7 Orthogonal Generating Systems 7.1 7.2
Introduction Generating Systems
7.3
Numerical Solutions
Two-Dimensional Regions • Curved Surfaces Discretized Equations • Boundary Conditions • Convergence Criteria • Two-Dimensional Regions • Curved Surfaces
Luís Eça
7.4
Summary
7.1 Introduction The generation of orthogonal grids is still one of the great challenges of grid generation. An orthogonal grid offers significant advantages in the solution of systems of partial differential equations: • The transformation of partial differential equations produces the smallest number of additional terms. • In general, the accuracy of the numerical differencing techniques is the highest in orthogonal grids. • The boundary conditions on rigid boundaries can be enforced in the simplest possible way. • The implementation of turbulence models, which often require information along perpendicular
directions, is simplified. However, for a three-dimensional complex geometry, a fully orthogonal grid may not exist. In fact, as noted in [1], the coordinate lines on the bounding surfaces of an orthogonal three-dimensional grid must follow lines in the direction of the maximum or minimum curvature of the surface. Therefore, this chapter will be limited to orthogonal generating systems for planes and curved surfaces. In an orthogonal grid, all the off-diagonal components of the metric tensor are equal to zero. This strong restriction on the grid construction is often in conflict with the possibility to have direct control of the grid line spacing. Conformal mapping is a well-known technique (see for example [2]) for orthogonal grid generation in two dimensions, which enforces all the grid cells to have the same aspect ratio.* Therefore, conformal mapping has no control of the grid line spacing. Although some successful applications of conformal mapping are still reported, for example [3], this chapter is mainly dedicated to orthogonal generating systems that allow control of the grid line spacing. As reported in [4] and [5], there are basically two types of orthogonal generating systems: • Trajectory methods, which generate an orthogonal grid from an existing nonorthogonal grid. • Field methods, which are based on the solution of a system of partial differential equations.
*Conformal mapping preserves the grid cell aspect ratio. In grid generation, the standard procedure is to adopt a uniform computational domain, which implies that in physical space all the grid cells have the same aspect ratio.
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In the first approach, the grid is constructed from a known nonorthogonal grid, where one set of coordinate lines is retained. In general, these methods use a marching process to recalculate the grid node distribution along the retained set of grid lines in such a way that the intersection between the new grid lines and the retained set of grid lines is orthogonal. The grid line spacing is determined by the retained set of coordinate lines of the nonorthogonal grid and by the grid node distribution on the boundary where the new set of grid lines starts. This type of methods allows the specification of the grid node distribution on three of the four boundaries of the domains. Several of these types of methods are discussed in references [1] and [4]. The main difficulties reported are the dependency of the orthogonal system on the nonorthogonal original grid and the requirement that in singly connected regions, the components of the boundary must be orthogonal; otherwise, the orthogonal trajectories may leave the physical domain. In the field approach, the grid is generated by the solution of a system of partial differential equations. Two types of generating systems have been used to generate orthogonal grids: elliptic systems and hyperbolic systems. Hyperbolic systems, which have some resemblances with the orthogonal trajectories methods, require that one of the boundaries must be left completely free. The solution is obtained by a marching procedure that starts from a known boundary and proceeds toward the free boundary. Hyperbolic generating systems are discussed in Chapter 5 of this book. This chapter will focus on orthogonal generating systems based on elliptic systems of partial differential equations, which require the knowledge of the boundary shape of all the domain. The control of the grid line spacing may be exercised by the specification of the boundary node distribution or by the specification of the grid cells aspect ratio. Elliptic systems of equations offer a wide range of possibilities for the generation of orthogonal grids. Unfortunately, there are only proofs of the existence and uniqueness of such orthogonal mappings for a restricted number of conditions [6]. Nevertheless, the numerical solution of elliptic systems of partial differential equations shows that it is possible to obtain orthogonal grids for a wide range of practical domains, with some control of the grid line spacing.
7.2 Generating Systems In an orthogonal grid, all the off-diagonal components of the metric tensor are identical to zero, which means that r r ∂x ∂x ∂y ∂y ∂z ∂z g = a ⋅a = + + = 0 with i ≠ j ij i j ∂ ξi ∂ ξ j ∂ ξi ∂ ξ j ∂ ξi ∂ ξ j
(7.1)
r where gij are the components of the covariant metric tensor, ai are the covariant base vectors, (x, y, z) are the coordinates in the physical domain, and (ξ 1, ξ 2, ξ 3) ≡ (ξ, h , z ) are the coordinates of the transformed plane. It is also known, [4] and [5], that any orthogonal grid has to satisfy the following system of partial differential equations:
∂ hηhζ ∂x i ∂ hξhζ ∂x i ∂ hξhη ∂x i + + =0 ∂ξ hξ ∂ξ ∂η hη ∂η ∂ζ hζ ∂ζ
(
)
(
)
(
)
(7.2)
where (x1, x2, x3) ≡ (x, y, z) and h x i are the scale factors defined by: 2
2
∂x ∂y ∂z hξ i = gii = i + i + i ∂ξ ∂ξ ∂ξ
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2
(7.3)
7.2.1 Two-Dimensional Regions In an orthogonal two-dimensional grid, Eq. 7.1 reduces to
g12 =
∂x ∂x ∂y ∂y + =0 ∂ξ ∂η ∂ξ ∂η
(7.4)
The ratio between the grid cell area in the physical and transformed domains is given by the Jacobian, g , of the transformation:
g
=
∂x ∂y ∂x ∂y − = g11g22 = hξ hη ∂ξ ∂η ∂η ∂ξ
(7.5)
From the orthogonality condition, Eq. 7.4, and the definition of the Jacobian in a 2D orthogonal grid, Eq. 7.5, it is easy to see that a 2D orthogonal grid must also satisfy the Beltrami equations
f
∂x ∂y = ∂ξ ∂η
f
∂y ∂x =− ∂ξ ∂η
(7.6)
where f is the so-called distortion function, which defines the grid cell aspect ratio
f =
hη = hξ
2
2
2
2
∂x ∂y + ∂η ∂η ∂x ∂y + ∂ξ ∂ξ
(7.7)
The equality of the second-order cross-derivatives of x and y and the Beltrami equations imply that
∂ ∂x ∂ 1 ∂x =0 f + ∂ξ ∂ξ ∂η f ∂η ∂ ∂y ∂ 1 ∂y =0 f + ∂ξ ∂ξ ∂η f ∂η
(7.8)
Eq. 7.8 are no more than the two-dimensional form of Eq. 7.2. If f is known, Eq. 7.8 are a set of linear elliptic partial differential equations. Otherwise, Eq. 7.8 becomes nonlinear, which implies that its solution must be iterative. The two equations are coupled through the specification of the boundary conditions or through the distortion function determination, if f is assumed to be unknown. It is interesting to note that Eq. 7.8 multiplied by the Jacobian of the transformation, g , may be rewritten as
∂ 2x ∂ 2x ∂x ∂x hη2 2 + P + hξ2 2 + Q = 0 ∂η ∂ξ ∂η ∂ξ ∂ 2y ∂ 2y ∂y ∂y hη2 2 + P + hξ2 2 + Q = 0 ∂η ∂ξ ∂η ∂ξ
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(7.9)
with
∂x ∂ 2 x ∂y ∂ 2 y ∂x ∂ 2 x ∂y ∂ 2 y + + 1 ∂f ∂ξ ∂ξ 2 ∂ξ ∂ξ 2 ∂ξ ∂η 2 ∂ξ ∂η 2 =− − P= f ∂ξ hξ2 hη2 Q= f
( )= −
∂ 1 ∂η f
∂x ∂ 2 x ∂y ∂ 2 y ∂x ∂ 2 x ∂y ∂ 2 y + + ∂η ∂ξ 2 ∂η ∂ξ 2 ∂η ∂η 2 ∂η ∂η 2 − hξ2 hη2
(7.10)
Equations 7.9 are the well-known elliptic generating system proposed by Thompson et al., [5], and the control functions P and Q, given by Eq. 7.10, are the control functions calculated iteratively at the boundaries with the GRAPE approach, [7], to obtain orthogonality at the boundaries (cf. Chapter 6). Although this result shows that Eq. 7.9 may also be used as an orthogonal generation system, for orthogonal grid generation it is better* to adopt Eq. 7.8 as the generating system. 7.2.1.1 Distortion Function and Boundary Conditions The specification of the distortion function and of the boundary conditions in Eq. 7.8 are closely related. In a closed domain, two types of boundary conditions may occur: • The coordinates of the boundary grid nodes are prescribed, which corresponds to Dirichlet
boundary conditions. • The shape of the boundary line is prescribed and the orthogonality condition Eq. 7.4 is satisfied,
which leads to a Neumann–Dirichlet boundary condition. The distortion function may be seen as a known function or as an unknown that has to be determined by the simultaneous solution of Eq. 7.8 and Eq. 7.7. If f is a known function, then Neumann–Dirichlet boundary conditions must be applied to ensure that the grid is orthogonal. The specification of x, y, and f at a boundary makes the problem overdetermined and will not guarantee that the orthogonality condition is satisfied. On the other hand, if f is assumed to be an unknown quantity to be determined in the solution procedure by Eq. 7.6 or Eq. 7.7, then the boundary grid coordinates should be prescribed. Unfortunately, it is only possible to prove that Eq. 7.8 has a unique solution [6] when f is given by an equation of the type
f (ξ, η) = ΜΠ(ξ )Θ(η)
(7.11)
where M is the conformal module of the physical domain, which guarantees that the four corners of the physical domain are mapped into the four corners of the transformed domain. The conformal module, M, is an intrinsic property of any quadrilateral domain which depends only on the boundary lines that define the domain. M may be calculated a priori, as in [6], or it may be calculated iteratively as suggested by Arina in [8] using
hη
M2 =
*See Section 7.3.1
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∫∫ hξ dξdη hξ
∫∫ hη
dξdη
(7.12)
If f is constant, and therefore equal to M, the grid is quasi-conformal,* which means that all the grid cells have the same aspect ratio. The functions Π ( x ) and Θ ( h ) represent one-dimensional stretching functions. Eq. 7.11 may be rewritten in an alternative way, where the one-dimensional stretching functions are determined iteratively from a prescribed boundary point distribution on two adjacent boundaries, x 0 and h 0 :
f (ξ , η) =
f (ξ0 , η) f (ξ , η0 ) f (ξ0 , η0 )
(7.13)
There is no analytical proof that the system of partial differential Eq. 7.8 has a unique solution, or even a solution, if f is not prescribed by a function of type Eq. 7.11, which is equivalent to specifying the boundary point distribution in two boundaries. However, it is possible to solve numerically the system of Eq. 7.8 with different approaches. Other forms of distortion functions may be used when Neumann–Dirichlet boundary conditions are applied on all the boundaries. It is also possible to generate orthogonal grids with the boundary point distribution prescribed on all the boundaries, if f is determined iteratively as a part of the solution. For complete boundary point correspondence, two different techniques have been attempted: • The distortion function is calculated at the boundaries from its definition equation, and the field
values are obtained from the boundary values by algebraic interpolation or by the solution of a partial differential equation. • The distortion function is calculated from its definition equation in the whole field. The first approach, which was introduced by Ryskin and Leal [9], allows control of the grid line spacing from the boundary point distribution and from the definition of the field values of f. However, this method is strongly dependent on the geometry of the physical domain and, in general, it is only able to produce nearly orthogonal grids [10]. More promising results can be obtained with the second approach, as reported in [11, 12, 13]. 7.2.1.2
Orthogonality Parameters
The off-diagonal metric terms of an orthogonal grid are equal to zero. In general, these terms are not calculated analytically. Therefore, in numerical solutions, it is important to quantify the orthogonality of a given grid. Usually, the deviation from orthogonality p--2- – q , where q is given by
cos(θ ) =
g12 hξ hη
(7.14)
is used to quantify the grid orthogonality. Another parameter which may also be used to quantify the grid orthogonality is the mean quadratic error of the Beltrami Eq. 7.6, which can be defined as 2 2 1 ∂x ∂y 1 1 ∂x ∂y φb = ∫∫ − + f + f dξdη g f ∂ξ ∂η f ∂η ∂ξ
7.2.2 Curved Surfaces On a curved surface, the orthogonality condition Eq. 7.1 reduces to
*M = 1 corresponds to a conformal mapping, where Eq. 7.6 become the Cauchy–Riemann equations.
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(7.15)
g12 =
∂x ∂x ∂y ∂y ∂z ∂z + + =0 ∂ξ ∂η ∂ξ ∂η ∂ξ ∂η
(7.16)
In a curved surface there are only two independent variables, which means that any curved surface may be described by a parametric representation with independent coordinates (u, v):
x = X (u, v) y = Y (u, v)
(7.17)
z = Z (u, v) As described in [14], an orthogonal grid must satisfy the following relations:
f
∂u a12 ∂u a22 ∂v = + ∂ξ a ∂η a ∂η
1 ∂u a ∂u a22 ∂v = − 12 − f ∂η a ∂ξ a ∂ξ
∂v a ∂u a12 ∂v f = − 11 − ∂ξ a ∂η a ∂η
1 ∂v a11 ∂u a12 ∂v = + f ∂η a ∂ξ a ∂ξ
(7.18)
where aij are the components of the metric tensor of the transformation between the physical domain, (x, y, z), and the parametric space (u, v):
∂x ∂y ∂z a11 = + + ∂u ∂u ∂u
2
∂x ∂y ∂z a22 = + + ∂v ∂v ∂v
2
2
2
2
a12 =
2
(7.19)
∂x ∂x ∂y ∂y ∂z ∂z + + ∂u ∂v ∂u ∂v ∂u ∂v
a = a11a22 − a122
(7.20)
As in the two-dimensional regions, f defines the grid cell aspect ratio, which in this case is defined by
2
f =
hη = hξ
2
2
2
2
∂z ∂x ∂y + + ∂η ∂η ∂η ∂x ∂z ∂y + + ∂ξ ∂ξ ∂ξ 2
(7.21)
Adding Eq. 7.18 differentiated with respect to x and h [14] it is possible to obtain the following elliptic system of partial differential equations:
∂ ∂u ∂ 1 ∂u ∂u ∂ a12 ∂v ∂ a22 ∂u ∂ a12 ∂v ∂ a22 + − − = f + ∂ξ ∂ξ ∂η f ∂η ∂η ∂ξ a ∂η ∂ξ a ∂ξ ∂η a ∂ξ ∂η a ∂ ∂v ∂ 1 ∂v ∂u ∂ a11 ∂v ∂ a12 ∂u ∂ a11 ∂v ∂ a12 + − − = f + ∂ξ ∂ξ ∂η f ∂η ∂ξ ∂η a ∂ξ ∂η a ∂η ∂ξ a ∂η ∂ξ a
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(7.22)
Eq. 7.22 is a coupled system of partial differential equations which, in general, are non-linear. Eq. 7.22 will become linear if f is assumed to be known and if the derivatives of the components of the aij metric tensor are independent of u and v. In the generating system defined by Eq. 7.22, the coefficients of the left-hand-side terms are functions of the transformation between physical domain, (x, y, z), and computational domain, ( x , h ), and the coefficients on the right-hand-side terms are functions of the transformation between the physical domain and the parametric space, (u, v). In [15] it is shown that it is possible to derive a generating system, which does not include explicitly the transformation between physical domain and computational domain, which is based on the orthogonality condition Eq. 7.16 written for the parametric coordinates:
∂u ∂u ∂v ∂v + +H=0 ∂ξ ∂η ∂ξ ∂η
(7.23)
where
H=
1 a11 + a22
∂u ∂v ∂u ∂v ∂v ∂v ∂u ∂u + − a22 − a11 a12 ∂ξ ∂η ∂ξ ∂η ∂ξ ∂η ∂η ∂ξ
(7.24)
Eq. 7.23 is written in a form similar to the off-diagonal component of the covariant metric tensor of a 2D coordinate transformation. Therefore, with an algebraic manipulation equivalent to the one which enables the derivation of the Beltrami equations in a 2D orthogonal transformation [4] it is possible to obtain the following equations:
∂u H ∂v b11 ∂v = + ∂ξ b ∂ξ b ∂η
∂u b ∂v H ∂v = − 22 − ∂η b ∂ξ b ∂η
∂v H ∂u b11 ∂u =− − ∂ξ b ∂ξ b ∂η
∂v b22 ∂u H ∂u = + ∂η b ∂ξ b ∂η
(7.25)
where bij stands for the component of the covariant metric tensor of the 2D coordinate transformation between parametric space and computational domain: 2
∂u ∂v b11 = + ∂ξ ∂ξ 2
2
∂u ∂v b22 = + ∂η ∂η b=
2
(7.26)
∂u ∂v ∂u ∂v − ∂ξ ∂η ∂η ∂ξ
From the equality of the cross-derivatives of the parametric coordinates, u and v with respect to x and h and Eq. 7.25, it is possible to construct the following generating system:
∂ b22 ∂u ∂ b11 ∂u ∂ H ∂u ∂ H ∂u + + + =0 ∂ξ b ∂ξ ∂η b ∂η ∂ξ b ∂η ∂η b ∂ξ ∂ b22 ∂v ∂ b11 ∂v ∂ H ∂v ∂ H ∂v + + =0 + ∂ξ b ∂ξ ∂η b ∂η ∂ξ b ∂η ∂η b ∂ξ
©1999 CRC Press LLC
(7.27)
Eq. 7.27 is a nonlinear set of partial differential equations that relate the parametric coordinates (u, v) to the computational domain coordinates ( x , h ). This system of Eq. 7.27 was suggested by Niederdrenk [16] as an alternative to the system proposed in [15], which is based on an equivalent form of Eq. 7.25 that led to a coupled system of equations. The generating system Eq. 7.27 does not include the distortion function f explicitly. Therefore, when the distortion function f is assumed to be known, it is better to adopt the generating system defined by Eq. 7.22. On the other hand, if f is assumed to be an unknown, then the numerical solution of Eq. 7.27 is the simplest. 7.2.2.1 Distortion Function and Boundary Conditions With the introduction of the parametric space (u, v), grid generation on a curved surface reduces to a two-dimensional transformation between the parametric space and the computational domain, ( x , h ). Therefore, in general, the specification of the distortion function f and of the boundary conditions is similar to what occurs in a two-dimensional region, which is described in Section 7.2.1.1. As in the two-dimensional regions, the boundary nodes must be allowed to move along the boundaries when the distortion function is specified, and f should be calculated iteratively when the coordinates of the boundary nodes are fixed. As shown by Arina [14], Eq. 7.22 reduces to a two-dimensional plane mapping when (u, v) are isothermic or conformal coordinates, [17], for which the right-hand side of Eq. 7.22 is zero. Therefore, the analytical proofs of existence and uniqueness of orthogonal mappings on curved surfaces are equivalent to the ones existing for two-dimensional plane regions [14]. The definition of f on a curved surface should also follow Eq. 7.11, Eq. 7.12, and Eq. 7.13, where the conformal nodule of the curved surface, M, also guarantees that the four corners of the physical domain are transformed into the four corners of the computational domain. As in the two-dimensional case, although there is no proof of existence and uniqueness of the solution, it is possible to solve numerically Eq. 7.22 with different types of distortion functions or with Dirichlet boundary conditions in more than two boundaries. For complete boundary point correspondence, it is better to solve Eq. 7.27 where the distortion function is not calculated explicitly. In this case, the metric coefficients of the transformation between parametric space and computational domain are calculated iteratively. Both generating systems Eq. 7.22 and Eq. 7.27, require the calculation of the covariant metric tensor components of the transformation between the physical and parametric domains, aij. In general, the best results are obtained when all the derivatives are discretized with the computational domain variables, x and h , as the independent variables. Therefore, the derivatives of x, y, and z with respect to u and v are obtained from
∂x i ∂x i ∂ξ ∂x i ∂η = + ∂u ∂ξ ∂u ∂η ∂u
(7.28)
∂x i ∂x i ∂ξ ∂x i ∂η = + ∂v ∂ξ ∂v ∂η ∂v with
∂ξ 1 ∂v = ∂u b ∂η
∂ξ 1 ∂u =− ∂v b ∂η
∂η 1 ∂v =− ∂u b ∂ξ
∂η 1 ∂u = ∂v b ∂ξ
(7.29)
7.2.2.2 Orthogonality Parameters On a curved surface, the deviation from orthogonality can be calculated in the same way as in a twop dimensional region, --- – q , with q give by Eq. 7.14. 2 On a curved surface, the relations between the first derivatives of the parametric coordinates, u and v, with respect to x and h may be written in several ways, Eq. 7.18 or Eq. 7.25. These equations may ©1999 CRC Press LLC
be seen as generalized forms of the Beltrami equations in a two-dimensional mapping. The definition of a mean quadratic error for these equations is not unique. However, the closest form to Eq. 7.6 is given by Eq. 7.18, which lead to a mean quadratic error, f c , given by
1 φc = ∫∫ (g1 (ξ,η) + g2 (ξ,η))dξdη b
(7.30)
where
∂u a12 ∂u a22 ∂v − − ∂ξ a ∂η a ∂η
1 ∂v a11 ∂u a12 ∂v − − f ∂η a ∂ξ a ∂ξ
∂v a11 ∂u a12 ∂v g2 (ξ , η) = f + + ∂ξ a ∂η a ∂η
1 ∂u a12 ∂u a22 ∂v + + f ∂η a ∂ξ a ∂ξ
g1 (ξ , η) = f
(7.31)
7.3 Numerical Solutions The generation of orthogonal grids on planes and curved surfaces with systems of partial differential equations is a nonlinear problem. In general, the nonlinearity is introduced by an unknown value of the distortion function f, which can be simply the conformal module of the domain, M. Even in the case where the distortion function is known, the orthogonality condition and the specified boundary shape will lead to a nonlinear equation at the boundary. Although there are methods to estimate a priori the unknown quantities, when f is defined by a product of two one-dimensional stretching functions [6,18], the following iterative algorithm may be applied to the generation of an orthogonal grid with a system of elliptic partial differential equations: 1. Construct an initial approximation for the grid. In general, linear transfinite interpolation provides an acceptable initial guess. 2. Calculate the metric coefficients that appear as coefficients of the generating system. 3. Solve the elliptic system of partial differential equations with fixed coefficients and the appropriate boundary conditions. 4. Go back to Step 2 if the convergence criteria are not satisfied.
7.3.1
Discretized Equations
There are several discretization techniques that can be applied to elliptic systems of partial differential equations. The advantages and drawbacks of the different discretization techniques are not discussed in this chapter. Although some of the basic ideas may be extended to other discretization techniques, the present discussion will be restricted to finite-difference discretizations. For the sake of simplicity, the discretization of the generating system of equations is exemplified for the x equation of a two-dimensional orthogonal mapping, Eq. 7.8. The integration of the x equation in a typical control volume with the unknowns collocated at the center of the control volume, as shown in Figure 7.1, leads to
f
1 i+ , j 2
∂x ∂x 1 ∂x 1 ∂x −f 1 + − =0 ∂ξ i + 1 , j i − 2 , j ∂ξ i − 1 , j f 1 ∂η i , j + 1 f 1 ∂η i , j − 1 2
2
i, j +
2
2
i, j −
2
(7.32)
2
The discretization of the first-order derivatives of x with central differencing schemes produces the following pentadiagonal system of algebraic equations:
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FIGURE 7.1 Typical control volume used in the discretization.
f
x
1 i +1, j i+ , j 2
+f
x
1 i −1, j i− , j 2
+
1
xi , j +1 +
f
1
xi , j −1 − Fi , j xi , j = 0
f
(7.33)
1 i, j − 2
1 i, j + 2
where
Fi , j = f
1 i+ , j 2
+f
1 i− , j 2
+
1
+
f
1 i, j + 2
1
(7.34)
f
1 i, j − 2
In each iteration of the solution procedure, Eq. 7.33 represent a linear algebraic system of equations, which, for example, can be easily solved with a successive line over-relaxation method. If the distortion function is an unknown quantity, its value at the boundaries of the control volume can be calculated using central differencing schemes in Eq. 7.7, where the (x, y) coordinates at the corners of the control volume are interpolated from the four surrounding nodes.
f
1 i+ , j 2
f
1 i− , j 2
≅
(x
) ( 2 (x − x ) + (y −x − x ) + (y 2 (x − x ) + (y 2 (x − x ) + (y −x −x ) + (y 2 (x − x ) + (y −x −x ) + (y 2
i +1, j +1
+ xi, j +1 − xi +1, j −1 − xi, j −1 + yi +1, j +1 + yi, j +1 − yi +1, j −1 − yi, j −1 2
i +1, j
≅
(x
i −1, j +1
+ xi, j +1
i +1, j
i, j
2
i −1, j −1
i , j −1
i, j
i −1, j
i −1, j +1
2
i, j
1 i, j + 2
≅
i , j +1
(x
i +1, j
+ xi +1, j +1
i , j +1
i, j
2
i −1, j
i −1, j +1
i +1, j
2
f
i, j −
1 2
≅
i, j
(x
i +1, j
+ xi +1, j −1
i , j −1
i, j
i −1, j −1
− yi, j
i +1, j
2
2
) )
)
2
2
(7.35) 2
+ yi +1, j +1 − yi −1, j − yi −1, j +1
− yi, j −1
2
i −1, j
)
+ yi, j +1 − yi −1, j −1 − yi, j −1
− yi −1, j
2
f
− yi, j
)
)
)
2
2
+ yi +1, j −1 − yi −1, j − yi −1, j −1
)
2
The accuracy of the calculation may be strongly affected by the determination of f at the faces of the control volume, or if Eq. 7.9 is adopted as the generating system of a two-dimensional orthogonal grid. The numerical errors that can be introduced by the discretization of the generating system are illustrated with a simple example. Consider a two-dimensional orthogonal mapping between two square domains. The computational domain has square grid cells defined by ∆x = ∆h = 1 . In the physical domain, a
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FIGURE 7.2 Two-dimensional orthogonal mapping with one-dimensional stretching applied in the x direction.
one-dimensional stretching function is applied in such a way that ∆ y is constant and ∆x i = ∆x i – 1 , where ∆x i = x i – x i – 1 . The two regions are illustrated in Figure 7.2. In this mapping, the distortion function f and the x coordinate are independent of h , and so Eq. 7.33 reduces to
f
i+
Eq. 7.36 is numerically satisfied if f
1 i + --2
1 2
∆xi +1 − f
and f
1 i− , j 2
1 i – --2
∆xi = 0
(7.36)
are calculated by Eq. 7.35. However, if the distortion
function at the boundaries of the control volume is calculated from the mean of f at the two surrounding grid nodes, Eq. 7.36 is not satisfied numerically, which means that the discretized equations indicate that the grid is not orthogonal! In the present example, it is easy to see that the application of central differencing schemes to Eq. 7.7 at a grid node produces
fi ≅
∆xi 2 ∆y 1 2 ∆y 2 ∆y ∆xi +1 . = = ∆xi +1 1 + ∆xi +1 ∆xi 1 + ∆xi ∆xi 1 + ∆xi +1
(7.37)
The mean values of f at the faces of the control volume are
∆xi +1 ∆y 1 + ∆xi +1 1 + ∆xi +1 1 + ∆xi +1
f
−˜
f
∆y ∆xi +1 1 −˜ + 2 ∆xi 1 + ∆xi +1 1 + ∆xi +1
1 i+ 2
i−
1 2
(7.38)
The substitution of Eq. 7.38 in Eq. 7.36 shows that with this approach, the discretized equations are not satisfied in an orthogonal grid!* A similar problem occurs with the generating system defined by Eq. 7.9, where the second-order derivatives have been expanded into two terms. Therefore, for the numerical generation of orthogonal grids, Eq. 7.8 is written in a more suitable form than Eq. 7.9.
*This result is in agreement with one of the first remarks made by Joe Thompson in the first Lecture Series on Grid Generation held at the von Kármám Institute in 1990: “Do not average metric coefficients! It is better to interpolate grid coordinates and to calculate the metric coefficients from the interpolated coordinates.”
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7.3.2 Boundary Conditions The generation of orthogonal grids on plane and curved surfaces may include two types of boundary conditions: • Dirichlet boundary conditions. The coordinates of the grid nodes are specified. • Neumann–Dirichlet boundary conditions. The orthogonality condition is directly satisfied at the boundary, and the grid nodes lie on a specified boundary shape. The numerical application of Dirichlet boundary conditions is straightforward. However, in general, the Neumann–Dirichlet boundary conditions lead to a nonlinear equation at the boundary. In general, it is easier to uncouple the solution of the system of algebraic equations that determines the coordinates of the interior grid nodes from the application of the Neumann–Dirichlet boundary conditions. Therefore, the linear algebraic system of equations obtained from the discretization of the generating system of partial differential equations is usually solved with Dirichlet boundary conditions. The orthogonality condition at the boundary is enforced a posteriori. However, if an iterative solver is adopted for the solution of the algebraic system of equations, the orthogonality condition at the boundary can be enforced after each iteration of the solver. The easiest way to implement the orthogonality condition at a boundary is to represent the boundary line in a parametric form. For example, in a x boundary, the derivatives of the grid coordinates with respect to x are obtained from the parametric representation of the boundary line. Using backward or forward differencing schemes for the derivatives in the h direction, the orthogonality condition becomes a nonlinear equation with x as the independent variable. This nonlinear equation may be solved by Newton iteration.
7.3.3 Convergence Criteria Any iterative solution procedure requires a convergence criterion to determine when to stop the iterative process. The maximum difference between grid coordinates of consecutive iterations, f x , can be used to define the convergence criterion.
(
)
φ xn = max x n − x n −1 , y n − y n −1 ,
(7.39)
where the superscript n refers to the iteration number. In surface grid generation (x, y) are substituted by the parametric coordinates (u, v). When f is calculated iteratively as part of the solution, it is also necessary to specify a convergence criterion for the determination of the distortion function. The examples presented in [13] show that it is difficult to specify a convergence criterion based on the maximum relative difference between the distortion function of consecutive sweeps,
f n − f n −1 φ nf = max . fn
(7.40)
However, the same results suggest that the difference between φf of consecutive iterations,
ψ n = φ nf −1 − φ nf ,
(7.41)
may be used as the convergence criterion of the determination of the distortion function. The application of this convergence criterion based on ψ n, allows the use of Neumann–Dirichlet boundary conditions when f is obtained from its definition equation, which, as shown in [13], leads to an unstable calculation if no convergence criteria is applied in the iterative determination of f.
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7.3.4
Two-Dimensional Regions
The generation of two-dimensional orthogonal grids with systems of elliptic partial different equations is exemplified for three types of domains: nonsymmetric, symmetric, and domains which do not have orthogonal boundary lines. In all these examples, the maximum and mean deviations from orthogonality, MDO and ADO, are calculated with the coordinate derivatives discretized by central differencing schemes. The mean quadratic error of the Beltrami equations, f b defined by Eq. 7.15, is calculated assuming that the integrand is constant in each control volume. The convergence criterion applied in these examples is f x ≤ 1.0 × 10 –6 . The convergence criterion of the iterative calculation of the distortion function is assumed to be ψ f ≤ 1.0 ≤ 10 –5 in more than two iterations. The boundary lines are represented by cubic splines, based on the initial boundary point distribution. The initial grids are generated with linear transfinite inter-polation. 7.3.4.1 Nonsymmetric Domains The different possibilities of orthogonal grid generation in a nonsymmetric region, with and without control of the grid line spacing, are illustrated in a very popular test case of orthogonal grid generation. The physical domain is defined by 0 ≤ x ≤ 1--2- + 1--3- cos ( py ) and 0 ≤ y ≤ 1 . The following options are considered: 1. Quasi-conformal mapping. Neumann–Dirichlet boundary conditions on all the boundaries and f = M. 2. Grid note distribution fixed on two boundaries and f given by the product of two one-dimensional stretching functions, Eq. 7.13. 3. Neumann–Dirichlet boundary conditions on all the boundaries and f given by the sum of linear and sine functions. 4. f obtained from Eq. 7.7 and grid note distribution fixed on three or four boundaries. The first two options correspond to situations for which there is an analytical proof of the existence and uniqueness of the solution. Although there is no proof that the solutions are unique for the remaining two options, the numerical solutions illustrate the versatility of the elliptic system of partial differential Eq. 7.8 in the generation of two-dimensional orthogonal mappings. Figure 7.3 presents the quasi-conformal grid and two grids where the one-dimensional stretching functions are iteratively determined from a fixed boundary node distribution on two boundaries. In both cases, Figures 7.3b and 7.3c, the boundary point distribution is prescribed on the boundary x = 1--2- + 1--3- cos ( py ) . In grid 7.3b, Dirichlet boundary conditions are also applied at the boundary y = 1, whereas, in grid 7.3c, an equidistant grid node distribution is prescribed on boundary y = 0. The quasi-conformal grid illustrates the lack of control of the grid line spacing of this technique, which, in this case, is caused by the boundary curvature. In grids 7.3b and 7.3c, the control of the grid spacing is determined by the two boundaries with fixed boundary nodes. The control of the grid line spacing can be achieved through the definition of the distortion function. As an example of such control, Figure 7.4 includes two grids where f is given by the sum of linear and sine functions of ξ and η. The definition of f in these examples is not included in the general class of distortion functions defined by Eq. 7.11. Therefore, there is no analytical proof of the existence of such mapping. Nevertheless, the numerical results show that, in practice, it is possible to adopt more general distortion functions to obtain a different grid line spacing. In the previous examples, the control of the grid line spacing is determined by the specification of the distortion function. In some cases, it may be useful to control the grid line spacing from the boundary point distribution. Figure 7.5 presents three grids where f is calculated iteratively from its definition equation. In grid 7.5a, the boundary nodes are prescribed on all the boundaries. In general, it is difficult to guess a boundary point distribution that produces a smooth orthogonal grid. In this example, there is a region where the grid line spacing tends to zero, which, in most cases, is unacceptable for numerical purposes. However, if the grid nodes are allowed to move in one of the boundaries, the grid becomes smooth, as illustrated in Figures 7.5b and 7.5c.
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FIGURE 7.3 25 × 25 orthogonal grids in a nonsymmetric region. Distortion function equal to constant, f = M, quasiconformal mapping, and f given as the product of two one-dimensional stretching functions.
FIGURE 7.4 25 × 25 orthogonal grids in a nonsymmetrical region. Distortion function equal to the sum of linear and sine functions, f1(ξ,η) = ξ – η and f2(ξ, η) = sin (πξ ) – sin (πη).
Table 7.1 includes the orthogonality parameters, maximum deviation from orthogonality (MDO), mean deviation from orthogonality (ADO), and the mean quadratic error of the Beltrami Eq. 7.15, of the 25 × 25 grids plotted in Figures 7.3, 7.4, and 7.5. The large values of MDO of the grids 7.3a and 7.4b are related to the lack of resolution at the lower right corner, whereas the large value of MDO of the grid 7.5a is originated by the distortion imposed by the orthogonality condition and the fixed boundary point distribution at the upper boundary. All the grids exhibit small values of ADO and f b . Figure 7.6 presents the variation of the orthogonality parameters with the number of grid nodes per direction, i.e., the effect of the discretization truncation error in the grid orthogonality. There are two different patterns in the variation of the orthogonality parameters, MDO, ADO, and f b , with the number of grid nodes per direction. As expected, in the mappings calculated with the distortion function equal to constant or given by the product of two one-dimensional stretching functions, grids 7.3, 7.3a, and 7.3b, the orthogonality parameters tend to zero with the increase in the number of grid nodes. The same ©1999 CRC Press LLC
FIGURE 7.5 25 × 25 orthogonal grids in a nonsymmetric region. Distortion function obtained from the definition equation.
TABLE 7.1 Orthogonality Parameters of Two-Dimensional Orthogonal Mappings in a Nonsymmetric Region (25 × 25 grids) Distortion Function
Boundary Conditions
Constant = M Two one-dimensional stretching functions Two one-dimensional stretching functions Linear and sine functions Linear and sine functions Definition Equation 7.7 Definition Equation 7.7 Definition Equation 7.7
Neumman–Dirichlet on the four boundaries Dirichlet on two boundaries Dirichlet on two boundaries Neumann–Dirichlet on the four boundaries Neumann–Dirichlet on the four boundaries Dirichlet on the four boundaries Dirichlet on three boundaries Dirichlet on three boundaries
MDO (degrees)
ADO (degrees)
φ b × 103
Figure
14.16 4.25
0.63 1.40
0.61 1.11
7.3a 7.3b
1.72
0.66
0.22
7.3c
4.38 22.76 6.79 1.19 0.41
0.46 1.43 0.82 0.21 0.07
0.17 2.49 0.56 0.03 0.002
7.4a 7.4b 7.5a 7.5b 7.5c
behavior is obtained when f is calculated by the definition equation and the coordinates of the boundary nodes are fixed in three boundaries, 7.5a and 7.5b. However, in the mappings calculated with f defined by a sum of linear and sine functions, grids 7.4a and 7.4b, and in the mapping with complete boundary point correspondence and f determined by the definition equation, grid 7.5a, the orthogonality parameters become almost independent of the number of grid nodes per direction. This result suggests that the conditions of grids 7.4a, 7.4b and 7.5a correspond only to a nearly orthogonal mapping. 7.3.4.2 Symmetric Domains In many cases of practical importance, the geometry exhibits one or more axes of symmetry. If the boundary point distribution is also symmetric, it is possible that there is more than one orthogonal mapping that satisfies the prescribed boundary point distribution. In fact, the grid orthogonality is completely independent of the grid node distribution along the symmetry line. Therefore, if different orthogonal mappings are generated in half-domain with fixed boundary point distributions, which only differ in the grid coordinates along the symmetry line, there is more than one orthogonal mapping for the full domain. This means that in these types of mappings, the distortion function should be specified to determine the mapping and, therefore, the boundary conditions should allow the grid notes to move
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FIGURE 7.6 Variation of the orthogonality parameters with the number of grid nodes per direction. Orthogonal mappings in a symmetric region.
along the boundary. However, as shown in [13], it is possible to specify the grid nodes in all the boundaries and to determine f from the definition Eq. 7.7. Although the solution of the problem may not be unique, it is possible to generate numerically a grid which may be useful for practical purposes. A widely used geometry has been selected to illustrate the results of orthogonal mappings in symmetric regions with f determined by its definition Eq. 7.7. It is a concave region limited by the lines x = 0, x = 1, y = 0, and y = 3--4- + 1--4- sin ( p ( 1--2- – 2x ) ) . Figure 7.7 presents three 25 × 25 grids generated with fixed boundary point distributions on all the boundaries. The corresponding orthogonality parameters are given in Table 7.2. The three grids have the same boundary point distribution on the top boundary, but very different grid node distributions along the remaining three boundaries. The orthogonality parameters of the three grids confirm the ability to generate orthogonal grids with a complete boundary point correspondence. The variation of MDO, ADO and f b with the number of grid nodes per direction is illustrated in Figure 7.8. The orthogonality parameters tend to zero with the increase in the number of grid nodes per
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TABLE 7.2 Orthogonality Parameters of Two-Dimensional Orthogonal Mappings in a Symmetric Region (25 × 25 grids) Distortion Function
Boundary Conditions
Definition Equation 7.7 Definition Equation 7.7 Definition Equation 7.7
Dirichlet on the four boundaries Dirichlet on the four boundaries Dirichlet on the four boundaries
MDO (degrees)
ADO (degrees)
φ b × 103
1.67 2.20 4.03
0.30 0.36 0.49
0.06 0.09 0.17
Figure 7.7a 7.7b 7.7c
FIGURE 7.7 25 × 25 orthogonal grids in a symmetric region. Boundary nodes fixed on the four boundaries. Distortion function obtained from the definition equation.
FIGURE 7.8 Variation of the orthogonality parameters with the number of grid nodes per direction. Orthogonal mappings in a symmetric region.
direction in the three cases, which are orthogonal mappings with complete boundary point correspondence. This result is not obtained in a nonsymmetric region, as illustrated in Figure 7.6. However, in a symmetric region, the symmetry line corresponds to a boundary with moving grid nodes, which means that this result is in agreement with the one obtained in the grids of the previous example, where the same behavior of the orthogonality parameters is obtained for a grid with fixed grid nodes on three boundaries. 7.3.4.3
Domains with Nonorthogonal Boundaries
The grid topology and/or the geometry of the domain may imply boundary lines which are not orthogonal. At the corners of the domain where the boundary lines are not perpendicular, the orthogonal
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FIGURE 7.9 25 × 25 orthogonal grids in domains with nonorthogonal boundaries. Boundary nodes fixed on the four boundaries. Distortion function obtained from the definition equation.
TABLE 7.3 Orthogonality Parameters of Two-Dimensional Orthogonal Mappings in a Domains with Nonorthogonal Boundaries (25 × 25 grids) Distortion Function
Boundary Conditions
Definition Equation 7.7 Definition Equation 7.7 Definition Equation 7.7
Dirichlet on the four boundaries Dirichlet on the four boundaries Dirichlet on the four boundaries
MDO (degrees)
ADO (degrees)
φ b × 103
Figure
4.47 6.95 1.42
0.51 0.39 0.24
0.18 0.14 0.04
7.9a 7.9b 7.9c
mapping becomes singular, which means that the Beltrami equations are not satisfied and so the elliptic generating system 7.8 cannot be applied. When Dirichlet boundary conditions are applied, it is not necessary to solve any differential equation at the boundary. Therefore, grid singularities can be handled very easily when f is determined iteratively from its definition equation. Grid singularities can also be dealt with when the distortion function is prescribed, as in quasi- conformal mapping. Examples of distortion functions appropriate to domains with grid singularities are given in [6]. To illustrate the possibilities of the elliptic generating system in geometries with nonorthogonal boundaries, three geometries with different types of singularities are considered: • A typical cross-section of a ship stern, where the intersection of the ship surface with the waterline
is not orthogonal. • An O-grid for a NACA 2412 airfoil, where the grid lines angle at the trailing edge is close to π. • A trilateral region, limited by the lines y = x, y = –x, and the line defined by x = rcos q , y = rsinθ , with r(θ ) = 1.0 – 0.15(1.0 – sinθ ). In this case, one of the sides of the computational domain is transformed into a single point in the physical domain. In these examples, Dirichlet boundary conditions are applied on all the boundaries, which means that f is determined iteratively from the definition equation. Figure 7.9 presents 25 × 25 orthogonal grids in the three domains, and the correspondent orthogonality parameters are given in Table 7.3. With the chosen boundary point distribution, the grid 7.9c is not symmetric. The orthogonality parameters of these grids are very similar to the ones obtained without grid singularities. The influence of the number of grid nodes per direction in the orthogonality parameters is illustrated in Figure 7.10. The three parameters tend to a constant value, which is the behavior obtained in a domain without grid singularities and fixed grid nodes in all the boundaries. It is also possible to consider mappings with moving grid nodes along the boundaries. However, the implementation of Neumann–Dirichlet boundary conditions in the vicinity of grid singularities may be troublesome.
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FIGURE 7.10 Variation of the orthogonality parameters with the number of grid nodes per direction. Orthogonal mappings in regions with nonorthogonal boundaries.
7.3.5 Curved Surfaces The generation of orthogonal grids on curved surfaces has the same possibilities as two-dimensional orthogonal mappings in plane regions, which have been described in the previous section. On curved surfaces, the grid coordinates are determined in a parametric space (u, v), where u and v are obtained from a mapping between parametric space and computational domain, which ensures that the mapping between physical space, (x, y, z), and computational domain, (ξ, η), is orthogonal. As in the twodimensional case, three types of domains are considered: nonsymmetric, symmetric and domains with non-orthogonal boundaries. In the present examples, Eq. 7.22 are solved when the distortion function is prescribed, whereas Eq. 7.28 are adopted when the distortion function is assumed to be unknown. The orthogonality parameters, MDO, ADO, and f c are calculated with the coordinate derivatives discretized by central differencing schemes. f c , defined by Eq. 7.32, is calculated assuming that the integrand is constant in each control volume. The initial grid is obtained with linear transfinite interpolation in the parametric space. In many practical problems, the surfaces do not have an analytical representation and some type of interpolation is required. In these examples, the surface geometry is represented by a cubic spline interpolation based on a fixed number of nodes. All the coordinate derivatives are discretized in the computational domain, which means that the derivatives of x, y, and z with respect to u and v are obtained from Eq. 7.30. The convergence criterion applied in these examples is f x ≤ 1.0 × 10 –5 . The convergence criterion of the iterative calculation of the coefficients of Eq. 7.27 is assumed to be y f ≤ 1.0 × 10 –4 in more than two iterations. 7.3.5.1 Nonsymmetric Domains On a curved surface it is possible to generate orthogonal grids with a prescribed distortion function, f, without control of the boundary point distribution, or with a specified boundary point distribution and an unknown f. The first case is equivalent to a specified boundary point distribution on two adjacent sides of the domain, when f is given by the product of two one-dimensional stretching functions. The following options are considered: 1. Quasi-conformal mapping. Neumann–Dirichlet boundary conditions on all the boundaries and f = M. 2. Boundary point distribution fixed on two boundaries and f given by the product of two onedimensional stretching functions, Eq. 7.13. 3. f obtained from Eq. 7.21 and boundary point distribution fixed on three or four boundaries.
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FIGURE 7.11 25 × 25 orthogonal grids in a nonsymmetric curved surface.
FIGURE 7.12 25 × 25 orthogonal grids in a nonsymmetric curved surface.
TABLE 7.4
Orthogonality Parameters of Mappings in a Nonsymmetric Curved Surface (25 × 25 grids)
Distortion Function
Boundary Conditions
Constant = M
Neumann–Dirichlet on the four boundaries Dirichlet on two boundaries Dirichlet on two boundaries Dirichlet on the four boundaries Dirichlet on three boundaries Dirichlet on three boundaries
Two one-dimensional stretching functions Two one-dimensional stretching functions Definition equation 7.21 Definition equation 7.21 Definition equation 7.21
MDO (degrees)
ADO (degrees)
φc × 104
Figure
2.06
0.09
0.14
7.11a
0.90 0.82 1.42 0.36 0.38
0.22 0.15 0.63 0.11 0.10
0.30 0.23 1.68 0.07 0.06
7.11b 7.11c 7.12a 7.12b 7.12c
In these options, only the first two have analytical proofs of the existence and uniqueness of the solution. However, as in the two-dimensional case, it is possible to obtain numerical solutions for the remaining option and, therefore, to increase the possibilities of control of the grid line spacing. The results of these mappings are illustrated for a surface defined by 0 ≤ x ≤ 1, y = 1.0 – 0.5 ( x 2 ( 3 – 2x )( 1.0 – sin p ( 1--2- – z ) ) and 0 ≤ z ≤ 1 . In this case, the parametric coordinates (u, v) are defined in the (x, z) plane. Figure 7.11 presents 25 × 25 grids correspondent to the first two options, which are calculated from the solution of Eq. (7.22). In the grids 7.11b and 7.11c, the boundary point distribution is prescribed on the boundary x = 1. In grid 7.11b, Dirichlet boundary conditions are also applied at the boundary z = 1, whereas, in grid 7.11c, an equidistant grid node distribution is prescribed on boundary z = 0. The 25 × 25 grids plotted in Figure 7.12 were obtained with the generating system (7.27). Grid 7.12a is a mapping with complete boundary point correspondence and grids 7.12b and 7.12c include moving grid nodes on one of the boundaries. Table 7.4 presents the orthogonality parameters of the grids plotted in Figures 7.11 and 7.12. The values of MDO, ADO, and f c of these grids confirm the ability to generate orthogonal grids on curved surfaces with different types of control of the grid line spacing.
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FIGURE 7.13 Variation of the orthogonality parameters with the number of grid nodes per direction. Orthogonal mappings in a nonsymmetric curved surface.
Figure 7.13 illustrates the influence of the number of grid nodes per direction on the orthogonality parameters of the mappings plotted in Figures 7.11 and 7.12. Although different values are obtained for each mapping, all the curves exhibit the tendency to converge to a constant value. This result may be unexpected for the mappings of Figure 7.11. However, it is important to note that the proof of existence and uniqueness of an orthogonal mapping on a curved surface, given in [14], is based on the use of isothermic parametric coordinates, for which the problem reduces to a two-dimensional orthogonal mapping between (u, v) and ( x, h ) . In the present example, u and v are not isothermic coordinates, which means that the right-hand side of Eq. 7.22 does not vanish and so the mapping between parametric space and computational domain is not orthogonal. Therefore, the proof presented in [14] is not applicable to the present example. 7.3.5.2
Symmetric Domains
If the surface exhibits an axis of symmetry and the boundary node distribution is also symmetric, the grid orthogonality becomes independent of the boundary point distribution along the symmetry line. Therefore, the grid cell aspect ratio should be specified. However, as in the two-dimensional case, it is possible to generate orthogonal grids on a symmetric domain assuming that the grid cell aspect ratio is unknown, as shown in [15]. The generation of orthogonal grids on symmetric curved surfaces is illustrated on a surface defined by 0 ≤ x ≤ 1, 0 ≤ z ≤ 3--4- + 1--4- sin ( p ( 1--2- – 2x ) ) and y = 1.0 – 1--4- ( 1.0 – sin ( p ( 1--2- – 2x ) ) (1,0 – sin(π ( 1--2- – z)), with the u and v parametric coordinates defined in the (x, z) plane. Figure 7.14 presents three 25 × 25 grids calculated with complete boundary point correspondence, which are obtained from the solution of the system of Eq. 7.27. The orthogonality parameters of these grids are given in Table 7.5 and the influence of the number of grid nodes in the orthogonality parameters
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FIGURE 7.14 25 × 25 orthogonal grids in a symmetric curved surface. Distortion function obtained from the definition equation. Complete boundary point correspondence.
TABLE 7.5
Orthogonality Parameters of Mappings in a Symmetric Curved Surface (25 × 25 grids)
Distortion Function
Boundary Conditions
Definition Equation 7.21 Definition Equation 7.21 Definition Equation 7.21
Dirichlet on the four boundaries Dirichlet on the four boundaries Dirichlet on the four boundaries
MDO (degrees)
ADO (degrees)
φc × 104
Figure
1.56 1.34 3.56
0.29 0.31 0.56
0.65 0.68 3.22
7.14a 7.14b 7.14c
FIGURE 7.15 Variation of the orthogonality parameters with the number of grid nodes per direction. Orthogonal mappings in a symmetric curved surface.
is illustrated in Figure 7.15. The results confirm the possibility to control the boundary point distribution of an orthogonal mapping on a curved surface, even in a symmetric domain. As in the nonsymmetric region, the values of MDO, ADO, and f c tend to a constant value with the increase in the number of grid nodes per direction. These constant values are almost independent of the specified boundary point distribution. 7.3.5.3
Domains with Nonorthogonal Boundaries
In many practical problems, a surface may exhibit boundary lines that are not orthogonal. At these locations, an orthogonal mapping becomes singular. However, when Dirichlet boundary conditions are applied, it is not necessary to solve any equation at the boundary. Therefore, non-orthogonal boundaries can be handled easily when Dirichlet boundary conditions are applied. The ability to generate orthogonal grids on curved surfaces with non-orthogonal boundaries is illustrated on two different geometries: the nose and cockpit of a fighter aircraft and a wing of elliptical planform
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FIGURE 7.16 25 × 25 orthogonal grids in curved surfaces with grid singularities. Distortion function obtained from the definition equation. Complete boundary point correspondence.
TABLE 7.6 Orthogonality Parameters of Orthogonal Mappings in Curved Surfaces with Nonorthogonal Boundaries (25 × 25 grids) Distortion Function
Boundary Conditions
Definition Equation 7.21 Definition Equation 7.21 Definition Equation 7.21
Dirichlet on the four boundaries Dirichlet on the four boundaries Dirichlet on the four boundaries
MDO (degrees)
ADO (degrees)
φc × 104
Figure
4.22 2.14 1.79
1.26 0.97 0.13
8.19 4.12 0.24
7.16a 7.16b 7.16c
FIGURE 7.17 Variation of the orthogonality parameters with the number of grid nodes per direction. Orthogonal mappings in curved surfaces with nonorthogonal boundaries.
with a NACA 4412 airfoil section. In both cases, one side of the parametric domain is transformed into a single point of the physical space. The present examples are restricted to mappings with Dirichlet boundary conditions on all the boundaries and, therefore, the generating system defined by Eq. 7.27. Three 25 × 25 grids are plotted in Figure 7.16 and the respective orthogonality parameters are given in Table 7.6. Figure 7.17 presents the influence of the number of grid nodes per direction on MDO, ADO, and f c . The results are equivalent to the ones obtained for curved surfaces without grid singularities. As in the two-dimensional mappings, the application of Neumann–Dirichlet boundary conditions with finite-difference discretizations in the vicinity of grid singularities may be troublesome.
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7.4 Summary This chapter presents an overview of orthogonal generating systems based on the solution of elliptic partial differential equations. In three-dimensional geometries, it is impossible to generate fully orthogonal grids in most of the cases, which implies that the main research effort in orthogonal grid generation is concentrated in two-dimensional regions and curved surfaces. The use of generating systems based on elliptic systems of partial differential equations allows the control of the grid line spacing from the definition of the grid cell aspect ratio or through the specification of the boundary point distribution. However, the number of situations for which there is a theoretical proof of the existence and uniqueness of an orthogonal mapping is rather small. In two-dimensional regions, it is possible to obtain such a proof when the grid cell aspect ratio is defined as the product of two one-dimensional stretching functions, which is equivalent to the specification of the boundary point distribution on two adjacent boundaries. This proof can be extended to curved surfaces when isothermic parametric coordinates are adopted to describe the surface. For practical purposes, it is possible to generate numerically orthogonal grids on two-dimensional regions with more general distributions of the grid cell aspect ratio or with the boundary point distribution fixed in more than two boundaries. In the latter case, the grid cell aspect ratio is determined iteratively as part of the solution. Although there is no theoretical proof that these mappings yield wellposed problems, the numerical solutions obtained in nonsymmetric and symmetric domains show that there are several possibilities to control the grid line spacing in orthogonal mappings. However, in mappings with complete boundary point correspondence, the orthogonality restriction may produce an interior grid line spacing which is unacceptable for numerical purposes. In general, in these cases, the use of moving grid nodes along one of the boundaries is sufficient to obtain a smooth interior grid line spacing. The numerical results also show that it is possible to generate orthogonal grids on curved surfaces adopting nonisothermic parametric coordinates. Overall, the properties of such mappings are similar to the ones of two-dimensional mappings. However, the numerical results suggest that complete orthogonality will only be achieved with isothermic parametric coordinates. Elliptic generating systems are also able to handle orthogonal mappings that include grid singularities at the boundary. The ability to specify Dirichlet boundary conditions at the boundaries allows the generation of orthogonal grids in domains with nonorthogonal boundaries with the same approach used in domains without grid singularities.
References 1. Eiseman, P. R., Orthogonal grid generation, Numerical Grid Generation, Thompson, Joe F., (Ed.), Elsevier Science, pp. 193–233, 1982. 2. Henrici, P., Applied and Computational Complex Analysis, Vol. III, John Wiley & Sons, NY, 1986. 3. Moretti, G., Orthogonal grids around difficult bodies, AIAA J., 30(4) pp. 933–938,1992. 4. Thompson, J. R., Warsi, Z. U. A., and Mastin, C. W., Boundary-fitted coordinate systems for numerical solution of partial differential equations — a review, J. Comput. Phys., 47(1), pp. 1–108, 1982. 5. Thompson, J. F., Warsi, Z. U. A., and Mastin, C. W., Numerical Grid Generation — Foundations and Applications, Elsevier Science, 1985. 6. Duraiswami, R. and Prosperetti, A., Orthogonal mapping in two dimensions, J. Comput. Phys., 98, pp. 254–268, 1992. 7. Sorensen, R. L., Grid generation by elliptic partial differential equations for tri-element augmentorwing airfoil, Numerical Grid Generation, Thompson, Joe F., (Ed.), Elsevier Science, pp. 193–233, 1982. 8. Arina, R., Orthogonal grids with adaptive control, Proceedings of the 1st International Conference on Numerical Grid Generation in CFD, Hauser, J. and Taylor, C., (Ed.), Pineridge Press, pp. 113–124, 1986.
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9. Ryskin, G. and Leal, L. G., Orthogonal Mapping, J. Comput. Phys., 50, pp. 71–100, 1983. 10. Chikhliwala, E. D. and Yortsos, Y. C., Application of orthogonal mapping to some two-dimensional domains, J. Comput. Phys., 57, pp. 391–402, 1985,. 11. Albert, M. R., Orthogonal Curvilinear Coordinate Generation for Internal Flows, Proceedings of the 2nd International Conference on Numerical Grid Generation in CFD, Hauser, J. and Taylor, C., (Ed.), Pineridge Press, pp. 113–124, 1988. 12. Allievi, A. and Calisal, S.M., Application of Bubnov-Galerkin formulation to orthogonal grid generation, J. Comput. Phys., 98, pp. 163–173, 1992. 13. Eça, L., 2D orthogonal grid generation with boundary point distribution control, J. Comput. Phys. 125, pp. 440–453, 1996. 14. Arina, R., Adaptive orthogonal surface coordinates, Proceedings of the 2nd International Conference on Numerical Grid Generation in CFD, Hauser, J. and Taylor, C., (Ed.),Pineridge Press, pp. 351–359, 1988. 15. Eça, L., Orthogonal grid generation with systems of partial differential equations, Proceedings of the 5th International Conference on Numerical Grid Generation in Computational Field Simulations, Soni, B. K., Thompson, J. F., Hauser, J. and Eiseman, P., (Ed.), Mississippi State University, 1996, pp. 25–36. 16. Niederdrenk, P., private communication, 1996. 17. Doubrovine, B., Novikov, S., and Fomenko, A., Géométrie contemporaine — méthodes et applications — géométrie des surfaces, des groupes de transformations et des champs, (French translation), Éditions MIR, Moscow, 1982. 18. Kang, I. S. and Leal, L. G., Orthogonal Grid Generation in a 2D Domain via the Boundary Integral Technique, J. Comput. Phys., 102, pp. 77–87, 1992.
Further Information The proceedings of the Conferences in Numerical Grid Generation in Computational Field Simulations include several papers dedicated to orthogonal grid generation. The conferences have been held since 1986. The monthly Journal of Computational Physics has reported most of the advances in orthogonal grid generation in the last few years.
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