chapter 9: surface grid generation systemsebrary.free.fr/mesh generation/handbook_of_grid... ·...

9Surface Grid

Generation Systems

9.1 Introduction9.2 Algebraic Surface Grid Generation

Distribution of Grid Points on the Boundary Curves • Interpolation of Grid Points Between Boundary Curves • NURBs Surface Grid Generation Examples

9.3 Elliptic Surface Grid GenerationConformal Mapping on Surfaces • Formulation of the Elliptic Generator • Numerical Implementation • Control Function

9.4 Summary and Research Issues

9.1 Introduction

Structured surface grid generation entails the generation of a curvilinear coordinate grid on a surface.It may be necessary to generate such a grid in order to perform a two-dimensional numerical simulationof a physical process involving the surface. Alternately, surface grid generation may represent a stage inthe generation of a volume grid, which itself would be used in a three-dimensional numerical simulationinvolving the volume or volumes bounded by the surface. We mention here that unstructured surfacemesh generation (wherein the surface is usually decomposed into a collection of triangles but no obviouscurvilinear coordinate system exists) is covered in Chapter 19. Unstructured surface meshes are arguablyeasier to construct and have found wide application in numerical simulation as well.

Grid quality is a critical area for many numerical simulation problems. The distribution of the gridpoints and the geometric properties of the grid such as skewness, smoothness, and cell aspect ratios havea major impact on the accuracy of the simulation. The solution of a system of partial differential equationscan be greatly simplified by a well-constructed grid. It is also true that a grid which is not well suited tothe problem can lead to an unsatisfactory result. In some applications, improper choice of grid pointlocations can lead to an apparent instability or lack of convergence. This chapter will cover techniquesfor the generation of structured surface meshes of sufficient quality for use in physical simulations.

Before a grid can be generated, the surface geometry itself must be created, usually by one of twomethods. In the first method, the object to be simulated has a shape that can be calculated from amathematical formula, such as a sphere. There are a wide variety of shapes in this class, including airfoils,missile geometries, and sometimes even complete wings. These types of shapes are very easy to define,and lead to an efficient grid generation process, with high-quality resulting grids.

The second manner in which surface geometries are specified involves representation of the initialgeometry as a computer-aided design (CAD) surface, where CAD systems typically represent the surfacesof a certain geometry with a set of structured points or patches. The CAD surface is then typicallyconverted to a nonuniform rational B-splines (NURBS) surface representation (cf. Part III).

Ahmed Khamayseh

Andrew Kuprat

©1999 CRC Press LLC


In any event, we presume that the surface geometry is available as a

parametrically defined surface

suchas a

quadric surface, Bezier surface, B-spline surface,

or

NURBS surface.

We thus presume the existence ofa surface geometry definition in the form of a mapping (

x

(

u,v

),

y

(

u,v

),

z

(

u,v

)) from a

parametric

(

u,v

)

domain

to a

physical

(

x,y,z

)

domain.

This mapping is assumed differentiable, and we assume that themapping and its derivatives can be quickly evaluated. We compactly denote this mapping as

x

(

u

), where

x

= (

x,y,z

), and

u

= (

u,v

).In structured surface grid generation, the actual grid generation process is the generation of a mapping

from the discrete rectangular

computational

(

ξ,η

)

domain

to the parametric (

u,v

) domain, which resultsin the composite map

x

(

ξ,η)

= (

x

(

ξ,η

),

y

(

ξ,η

),

z

(

ξ,η

) (see Figure 9.1). As seen in the figure, the physical space is a subset of

IR

3

; the parametric space is a subset of

IR

2

, whichis taken to be the [0,1]

×

[0,1] unit square. Technically speaking, the computational space is a discreterectangular set of points (

ξ,η

), , . However, in order for us to be able toapply the powerful machinery of differentiable mappings between spaces, we extend the computationalspace to be a continuum, so that it is the rectangle [0,

m

]

×

[0,

n

]. This is what is depicted in Figure 9.1.(Note:

In this chapter the coordinates of a point in computational space are sometimes denoted by (

ξ,η

),and other times (

i,j

). The (

i,j

) notation is usually used in algorithms where

i,j

take on only integer values,while the (

ξ,η

) notation is usually used in mathematical derivations where

ξ,η

can take on continuumvalues.)

With regard to the composite map

x

(

ξ,η

)

or

the mapping

u

(

ξ,η

), we define

grid lines

to be lines ofconstant

ξ

or

η

,

grid points

to be points where

ξ,η

are integers, and

grid cells

to be the quadrilateralsformed between grid lines. It will always be clear if by

grid lines

,

grid points

, or

grid cells

we are referringto objects on the gridded surface or to objects in the parametric domain.

The surface geometry

x

(

u

) may contain some singularities (e.g., the mapping of a line to a point in acertain parameterization of a cone). We require that the composite map

x

(

ξ,η

) =

x

(

u

)

o

u

(

ξ,η

) notcontain any additional degeneracies. This leads to the requirement that

u

(

ξ,η

) be one-to-one and onto.If a

u

(

ξ,η

) mapping is generated which is not one-to-one and onto, quite often the problem will bedetected as a visible “folding” of grid lines when the gridded surface is viewed using computer graphics.

That

u

(

ξ,η

) should be an isomorphism is a “bare bones” requirement. It is usually also required thatthe

u

(

ξ,η

) map be constructed such that the composite map

x

(

ξ,η

) have the following properties in theinterest of reducing errors occurring in numerical simulations that use the grid:

1. Grid lines should be smooth to provide continuous transformation derivatives.2. Grid points should be closely spaced in the physical domain where large numerical errors are expected.3. Grid cells should have areas that vary smoothly across the surface.4. Excessive grid skewness (nonorthogonal intersection of grid lines) should be avoided, since it

sometimes increases truncation errors.

FIGURE 9.1

Mapping from computational (“

ξ

,

η

”) space to physical (“

x

,

y

,

z

”) space via parametric (“

u

,

v

”) space.

x 0 1 … m, , ,{ }Œ h 0 1 … n, , ,{ }Œ

In order to generate surface grids with the above requirements, two approaches,

algebraic

and

elliptic,

have been most popularly embraced in the mesh generation community. This chapter covers these two


techniques in some detail, presenting practical algorithms as well as theoretical development. Both thesemethods complement each other and both are typically used in a complete grid generation system.

Algebraic mesh generation proceeds in stages. The grid is first constructed on the boundary curves,and a surface grid is then constructed by algebraic interpolation between the boundary curves. In fact,one could then continue further by constructing an interpolated volume grid between bounding surfacegrids. This process can itself be a complete method for the generation of meshes. Indeed, a certaininterpolation method that we describe — cubic Hermite interpolation — can be used to generate surfacemeshes that possess boundary orthogonality required in certain numerical simulations. Usually, however,the simplest form of algebraic mesh generation — linear transfinite interpolation — is used to producea valid “initial” mesh that can then be smoothed by another method to satisfy possible requirements ongrid line orthogonality or grid point distribution.

Elliptic mesh generation is the natural complement to the above process. An initial grid, usuallyproduced by algebraic methods, is smoothed by iteratively solving a system of partial differential equationsthat relate the physical (x,y,z) and computational (ξ,η ) variables. Desired orthogonality properties anddesired grid point distributions in the physical domain are effected by imposing appropriate boundaryconditions and/or source terms in the elliptic system of equations. An alternative technique for smoothinginitial grids to produce desired properties are the variational methods in Brackbill and Saltzman [5],Castillo [6], and Saltzman [18]. They will not be covered in this chapter.

Related surveys on algebraic methods and the use of transfinite interpolation in grid generation canbe found in Abolhassani and Stewart [1], Chawner and Anderson [7], Smith [19], and Soni [20]. Forsurveys on elliptic methods in grid generation, we refer the reader to Khamayseh and Mastin [12],Sorenson [21], Spekreijse [22], Thomas and Middlecoff [26], Thompson et al. [27], Thompson [29],Warsi [30], and Winslow [33]. For further study on the foundations and fundamentals of grid generation,we refer to Knupp and Steinberg [13] and Thompson et al. [28].

Finally, we refer the reader to other related chapters in this book; these are Chapter 3 on TFI generationsystems, Chapter 4 on elliptic generation systems, Chapter 6 on boundary orthogonality, Chapter 7 onorthogonal generation systems, and Part III on surface generation. Although we cite individual papersthroughout this chapter, in most cases referral to these chapters will suffice.

9.2 Algebraic Surface Grid Generation

Algebraic surface grid generation involves (1) distribution of grid points along the boundary curves and(2) bidirectional interpolation usually called transfinite interpolation (TFI), which defines the remainingpoints, while simultaneously matching all four boundary curves (cf. Chapter 3). Step (2) can be doneby unidirectional interpolation between boundaries, but this is not as reliable or popular an approach.The transfinite interpolation will incorporate the specified spacing at the boundaries, and possiblyorthogonality conditions as well. Grid orthogonality at the boundaries, wherein the grid intersects theboundaries as close as possible to a 90° angle, can be crucial in certain numerical applications.

Since interpolation is fundamentally projection from boundaries, problems can arise in configurationsin which the line of sight to boundaries in the parametric plane is not present. In this case, the user mustbreak the surface into a sufficient number of subsurface patches to alleviate the problem. In the following,we assume that we are to generate a grid on a reasonably well-behaved subsurface patch.

9.2.1 Distribution of Grid Points on the Boundary Curves

The methodology of constructing an (m + 1) × (n + 1) algebraic grid on a physical surface starts withthe specification of the boundary distribution along the physical boundaries of the surface. This isequivalent to specifying the distribution of the four boundary curves in the parametric domain:

u u n v v m m nξ ξ η η ξ η, , , , , , ,0 0 0 0( ) ( ) ( ) ( ) ≤ ≤ ≤ ≤{ }


Without loss of generality, let us generate the points on the “lower” boundary curve {u(ξ,0)|0 ≤ ξ ≤ m}.This curve in parametric space corresponds to the curve {x(u,0)|0 ≤ u ≤ 1} in physical space. The treatmentof the other three (“upper,” “left,” and “right”) boundary curves will be similar. For convenience, wesuppress the constant second arguments of x and u, so that we have

and our task is to find so that is a “good” parameterizationof the boundary curve x(u).

The task of finding u(ξ ) is of course equivalent to finding ξ (u). Now let us define . Then

We see that finding ρ is equivalent to obtaining ξ. However, ρ is readily seen to be the desired grid pointdensity, which can be dictated in a straightforward manner from physical considerations.

Indeed, physical considerations may guide the user to desire

1. Equal arc length spacing wherein points are spaced at equal distances in physical space. In thiscase, grid point density should be proportional to the rate of change of arc length. That is, .

2. Curvature-weighted arc length spacing, wherein points are connected in areas of large curvature.In this case, we have

where κ (u) is the curvature of the boundary curve x(u) at u.

3. Grid attraction to an attractor point u* in parametric space corresponding to a point x* = x(u*)in physical space. A typical case is u* = 0 or u* = 1, when one has interesting physical phenomena(such as a Navier–Stokes boundary layer) at one end of the boundary curve. Or perhaps we mighthave 0 < u* < 1, with a point in the interior of the curve being of interest. In either case, a goodchoice for ρ is

where κ is a strength factor that determines the degree of attraction to u*.

9.2.1.1 Hybrid Grid Density Functions

In practice, the user will likely desire a hybrid grid density function that is a linear combination of severalother grid density functions. Assume we have grid density functions ρi, each normalized so that

. Then if we have positive constants λ i such than Σλi = 1, we have that ρ = Σ λ i ρi

is a grid density function with suitable normalization. This hybrid density function will attempt to movegrid points into regions where any one of the functions ρi desires grid points. Thus one could distribute

u u

x u x u

ξ ξ( ) ≡ ( )( ) ≡ ( )

,

,

0

0

u x( ) 0 x m≤ ≤{ } x u x( )( ) 0 x m≤ ≤{ }

r u( ) dxdu------≡

ξ ρu w dwu

( ) = ( )∫0

r x ′∝

ρ κ∝ ( ) ′u x

ρκ

u uu u

∗

∗( ) ∝

−( )( ) +

1

12

ri ud0

1

∫ x 1( ) x 0( )– m= =

grid points based on the hybrid criteria of arc length, curvature, and attraction to a set of distinct

points.

ui∗{ }


This hybrid approach is the most useful, since it can accommodate many different situations that arisein practice. In this section, we will present an algorithm for grid point distribution along boundary curvesbased on a hybrid grid density function. The general principle of the algorithm is that (1) we construct

ρ(u) on a relatively fine grid of points where M is 5–10 times m, (2) the grid function

ξ(u) = dw is evaluated by integrating ρ on the fine grid, and (3) the curve points u(ξ ) are

generated in the parametric space of the curve by inverting the grid function ξ(u). Note: Withoutcomputing ξ(u) on a finer grid than that desired for u(ξ ), step (3) would be prone to inaccuracy, possiblyleading to an unacceptable grid distribution.

Before we present the algorithm, we touch on a few technical points.

1. The grid density function for arc length is given by

Here is the normalization required so that . If u = and

, we use the approximation.

2. The grid density function for curvature-weighted arc length is

By definition κ (u) = where dθ is the angular change in the direction of the tangent of the

curve during a small traversal of arc length ds along the curve. Thus

If u = we use the approximation

where is the unit tangent vector to the physical curve at If the total integrated

curvature is less than some minimal angular tolerance (say εκ =

.01 radian), then we remove curvature weighted arc length as a criterion for grid point distributionand replace it with a simple arc length criterion. We do this to avoid distributing points based ona quantity which is essentially absent, which can lead to a nonsmooth distribution.

ui

˜ im---- 0 i M,≤ ≤,=

r w( )0

u

∫

ρs um u

w dw( ) =

′( )( )∫x

x0

1

m

x w( ) wd0

1

∫------------------------------ rs u( ) ud

0

1

∫ m= ui

du ui ui 1––=

′( ) ≈ ( ) − ( )−x x x˜ ˜ ˜u du u ui i i 1

ρκ

κκ u

m u u

w w dw( ) =

( ) ′( )( ) ′( )∫

x

x0

1

dqds------

κ θ θu u

d

ds

ds

du

d

du( ) ′( ) = =x

ui

κ θ θ˜ ˜u u du t ti i i i i i( ) ′( ) ≈ − = −− −x 1 1

ti

x' ui( )x' ui( )

------------------≡ ui

k u( ) x' u( ) u ti t– i 1–

i 1=

M

∑≈d0

1

∫

3. The grid density function for attraction (with strength k) to a point u* is given by


(9.1)

If u* = 0, we have

This leads to a grid distribution of the form

It has been noted that the smoothness of this distribution in the vicinity of u* = 0 results in smallertruncation errors in finite difference discretizations than “exponential” distributions that approach thepoint of attraction in a more severe fashion, see Chapter 32 and Thompson et al. [28].

Algorithm 2.1 Hybrid Curve Point Distribution Algorithm

Assume physical curve x(u), . Given weights λs,λκ, points {u *i | }, weights{λi | } and strengths { } with λs + λκ + Σp

i=1λ i = 1, we create a distribution ofm + 1 points u0,u1,K,um that are simultaneously attracted to each of the points in {ui*}, placed in regionsof high curvature, and placed to avoid large gaps in arc length. User also specifies a parametric grid size

and minimum integrated curvature tolerance εκ . (We suggest M = 5m and εκ =.01)

1. Initialize grid function ξ to zero.

2. Compute arc lengths. Rescale so that maximum scaled arc length is m. Add to ξ, weighted by λs.

ρ

ρ

u

u

u

u mk u u k w u

dw

w dw mk u u ku

k 1 u ku

∗

∗

( ) =−( )( ) + −( )( ) +

( ) =−( )( ) + ( )−( )( ) + ( )

∗

∗ ∗

∗ ∗

∫

∫

1

1

1

12 20

1

0

*

arcsinh arcsinh

arcsinh arcsinh

ρ00w dw m

ku

k

u

( ) = ( )( )∫ arcsinh

arcsinh

umξ

αξ

α( ) =

sinh

sinh

0 u 1≤ ≤ 0 ui∗ 1 1 i p≤ ≤,≤ ≤1 i p≤ ≤ ki ki 0 1 i p≤ ≤,≥

M m≤

Do

i

i M=←

1

0

,...,

ξ

Do

+

Do

s

i M

s si

M

i

M

i M

s ms

s

s

i i

ii

M

i i s i

0

1

0

1

1

1

←=

←

− −

=

←

← +

−

,...,

,...,

x x

ξ ξ λ

3. Compute curvature weighted arc lengths on fine grid. Check if curve has nontrivial amount ofcurvature. If so, normalize to m, and add into ξ, weighted by λκ . Otherwise, use arc length instead.


4. Add in contributions to grid function due to attractor points.

5. Obtain point distribution by inverting grid function.

9.2.1.2 Determination of Weights �s,�� ,�1 and Strengths ki

When using the boundary point distribution algorithm, one must choose weights λs,λκ ,λi and strengthski. As a rough guide, we find it is sufficient to set the weights for each desired criterion to be equal (andto add to 1). So for example, if we desire distribution on arc length and two attractor points, we wouldset λs = λ1 = λ2 = . (In this case, we would set λκ = 0.)

Do

Do

If then

Do

Else

Do

i M

ii

M

i

M

i M

i i

i M

m

i M

i i

M

ii

M

i i i

i i

=

( ) ← ′ ′

←=

← + ( ) − −( )≥( )=

←

← +

=← +

−

0

0

1

1

1

1

0

1

,...

,...

,...

,...

t x x

t t

θ

θ θ

θ ε

θ θθ

ξ ξ λ θ

ξ ξ λ

κ

κ

κκ si

Do

arcsinh arcsinh(

arcsinh k arcsinh

j p

Do i M

mk

iM

u k u

u k ui i j

j j j j

j j j j

==

← +−

+

−( )( ) +

∗ ∗

∗ ∗

1

1

1

,...,

,...,

)

( )ξ ξ λ

ξ

ξξ

ξ ξ

M

i

ji

i ij

m m

u

j

i M

j

ui

M

i

M

j

Mu

i

M

j j

← ( )←

←=

≤

← − −−

< ≤

← +−

Force final grid function value to be exactly ,

Do

Do while

Obtaini -1

using linear interpolation.

0

1

0

1

1

1

,...,

( )

13---

As far as setting the strengths ki on the attractor points ui*, one needs to consider the degree ofconcentration required by the particular application. We consider the case of a single attractor point u*


= u1*. From Eq. 9.1 we have that

So

(9.2)

Thus, for example, setting k = 100 would give us ρ( , which means that the grid lines arepacked in the neighborhood of u* at a density in excess of 10 times of the average grid density ρave = m.Now suppose that the user is required to construct a grid with a specified value of – that is,a specified excess grid density at the attractor u*. As a rough guide, we recommend trying the heuristic

(9.3)

and adjusting it as needed. Although one could solve the nonlinear Eq. 9.2 for k exactly, the presence ofother criteria (such as arc length, curvature, or other attractor points) muddles the analysis, so that onein practice tries Eq. 9.3 and adjusts k as necessary.

If one desires a certain grid spacing ∆x in the region near x* = x(u*), we note that

Using Eq. 9.3, we conclude that

is a rough estimate for the strength k required to obtain a grid with the desired spacing ∆x near theattractor x* = x(u*) on the physical curve x(u(ξ )).

9.2.2 Interpolation of Grid Points Between Boundary Curves

The second step in algebraic grid generation involves interpolation from the boundary curve distributionsonto the interior of the surface. This is equivalent to finding the interior points in parametric space:

given that we know the boundary distributions in parametric space:

p u mk k u u

k u ku( ) =

−( )( ) +

−( )( ) + ( )∗

∗ ∗

21

1arcsinh arcsinh

ρ u mk

k u kum

kk

∗∗ ∗( ) =

−( )( ) + ( ) ≥

arcsinh arcsinh arcsinh1 22

u∗ ) 10m≥

r u∗( ) m⁄

ku

m=

( )∗

15ρ

∆x uu

uu u u u= = ′ ⋅ =

′( )( )= =

∗

∗∗ ∗x x

xξ ξ ρ

ku

m x=

′( )∗

15x

∆

u v m nξ η ξ η ξ η, ,( ) ( ) < < < <{ }, 0 0

u u n v v m m nξ ξ η η ξ η, , , , , , ,0 0 0 0( ) ( ) ( ) ( ) ≤ ≤ ≤ ≤{ }


The technique for accomplishing this is called transfinite interpolation (Chapter 3), which generatesan interpolated grid while matching all four boundaries at all points. When performing interpolationcalculations, it is mathematically convenient to rescale the domain (ξ,η) space to be the unit square. Wethus define

and our task is made equivalent to finding

given that we know the boundary curves

(9.4)

As always, “i,j” will denote coordinates in computational space. So, for example (us)i,j means evaluated at ξ = i,η = j, or equivalently at .

Transfinite interpolation involves the sum of unidirectional interpolation in both the “s” and “t”directions, minus a tensor product interpolation that ensures the simultaneous matching of all fourboundaries. Symbolically, this is written as

(9.5)

Here usi,j is obtained by interpolation in s between the uo,j and um,j and ut

i,j is obtained by interpolation int between ui,0 and ui,n. ust

i,j is obtained by the composite operation of (1) interpolation in t between thefour corners u0,0,u0,n,um,0,um,n to produce interpolated u0,j,um,j values, and (2) interpolation in s betweenthe interpolated u0,j,um,j values. (Note: It will be seen in the expressions that follow that the order of s-and t-interpolation in the evaluation of ust

i,j could be interchanged with no change in the result.) In thissection, we give explicit formula for two kinds of transfinite interpolation schemes corresponding to twodifferent choices for the underlying unidirectional interpolation scheme.

Our first set of transfinite interpolation formulas assume that the underlying unidirectional interpo-lation scheme is simply linear interpolation. The formulas for this kind of interpolation are given by

(9.6)

The (u,v) values computed by the above formula may produce a surface grid suitable for manyapplications. However, it is possible that the grid might be unsuitable due to nonorthogonality of the grid

s m t nξ η ξ ξ η η, ,( ) ≡ ( ) ≡

u s t v s t s t, , ,( ) ( ) < < < <{ } 0 1 0 1

u s u s v t v t s t, , , , ,0 1 0 1 0 1 1( ) ( ) ( ) ( ) ≤ ≤ ≤ ≤{ , , 0

du ds⁄s si

im---- t,≡ t j

jn---≡= =

u u u ui j i js

i jt

i jst

, , , ,= + −

uu

u

uu

u

uu u

u u

i js i

i

T

m j

i jt i

i n

Tj

j

i jst i

i

Tn

m m n

j

j

s

s

t

t

s

s

t

t

,,

,,

,

,

, ,

=−

=

−

=−

−

1

1

1 10

0

0, j

,0

0,0

lines. In this case, the grid is still suitable as a starting grid for elliptic smoothing iterations which canimpose orthogonality of the grid lines at the boundaries.

Alternately, if the surface grid generated using Eq. 9.6 is unacceptable due to nonorthogonality at theboundaries, one may rectify the problem by using Hermite cubic transfinite interpolation. The formulasfor this kind of interpolation allow the direct specification of derivatives at the boundaries, which meansthat orthogonality can be imposed.

Cubic Hermite transfinite interpolation is given by Eq. 9.5, where now

(9.7)

Here

which obey the conditions

Note: The above expressions for usi,j, ut

i,j, usti,j can be also used in the context of surface generation, rather

than grid generation. In other words, by viewing u(s,t) as a mapping from parametric space to physicalspace, one could use these expressions to generate a surface patch that matches the specified physicalboundary curves. This type of surface patch is known as a Coons patch, see Part III and Farin [9] andYamaguchi [34].

u

u

u

u

u

u

u

u

u

u

i js

i

i

i

i

T

j

s j

s m j

m j

i jt

i

t i

t i n

i n

H s

H s

H s

H s

,

,

,

,

,

,

,

,

,

,

=

( )( )( )( )

( )( )

=( )( )

00

01

11

10

0

0

0

0

( )( )( )( )

=

( )( )( )( )

( ) ( )

H t

H t

H t

H t

H s

H s

H s

H s

j

j

j

j

i jst

i

i

i

i

T

t t n

00

01

11

10

00

01

11

10

0 0 0 0 0

u

u u u u

,

, , , 00

0 0 0 0 0 0

0 0

0 0

00

01

11

,

, , , ,

, , , ,

, , , ,

n

s st st n s n

s m st m st m n s m n

m t m t m n m n

j

j

j

H t

H t

H t

u u u u

u u u u

u u u u

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

( ) ( )

( )( )( )

HH tj10( )

H t t t

H t t t

H t t t

H t t t

00 2

01 2

11 2

10 2

1 2 1

3 2

1

1

( ) = −( ) +( )( ) = −( )( ) = −( )

= −( )( )

d H

dtt

αβα

α β βα αβ δ α α β β′

′

′′( ) = ′ ′ { }∈,

, , , , , 0 1


The above formulas are not complete until we can supply the normal derivatives us at the left andright boundaries and ut at the bottom and top boundaries. We also need the “twists” ust at the four


corners. It turns out that the assumption of orthogonality of grid lines at the boundaries in the physicaldomain will allow us to supply the normal derivatives in the parametric domain. The twists will then bechosen to be consistent with these normal derivatives.

We now consider computation of the normal derivative us at the left and right boundaries, and ut atthe top and bottom boundaries. The computation of these derivatives is equivalent to the computationof uξ and uη, since us = muξ and ut = nuη. To compute uξ , at the left and right boundaries in parametricspace, we first assume boundary orthogonality in physical space. That is, we assume

Thus,

Now on these boundaries we know that uη = 0. We also know that vη ≠ 0 because the density of gridpoints on the boundaries is finite everywhere. Using this, we easily derive

Denoting the metric tensor components by = , 12 = 22 = , this is equivalentlywritten as

(9.8)

This determines the normal derivatives uξ to within a constant. To determine the magnitudes of thederivatives, we need to add one more piece of data, which is the spacing off of the boundary:

We have found that a good spacing is obtained from linear transfinite interpolation as follows. Wecompute Eq. 9.5 using Eq. 9.6, denoting the normal derivatives computed at the boundary by

Here, for the left boundary, (uoξ)0,j = u1,j – u0,j, where u1,j was computed by Eq. 9.5 and Eq. 9.6. For the right

boundary (uoξ) m,j = um,j – um–1,j where again um–1,j was computed by Eq. 9.5 and Eq. 9.6. Now we specify that

the new grid spacing ||xξ || should be equal to xoξ projected onto the orthogonal direction xξ / ||xξ || off the

boundary. The idea is that the correct positions of the interior grid points in our final grid will be obtainedby having the interior grid points of the linear TFI grid slide along the first interior grid line until theyare in orthogonal position (see Figure 9.2). This condition is

(9.9)

x xξ η⋅ = 0

x x x xu v u vu v u vξ ξ η η+( ) ⋅ +( ) = 0

x x x xu v v vu v⋅( ) + ⋅( ) =ξ ξ 0

g11 xu xu⋅ g xu xv⋅ g xv xv⋅

g u g v12 22 0ξ ξ+ =

xξ ξ ξ ξ ξ= + +v

g u g u v g v112

12 2222

x x xξ ξ ξ0 0 0= +u vu v

x xx

xξ ξξ

ξ

= ⋅0


Solving both Eq. 9.8 and Eq. 9.9, we obtain

Using similar reasoning at the “bottom” and “top” boundaries, we obtain

Here, for the bottom boundary, (uoη) i,0 = ui,1 – ui,0, where ui,1 was computed by Eq. 9.5 and Eq. 9.6. For

the top boundary, (uoξ) i,n = ui,n – ui,n–1 where again ui,n–1 was computed by Eq. 9.5 and Eq. 9.6. Thus, the

desired normal derivatives are given by

(9.10)

Thus it appears that we can use substitution of Eq. 9.10 into Eq. 9.7 to obtain algebraic surface gridswith perfectly orthogonal grid lines at the boundary. Unfortunately, our normal derivatives will in generalnot satisfy the following compatibility conditions:

(9.11)

FIGURE 9.2 Derivation of grid spacing off boundary from linear TFI.

uξ ξ ξ=

ug

gu0 0, - 12

22

uη η η= −

g

gv v12

11

0 0,

u

u

s

t

m ug

gu

ng

gv v

= −

= −

ξ ξ

η η

0 12

22

0

12

11

0 0

,

,

lim

lim

ts s

st t

t

s

→

→

( ) = ( ){ }

( ) = ( )∈

β

α

α α β

α ββ α β

u u

u u

, ,

,

, , .

,0 1

This is because the right-hand side values are determined by the boundary data Eq. 9.4, while the left-hand side values are determined by the orthogonality conditions Eq. 9.10, and these can be very easily


inconsistent.Since Eq. 9.11 is violated, it is necessary to relax the orthogonality conditions in some vicinity of the

corners. Although elliptic methods in the next section allow this vicinity to be quite small, algebraicmethods are quite fragile, and so it is in practice best to impose exact orthogonality Eq. 9.10 only at themidpoint positions

Normal derivatives between the midpoints and the corners are then computed using cubic Hermiteinterpolation. Thus for the derivatives along the “left” and the “right” boundaries,

(9.12a)

Similarly, for the “top” and “bottom” boundaries, we have

(9.12b)

Note: In case one or more of the four boundaries does not require orthogonality (e.g., the boundary isan internal boundary dividing two subsurface patches), we can use a Hermite interpolation schemesimilar to Eq. 9.12 to interpolate all the derivatives on the curve. So for example, for the bottom curve,a purely interpolated (nonorthogonal) derivative would be

Violation of consistency conditions also causes problems for the twists ust, see Farin [9]. In general,neither the orthogonal derivatives Eq. 9.10 nor the interpolated derivatives Eq. 9.12 will satisfy

(9.13)

This means that the twists ust(α,β ) are not necessarily well-defined. (Indeed, if Eq. 9.11 is also false, theone or both sides of Eq. 9.13 may be infinite!)

A practical resolution of this is to compute the twists ust(α,β ) using a finite difference formula witha sufficiently large finite difference increment to “blur” the inconsistencies. For the twist ust(0,0), such aformula is suggested by Figure 9.3. Here

s t, , , , , , , ,( ) =

0

12

112

12

012

1

u

u u

u u

s

s s

s s

t

H t H t

H t H t t

α

α α

α

α α

,

( ) , ,

, ,

( ) =

+ ( ) ( )

{ }

−( )

+ −( ) ( ) < <

∈

10

00

10

00

212

2 0

2 212

2 2 112

1

0 < t <12

0,1

u

u u

u u

t

t t

t t

s

H s H s s

H s H s s

,

( ) , ,

, ,

β

β β

β

β β

( ) =

+ ( ) ( )

{ }

−( )

+ −( ) ( ) < <

∈

10

00

10

00

212

2 0

2 212

2 2 112

1

0 < <12

0,1

u u ut t ts H s H s s, , ,0 1 0 0 0 0 110

00( ) = ( ) ( ) + ( ) ( ) < <

lim lim s

t t

t

s ss

s

t

t→ →

( ) − ( )−

= ( ) − ( )−

{ }∈α β

β α βα

α α ββ

α βu u u u, , , ,, ,0 1


where u*0,0 is the intersection point between (1) the line with direction ut(1/2,0) passing through u(1/2,0)and (2) the line with direction us(0,1/2) passing through u(0,1/2).

For the general twist ust(α,β ), we thus use

where u*α,β is the intersection point between (1) the line with direction ut(1/2,β ) passing through u(1/2,β )and (2) the line with direction, us(α,1/2) passing through u(α,1/2).

9.2.3 NURBS Surface Grid Generation Examples

After generation of a grid {uij | } in parametric space, the actual grid in physical spaceis simply {x(uij) | }. The examples in this chapter all utilize a NURBS surface repre-sentation (cf. Chapter 30):

FIGURE 9.3 Heuristic scheme for computing a reasonable twist ust(0,0).

u

u u u u

u u u ust 0 0

0

12

0 0

12

12

412

0 0 0,

, ,

, ,( ) ≈

−

−

− ( )

= −

−

+ ( )

∗

∗

0,0

0,0

12

0,12

0,12

u u u u ust α β β α α βα β, , , ,( ) = −

−

+ ( )

∗412,

12

0 i m 0 j n≤ ≤,≤ ≤0 i m 0 j n≤ ≤,≤ ≤

x

d

u v

N u N v

N u N v

i j i j ik

jl

i

m

j

n

i j ik

jl

i

m

j

n,, ,

,

( ) =( ) ( )

( ) ( )==

==

∑∑

∑∑

ω

ω

00

00


defined by

• Two orders k and l,

• Control points di,j = (xi,j,yi,j,zi,j), i = 0, K, m,j = 0,K,n,

• Real weights ωi,j,i = 0,K,m,j = 0,K,n,

• A set of real u – knots, {u0,K,um+k | ,K,(m + k – 1)},

• A set of real v – knots, {v0,K,vn+l | ,K,(n + l – 1)},

• B-spline basis functions Nki(u), ,i = 0,K,m,

• B-spline basis functions Nlj(v), ,j = 0,K,n, and

• Surface segments xi,j(u,v), ,i = (k – 1),K, ,j = (l – 1),K,n.

The advantage of using a NURBS-based geometry definition is the ability to represent both standardanalytic shapes (e.g., conics, quadrics, surfaces of revolution, etc.) and free-form curves and surfaces.Therefore, both analytic and free-form shapes are represented precisely, and a unified database can storeboth. Another potential advantage of using NURBS is the fact that positional as well as derivativeinformation of surfaces can be evaluated analytically. For the use of NURBS in grid generation we referto Khamayseh and Hamann [11]. For a detailed discussion of B-spline and NURBS curves and surfaceswe refer the reader to Bartels et al. [3], de Boor [8], Farin [9], and Piegl [16]. We also refer the readerto Part III on CAGD techniques for surface grids.

In our first example (Figure 9.4), we use linear TFI to generate a surface grid on a portion of a surfaceof revolution. The boundary point distribution on these curves was generated by using Algorithm 2.1with λs = λκ = 1/2. That is, the points are distributed equally according to both arc length and curvatureconsiderations. The effect of curvature distribution is clearly seen: boundary grid points are clustered inareas of high curvature. The fact that arc length is still considered to some degree is seen in the fact thata nonzero density of grid points is still distributed where curvature is small or absent. The linear TFIuniformly propagates these boundary distributions into the interior of the grid.

In the next example (Figure 9.5), we again use linear TFI to generate a grid on a similar surface ofrevolution. However, in addition to distribution on arc length and curvature, we instruct Algorithm 2.1

FIGURE 9.4 Linear TFI surface grid with boundary point distribution based on arc length and curvature.

ui ui 1+ i,≤ 0=

v j v j 1+ j,≤ 0=

u ui ui k+,[ ]Œv v j v j 1+,[ ]Œ

u ui ui 1+,[ ]∈ m,v v j v j 1+,[ ]Œ


to heed the influence of four attractor points on the “top” and “bottom” boundary curves. These attractorscan be seen to be at both endpoints and at two interior points. (The concentration of the grid at thecenter, however, is not due to any attractor, but is due to the physical curvature of the surface.) Theparameters used for the top and bottom curves were λs = λκ = λ1 = λ2 = λ3 = λ4 = 1/6,k1 = k2 = k3 = k4 =120, and u*1 = 0,u*2 = .1,u*3 = .9, and u*4 = 1. By Eq. 9.3, ki = 120 implies that the grid should be packedin the neighborhood of the attractors u*i at a density approximately 8 times higher than the average gridline density. This is consistent with the appearance of Figure 9.5.

The final example of algebraic surface grid generation in this section (Figure 9.6) uses cubic HermiteTFI in conjunction with uniform arc length boundary point distribution. Orthogonality at the boundariesis clearly visible on this surface. However, boundary orthogonality can easily cause cubic Hermite gridsto “fold” in the interior on more challenging geometries. In practice, a more robust approach to enforcingboundary orthogonality is to generate an initial linear TFI grid and then use it as a starting grid for theelliptic grid generation system described in the next section.

9.3 Elliptic Surface Grid Generation

Elliptic grid generation is a technique of smoothing an initial (usually algebraic) mesh to improve gridquality. Grid improvement may involve forcing grid line orthogonality, forcing smooth grading of cellsizes, etc. What makes elliptic grid generation challenging is that grid smoothing must always ensure thatthe resulting grid points stay on the surface. With this constraint in mind, the efficiency approach ofconstructing a smooth grid is to work in the parametric space rather than on the physical surface.However, there are some disadvantages associated with this approach. The differential equations becomemore complicated and contain two sets of derivatives, the derivatives of the physical variables with respectto the parametric variables (xu,xv,yu,yv,zu,zv,xuu,xuv,xvv,K) and the derivatives of the parametric variableswith respect to the computational variables (uξ,uη,vξ ,vη ,uξξ,uξη,uηη,K).

The elliptic system may preserve the original distribution of grid points or redistribute points basedupon the choice of the control functions that are commonly used in adaptive grid generation. The controlfunctions are evaluated either directly from the initial algebraic grid, or by interpolation from theboundary point distributions and then smoothed. Orthogonality of the grid may be imposed alongcertain boundary components of the physical region. Boundary orthogonality can be achieved throughNeumann boundary conditions, which allow the boundary points to float along the boundary of the

FIGURE 9.5 Linear TFI surface grid with boundary point distribution based on arc length, curvature, and four attractor points.


surface. Alternatively, the control functions can be determined to provide orthogonality at boundarieswith specified normal spacing.

The use of elliptic models to generate curvilinear coordinates is quite popular, see Chapter 4 andThompson et al. [28]. Since elliptic partial differential equations determine a function in terms of itsvalues on the entire closed boundary of a region, such a system can be used to generate the interiorvalues of a surface grid from the values on the sides. An important property is the inherent smoothnessin the solutions of elliptic systems. As a consequent of smoothing, slope discontinuities on the boundariesare not propagated into the field.

Early progress on the generation of surface grids using elliptic methods was made by Takagi et al. [25],Warsi [31], and Whitney and Thomas [32]. The elliptic grid generation system and the surface equationsobtained by Warsi [30, 31] were based on the fundamental theory of surfaces from differential geometry,which says that for any surface, the surface coordinates must satisfy the formulas of Gauss and Weingarten,see Struik [24]. On the other hand, the same generation system was derived by Takagi et al. [25] andWhitney and Thomas [32] based on Poisson’s differential equation in three dimensions.

Distinct from the previous two approaches to deriving a system of elliptic equations for grid generation,there is the approach based on conformal mappings. Mastin [15], Thompson et al. [27], and Winslow[33] developed elliptic generation systems based on conformal transformations between the physical andcomputational regions. Planar two-dimensional smooth orthogonal boundary-fitted grids were producedusing these techniques. These methods have been extended by Khamayseh and Mastin [12] to developthe analogous elliptic grid generation system for surfaces.

In this section we present the derivation of the standard elliptic surface grid generation system usingthe theory of conformal mappings. First, conformal mapping of smooth surfaces onto rectangular regionsis utilized to derive a first-order system of partial differential equations analogous to Beltrami’s systemfor quasi-conformal mapping of planar regions. Second, the usual elliptic generation system for three-dimensional surfaces, including source terms, is formulated based on Beltrami’s system and quasi-conformal mapping. We conclude this section with a detailed description of how the elliptic grid gener-ation system is implemented numerically.

9.3.1 Conformal Mapping on Surfaces

A surface grid generated by the conformal mapping of a rectangle onto the surface is orthogonal and has aconstant aspect ratio. These two conditions can be expressed mathematically as the system of equations

FIGURE 9.6 Cubic Hermite TFI surface grid with uniform arc length boundary point distribution.

(9.14)x xξ η⋅ = 0


where F is the grid aspect ratio. These two equations can be rewritten as

Using the chain rule for differentiation, the physical derivatives are expanded as

Thus, the above system is equivalent to

and

The above equations are combined to give the complex equation

x xξ η=F

x x y y z z

F x y z x y z

ξ η ξ η ξ η

ξ ξ ξ η η η

+ + =

+ +( ) = + +

0

2 2 2 2 2 2 2

x x x

x x x

x x x x x x

x x x x x x

x x x

ξ ξ ξ

η η η

ξξ ξ ξ ξ ξ ξξ ξξ

ξη ξ η ξ η η ξ ξ η ξη ξη

ηη η

= +

= +

= + + + +

= + +( ) + + +

= +

u v

u v

uu uv vv u v

uu uv vv u v

uu uv

u v

u v

u u v v u v

u u u v u v v v u v

u

2 2

2

2

2

and

uu v v u vvv u vη η η ηη ηη+ + +x x x2

x u u x x u v u v x v v

y u u y y u v u v y v v

z u u z z u v u v z v v

u u v v

u u v v

u u v v

2 2

2 2

2 2 0

ξ η ξ η η ξ ξ η



+ +( ) +

+ + +( ) +

+ + +( ) + =

F x u x x u v x v

y u y y u v y v

z u z z u v z v

x u x x u v x v

y u y y u v

u u v v

u u v v

u u v v

u u v v

u u v

2 2 2 2 2

2 2 2 2

2 2 2 2

2 2 2 2

2 2

2

2

2

2

2

ξ ξ ξ ξ

ξ ξ ξ ξ

ξ ξ ξ ξ

η η η η

η η η

+( +

+ + +

+ + + )= + +

+ + + yy v

z u z z u v z v

v

u u v v

2 2

2 2 2 22

η

η η η η+ + + .

x y z Fu iu

x x y y z z Fu iu Fv iv

x y z Fv iv

u u u

u v u v u v

v v v

2 2 22

2 2 22

2

0

+ +( ) +( )+ + +( ) +( ) +( )+ + +( ) +( ) =

ξ η

ξ η ξ η

ξ η

This equation can be put into a compact form:


(9.15)

where

(9.16)

Solving the quadratic Eq. 9.15 either for Z or W, say Z, we have

or in terms of u and v,

where and is the Jacobian of the mapping from the parametric space to thesurface. We equate the real and the imaginary parts of the above equation to obtain

The above system of equations can be expressed in the form of a first-order elliptic system:

(9.17)

where

Note that ac – b2 = 1 which is sufficient for ellipticity. The sign ± needs to be chosen such that the Jacobian

g g W g W112

12 2222 0Ζ Ζ+ + =

g x y z

g x x y y z z

g x y z

Z Fu iu W Fv iv

u u u u u

u v u v u v u v

v v v v v

112 2 2

12

222 2 2

= ⋅ = + += ⋅ = + +

= ⋅ = + += + = +

x x

x x

x x

ξ η ξ ηand

Zg g g g

gW=

− ± −12 122

11 22

11

Fu iug iJ

gFv ivξ η ξ η+ = − ± +( )12

11

J g= g g11g22 g212–=

Fug

gFv

J

gv

uJ

gFv

g

gv

ξ ξ η

η ξ η

= − ±

= ± −

12

11 11

11

12

11

Fu av bu

Fv bv cu

ξ η η

ξ η η

= −

= −

ag

J

bg

J

cg

J

= −±

=±

= −±

22

12

11

and

J u v u v= − >ξ η η ξ 0

We have that by definition and , so choosing the negative sign will make and .From the system Eq. 9.17, we see that FJ = F(uξvη – uηvξ) = av2

η – 2buηvη + cu2η. Noting b2 = ac – 1 implies

g11 0≥ g22 0≥ a 0≥ c 0≥

2


that b = , we have that FJ > av2η – 2 and hence J

> 0.

9.3.2 Formulation of the Elliptic Generator

In this subsection, we will see that conformal mappings on surfaces produce an elliptic system equivalentto that produced by quasi-conformal mappings of planar regions. The inhomogeneous form of thissystem will be our elliptic grid generator for surfaces.

A quasi-conformal mapping is a homeomorphism:

that maps the (u,v) space onto (ξ,η) space so that the real and the imaginary parts of ψ satisfy Beltrami’ssystem of equations:

(9.18)

where p,q, and r are functions of u and v with p,r > 0 and satisfy the equation pr – q2 = 1. The quasi-conformal quantity M is invariant and often referred to as the module or the aspect ratio of the regionof consideration. For further study of the theory and application of quasi-conformal mappings, we referto Ahlfors [2] and Renelt [17].

It is the system Eq. 9.18 that forms the basis of general elliptic grid generation for the planar two-dimensional case, see Mastin and Thompson [14]. An earlier approach was proposed by Belinskii et al.[4] and Godunov and Prokopov [10] to handle the problem of quasi-conformal mappings to constructcurvilinear grids.

In fact, Eq. 9.18 forms the basis of a general elliptic grid generator for surfaces as well. We first invertthe system Eq. 9.17 so that the computational variables (ξ,η) are the dependent variables and theparametric variables (u,v) become the independent variables.

Assume that ξ and η are twice continuously differentiable and the Jacobian of the inverse transfor-mation J = uξvη – uηvξ is nonvanishing in the region under consideration. Then the metrics (uξ,uη,vξ,vη)and (ξu,ξv,ηu,ηv) are uniquely related by the following:

(9.19)

Using these quantities the system Eq. 9.17 so that the parametric variables become the independentvariables, the system can be expressed either in the form

(9.20)

or

(9.21)

ac 1– ac< acuhvh cu2h+ avh cuh–( ) 0≥=

ψ ξ ηu v u v i u v, , ,( ) = ( ) + ( )

M p q

M q r

v u v

u u v

η ξ ξη ξ ξ

= +− = +

ξ ξ

η η

η η

ξ ξ

u v

u v

v

J

u

Jv

J

u

J

= = −

= − =

and

F a b

F b cv u v

u u v

η ξ ξη ξ ξ

= +− = +

ξ η η

ξ η ηu v u

v u v

F c b

F a b

= +( )− = +( )

These first-order elliptic systems (which represent conformal mapping of a parametric surface onto asquare) are thus in the form of Beltrami’s system of equations for quasi-conformal mapping of planar


regions.The elliptic system of equations actually used for surface grid generation is a straight-forward gener-

alization of the above systems. Indeed these systems are equivalent to the following uncoupled second-order elliptic system:

This implies ξ and η are solutions of the following second-order linear elliptic system with Φ = Ψ = 0:

(9.22)

where ∆2u and ∆2v are defined by

It is this system which forms the basis of the elliptic methods for generating surface grids. The sourceterms (or control functions), Φ and Ψ, are added to allow control over the distribution of grid pointson the surface. In the computation of a surface grid, the points in the computational space are given andthe points in the parametric space must be computed. Therefore, in an implementation of a numericalgrid generation scheme, it is convenient to interchange variables again so that the computational variableξ and η are the independent variables. Introducing Eq. 9.19 in Eq. 9.22, the transformation is reducedto the following system of equations:

(9.23)

where

a b c a b b c

a b c a b b c

uu uv vv u v u u v u

uu uv vv u v u u v v

ξ ξ ξ ξ ξ

η η η η η

+ + + +( ) + +( ) =

+ + + +( ) + +( ) =

2 0

2 0

g g g u v

g g g u v

uu uv vv u v

uu uv vv u v

22 12 11 2 2

22 12 11 2 2

2

2

ξ ξ ξ ξ ξ

η η η η η ψ

− + +( ) + ( ) =

− + +( ) + ( ) =

∆ ∆

∆ ∆

Φ

∆

∆

222 12

211 12

u J a b Ju

g

J v

g

J

v J b c Jv

g

J u

g

J

u v

u v

= +( ) =

−

= +( ) =

−

∂∂

∂∂

∂∂

∂∂

− + =

− =

Au Bv

Au Bv

ξ ξ

η η

ψ

Φ

Ag v g v g v

JJ

v

JJ

Bg u g u g u

JJ

u

JJ

g g u g u v g v

g g u u g u v u v

=− +

−

=− +

−

= ⋅ = + +

= ⋅ = + +( ) +

22 12 11

32

22 12 11

32

11 112

12 222

12 11 12

2

2

2

ξξ ξη ηη

ξξ ξη ηη

ξ ξ ξ ξ ξ ξ

ξ η ξ η ξ η η ξ

∆

∆

x x

x x gg v v

g g u g u v g v

22

22 112

12 2222

ξ η

η η η η η η

x x

and

= ⋅ = + +

Solving the system Eq. 9.23 for A and B, we have


From the above equations, we see that u and v are solutions of the following quasi-linear elliptic system:

(9.24a)

(9.24b)

where

We thus have completed our derivation of the standard elliptic generation system Eq. 9.24 from theconformal mapping conditions for surfaces Eq. 9.14. This system is solved for the parametric functionsu(ξ,η) and v(ξ,η) at the grid points using the techniques of the next section.

Note that if , then , and ∆2u = ∆2v = 0, makingthe generation system identical to the well-known homogeneous elliptic system for planar grid generationpresented in Thompson et al. [28].

9.3.3 Numerical Implementation

In this subsection, we deal with the numerical discretization and implementation of the elliptic generationsystem derived in this chapter. We first examine the basic concept of finite difference approximation, andthe derivation of the difference schemes for the elliptic equations. Later we present the effect and themethodology of computing control functions in elliptic surface grid generation.

We begin our discussion of finite difference schemes for the elliptic generation system Eq. 9.24. Thebasic idea of finite difference schemes is to replace derivatives by finite differences. As before, ui,j denotesu(ξ,η) evaluated at the ξ = i, η = j grid point, and similarly for vi,j. The first derivatives are computedusing difference approximations of the form

AJ

v v

u u

n= − +[ ]= − +[ ]

1

1

Φ

Φ

ξ

ξ η

ψ

ψ

and

BJ

g u Pu g u g u Qu J u22 12 112

22ξξ ξ ξη ηη η+( ) − + +( ) = ∆

g v Pv g v g v Qv J v22 12 112

22ξξ ξ ξη ηη η+( ) − + +( ) = ∆

PJJ

g

QJJ

g

=

=

2

22

2

11

Φ and

ψ

x u y v z 0≡,≡,≡ g11 1 g12, 0 g22, 1 J, 1= = = =

∂∂ξ

ξ ηξ

∂∂ξ

ξ ηξ

∂∂ξ

ξ ηξ

u u u

u u u

u u u

i j i j

i j i j

i j i j

,

,

,

, ,

, ,

, ,

( ) ≈−

( ) ≈−

( ) ≈−

( )

+

−

+ −

1

1

1 1

2

∆

∆

∆

where ∆ξ is the computational grid spacing in the ξ-direction. The above discretizations are known asforward, backward, and central differences, respectively. The second derivatives are approximated with


central difference expressions of the form

and expressions of the form

for the mixed partial derivatives. Now we apply central difference discretization to approximate thesolution of the elliptic system Eq. 9.24 for ui,j and vi,j. Knowing that ∆ξ = ∆η = 1, we obtain the followingfinite difference schemes:

(9.25a)

(9.25b)

The quantities gi,j,J,∆2u, and ∆2v in the difference equations involve two types of approximations. Thederivative of the parametric variables with respect to the computational variables are approximated usingfinite difference approximation, whereas the derivative terms of the physical variables with respect to theparametric variables are computed analytically from the surface definition x(u). For ease of notation, quan-tities with subscripts omitted are assumed evaluated at (i,j), so that for example g11 = (g11)i,j = g11(ui,j,vi,j).

As a convenience, we present the expanded forms of ∆2u and ∆2v, which must be evaluated in thenumerical scheme

with

∂∂ξ

ξ ηξ

2

21 1

2

2u u u ui j i j i j, , , ,( ) ≈− +

( )+ −

∆

∂∂ξ∂η

ξ ηξ η

21 1 1 1 1 1 1 1

4u u u u ui j i j i j i j, , , , ,( ) ≈

− − +( )( )

+ + − + + − − −

∆ ∆

g u u ug P

u u

g u u ug Q

u u

gu u u

i j i j i j i j i j

i j i j i j i j i j

i j i j i j

22 1 122

1 1

11 1 111

1 1

121 1 1 1 1 1

22

22

2

+ − + −

+ − + −

+ + − + + −

− +( ) + −( ) +

− +( ) + −( ) =

− −

, , , , ,

, , , , ,

, , , ++( ) +− −u J ui j1 12

2, ∆

g v v vg P

v v

g v v vg Q

v v

gv v v

i j i j i j i j i j

i j i j i j i j i j

i j i j i j

22 1 122

1 1

11 1 111

1 1

121 1 1 1 1 1

22

22

2

+ − + −

+ − + −

+ + − + + −

− +( ) + −( ) +

− +( ) + −( ) =

− −

, , , , ,

, , , , ,

, , , ++( ) +− −v J vi j1 12

2, ∆

∆

∆

2 22 12 22 12

2 11 12 11 12

1

1

uJ

J g g g J g J

vJ

J g g g J g J

u v u v

v u v u

= ( ) − ( )[ ] − ( ) − ( )[ ]{ }= ( ) − ( )[ ] − ( ) − ( )[ ]{ }

JJ

g g g g g g

JJ

g g g g g g

u u u u

v v v v

( ) = ( ) + ( ) − ( )[ ]( ) = ( ) + ( ) − ( )[ ]

12

2

12

2

11 22 22 11 12 12

11 22 22 11 12 12

gu u uu11 2( ) = ⋅x x ,


Now we consider the iterative method known as successive overrelaxation (SOR) to solve the ellipticgeneration system 9.24. This method is relatively easy to implement and requires little extra computerstorage when we use the Gauss–Seidel methodology of immediate replacement of the “old” values by the“new” values at each iteration. For these reasons, this technique is very widely used in the numericalsolution of elliptic equations.

Solving for ui,j from Eq. 9.25a, and for vi,j from Eq. 9.25b, we have

(9.26a)

(9.26b)

To update the solution through an iterative method, SOR is used so that the values of the parametriccoordinates given by Eq. 9.26 are taken as intermediate values, and the acceleration process yields thenew values at the current iteration as

where ωi,j is the acceleration parameter. It is well known, see Strikwerda [23], that for linear systems anecessary condition for convergence is that the acceleration parameter ωi,j should satisfy

(9.27)

However, Eq. 9.27 does not in general imply convergence for linear systems, or our system Eq. 9.26which is usually nonlinear. In practice, we have found that for most geometries, the choice of ωi,j = 1leads to convergence. This is the usual Gauss–Seidel relaxation scheme. For certain highly curved geom-etries, the system is highly nonlinear, and underrelaxation (choosing 0 < ωi,j < 1) may be required toensure convergence. In practice, we have never used overrelaxation (1 < ωi,j < 2) for the solution ofEq. 9.26.

g

g

g

g

g

v u uv

u v uv

u v vv

u u uv v uu

v u vv v uv

11

22

22

12

12

2

2

2

( ) = ⋅

( ) = ⋅

( ) = ⋅

( ) = ⋅ + ⋅

( ) = ⋅ + ⋅

x x

x x

x x

x x x x

x x x x

,

,

,

,

.

and

ug g

g u u g P u u

g u u g Q u u

g u u

i j i j i j i j i j

i j i j i j i j

i j i j

, , , , ,

, , , ,

, ,

=+( ) +( ){ + −( ) +

+( ) + −( ) −

−

+ − + −

+ − + −

+ + − +

14

2

2

2

11 2222 1 1 22 1 1

1 1 11 1 1

1 1 1 1

11

12 −− +( ) − }+ − − −u u J ui j i j1 1 1 122, , ∆2

vg g

g v v g P v v

g v v g Q v v

g v v

i j i j i j i j i j

i j i j i j i j

i j i j

, , , , ,

, , , ,

, ,

=+( ) +( ){ + −( ) +

+( ) + −( ) −

−

+ − + −

+ − + −

+ + − +

14

2

2

2

11 2222 1 1 22 1 1

11 1 1 11 1 1

12 1 1 1 1

−− +( ) − }+ − − −v v J vi j i j1 1 1 122, , ∆2

u u ui jk

i j i jk

i j i jk

, , , , ,+ += + −( )1 1 1ω ω

0 2< <ωi j,

9.3.4 Control Function Computation

For the elliptic generation system, the source terms or control functions P and Q are used to control the


specified distribution of grid points on the surface. In the computation of the elliptic surface grid, thecontrol functions are evaluated once and then used in the iterative technique to update the grid. Thecontrol functions must be selected so that the grid has the required distribution of grid points on thesurface.

In the absence of control function, i.e., P = Q = 0, the generation system tends to produce the smoothestpossible uniform grid, with a tendency of grid lines to concentrate over convex boundary regions andto spread out over concave regions.

The elliptic system Eq. 9.24 can be solved simultaneously at each point of the algebraic grid for thetwo functions P and Q by solving the following linear system:

(9.28)

where

The derivatives here are represented by central differences, except at the boundaries where one-sideddifference formulas must be used. This produces control functions that will reproduce the algebraic gridfrom the elliptic system solution in a single iteration. Thus, evaluation of the control functions in thismanner would be of trivial interest except when these control functions are smoothed before being usedin the elliptic generation system. This smoothing is done by replacing the control function at each pointwith the average of the nearest neighbors along one or more coordinate lines. However, we note that theP control function controls spacing in the ξ-direction and the Q control function controls spacing in theη-direction. Since it is usually desired that grid spacing normal to the boundaries be preserved betweenthe initial algebraic grid and the elliptically smoothed grid, it is advisable to not allow smoothing of theP control function along ξ-coordinate lines or smoothing of the Q control function along η-coordinatelines. This leaves us with the following smoothing iteration where smoothing takes place only alongallowed coordinate lines:

Smoothing of control functions is done for a small number of iterations.The effect of using smoothed initial control functions is that the final elliptic grid is smoother as well

as more orthogonal than the initial grid, while essentially maintaining the overall distribution of gridpoints.

As presented up to this point, the elliptic smoothing scheme with nonzero control functions is well-defined only if the Jacobian of the transformation from computational to parametric variables for theinitial grid is non-vanishing. If, for example, the initial grid was produced by linear TFI and contains“folded” grid lines, the system Eq. 9.28 for generating control functions Pi,j, Qi,j will in fact be singular.If the “folding” of initial grid lines occurs at the boundary, this is a fatal flaw and the surface patch must

g u g u

g v g v

P

Q

R

R22 11

22 11

1

2

ξ η

ξ η

=

R J u g u g u g u

R J v g v g v g v

12

2 12 22 11

22

2 12 22 11

2

2

= + − −

= + − −

∆

∆ξη ξξ ηη

ξη ξξ ηη

and

P P P

Q Q Q

i j i j i j

i j i j i j

, , ,

, , ,

= +( )= +( )

+ −

+ −

1212

1 1

1 1


be divided into sufficiently small subsurface patches for which we can generate nonfolded initial meshesin the vicinities of the patch boundaries. If, however, the initial mesh has valid Jacobians at the boundaries,with folding restricted to the interior, then the surface patch need not be subdivided. In this case, controlfunctions can be computed at the boundaries from Eq. 9.28 using one-sided derivatives, and then lineartransfinite interpolation (discussed in Section 9.2.2) can be used to define the control functions in theinterior of the grid.

Figure 9.7 shows the effect of elliptic smoothing (with zero control functions) applied to an aircraftgeometry. The initial algebraic mesh computed using linear TFI with uniform arc length distributionclearly exhibits kinked grid lines in front of the aircraft engine inlet, as well as a nonuniform distributionof grid points in this region. These grid defects could conceivably lead to unacceptable artifacts in aNavier–Stokes flow computation involving the grid. The elliptically smoothed grid has created orthog-onality of grid lines and uniformity of grid point distribution. Of course the shape of the gridded surfacehas not been affected whatsoever, since all smoothing is done in the parametric domain.

We close this section by noting that our derivation of the elliptic grid generation equations from theconformal mapping conditions for surface Eq. 9.14 did not take boundary conditions into account. Aconsequence of this is that even with zero control functions (P = Q = 0) the elliptic generator Eq. 9.24may produce nonorthogonal grids in the vicinity of the surface boundaries, especially if a highly non-uniform grid point distribution is specified on the boundary curves. Grid orthogonality at the boundariesis often necessary for accuracy of numerical simulations.

In this book, Chapter 6 covers in detail two techniques for achieving grid orthogonality at the bound-aries. The first technique allows the grid points to move along the boundary. This technique involvesderivative boundary conditions for the elliptic grid generation equations and is referred to as Neumannorthogonality. The second technique leaves the boundary points fixed, but modifies the elliptic equationsthrough the control functions to achieve orthogonality and a specified grid spacing off the boundary.This technique is referred to as Dirichlet orthogonality, since the boundary conditions for the ellipticsystem are of Dirichlet type.

FIGURE 9.7 Aircraft geometry — algebraic grid (top) and elliptic grid (bottom).

9.4 Summary and Research Issues


Algebraic and elliptic techniques for the efficient construction of high-quality structured surface gridshave been presented in this chapter. We have seen that surface grids are first generated using algebraicmethods, and usually improved by applying elliptic smoothing iterations.

Algebraic techniques start with the distribution of points along the boundary curves of the surface.For this, we have presented a sophisticated algorithm which takes into account arc length, curvature, andattraction to an arbitrary set of “attractor” points.

Linear and cubic Hermite transfinite interpolation methods are presented for algebraic surface gridgeneration. We have described the simplest and most widely used algebraic grid generator — lineartransfinite interpolation. This method is usually sufficient for producing the initial grids required byelliptic methods. We have also presented a detailed algorithmic description of cubic Hermite transfiniteinterpolation, which is an algebraic method capable of imposing boundary orthogonality — a commonrequirement for the success of numerical simulations. However, in practice cubic Hermite TFI is not veryrobust and might force the user to subdivide the surface into an excessive number of subsurface patchesin order to achieve the desired result.

A complete development of elliptic surface grid generation with control functions has been presented.Our development follows from the properties of conformal mappings of surfaces. Elliptic smoothing isa robust method of enforcing desired grid properties such as orthogonality and smoothness of grid lines.Elliptic smoothing is especially useful when the surface is “poorly parameterized” and the algebraicinterpolation of parametric values does not give a satisfactory grid. Since this situation arises frequentlywhen surfaces are defined by CAD packages, the capability to smooth and improve surface grids isessential in any state-of-the-art grid generation code.

The techniques covered in this chapter have been incorporated into several grid generation packagesthat have the capability of producing high-quality surface grids on complex design geometries. Never-theless, research issues still exist.

The iterative solution of the nonlinear elliptic system Eq. 9.24 is considerably more expensive than theanalogous system for planar two-dimensional grids. This is because of the presence of the geometry-dependent terms ∆2u,∆2v,g11,g12,g22 which must be reevaluated every iteration. These terms requireevaluation of the geometry definition x(u), which can be relatively expensive. Thus very large surfacemeshes may require a nontrivial amount of computer time to smooth elliptically. Multigrid or ad hocgrid sequencing methods are a promising avenue of research addressing this problem.

Much more daunting than any amount of computer time required to generate a mesh is the muchlarger amount of “people time” required to “block” complex surface geometries. “Blocking” of a complexsurface is the task of decomposing a surface into an adequate set of subsurface patches. Subsurface patchesmust for the most part be four-sided, although some degeneracies are allowed. Moreover, it is better (forgood performance of the algebraic and elliptic techniques covered in this chapter) if the subpatchboundaries are aligned in a natural way with the distinctive geometrical features of the overall surface.This process thus represents an area of expensive human intervention and is usually the most time-consuming component of the grid generation process. “Autoblocking” — the automation of the blockingtask — is thus a hot area of research. For a description of progress in this area, see Chapter 10.

Finally, we mention that adaptive surface grid generation is very much an open problem. Given acomputational field (such as temperature, pressure, etc.) defined over a surface grid, it may be desiredto concentrate grid lines in areas where the field has a large gradient or second derivative. This problemhas been addressed in planar two-dimensional grid generation by modifying the control functions in theelliptic grid generation system to force adaptation of grid lines to the field being simulated. Analogousmodification of control functions for surface grid generation has not been undertaken to our knowledge.We note that the rewards of adaptive grid generation are potentially large, especially in time-dependentsimulations where it is desirable to have a dense region of grid lines track moving solution features.

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