and 
matrix form of the inetric relations. 
are evaluated as finite differences. Eyua,tion (6)  is t81ie 
3. Iterative Interpolation Procedure 
The iterative scheme developed by Benek et al. (ref. 2) is applicable to three- 
diiiiensional generalized curvilinear coordinates. Trilinear interpolation is used. Yarrow 
and Mehta (report to be published as a NASA Technical Memorandum) ut,ilize tricubic 
interpolants within tlie same iterative schenie. In this section, the iterative procedure is 
presented for a two-dimensional space. 
Since the grid is uniform and rectangular in computational space, the data can be 
readily interpolated as a function of tlie generalized coordinates by conventional niultivari- 
ate interpolation techniques. The interpolant for the function f can be written as 
The iterative procedure is used to deternine the generalized coordinates ((* , q*)  which 
correspond to the physical point (z*,y*) at which tlie interpolation is to be performed. 
Since each point in computational space is mapped to only one physical point, we can form 
tlie following interpolants for the Cartesian coordinates: 
For a two-dimensional space, bivariate Newton-Raphson iteration is used to solve for 
((*, q*)  such that 
The value of the function f at the point (z*,y*)  is then determined from 
4. Direct Interpolation Procedure 
The direct interpolation procedure proceeds in three steps. First, required partial 
derivative information in physical space is determined using finite-difference approxima- 
tions to the partial derivatives in coinputational space. The individual cells are then 
analytically transformed into unit squares by a bilinear transformation. Finally, a bicubic 
spline interpolant is determined within each unit square. 
For a function f known at the grid nodes, we require fz, fy , fzz, f zy  , fyy . Using finite 
differences in computational space, we can determine f t  , f v ,  f t t ,  f,tv, f v v .  Applying the 
chain rule to equation (3) yields 
3 
Equations (3) and (11) may be assembled as 
where 
B =  
Similarly, 
8& = CdZY, 
where 
C =  
Therefore B = C-I. 
In block matrix form, B can be written as 
where 
Si l i d  arl y , 
4 
where 
Now B = C-' gives 
and 
B1 = L'I-' (the metric relations), 
B3 = C'3-l, 
B2 = -C'3-1C2C1-1. 
Substituting equations (16) and (17) into equation (18) gives 
Finally, use of the metric relations gives 
2 2 
XCC - 2Yll  YCXCl, + Y1,YC2X,, - X,Y? YC€ + 2X,YllY€YCll - "CYt2Y?7?ll 1 3  t x x  = -- [Yll 
72, = -- [-YtYll X C t  + 2YC2Y, - Yt3X,, + XCY?l2YCC - 2XCYVYCYCll + XCYC Y,,] 
J 3  
1 
J3 
1 
2 2 
t "y = -- J3 [-Y?12x,xCt + Y d Y ? l r t  + XllYC)XC, - YllYCXCX,, + Xl12Y1Y€€ 
%Y = -J3 [ Y t Y l l X l l W  - Y d Y l l X C  + XllYC)XC, + YC2XCX,11 - "€YvXC17YC€ 
tYY = -J3 [Y,XV2XC€ - 2Yl lXl lXtXCl l  + YllXt2Xl l ,  - Xl13YCt + 2 X V 2 X t Y € 7 l  - X1,XC2Y?,] 
17YY = -- [-YCxa2x€€ + 2YCX,XtX€l l  - YCXt2X,1, + ~C"Tl2Y€t  - 2X,XC2YCll + XC3Yll?,] * 
- Xll(Y?lXC + X l l Y d Y C l l  + ~ , Y C ~ € Y l l l l l  
+ X C ( Y l l X t  + X l l Y d Y € l l  - "€3Y€Ys,J 
1 
1 
1 
J 3  
(20) 
Therefore, all of the elements of matrix B can be expressed in terms of the elements 
of matrix C. As a result, dZyf can be determined given atsf. The elements of C' and 
can be approximated using finite differences. An alternative procedure is to calculate f z  
and f y  given f~ and f, from equation (3) and then to use finite differences to determine 
(. fz)t ,  ( fy) t ,  (fZ) , ,  (fv)ll. The required values of f z z ,  fxy, and fyy can then be calculated 
using equation (3) again. The first method involves more computer storage but it also 
involves a smaller stencil. 
Given f z ,  f y  , f x z ,  fzy, fyy at the corners of a given cell, the problem remains to inter- 
polate f on an arbitrary quadrilateral. We now consider the new two-dimensional space 
given by ( p , q )  shown in figure 1. The quadrilateral in physical space is related to a unit 
5 
square in ( p ,  q )  space by a bilinear mapping as follows. (Without loss of generality, we will 
assume the lower-left-hand corner to be at (O,O).)  
x = a p  + bq + cpq 
Y = dP + eq + f p q  
Note that a different mapping is used for each quadrilateral, or cell, in the grid. This con- 
trasts with the generalized curvilinear coordinate transformation in which a single smooth 
iiumerically generated mapping is applied to the entire grid. Consequently, analytical 
mappings can be utilized. 
Figure 1. - Two-dimensional space given by ( p , q ) .  
The coefficients of the bilinear mapping are given by 
a = x4, b = 22, c = x3 - 2 4  - 2 2 ,  
d = y 4 ,  e = y 2 ,  f = y 3 - y 4 - y 2 *  
The required derivatives in ( p ,  q )  space are given (as in equation ( 1 2 ) )  by 
a p  = xpax + Ypdy 
aq = xqax + Yqdy 
a,, = xpqax + Ypqdy + x p x q a x x  + (XpYq + x9Yp)dxy + YpYq d Y Y ,  
where 
- and we have used x p p  = xqg - ypp = yqq = 0. 
Froiii equation ( 2 1 a ) ,  we have 
Given a location in physical space (20, yo), we must find the corresponding ( P O ,  yo ). 
G 
Substituting this into equation ( 2 l b )  as follows: 
) Yo = duo + ( e  + f p o ) (  2 o  - U P 0  b + cpo 
gives the following quadratic for po 
( 2 5 )  
po2(cd  - uf) + po(-cyo + bd + .of - ae)  + ( - y o b  + e x o )  = 0. (26) 
The solution of equation (26) gives two values of p o .  The corresponding values of qo are 
determined from equation (24). One location (p0,qO) will not lie within the unit square 
and hence can be discarded. 
Using fz, fy , fzz  , fry, fyy as calculated from equation (12)) we can determine fp, fq, 
and fpq from equat,ion (23) at each corner of the unit square in ( p , q )  space. We then 
use the standard formula for bicubic interpolation within a unit square (see Appendix). 
The bicubic is finally evaluated at ( p o , q o ) ,  as calculated from equations (26) and (24)) 
respectively. 
5 .  Conclusions 
A procedure has been presented for interpolation on a structured, curvilinear, two- 
dimensional grid. In contrast to existing methods (ref. 2), the procedure avoids the need for 
iteration. A piecewise bicubic spline is used, leading to a smooth and accurate iiiterpolaiit 
which is capable of preserving derivative information generated by a flow solver based on 
finite differences. Potential applications include multiblock algorithms using overlapping 
grids in the solution of problems in computational fluid dynanlics, surface definition, and 
graphical representation of data. Extension to three dimensions is straightforward but 
nontrivial . 
7 
Appendix 
Locally Bicubic Spline Interpolation on a Unit Square 
In this appendix, we give a piecewise bicubic spline forinulation for interpolation 
within a unit square. De Boor (ref. 5 )  presents a technique for determining the required 
partial derivatives at the corners of each cell in a rectangular grid such that G2 continuity 
is obtained for the domain. In the formulation used here, the required partial derivatives 
at the corners of the unit square are given by centered difference formulas, leading to G1 
continuity, analogous to the univariate cubic Bessel spline given by de Boor (ref. 6). This 
formulation has the advantage of producing a local interpolant. 
The bicubic spline interpolant on a unit ( p , q )  square can be written as 
where 
bmn] = A K A t ,  
1 0 0 0  
A =  [+ ;2 1 0  3 :1]
-2 1 
and 
8 
Refer e nc es 
1. Prilliaiii, T .  H.: Efficient, Solution Methods for the Navier-St,okes Equations. Lecture 
Notes for the Von Kariiian Institute for Fluid Dynaiilics Lecture Series: Numerical 
Techniques for Viscous Flow Computation in Turbomachinery Bladings, Brussels, 
Belgium, Jan. 1986. 
2. Benek, J. A.; Buning, P. G.; and Steger, J .  L.: A 3-D Chimera Grid Embedding 
Technique. AIAA Paper 85-1523, June 1985. 
3. Carlson, R. E.; and Fritsch, F. N.: Monotone Piecewise Bicubic Interpolation. SIAM 
J.  Numer. Anal., vol. 22, no. 2, April 1985. 
4. Schiess, J. R.: Two Algorithms for Rational Spline Interpolation of Surfaces. NASA 
TP 2536, 1986. 
5 .  De Boor, C.: Bicubic Spline Interpolation. J .  Math. and Phys., vol. XLI, no. 3, 
Sept. 1962. 
6. De Boor, C.: A Practical Guide to Splines. Springer-Verlag, New York, 1978. 
c 
9 
Report Documentation Page 
1. Report No. 
NASA TM-102213 
2. Government Accession No. 
A Direct Procedure for Interpolation on a Structured Curvilinear 
Two-Dimensional Grid 
17. Key Words (Suggested by Author(s)) 
Interpolation 
Bicubic interpolation 
Generalized curvilinear coordinates 
7. Author@) 
David W. Zingg* and Maurice Yarrow** 
18. Distribution Statement 
Unclassified - Unlimited 
Subject Category - 64 
9. Performing Organization Name and Address 
Ames Research Center 
Moffett Field, CA 94035 
12. Sponsoring Agency Name and Address 
National Aeronautics and Space Administration 
Washington, DC 20546-0001 
3. Recipient's Catalog No. 
5. Report Date 
July 1989 
6. Performing Organization Code 
8. Performing Organization Report No. 
A-89210 
10. Work Unit No. 
505-60 
1 1. Contract or Grant No. 
13. Type of Report and Period Covered 
Technical Memorandum 
14. Sponsoring Agency Code 
15. Supplementaty Notes 
Point of Contact: Harvard Lomax, Ames Research Center, MS 202A-1, Moffett Field, CA 94035 
(4 15) 694-5 124 or FTS 464-5 124 
*University of Toronto, Downsview, Ontario, Canada M3H 5T6 
**Sterling Software, Ames Research Center, Moffett Field, CA 94035 
16. Abstract 
A direct procedure is presented for locally bicubic interpolation on a structured, curvilinear, two- 
dimensional grid. The physical (Cartesian) space is transformed to a computational space in which the grid 
is uniform and rectangular by a generalized curvilinear coordinate transformation. Required partial 
derivative information is obtained by finite differences in the computational space. The partial derivatives 
in physical space are determined by repeated application of the chain rule for partial differentiation. A 
bilinear transformation is used to analytically transform the individual quadrilateral cells in physical space 
into unit squares. The interpolation is performed within each unit square using apiecewise bicubic spline. 
Structured grids 
19. Security Classif. (of this report) I 20. Security Classif. (of this page) I 21. NO. of Pages I 22. Price 
A02 I lo  I Unclassified I Unclassified 
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A direct procedure for interpolation on a structured curvilinear two-dimensional grid

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