3 Transfinite Interpolation (TFI) Generation Systems 3.1 3.2 3.3 3.4

Introduction Grid Requirements Transformations and Grids Transfinite Interpolation (TFI) Boolean Sum Formulation • Recursion Formulation • Blending Function Conditions

3.5

Practical Application of TFI Linear TFI • Langrangian TFI • Hermite Cubic TFI

3.6

Grid Spacing Control Single-Exponential Function • Double-Exponential Function • Hyperbolic Tangent and Sine Control Functions • Arclength Control Functions • Boundary-Blended Control Functions

Robert E. Smith

3.7 3.8

Conforming an Existing Grid to New Boundaries Summary

3.1 Introduction This chapter describes an algebraic grid generation produced called transfinite interpolation (TFI). It is the most widely used algebraic grid generation procedure and has many possible variations. It is the most often-used procedure to start a structured grid generation project. The advantage of using TFI is that it is an interpolation procedure that can generate grids conforming to specified boundaries. Grid spacing is under direct control. TFI is easily programmed and is very computationally efficient. Before discussing TFI, a background on grid requirements and the concepts of computational and physical domains is presented. The general formulation of TFI is described as a Boolean sum and as a recursion formula. Practical TFI for linear, Lagrangian, and Hermite cubic interpolation is described. Methods for controlling grid point clustering in the application of TFI are discussed. Finally, a practical TFI recipe to conform an existing grid to new specified boundaries is described.

3.2 Grid Requirements Grids provide mathematical support for the numerical solution of governing field equations in a continuum domain. The physics is expressed as a system of differential or integral equations subject to initial and boundary conditions. A numerical solution is obtained by superimposing a grid onto the continuum

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domain, discretizing the governing equations relative to the grid, and applying a numerical solution algorithm to the discrete approximation of the governing equations. The result is an evaluation of the solution at the grid points. Two key ingredients necessary for obtaining an accurate and efficient solution are (1) the numerical solution algorithm, and (2) the grid. A grid generation technique should be as efficient as possible to achieve the desired characteristics. However, the importance of a particular characteristic or combination of characteristics can outweigh alone in determining which grid generation technique is applied to a particular problem. The most efficient grid generation techniques are algebraic and are based on the application of interpolation formulas. Algebraic grid generation techniques relate a computational domain, which is a rectangular parallelepiped (a square in two dimensions and a box in three dimensions), to an arbitrarily shaped physical domain with corresponding sides. The computational domain is a mathematical abstraction. The physical domain is the bounded continuum domain where a numerical solution to a system of governing equations of motion is desired. A side in the computational domain can map into a line or a point in the physical domain, in which case a singularity occurs in the mapping. Singularities can pose problems for the computation of numerical solutions when the governing equations are expressed in differential form. However, grid singularities usually do not cause problems when the governing equations are expressed in integral form. A single block (square or box in the computational domain and deformed block in the physical domain) is not usually sufficient to fit to boundaries of a complex solution domain. Therefore, the complex domains must be divided into subdomains and multiple blocks used to cover the subdomains. Depending on the solution technique used to solve the governing equations, the grid points at the boundaries of adjoining blocks must be contiguous. TFI is a multivariate interpolation procedure. When TFI is applied for algebraic grid generation, a physical grid is constrained to lie on or within specified boundaries. TFI is a Boolean sum of univariate interpolations in each of the computational coordinates. Virtually any univariate interpolation (linear, quadratic, spline, etc.) can be applied in a coordinate direction. Therefore, there are a limitless number of possible variations of TFI that can be created by using different combinations and forms of the univariate interpolations. Often for a particular application, a high order and more sophisticated interpolation is used in one coordinate direction, which we will call the primary coordinate direction, and a low-order interpolation, such as linear interpolation, is used in the remaining coordinate directions.

3.3 Transformations and Grids Algebraic grid generation techniques are transformations from a rectangular computational domain to an arbitrarily shaped physical domain. This is shown schematically in Figure 3.1 and as a general equation  x(ξ , η,ζ )   X (ξ , η,ζ ) =  y(ξ , η,ζ )     z(ξ, η,ζ )  0 ≤ ξ ≤ 1 0 ≤ η ≤ 1 and 0 ≤ ζ ≤ 1 A discrete subset of the vector-valued function X, (ξΙ , ηJ , ζK) is a structured grid for 0 ≤ ξI =

I −1 J −1 K −1 ≤1 ≤ 1 0 ≤ ζI = ≤ 1 0 ≤ ηJ = Iˆ − 1 Jˆ − 1 Kˆ − 1

where I = 1, 2, 3,..., Iˆ J = 1, 2, 3,..., Jˆ K = 1, 2, 3,..., Kˆ ©1999 CRC Press LLC

(3.1)

FIGURE 3.1 Transformation between computational and physical domains.

FIGURE 3.2 Grids in computational and physical domains.

The relationships between the indices I, J, and K and the computational coordinates (ξ ,η ,ζ ) uniformly discretize the computational domain and imply a relationship between discrete neighboring points. The transformation to the physical domain produces the actual grid points, and the relationship of neighboring grid points is invariant under the transformation (Figure 3.2). A grid created in this manner is called a structured grid. TFI provides a single framework creating the function X(ξ ,η ,ζ ).

3.4 Transfinite Interpolation (TFI) Transfinite interpolation (TFI) was first described by William Gordon in 1973 [1]. TFI has the advantage of providing complete conformity to boundaries in the physical domain. In the early 1980s, Lars Eriksson described TFI for application to grid generation for computational fluid dynamics (CFD) [2,3,4]. Variants of TFI have since been described many times [5,6,7].

3.4.1 Boolean Sum Formulation The essence of TFI is the specification of univariate interpolations in each of the computational coordinate directions, forming the tensor products of the interpolations, and finally the Boolean sum. The univariate

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interpolation functions are a linear combination of known (user-specified) information in the physical domain (positions and derivatives) for given values of the computational coordinate and coefficients that are blending functions whose independent variable is the computational coordinate. The general expressions of the univariate interpolations for three dimensions are L

P

U(ξ , η,ζ ) = ∑ ∑ α in (ξ )

∂ n X (ξi , η,ζ ) ∂ξ n

i =1 n = 0 Q

M

V(ξ , η,ζ ) = ∑ ∑ β m j (η )

(

∂ m X ξ , η j ,ζ ∂η m

j =1 m = 0 N

W(ξ , η,ζ ) =

R

∑ ∑ α in (ξ )

)

(3.2)

∂ l X (ξ , η,ζ k ) ∂ζ l

k =1l = 0

Conditions on the blending functions are ∂ n ∂ in (ξi ) ∂ξ n

( ) =δ

ηj ∂m β m j

= δ ii δ nn

∂η m

δ jj mm

∂ lγ kl (ζ k ) ∂ζ l

= δ kk δ ll

(3.3)

i = 1, 2,..., L j = 1, 2,..., M k = 1, 2,..., N n = 0,1,..., P m = 0,1,..., Q l = 0,1,..., R The tensor products are L

N

R

P

UW = WU = ∑ ∑ ∑ ∑ α in (ξ )γ kl (ζ )

∂ ln X(ξi , η,ζ k )

i =1 k =1 l = 0 n = 0

L M

Q

R

∑ (ξ ) (η)

UV = VU = ∑ ∑ ∑

α in i =1 j =1 m = 0 n = 0 Q

M N

VW = WV = ∑ ∑ ∑

R

∑ (ξ ) (ζ ) βm j

j =1 k =1 m = 0 l = 0

L M N

R

Q

βm j

UVW = ∑ ∑ ∑ ∑ ∑

P

γ kl

∂ζ∂ξ n

(

∂ nm X ξ , η j ,ζ ∂η ∂ξ m

(

∂ lm X ξ , η j ,ζ k ∂ζ l ∂η m

∑ α in (ξ )β mj (η)γ kl (ζ )

i =1 j =1 k =1 l = 0 m = 0 n = 0

)

n

)

(3.4)

(

∂ lmn X ξi , η j ,ζ k l

∂ζ ∂η ∂ξ m

)

n

The commutability in the above tensor products is assumed in most practical situations, but in general, it is not guaranteed. It is dependent upon the commutability of the mixed partial derivatives. The Boolean sum of the three interpolations is X(ξ , η,ζ ) = U ⊕ V ⊕ W = U + V + W − UV − UW − VW + UVW

(3.5)

3.4.2 Recursion Formulation The application of TFI as a Boolean sum of univariate interpolations in the computational coordinate directions implies that each of the terms in the sum be evaluated and then the sum is evaluated.

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Alternately, TFI can be expressed as a three-step recursion formula. The first step is to express the univariate interpolation in one coordinate direction L

P

X1 (ξ , η,ζ ) = ∑ ∑ α in (ξ )

∂ n X (ξi , η,ζ )

(3.6)

∂ξ n

i =1 n = 0

The second and third steps use the preceding step. That is

(

)

(

M Q  ∂ m X ξ , η j ,ζ ∂ m X1 ξ , η j ,ζ  − X2 (ξ , η,ζ ) = X1 (ξ , η,ζ ) + ∑ ∑ β m j (η )  ∂η m ∂η m j =1 m = 0 

)   

N R  ∂ l X (ξ , η,ζ k ) ∂ l X2 (ξ , η,ζ k )  − X (ξ , η,ζ ) = X2 (ξ , η,ζ ) + ∑ ∑ γ kl (ζ )  ∂ζ m ∂ζ l   k =1 l = 0

(3.7)

(3.8)

3.4.3 Blending Function Conditions In the above equations, a in ( x ), b mj ( h ), and g kl ( z ) are blending functions subject to δ function con∂ lmn X ( x i, h j, z k ) - in the equations are positions and partial derivatives ditions. The defining parameters --------------------------------------------∂ zl ∂ hm ∂ xn in the physical domain and are user-specified. In this definition, the implicit assumption is that coordinate curves are to be interpolated along with their derivatives. This occurs through a network of intersecting surfaces and derivatives that must be specified.

3.5 Practical Application of TFI In the practical process of generating grids, it is necessary to minimize, or at least keep to a manageable level, the amount of input geometry data (position and derivatives along curves or surfaces). At the same time, it is necessary to maintain a high degree of control, particularly near boundary surfaces for which there may be high gradients in the solution of the governing equations.

3.5.1 Linear TFI The simplest application of TFI is to use linear interpolation functions for all coordinate directions and specify the positional data on the six bounding surfaces (Figure 3.3). P = Q = R = 0 and L = M = N = 2 in Eq. 3.2.

The linear blending functions that satisfy the δ function conditions in Eq. 3.3 are α10 (ξ ) = 1 − ξ α 20 (ξ ) = ξ β10 (η) = 1 − η β 20 (η) = η γ 10 (ζ ) = 1 − ζ γ 20 (ζ ) = ζ

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FIGURE 3.3

Boundary surfaces for linear TFI.

The univariate interpolations and tensor products are U(ξ I , η J ,ζ K ) = (1 − ξ I )X(0, η J ,ζ k ) + ξ I X(1, η J ,ζ K ) V(ξ I , η J ,ζ K ) = (1 − η I )X(ξ , 0,ζ k ) + η I X(ξ I ,1,ζ K )

W(ξ I , η J ,ζ K ) = (1 − ζ K )X(ξ I , η J , 0) + ζ I X(ξ I , η J ,1)

UW(ξ I , η J ,ζ K ) = (1 − ξ I )(1 - ζ K )X(0, η J , 0) + (1 − ξ I ,ζ K )X(0, η J ,1) \

+ ξ I (1 − ζ K )X(1, η J , 0) + X(1, 0,ζ K ) + ξ I η J X(1,1,ζ K )

UV(ξ I , η J ,ζ K ) = (1 − ξ I )(1 - η J )X(0, 0,ζ K ) + ξ Iζ K X(1, η J ,1)

+ ξ I (1 − η J )X(1 - η J )X(1, 0,ζ K ) + ξ I η J X(1,1,ζ K )

VW(ξ I , η J ,ζ K ) = (1 − η J )(1 - ζ K )X(ξ I , 0, 0) + (1 − η J )ζ K X(ξ I ,1, 0) + η J (1 − ζ K )X(ξ I , 0,1) + η J ζ K X(ξ I ,1,1)

UVW(ξ I , η J ,ζ K ) = (1 − ξ I )(1 − η J )(1 − ζ K )X(0, 0, 0) + (1 − ξ I )(1 − η J )ζ K X(0, 0,1) + (1 − ξ I )η J (1 − ζ K )X(0,1, 0) + (1 − ξ I )η J ζ K X(1, 0,1) + ξ I η J (1 − ζ K )X(1,1, 0) + ξ I η J ζ K X(1,1,1)

The expression for a TFI grid ( I = 1, …, Iˆ, interpolation functions (Eq. 3.5) is

J = 1, …, Jˆ ,

K = 1, …, Kˆ ) with linear

X(ξ I , η J ,ζ K ) = U(ξ I , η J ,ζ K ) + V(ξ I , η J ,ζ K ) + W(ξ I , η J ,ζ K ) − UW(ξ I , η J ,ζ K ) − UV(ξ I , η J ,ζ K ) − VW(ξ I , η J ,ζ K ) + UVW(ξ I , η J ,ζ K )

(3.9)

3.5.2 Lagrangian TFI When additional surfaces corresponding to the interior of the computational box can be provided (see Figure 3.4 for the case of two interior surfaces that would correspond to cubic Lagrangian interpolation),

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FIGURE 3.4

Transfinite interpolation with Lagrangian blending functions.

a general formula for the blending functions can be used. The formula for a computational coordinate, for instance, the ξ coordinate is

∏ i ≠ i (ξ − ξi ) α i0

i =1

(ξ ) =

(3.10)

L

∏ i ≠ i (ξi − ξi ) i =1

The univariate interpolation function in the ξ computational coordinate direction is L

U(ξ , η,ζ ) = ∑ α i0 (ξ )

∂ 0 X(ξ , η,ζ )

i =1

∂ξ 0

L

= ∑ α i0 (ξ )X(ξ , η,ζ )

(3.11)

i =1

The Lagrange blending function allows a polynomial interpolation of degree L – 1 through L points and satisfies the cardinal condition a i0 ( x i ) = d ii . It is not recommended that high-degree Lagrangian blending functions be used for grid generation because of the large quantity of geometric data that must be supplied and the potential excessive movement in the interpolation. Using L = 2 results in the linear interpolation above being a special case of Lagrangian interpolation.

3.5.3 Hermite Cubic TFI Often in grid generation, the outward derivative at one or more sides of the physical domain corresponding to sites of the computational domain can be specified. It is then feasible to use Hermite blending functions in the coordinate direction in which derivative information can be specified. For example, if ξ is the coordinate direction, the univariate Hermite interpolation (L = 2, P = 1) corresponding to Eq. 3.2 is 2

1

U(ξ , η,ζ ) = ∑ ∑ α in (ξ ) i =1 n = 0

α10

(ξ )X(ξ1 ,η,ζ )

©1999 CRC Press LLC

+ α 11

∂ n X(ξi , η,ζ ) ∂ξ n

=

∂X(ξ , η,ζ ) ∂X(ξ , η,ζ ) (ξ ) ∂1ξ + α 20 (ξ )X(ξ2 ,η,ζ ) + α 12 (ξ ) ∂2ξ

(3.12)

FIGURE 3.5

FIGURE 3.6

Transfinite interpolation with Hermite cubic blending functions.

Outward derivatives obtained from cross-product of surface derivatives.

where

α10 (ξ ) = 2ξ 3 − 3ξ 2 + 1 α11 (ξ ) = ξ 3 − 2ξ 2 + ξ α 20 (ξ ) = −2ξ 3 + 3ξ 2 α 12 (ξ ) = ξ 3 − ξ 2 The outward derivatives in the ξ coordinate direction can be specified by the cross-product of the tangential surface derivatives in the η and ζ coordinate directions at ξ = 0 and ξ = 1. This effectively creates the trajectories of grid curves that are orthogonal to the surfaces X(ξ1, η, ζ ) and X(ξ2, η, ζ ). That is, ∂X(ξ1 , η,ζ ) ∂ξ

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 ∂X(ξ1 , η,ζ ) ∂X(ξ1 , η,ζ )  = × ψ 1 (η,ζ ) ∂η ∂ζ  

(3.13)

and ∂X(ξ2 , η,ζ ) ∂ξ

 ∂X(ξ2 , η,ζ ) ∂X(ξ2 , η,ζ )  = × ψ 2 (η,ζ ) ∂η ∂ζ  

(3.14)

The scalar functions ψ1( h , z ) and ψ2( h , z ) are magnitudes of the outward derivatives in the ξ direction at X ( x 1 h, z ) and X ( x 2 h, z ). The derivative magnitude parameters can be constants or surface functions. Increasing the magnitudes of the derivatives extends the orthogonality effect further into the physical domain between the two opposing surfaces. However, the magnitudes can be excessively large, resulting in the interpolations equation being multivalued. This is manifested by grid crossover and is remedied by lowering the magnitudes. Note that when the interpolations in the η and ζ directions are applied, the orthogonality effect achieved with the above application of Hermite interpolation in the ξ direction can be altered.

3.6 Grid Spacing Control TFI transforms a rectangular computational domain to a physical domain with irregular boundaries. A uniform grid in the computational domain is obtained by partitioning each computational coordinate into equal increments. With the transformation, the discrete points in the computational domain map into irregular spaced points in the physical domain creating a physical grid. The spacing between points in the physical domain is controlled by the blending functions a in ( x ), b mj ( h ) and g kl ( ζ ). Blending functions that produce the desired shape of a grid (i.e., relative orientation between points) may not produce the desired spacing between points. In order to create grids with desired grid concentrations, additional information must be provided. One approach is to design or modify the blending functions to exactly produce the desired concentrations. Another approach, which is effective and practical, is to define an intermediate control domain between the computational domain and the physical domain. An intermediate control domain is defined to be a rectangular domain where each intermediate coordinate is related to the computational coordinates by u = f (ξ , η,ζ ) v = g(ξ , η,ζ ) w = h(ξ , η,ζ )

(3.15)

Under the application of these functions, uniformly spaced grid points in the computational domain map to nonuniformly spaced grid points in the control domain enclosed by the unit cube (Figure 3.7). The intermediate coordinates u, v, and w must be single-valued functions of f(ξ,η,ζ ), g(ξ,η,ζ ), and h(ξ,η,ζ ), respectively. The blending functions are redefined with the intermediate coordinates as the independent variables. That is a in ( u ), b mj ( v ) and g kl ( w ) . There are many practical considerations to be exercised at this point. The overall TFI formulation will shape a grid to fit the six boundary surfaces. Control functions that manipulate the grid point spacing are applied. These functions can be simple and be applied universally, or they can be complex and blend from one form to another, transversing from one boundary to an opposite boundary. It may be desirable for a control function to cause concentration of grid points at the extremes of the computational coordinate or somewhere in between. A low slope in a control function leads to grid concentration and high slope leads to grid dispersion. Several control functions are described.

3.6.1 Single-Exponential Function A useful function that maps an independent variable, r, 0 ≤ r ≤ 1, to a monotonically increasing dependent variable, r, 0 ≤ r ≤ 1, is

©1999 CRC Press LLC

FIGURE 3.7

FIGURE 3.8

Intermediate control domain.

Single-exponential control function example.

r=

e Aρ − 1 eA − 1

(3.16)

where ρ is assumed to be a computational coordinate and r is assumed to be an intermediate variable. The sign and magnitude of the parameter A specifies whether the lowest slope is near (0, 0) or (1, 1) and the magnitude of the slope (Figure 3.8). For A = 0 the single exponential function is singular and is not useful for producing an exact straight line between (0, 0) and (1, 1). This would correspond to a uniform clustering of the dependent variable. However, a magnitude of A = .0001 will produce a very near straight line. A uniform discrete spacing of the independent variable evaluation of the control ©1999 CRC Press LLC

function produces concentration or dispersion in the discrete values of the dependent variable. Often the r2 value (r1 = 0) at ∆r or the r Nˆ – 1 value ( r Nˆ = 1 ) at 1 – ∆r is specified, and the value of A that causes the function to pass through the point ( ∆r, r 2 ) or ( 1 – ∆r, r Nˆ – 1 ) is determined with a Newton–Raphson iteration. This creates a control function that specifies the spacing between the first and second grid point or the next to last and last grid point in a coordinate direction. Nˆ is the index for the last grid point.

3.6.2 Double-Exponential Function Another function that maps an independent variable, r, 0 ≤ r ≤ 1, to a monotonically increasing dependent variable r, 0 ≤ r ≤ 1, and provides more flexibility than the single exponential is A2 ρ A3

e −1 e A2 − 1 0 ≤ ρ ≤ A3 0 ≤ r ≤ A1

r = A1

A4

ρ − A3 1− A3

−1 e −1 A1 ≤ r ≤ 1

r = A1 + (1 − A1 ) A3 ≤ ρ ≤ 1 A4 chosen ∋

e

A4

Dr( A3 )

(3.17)

⊂ C1

Dρ

The user-specified parameters in Eq. 3.17 are A1, A2, and A3. The parameter A4 is computed. A3 and A1 are the abscissa and ordinate of a point inside the unit square through which the function will pass. A2 and A4 are exponential parameters for the two segments. The derivative condition at the joining of the two exponential functions is satisfied by applying a Newton–Raphson iteration that adjusts the value of the parameter A4. The double exponential control function provides added spacing control as compared to the single exponential function for concentrations near (0, 0) or (1, 1). Also, the double-exponential function allows a grid concentration in the interior or the domain (Figure 3.9). The concept of the doubleexponential function can be extended to an arbitrary number of segments, but it is recommended to keep the number of segments small.

3.6.3 Hyperbolic Tangent and Sine Control Functions Two other single-segment control functions that are used for grid clustering are the hyperbolic tangent (tanh) control function and the hyperbolic sine (sinh) control function. They are r = 1+

r = 1+

(

)

tanh B( ρ − 1) tanh B

(

)

(3.18)

sinh C(1 − ρ )

sinh C 0 ≤ ρ ≤1 0 ≤ r ≤1

(3.19)

where the parameters B and C govern the control functions and their derivatives. The hyperbolic tangent function in many references is a preferred control function for clustering grid points in a boundary-layer for computational fluid dynamics applications.

©1999 CRC Press LLC

FIGURE 3.9 Double-exponential control function example.

FIGURE 3.10 Arclength control function example.

3.6.4 Arclength Control Functions Very often an existing sequence of grid points along a coordinate curve, for instance, along a boundary curve, is known (Figure 3.10). It is desirable to use the sequence of points to create a control function. This can be done by normalizing the indices of the points to create the independent variable and computing the normalized accumulated chord lengths along the sequence of points to create the dependent variable. This process approximates the normalized arclength along the curve. A sequence of points is {xI,J,K, yI,J,K, zI,J,K, I = 1, 2, …Nˆ } and J and K are fixed, the formulae for the independent variable r, 0 ≤ r ≤ 1, and the dependent variable r, 0 ≤ r ≤ 1, are

©1999 CRC Press LLC

FIGURE 3.11

Boundary-blended control function example.

ρI = sI =

I −1 Nˆ − 1

( x I, J , K − x I −1, J , K )2 + ( yI, J , K − y1, J , K )2 + (z I, J , K − z I −1, J , K )2 + sI −1 rI =

(3.20)

sI s Nˆ

Note that if the number of grid points to be used in the grid generation formula (i.e., TFI) is Nˆ , there is no need to compute the independent variable ρI. If, however, the number of grid points in the coordinate direction is different from Nˆ , then the dependent variable rI must be interpolated from the normalized approximate arclength evaluation, and the independent variable values ρI are necessary.

3.6.5 Boundary-Blended Control Functions One of the practical problems that occurs in grid generation is the need to have different control functions specified along each edge of the intermediate domain and compute blended values of the intermediate variables interior to the domain. Soni [8] has proposed a blending formula for arclength control functions along the boundary edges that is very useful. This formula also is applicable for other control functions defined along the edges (Figure 3.11). A two-dimensional description of this type of blending is shown. Let s1(ξ ), 0 ≤ s 1 ( x ) ≤ 1, and s2(ξ ), 0 ≤ s 2 ( x ) ≤ 1, be control functions along the edges spanning between t1(η = 0), t2(η = 0) and t1(η = 1), t2(η = 1). Let t1(η), 0 ≤ h ≤ 1, 0 ≤ t 1 ( h ) ≤ 1 and t 2 ( h ), 0 ≤ h ≤ 1, 0 ≤ t 2 ( h ) ≤ 1 , be control functions along the edges spanning between s1(ξ = 0), s2(ξ = 0) and s1(ξ = 1), s2(ξ = 1). The blended values of intermediate control variables are u=

(1 − t1 (η))s1 (ξ ) + t1 (η)s2 (ξ ) 1 − ( s2 (ξ ) − s1 (ξ ))(t2 (η) − t1 (η))

(1 − s1 (ξ ))t1 (η) + s1 (ξ )t2 (η) v= 1 − (t2 (η) − t1 (η))( s2 (ξ ) − s1 (ξ ))

©1999 CRC Press LLC

(3.21)

3.7 Conforming an Existing Grid to New Boundaries TFI is normally used to generate a grid given three pairs of defined opposing boundaries. A variation of TFI can also be used to adjust an existing grid to three new pairs of opposing boundaries. This TFI variation can be stated in the following way. Note that x I , h J , and z K are replaced with the indices I, J, and K. ˆ (I, J, K), I = 1, 2…Iˆ, J = 1, 2, …Jˆ , K = 1, 2, …Kˆ and boundary surface grids Given a grid X X(1, J, K), X( Iˆ , J, K), X(I, 1, K), X(I, Jˆ , K), X(I, J, 1), and X(I, J, Kˆ ), an adjusted grid X(I, J, K), can be produced by

[

X1 ( I , J , K ) = Xˆ ( I , J , K )

[(

) (

)]

[(

)

(

)]

[(

)

(

)]

]

+α 10 (ξ ) X (1, J , K ) − Xˆ (1, J , K ) + α 20 (ξ ) X Iˆ, J , K − Xˆ Iˆ, J , K X2 ( I , J , K ) = X1 ( I , J , K )

+ β10 (η)[ X ( I ,1, K ) − X1 ( I ,1, K )] + β 20 (η) X I , Jˆ, K − X1 I , Jˆ, K X ( I , J , K ) = X2 ( I , J , K )

+γ 10 (ζ )[ X ( I , J ,1) − X2 ( I , J ,1)] + γ 20 (ζ ) X I , J , Kˆ − X2 I , J , Kˆ

(ξ ) = 1 − u1 (ξ ) α 20 (ξ ) = u2 (ξ )

α10

β10 (η) = 1 − v1 (η) β 20 (η) = v2 (η) γ 10 (ζ ) = 1 − w1 (ζ ) γ 20 (ζ ) = w2 (ζ ) u1 (ξ ) =

e C1ξ − 1 e C1 − 1

u2 (ξ ) =

e C2 ξ − 1 e C2 − 1

v1 (η) =

e C3η − 1 e C3 − 1

v2 (η) =

e C4η − 1 e C4 − 1

w1 (ζ ) =

e C 5ζ − 1 e C5 − 1

w2 (ζ ) =

e C6ζ − 1 e C6 − 1

where the constants C1, C2, …C6 specify how far into the original grid the effect of the six boundary surfaces is carried.

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3.8 Summary TFI generates grids that conform to specified boundaries. The recipe is a Boolean sum of univariate interpolations, and it is also expressed as a recursion formula. Since any univariate interpolation subject to δ conditions can be applied in a coordinate direction, there are an infinite number of variations of TFI. However, low-order univariate interpolation functions are the most practical. Lagrangian and Hermite cubic formulae have been presented. Grid spacing control can be best achieved by creating intermediate variables to be used in the interpolation functions. The intermediate variables are computed with control functions whose independent variables are computational coordinates and have adjustable parameters affecting spacing. Several examples of practical control functions have been presented. A variation of TFI to conform an existing grid to new specified boundaries has also been represented. This minor variation is highly useful in a practical grid generation environment.

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