36 Variational Methods of Construction of Optimal Grids 36.1 36.2

Introduction Constructions of the Functionals Formalizing the Optimality Criteria Analysis of the Functionals (U) and (A) in One-Dimensional Case • Construction of Two-Dimensional and ThreeDimensional Functionals (U), (O), (A) • Boundary Conditions. The Analysis of Boundary Value Problems in the Two-Dimensional Case

36.3

Effective Algorithms of Optimal Grid Generation Organization of the Iterative Process • Multiply-Connected Optimal Grids in Two-Dimensional Domains. The Program MOPS-2a • Algorithm of Two-Dimensional Optimal Adaptive Grid Generation. The Program LADA

O.B. Khairullina A.F. Sidorov

36.4

O.V. Ushakova

36.5

Simulation of Rotational Flows of Gas in Channels of Complex Geometries by Means of Optimal Grids Conclusion

36.1 Introduction Although the variational methods of construction of curvilinear grids in complex domains require realization of the solution of rather laborious problems (minimization of functionals for functions of many variables or solution of the appropriate Euler–Ostrogradsky equations (E-O)), nevertheless they give an opportunity to generate grids with good computational properties. As a rule, with the help of the variational approaches structured or block-structured grids in simply connected and multiply connected domains can be generated with distinct grid topology. The following criteria of grid optimality are mostly used in the solution of the boundary value problems associated with the pertinent partial differential equations. 1. Closeness to uniformity (U). The volumes of the neighboring elementary cells of a grid should be of the same size. Otherwise, it is difficult to build difference approximations of sufficient accuracy for the differential equations. Besides, the conditionality of the systems of difference equations approximator on the constructed grid a system of differential equations is sharply worsened.

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2. Closeness to orthogonality (O). The coordinate lines or surfaces of various families in each block should not cross at angles close to 0 or π. Otherwise, again the conditionality of systems of difference equations is worsened. 3. Adaptation (A). The curvilinear grid should follow the properties of a given function (family of functions) or should change in iterative or nonstationary processes in accordance with the solution of boundary value problems. The concentration of grid lines should take place, in particular, in zones of large gradients, for which adaptive grid is generated. These criteria, especially (U) and (A), are contradictory. As a rule, they are applied by means of weight parameters determining the values of optimality criteria. The most widely used is the approach where smooth nondegenerate mapping of some simple domain in the space of parameters (rectangle, parallelepiped, their combinations) onto the given domain in the space of initial variables is searched. A set of functions that define the required mapping should minimize some variational functional with a given boundary or natural conditions. The set of such functionals is rather wide (some examples can be found in Chapter 35.) In the overwhelming majority of cases, integral variational functionals, formalizing the optimality criteria, contain first partial derivatives of functions realizing the mapping. The E-O equations for them is the system of partial differential equations of the second order, as a rule, of elliptic type. These approaches in the literature have gotten enough attention, and they will be described in this chapter very briefly, by way of review. The main contents of the chapter are concerned with the presentation of another concept of constructing grids, developing mainly in works of Russian scientists during the past 30 years [25]. The main feature of the approach is associated with the special way of formalization of criterion (U) which gives a nonlinear variational functional containing both first and second partial derivatives of the functions realizing the mapping. This continuous functional arises naturally after the consideration of a discrete functional minimization of the measure of a relative error of a nonuniform grid in comparison with uniform grid. Such formalization leads to a system of E-O equations of the fourth order, hyperbolic in a wide sense. It has enabled consideration of new wider types of boundary conditions, as well as development of effective algorithms and programs of grid generation for the complex domains. The economic and effective procedures of calculation of grids are connected with the use of iterative processes based on the special nonstationary modification of E-O equations, as well as on the direct geometrical ways of minimization of discrete functionals formalizing all three optimality criteria. In Section 36.2 of this chapter, a brief review of variational functionals for constructing structured grids is presented. The deduction of discrete functionals formalizing criteria (U), (O), (A) is carried out, and the analysis of their properties in one-dimensional cases is given. Section 36.3 is devoted to the description of effective algorithms that allow the construction of twodimensional optimal smooth grids with simple and complex topology in simply connected and multiply connected domains. The description of capabilities of two programs MOPS-2a and LADA for generation of optimal and adaptive grids is given. A new way of automatic generation of an initial approximation of a grid is considered. Examples of grids and results of their testing are shown. In Section 36.4 a number of applications of geometrically optimal grids to the numerical solution of problems of hydrodynamic and gasdynamic flows in axially symmetric channels involving complex geometries is described. In the construction of fast iterative processes of the solution of these stationary problems, the requirements on grids are very high, since the parameters of flows change in a wide range. Examples of such calculations are given. In the conclusion of this chapter the capabilities of the approach under development for generation of three-dimensional grids and problems arising here as well as for parallelizing the algorithms for computing optimal grids are briefly described.

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36.2 Constructions of the Functionals Formalizing the Optimality Criteria 36.2.1 Analysis of the Functionals (U) and (A) in One-Dimensional Case We shall consider first the possibilities of representations of the functional (U). Let on the segment L = [0, M] it be required to construct grid nodes xi (i = 0, 1, …, N) with given lengths of boundary intervals A and B at the ends. The grid should be closest in the some metric to the uniform grid. For evaluation of a measure of deviation of grids from uniform grids we shall use two functionals, 2

h  = ∑  i +1 − 1 ,  i =1  hi N −1

(1)

JU

N −1

(36.1)

JU( 2 ) = ∑ (hi +1 − hi ) , 2

(36.2)

i =1

(hi = xi – xi–1, i = 1,2, …, N, h1 = A, hN = B, x0 = 0, xN = M, M > A + B), which need to be minimized. Usually it is more convenient to use the continuous formulation of these problems. Let x = x(ξ ) transform the parametric segment ξ ∈ [0, N] into the segment L so that xi = x(i), i = 0, …, N. We shall consider

hi ≈ y(i ), hi +1 − hi ≈ y ′(i ),

hi +1 − hi y ′(i ) ≈ , i = 0,..., N − 1 hi y( i ) N

where y(ξ ) = xξ (ξ ), ξ ∈ [0, N]. Obviously, the relation

∑h

i

= M must be satisfied. Then instead of

i=1

the discrete functionals 36.1 and 36.2 it is possible to consider the continuous functionals N

IU = ∫ (1)

xξξ2

dξ,

(36.3)

Iu( 2 ) = ∫ xξξ2 dξ.

(36.4)

0

xξ2

N

0

The minimization of the functionals I (1) and I (2)U should be considered under the conditions U N

∫ x dξ = M, ξ

xξ (0) = A, xξ ( N ) = B.

(36.5)

0

Thus, isoperimetric variational problems arise with analytical solution. The extremal functions for Eq. 36.3 have one of three possible forms:

y(ξ ) = a1 cos −2 (a2ξ + a3 ),

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(36.6)

y(ξ ) = b1ch −2 (b2ξ + b3 ),

(36.7)

y(ξ ) = c1 (ξ + c2 ) ,

(36.8)

−2

where ch is a designation of a hyperbolic cosine. The constants ak, bk, ck are defined from the conditions Eq. 36.5. If in this case the value

M ABN

q=

is less than 1, the representation Eq. 36.6 applies, if q > 1 – Eq. 36.7, and, finally, if q = 1 – Eq. 36.8 applies. The positive solution exists at any N, A > 0, B > 0, A + B < M. The problem can be solved analytically also for the functional Eq. 36.4, but the condition of the positiveness of the solution (h k > 0) is not always satisfied here. For example, at A = B it is satisfied only under the condition

M 1 − A > 0. N 3 For this reason, hereinafter in constructing the multidimensional functional during the generalization, preference is given to the functional Eq. 36.1 and is analog Eq. 36.3, though in the literature the generalization of the functionals Eqs. 36.2, 36.4, which leads to linear E-0 equations in the parametric spaces, is very frequently used. It turns out that the grids constructed on the basis of Eqs. 36.6–36.8 [29] have a number of useful properties. Thus in [28] it has been shown that hi+1 – hi ≈ 0(N–2) at large N and it is possible to approximate more precisely the derivatives of high orders. In [40, 41] it has been shown that at the expense of choice only of the boundary values A(ε , N), B(ε, N) constructed on the basis of such grids, usual difference schemes for the solution of boundary value problems for ordinary equations containing the small parameter ε have the property of uniform convergence on parameter ε at N → ∞. Thus, this construction of the functional in a number of cases allows adaptation of grids to the properties of the boundary value problem solution at the expense only of choice of boundary intervals. Let us consider now some ways of formalization of criterion (A), when the grids should automatically concentrate in the zones of large gradients of a given function Φ(x) or system of functions {Φ i(x)}. Let us use as a discrete measure of adaptation N

[

]

J A = ∑ Φ( xi ) − Φ( xi −1 ) hi2 . i =1

2

(36.9)

The functional JA presents a sum of squares of the areas of rectangles (Figure 36.1), the vertices of which belong to the curve f = Φ (x). The minimization of JA with the choice of the nodes xi results in concentrations of a grid in zones of large gradients of the function Φ. If x = x(ξ ), the continuous counterpart of the functional JA will be of the form N

I A = ∫ Φ 2x xξ4 dξ. 0

Let λU ≥ 0 and λA ≥ 0 — some constant weight coefficients. The general functional for construction of a grid satisfying criteria (U) and (A) will have the form

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FIGURE 36.1

N

I = λU ∫ 0

xξξ2 xξ2

N

dξ + λ A ∫ Φ 2x xξ4 dξ.

(36.10)

0

Note that if λA ≠ 0, we do not manage to get rid of second derivatives in Eq. 36.10 in the first integral by means of function xξ . Two boundary conditions for the function x(ξ ) are obvious:

x(0) = 0, x( N ) = M.

(36.11)

At λ U = 0 and Φ′ ≡/ 0 from Eqs. 36.10, 36.11 we get the solution in the implicit form: x

ξ( x ) =

N ∫ 4 Φ 2x (ζ )dζ 0 M

∫

4

Φ (ζ )dζ

.

2 x

0

The analogs of this solution are used frequently (see [33]) for construction of adaptive grids. Instead of Eq. 36.10) it is possible to use functionals of a more general form (k = 1, 2); N

Ik = λU ∫ 0

xξξ2

N

dξ + ∫ wk ( x(ξ )) xξ4 dξ, 2

xξ

0

 d k Φ( x )  w1 ( x ) = b0 + ∑ bk   ,  dx k  k =1 2

s

 dΦ  w2 ( x ) = c0 + ∑ c j  j   dx  j =1 l

2

(36.12)

(for a system of functions)

where bk, cj – nonnegative weight constants. At bk = 0, k = 1, …, s the minimization of Eq. 36.12 gives the uniform grid x(ξ ) = Mξ /N. Besides the conditions Eq. 36.11 for I Eq. 36.10, it is necessary to set two more boundary conditions. These can be, for example, conditions xξ (0) = A, xξ (N) = B (see Eq. 36.5) or natural boundary conditions xξξ (0) = xξξ (N) = 0.

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The E-O equation for the functional Eq. 36.12 has the form

3 λU ( µ ′′′µ 2 − 4 µ ′′µ ′µ + 3µ ′ 3 ) − wk′ µ 8 − 6 wk µ 6 µ ′ = 0, µ = xξ . 2 Without an analytical solution here we need to use numerical methods, in particular, the method of reaching the steady-state condition during the solution of appropriate boundary value problems [34]. In [35] the theorem of existence and uniqueness of the solution of the boundary value problems for wide classes of functions wk (x) has been proven.

36.2.2 Construction of Two-Dimensional and Three-Dimensional Functionals (U), (O), (A) We shall consider, at first, an elementary situation in the two-dimensional case. Let G be a simply connected domain in the plane x, y that is considered as a curvilinear quadrangle ABCD with the given vertices. We shall seek the functions

x = x(ξ, η), y = y(ξ, η)

(36.13)

mapping at integers N, M a parametric rectangle P = {[0, N] × [0, M]} onto a given domain G. Eq. 36 13 determine at ξ = i, η = j (i = 0, …, N, j = 0, …, M) the equations of coordinate lines in the parametric form, if the Jacobian D of the mapping is nondegenerate. The variational approach by Brackbill and Saltzman [3], generalizing the Winslow approach [42] for generation of grids, consists of minimization of the functional

∫∫ D ( x 1

2 ξ

)

+ yξ2 + xη2 + yη2 dξdη.

P

(36.14)

As a rule, it is assumed that the functions x, y on ∂P are given, i.e., the arrangement of nodes on the boundary ∂G is given. The E-O equations for Eq. 36.14 give rise to elliptic generators of grids. Algorithms for construction of such grids are described in Chapter 4. In [3] to the functional Eq. 36.14 the functionals

(

)

2

I0 = ∫∫ xξ yξ + xη yη dξdη, P

I A = ∫∫ D2W ( x, y)dξdη, P

responsible for criteria (O) and (A) were also added. Here W = W (x, y) — some positive weight function, dependent on the solution, under which the adaptation of a grid is carried out. Note that earlier in [39] the variational principles for construction of a moving grid, adapted to the solution of gas dynamics problems were formulated. In [6] one can find the algorithm for the solution of the variational problem of minimizing the functional

∫∫ l( x 

P

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2 ξ

) 1l ( x

+ yξ2 +

2 η

)

+ yη2 dξdη 

FIGURE 36.2

where I is a parameter. In [2] consideration was given to the functional

∫∫ sin[β (η) − α (ξ )] {exp[q (η) − q (ξ )]( x 1

1

2 ξ

2

P

](

[

)

[

+ yξ2

) )}

](

(36.15)

+ exp q2 (ξ ) − q1 (η) x + y − 2cos β (η) − α (ξ ) xξ xη + yξ yη dξdη 2 η

2 η

where functions q1(η ), q2(ξ ), α(ξ ), β(η ) have to be found in the process of minimizing the functional Eq. 36.15 on the class of functions x(ξ, η ), y(ξ, η ) with given values on the boundary ∂P. These present construction of continuous functionals, as well as a wide range of other possible representations and other principles in the background of grid generation, are described in detail in [32, 33] and in the recently published survey [20]. Note that very often for variational methods of optimal grid generation, not only continuous functionals but their discrete counterparts are used. Let us introduce some of them. In [7] for optimization of three-dimensional grids, the sum for all inner nodes of corresponding local measures has been chosen as the measures of uniformity and orthogonality. The local measure of uniformity for each inner node is a sum of squares of lengths of the vectors connecting each node with neighboring nodes, and the local measure of orthogonality is a sum of scalar products of those vectors. In [4] the sum of squares of cell areas has been considered as a measure of uniformity. The base for construction of discrete measures of adaptation is the equidistribution principle formulated in [32]. Let us introduce discrete functionals used in the given approach. Let the grid with nodes Hij be constructed in a curvilinear quadrangle ABCD. We shall denote by ri±1, j , ri, j±1 the Euclidean distances between nodes Hij and Hi±, j , Hij and Hi, j±1, by α (k) ij angles between lines connecting the node Hij sequentially with the nodes Hi+1, j , Hi, j+1, Hi–1, j , Hi, j–1, by Φ(Hij ) the value in the node Hij of a given function Φ(x, y) under which the adaptation is carried out and by Sij area of a cell defined by the nodes Hij , Hi+1,j , Hi, j+1, Hi+1, j+1 (Figure 36.2). The functionals

JU =



∑ (r

( i , j )∈Ph



JA =

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i +1, j

∑

2 1 2 1 1  1  − ri −1, j  2 + 2  + ri , j +1 − ri , j −1  2 + 2  , r r r r  i +1, j  i , j +1 i −1, j  i , j −1   

( i , j )∈Ph

)

[

( ) (

(

)

) ( ) (

Sij Φ Hij − Φ Hi +1, j + Φ Hij − Φ Hi, j +1

)]

(36.16)

(36.17)

are direct generalizations of the one-dimensional functionals Eq. 36.1 and Eq. 36.9. Minimization of the functional

Jo =

4

∑ ∑ sin

( i , j )∈Ph

−2

α ij( k )

(36.18)

k =1

should ensure the closeness of grids to orthogonality (α (k) ij ≠ 0, π ). The general discrete functional has the form

J = λU JU + λo Jo + λ A J A

(36.19)

where λ U , λO, λA are weight coefficients. Using the functions x, y from Eq. 36.13, mapping the domain G onto a parametric rectangle P, we shall write continuous counterparts of the discrete functionals Eqs. 36.16–36.19 in the form

1 1 2 2  IU = ∫∫  2 ( g11 )ξξ + 2 ( g22 )ηη dξdη, g22  g11 

(36.20)

g g Io = ∫∫  11 222 dξdη,  D 

(36.21)

I A = ∫∫ WD2 dξdη, W = Φ 2x + Φ 2y + α , α = const > 0,

(36.22)

I = λU IU + λo Io + λ A I A .

(36.23)

g11 = xξ2 + yξ2 , g22 = xη2 + yη2 , D = det{W } = xξ yη − xη yξ .

(36.24)

Similarly, it is possible to construct functionals JU (Ω), JO(Ω), JA(Ω) for generation of grids in a curvilinear quadrangle G(Ω) on a surface S determined in R3 by the parametric equations

xi = xi ( µ1 , µ2 ), i = 1, 2, 3, ( µ1 , µ2 ) ∈ Ω

(36.25)

(Ω is a limited area in a parametric plane µ1, µ2). The form of functionals Eqs. 36.20–36.22 is retained, if instead of x, y we use functions µ1 = µ1(ξ, η ), µ2 = µ2(ξ, η ), (ξ, η ) ∈ P and for Eq. 36.24 substitute the expressions

gii +

∑

j , k =1, 2

γ jk

3 ∂µ j ∂µ k ∂x ∂xl , γ jk = ∑ l , i = 1, 2, ∂pi ∂pi ∂ l =1 µ j ∂µ k

 ∂µ  D = D1 ⋅ det  j  ≠ 0, D1 = γ 11γ 22 − γ 122 , p1 = ξ, p2 = η  ∂pk  j ,k =1,2 After determining the functions µ1, µ2 the relations xi(ξ, η ) = xi(µ1 (ξ, η ), µ2(ξ, η )) Eq. 36.25 will define at ξ = const. and η = const. two sets of coordinate lines lying on the surface S.

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We shall consider now a three-dimensional domain G representing a three-dimensional curvilinear hexahedron with 8 given vertices, 12 curvilinear edges and 6 curvilinear sides. We shall search for functions

xi = xi (ξ1 , ξ2 , ξ3 ), i = 1, 2, 3,

(36.26)

mapping a rectangle parallelepiped P{[0, N1] × [0, N2] × [0, N3]} onto a given domain G with preservation of the correspondence of vertices, edges, and sides. The generalization of functionals Eqs. 36.20–36.22 in the three-dimensional case is based on the consideration of the discrete counterparts Eqs. 36.16 and 36.17. The general functional with weight coefficients λU , λO, λA has the form 3  3 1  ∂g  2   1 Gi G j  I = λU ∫∫∫ ∑ 2  kk  dξ1dξ2 dξ3 + λo ∫∫∫ ∑  dξ1dξ2 dξ3 + 2  g ∂ ξ g D   k k ij k 1 1 , = = ≠  kk k kk   P  P  

 2  3  ∂Φ  2 + λ A ∫∫∫ ∑   + α D dξ1dξ2 dξ3 , α = const. > 0.  P   k =1  ∂xk 

(36.27)

At ξ j = const., j = 1,2,3, the formulas Eq. 36.26 determine the families of coordinate surfaces in the domain G.

36.2.3 Boundary Conditions. The Analysis of Boundary Value Problems in the Two-Dimensional Case In the algorithm of grid generation, various ways of boundary node arrangement are possible. Most frequently the nodes on the boundary of the domain are considered to be given and fixed. This way is used in generation of block-structured grids, when the domain is cut on subregions and on their common boundaries the nodes should coincide. If grids in separate blocks are calculated independently from each other, the smoothness of grid lines on the interfaces of blocks is broken. The smoothness of grid lines and movement of nodes on lines of block interfaces in correspondence with the considered optimality criteria are achieved by special organization of overlapping of blocks, as realized in the program MOPS-2a. In construction of adaptive grids it is more natural to determine the boundary nodes in the process of calculation from some requirements on the grid at the boundary, i.e., to consider moving boundary nodes. In some methods the algorithm of grid generation allows fixed and given slopes of coordinate lines to the boundary of the domain to be considered. In [33] it is remarked that the application of grids very much different from orthogonal ones near boundaries can cause additional difficulties in approximation of boundary conditions during the solution of the problems on such grids. Therefore, frequently grids orthogonal or near-orthogonal or near-orthogonal at the boundary are considered (see Chapter 7). In the suggested approach, it is possible to realize all boundary conditions listed above. As was already mentioned in the introduction, the summand λU IU is leading here. It has the second order of the integrated expression in the functional I. This allows in the variational problem arbitrariness in the choice of unknown functions and their first derivatives on the boundary of the domain. It is possible to fix or to leave free both the location of boundary nodes and the slope of coordinate lines at the boundary. In the programs MOPS-2a and LADA, the nodes on the boundary of the domain are considered to be given and fixed:

xi

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∂P

()

= li ξ , s ∈ ∂P, i = 1,2

(36.28)

( l i ( x ) — given functions of node coordinates on the boundary ∂P). In addition to Eq. 36.28 it is possible to consider also necessary boundary conditions originating from minimization of the functional Eq. 36.23 on the class of functions satisfying on the boundary ∂P the conditions Eq. 36.28. These are natural boundary conditions for derivatives:

Vi

∂x j ∂ξi

= 0, Vi = ξi = 0 , N i

1 ∂gii gii2 ∂ξi

j = 1,2, i = 1,2.

(36.29)

ξi = 0 , Ni

Other variants of the boundary conditions were considered in [36], where the algorithm with moving boundary nodes and coordinate lines orthogonal to the boundary was described. Unfortunately, theorems of existence of the solution, uniqueness of it, and the correctness of the posed problems in contrast to the one-dimensional case are at the moment unknown. Only formal reasons (eight functions l i ( x ) are given: there is the arbitrariness in eight functions) and the large experience of calculations of grids confirms a hypothesis about the existence of such theorems. The summand lU not only determines boundary conditions, but also the type of a system of the E-O equations. They system of E-O equations for functionals Eq. 36.27 in the two-dimensional and threedimensional cases is too cumbersome. The structure can be presented in the form

∂xk ∂ 4 xk + Li ( x1 ,..., xn ) = 0, i = 1,..., n, n = 2,3 ∑ 4 k =1 ∂ξi ∂ξi n

(36.30)

where Li(xi, …, xn) — nonlinear forms containing partial derivatives of functions xk not higher than third order. Let the equation

Ψ(ξ1 ,..., ξn ) = 0 be the equation of characteristic variety for the system of Eq. 36.30. From 36.30 it follows that the differential equation for Ψ has the form

Ψξ41 ⋅ ... ⋅ Ψξ4n = 0. Thus, the system of Eq. 36.30 is hyperbolic in a wide sense [19], and the lines or planes ξi = const. are characteristics. If in Eq. 36.27 we put λU = λA = 0 and consider only the functional responsible for the closeness of grids to orthogonality, then the direct analysis of the system of E-O equations [30] shows that this system is on the second order of a mixed elliptic–hyperbolic type so that the boundary problem with data Eq. 36.28 is incorrectly formulated. Thus, the introduction of the summand with λU ≠ 0 plays the important regularizing role.

36.3 Effective Algorithms of Optimal Grid Generation The variation methods are the most natural for generation of optimal grids. The implementation of effective algorithms, however, involves overcoming of a lot of difficulties. Numerical procedures for grid generation based only on the solution of the E-O equations frequently [17, 37] suffer from several problems: 1. The bulky form of the E-O equations results in large numbers of arithmetical operations. 2. For stability of calculations in the iterative schemes, the small time step should be selected, and that has an effect on the number of iterations required to reach the steady-state condition.

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3. The contradictoriness of the requirements included in the basis of a variational method leads to natural difficulties in the choice of control parameters defining the value of one or another criterion of optimality. Variation of the weight coefficients in a wide range can cause instability of the numerical procedure in the solution of the equations [3]. In the approach here, at any positive weight coefficients the type of the E-O system does not vary. However, since at λ U = 0, λO ≠ 0 the system becomes of a mixed elliptic–hyperbolic type, then for stability of calculations in the solution of the equations the weight coefficients should be selected so that the contribution of summands corresponding to IO and IA does not exceed IU. Otherwise in a discrete solution the problem can turn out to be unstable. The detailed recommendations for choice of weight coefficients in the variational methods based on the solution of the E-O equations, for the example of the Brackbill–Saltzman equations, are given in [17, 33]. Note that numerical solution of the E-O equations is not the only way for implementation of the variational principles. The direct methods of minimization of discrete functionals [7] and [21] can be more effective in generation of grids (see also Chapter 33). In the approach here, the effective procedures of calculation of grids are realized by special iterative processes that uses a solution of special nonstationary modifications of the E-O equations and direct geometric minimization of discrete functionals (Sections 36.3.1 and 36.3.2). In Section 36.3.3 an algorithm for two-dimensional optimal adaptive grid generation in simplyconnected domains using only direct methods of minimization of functional is described.

36.3.1 Organization of the Iterative Process The algorithm for optimal curvilinear grids generation was developed according to the requirements for the automatic generation of grids (universality, cost-effectiveness, reliability, minimum of a used information) [25] and optimally criteria of grids (U), (O), when the functional Eq. 36.23

I = λU IU + λo Io

(36.31)

is minimized at a given arrangement of nodes Eq. 36.28 on the boundary with λA = 0. For organization of the iterative process, we use the solution of an auxiliary nonstationary system for the E-O Eq. 36.30,

α11 xt + α12 yt = xξ xξξξξ + yξ yξξξξ + L 1( x, y) , α 21xt + α 22yt = x ηx ηηηη + y η yηηηη + L 2( x, y) ,

(36.32)

where αij (i, j = 1, 2) are parameters, x = x(ξ, η, t), y = y(ξ, η, t). If a matrix A = {αij} is taken in the form A = – W* where W * is the matrix conjugate to a matrix W Eq. 36.24 [26], in the approximation of “frozen” coefficients the analysis of a short linear system with constant coefficients obtained from Eq. 36.32 shows that the Cauchy problem with periodic initial data is correct. The set of equations 36.32 at A = –W* can be used for the calculation of moving grids varying in time, when the form of the domain G(t) varies. Instead of boundary conditions Eq. 36.28 it is then necessary to use nonstationary boundary conditions

( )

( )

x ∂P = l1 ξ , t , y ∂P = l2 ξ , t , ξ ∈ ∂P determining the deformation of the boundary ∂G(t) in time. The functions li(s, t) should be defined beforehand or during the solution of a nonstationary system of the differential equations. The parametric domain P remains constant.

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Using new unknown functions Vi from Eq. 36.29 and a designation

(

)

(

)

K1 = g112 V1ξξ − F1 , K2 = g222 V2ηη − F2 , (F1, F2 are functions dependent on L1, L2 Eq. 36.32, we write Eq. 36.32, steady to perturbations, in the form

xt =

(

)

(

)

1 1 K1 yη − K2 yξ , yt = K2 xξ − K1 xη , D D

(36.33)

The formulated problem is reduced to the problem of search for x = x(ξ, η, t), y = y(ξ, η, t) defined together with their partial derivatives at each moment of time t in the rectangle P = {[0, N] × [0, M]) satisfying the set of Eq. 36.33, boundary conditions Eq. 36.28 and some initial conditions x(ξ, η, 0) = x0(ξ, η ), y(ξ, η, 0) = y0(ξ, η ). In [30] a sign of the first variation of a functional Eq. 36.31 is investigated. It turns out that functions

x τ = x t + xttτ , yτ = y t + yttτ give to the functional I the value no greater than xt, yt;

I ( x τ , yτ ) ≤ I ( x t , y t ). On the basis of this, the explicit difference iterative scheme for calculation of the coordinates of a grid [26] is developed as

x n +1 (ξ, η) = x n (ξ, η) + τQ1n , y n +1 (ξ, η) = y n (ξ, η) + τQ2n where τ is time step, xn(ξ, η ), y n(ξ, η ) are the coordinates of the grid node on the nth iteration (n = 0, 1, …) at the moment of time t = nτ ; Qn1 , Qn2 are discrete approximations of right sides of the system Eq. 36.33 in the corresponding point (ξ, η ) ∈ P. In the calculation of any point of a grid the pattern of nine nearest points (Figure 36.3) is considered. There the problem of the choice of a step τ emerges. Numerous calculations have shown that for organization of movement of all points of a grid on each iteration and in each point, the step should be variable and such that the calculated point should not leave the pattern and self-crossing cells should not arise. It has been found that when the value of the functional I(ξ, η ) at the point (ξ, η ) is large, then the value of τ is small. Movement of all points is ensured if τ (ξ, η ) at the point (ξ, η ) is selected so that

τ (ξ, η) I (ξ, η) < B = 0.5 min( d1 , d2 ) where d1, d2 – are diagonals of the quadrangle PQRT. The recalculation of points at each iteration by this method even in the case of a poor initial approximation (with patterns where the angle at point (ξ, η ) is small, with nonconvex patterns or stretched along one dimension) provides the movement of all points, but leads to slow stabilization of all nodes. For the faster stabilization of nodes the iterative process, realized in the program MOPS-2a ([12–15]). has been constructed by the following way. The calculation of optimal grids is ordered on bordering lines (Figure 36.4). On odd iterations the calculation is carried out from the central bordering line up to that near the boundary, and on even from bordering line near boundary up to central. Thus on odd iterations the grid is stretching faster in the direction to the center of the domain, and on even iterations information about geometry of the boundary is transmitted more completely inside of the domain.

©1999 CRC Press LLC

FIGURE 36.3

FIGURE 36.4

On each iteration for calculation of coordinates of a point (ξ, η ) there are considered three points A1, A2, A3 located uniformly on the segment connecting the point (ξ, η ) with the center of gravity of the pattern. In the case, for example, of nonconvex patterns it can turn out that A2 or A3 get out of the pattern, and then the coordinates of these points are recalculated with a half step; if recalculated points get out of the pattern again, the movement of a point is organized in the direction of an interior diagonal toward pattern concavity. For each point Ai (i = 1, 2, 3), the coordinates of points Aτi are calculated from E-O equations under the explicit difference scheme with a variable step. At six points, values of the functional I(ξ, η ) are calculated and the minimal value is selected. At a new point (ξ, η ) a point corresponding to this value of a functional is selected. After a given number of iterations l the correction of a grid is carried out, i.e., at iterations, the number of which is multiple of the number l, movement of points is organized not toward the center of gravity of the pattern but toward the point of intersection of diagonals of the quadrangle P1 Q1R1T1. On each iteration a summarized value of a functional I for all calculated points of a grid is computed. The calculation of a grid is considered complete if a relative variation of I on two adjacent iterations is no more than 0.1%. The calculation of a grid can be continued at other values of weight λ = λU / λO and the number of corrections l.

36.3.2 Multiply Connected Optimal Grids in Two-Dimensional Domains. The Program MOPS-2a On the basis of the algorithm described in Section 36.3.1, optimal curvilinear block-structured grid in simply connected and multiply connected domains with simple and complex topology are constructed, ©1999 CRC Press LLC

FIGURE 36.5

but the mapping of a given domain G in the plane (x, y) onto a set of rectangles P in a parametrical plane (ξ, η ) and inverse mapping can be ambiguous. Such grids contain the elements of basis grids of O, C, H type [33]. The grids generated by MOPS-2a are characterized by smoothness of grid lines on the boundaries of block interfaces. To realize that we use the method of overlapping of blocks. The automatic organization of a method allows a reduction and simplification of the volume of input information for calculation of grids. 36.3.2.1 Initial Approximation of Grid The process of the construction of grids includes some preliminary stages: first of all the choice of topology of grid, which specifies the direction of coordinate lines of a curvilinear grid, i.e., the structure and to a large degree the quality of grids. This process is carried out by the performer of the calculation. In the proposed method the algorithms for dividing the domain into blocks, describing the boundary of blocks, constructing the initial approximation of a grid, and overlapping of blocks are formalized and automated by the program. At the construction of the initial approximation, the boundary of the domain is represented by a single or several closed curves, each of which is described by a set of specific nodes connected by straight lines or arcs of circles of given radii in a specific direction. The initial approximation of a grid is automatically generated for different input information: • For given coordinates of intersection points of typical horizontal and vertical lines that divide

blocks into convex or rather close to convex subblocks, the opposite sides are automatically divided into a given number of equal segments. The points of a partition are connected by straight segments (three points in Figure 36.5a) [12]. • If the block is of a star-shaped typed, it is possible to insert in it the corner of some quadrangle with a uniform grid, which is simultaneously a near-boundary bordering line and a fictitious interior boundary (Figure 36.5b) [13]. • For minimal information (specific vertices of blocks and number of points on both sets of coordinate lines) with application of method of R-functions (Figure 36.5c) [5]. For construction of grids in multiply connected regions, the domains are divided into blocks — curvilinear quadrangles, the vertices of which belong to the boundary of domain. We shall name the dividing lines as the interior boundaries of the domain. If the domain contains the elements of grids of H, O, C types, as slits (O–, C–grids), and splits (H–grids) should coincide with coordinate lines in plane (ξ, η ) and be grid lines in plane (x, y). The domain is divided into blocks for the purpose of selection of simply connected subregions from multiply connected, in which structured grids are generated, or with the purpose of selection of subregions with simple configurations, in which for generations of the grid initial approximation the minimum of information is required. The points of a grid are numbered on horizontal lines and vertical lines; thus, in each block k the grid is determined by a set of coordinates {x(ξ, η ), y(ξ, η )} where ξ = N1k, …, NNk, η = M1k, …, MMk. N1k, NNk, M1k, MMk should be matched with appropriate N1l , NNl, M1l, MMl (l = 1,2, …) of adjacent blocks. The common block grid in the domain is obtained at the expense of the combination of grids in all ©1999 CRC Press LLC

FIGURE 36.6

blocks covering this domain. if the least values N1l, M1l (l = 1,2, …) are equal to unity, the greater values NNl, MMl (l = 1,2, …) define the size (M × N) of block grid of the domain. The coordinates of grid nodes are stored in a matrix that is filled by a “flag” method. The image of the domain is inscribed in the rectangle of size M × N. If its point does not belong to a specific domain, then a “flag” (a large number) is inserted into the corresponding element of the matrix. Thus, the structure of the matrix is determined by the geometry of the domain P in the parametric plane (Figure 36.6c). There are two columns in the matrix to store the coordinates of the boundaries of a split, if this line is vertical, or two lines, if the line is horizontal (ab, cd). The cut in the plane (ξ, η ) has two images, so that two matrix elements (for example, the elements of the columns q1m1, q3m2) correspond to each point (the cut Qm) of a slit in plane (x, y). One point can carry out a few slits (a point Q (Figure 36.6a) and three slits); therefore, more than two matrix elements (qi , i = 1, 2, 3) may correspond to endpoints of the slit. In Figure 36.6c the arrows indicate the correspondence between cuts that are singled out by bold lines. In Figure 36.6a the given boundary is presented and six blocks are marked, in which a grid of an initial approximation is generated by one of the above described methods. Then it is symmetrically mapped over the axis mn (Figure 36.6b). The markers select grid lines on which splits are located. 36.3.2.2 Automatic Overlapping of Subdomains To construct a block-structured optimal grid we consider each of the blocks as a given simply connected domain. In the blocks the grid is generated by the method with a prescribed node arrangement on the boundary. If in each block the grid is built independently, not connected with coordinates of grids of adjacent adjoining blocks, on the boundaries of block interfaces a smoothness of coordinate lines will be lost. For the solution of the problems, the unknown quantities of which have large gradients in the neighborhood of the boundaries of interfaces, the grid lacking smoothness is considered to be unsuitable. ©1999 CRC Press LLC

FIGURE 36.7

Let us apply a method of overlapping. Each block, which has as its boundary a part of the interior boundary of the domain, is extended beyond this boundary on one coordinate strip. Thus we take as the boundary of the block the vertical or horizontal line from the adjacent block. When we perform the calculation on each iteration in all blocks successively, we calculate the grid points on the interior boundaries of blocks in the correspondence with the given optimality criteria. It is rather difficult to realize this method (in a logic sense) for multiply connected domains with complex topology when in the domain there are slits and splits, and on which it is also necessary to provide the movement of grid nodes. In this case we are to analyze a large number of geometric possibilities of block interfaces. The solution of this problem has allowed the volume of input data to be reduced and quality of calculated grids to be improved. The split is two parts of the boundary (AB, CB in Figure 36.7a), the points of which have different coordinates in initial plane (x, y) but identical in curvilinear coordinates (ξ, η ): the slit has identical coordinates in (x, y) and different in (ξ, η ). The presence of a split is determined by the program in generating the boundary. If the coordinates of two different points of the boundary fall on one element of the matrix and there are more than two such adjacent points on horizontal or vertical lines (and correspondingly matrix elements), this line is a split. After determining these lines are storing the ambiguity of their mappings, the matrix of grid coordinates is extended on the appropriate number of lines, if the splits are horizontals (Figure 36.6c), or on an appropriate number of columns if they are verticals (one column in Figure 36.7b). Coordinates of the boundary of a grid (Figure 36.6a) are enumerated, and the initial approximation of grid (Figure 36.6b, 36.7a) is constructed. In order to reveal the slits, the parts of the boundary of the domain are automatically analyzed by the coordinates of their endpoints after the initial approximation of grid is constructed in the whole given domain. All slits are numbered by a certain way, and the splits are labeled by a special marker. In order to organize block overlapping we determine the type of the boundary. A block is topologically equivalent to a rectangle and has four sides. A side may be rigid, when its points belong to the boundary of the domain and the coordinates of these points are specified; it may be movable, when grid points can move during calculation; it may be a slit; it may be rigid or movable, but lying on a coordinate line on which split is located; it may be mixed, when the boundary is a combination of parts of different types. In order to construct a smooth grid the overlapping of blocks is organized through movable sides and slits (Figure 36.8d, shaded strips).

©1999 CRC Press LLC

FIGURE 36.8

FIGURE 36.9

The analysis of blocks is carried out by the program. If the boundary of the block is rigid, the coordinates of all its points to this moment of time are calculated and are written in the matrix. If the boundary of the block is movable (KO (Figure 36.7a)), two of its endpoints are connected by straight lines, and grid points of the boundary of the block are calculated by the method of linear interpolation. On the mixed boundary (ABO (Figure 36.7a)), the parts of rigid (AB) and movable (OB) boundaries are selected and the block is automatically divided into two (Figure 36.7a) or more blocks (Figure 36.8a) so that through the chosen movable boundaries hereinafter to realize overlapping of them (dashed lines are the line of decomposition). All cuts are enumerated. The number of a sit is assigned to the corresponding side. It is common for some two blocks to have sides with the same numbers (p1a1, p2a2 (Figure 36.7a), l1k1, l2k2 (Figure 36.8a)). If a grid is calculated in the block with slits, another block with a slit of the same number is searched for to organize block overlapping. The first block is extended on one coordinate strip beyond the slit and coordinates of points of the slit, and the adjacent grid line from another block are transferred to the strip (Figure 36.9a). The next step in the analysis of block boundaries is to check for the possibility of the blocks overlapping. For example, if one of the block sides is a slit, its adjacent sides cannot be movable (pq in Figure 36.8d); if two adjacent sides are movable, the point at the intersection of coordinate lines bordering these sides should belong to given domain (point A in Figure 36.9b). If block sides belong to one side of a slit, the automatic check of a possibility of organization of overlapping of blocks is carried out similarly, but with the use of working columns (lines) of matrices.

©1999 CRC Press LLC

As a result of the above discussions on automatic organization of overlapping blocks, the volume of input information for the calculation of a grid in comparison with hand organization of overlapping was reduced by 4–20 times, depending on the complexity of the configuration of the domain and its topology. So the domain represented on Figure 36.6b, after the analysis of the boundary and description of six blocks for input data, was divided automatically into 42 subregions to organize the block overlapping. Testing of the algorithm and program MOPS-2a according to criteria from work [22] has shown that for construction of grids closer to uniform, it is necessary to select the weight λ in the range 0.1–0.3 and for grids closer to orthogonal — from 1 up to 10. The optimal numbers of correction are l = 2, 3. For calculation of grids on average 4–20 iterations are required. The number of iterations depends on the initial approximation, number of correction l and the weight λ. The quality of grids essentially depends on the choice of its topological image. The computation time for the grid (Figure 36.6) of size 72 × 54 on PC/486 (40 MHz) (nine iterations) is ≈ 0.5 min.

36.3.3 Algorithm of Two-Dimensional Optimal Adaptive Grid Generation. The Program LADA This algorithm represents the iterative procedure of minimization of the functional J (Eq. 36.19). The calculation begins with some initial approximation — a non-self-intersecting grid. At each time step the calculations are carried out along bordering lines in the counterclockwise direction moving from the boundary (Figure 36.4). While defining the node (i, j), the other nodes are fixed, and the position of a node is found from the condition of nondegeneracy of a grid and the condition of a minimum of the functional J on a special set of points Ω1 or Ω2. During the calculations the coordinates of nodes are replaced by new ones. 36.3.3.1 Set of Points for Minimization of the Functional J Two sets of points determined by means of special points Cω +, Cω –, C ω* – are considered. To construct them we use the equidistribution principle [32] for weight function ω = α + Φ 2x + Φ 2y , α = const. > 0 n for each point on its own segment. We find the point Cω + , if the cell Cij determined by points H i,j–1 , n n n H i,j+1, H i–1,j, H i+1, j (n is the number of iteration) is convex. For its determination the equidistribution principle is applied on the segment [Ci , Cj] where the points Ci , Cj are found from the same principle [32] on intervals [H ni–1, j+1, H ni–1,j–1, H ni+1, j–1, H ni+1, j+1] correspondingly. For a nonconvex cell Cij we find the point Cω – on its interior diagonal. Similarly, for the point C *ω – we shall find for nonconvex cell C *ij = {H ni–1, j+1, H ni–1, j–1, H ni–1, j–1, H ni+1, j+1}. Then we shall construct the set of points H+ , H–, H–* .

→ →

    k n k k n k = 0,1, 2, 3  H Hij H = Hij Cω 3     where points Cω coincides with corresponding point Cω + , Cω – , C ω* –. After this we shall define the sets Ω1 and Ω 2 by means of Table 36.1.

TABLE 36.1

Sets of Points Ω1 and Ω2

Cell Cij

Cell C*ij

Set Ω1

Set Ω2

Convex

Nonconvex Nonconvex Convex Nonconvex

H+

H+ H+ ∪ Η −∗ H+ ∪ Η −

Nonconvex

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H+ ∪ Η −

FIGURE 36.10

Note, that if ω = const., the point Cω + coincides with the center of gravity of a cell Cij, and Cω – , Cω* – — with the middles of interior diagonals of cells Cij, C *ij respectively. The values of functions Φ(x, y) at nodes Ci , Cj, Cω + , Cω – , C ω* –, Hk are calculated by a linear interpolation, and the derivatives Φx, Φy ; according to formulas

Φx =

(

)

(

)

1 1 Φ yη − Φη yξ , Φ y = Φη xξ − Φξ xη , D ξ D

where Φ = Φ(x(ξ, η ), y(ξ, η )). The derivatives xξ , yξ , xη , yη , Φξ , Φη inside the domain are approximated by central differences and on the boundary ∂P by one-sided differences. 36.3.3.2 Organization of Calculations In the program LADA the following two ways of calculations are utilized: • Global search for Minimum. Sets of points Ω1 or Ω2 are calculated. We choose a new node H

n+1 ij

from a selected set of points that give the minimal contribution to the functional J and that with other nodes form noncrossing grid. In this case in the whole domain all optimality criteria are taken into account. n+1 • Local Search for Minimum. If cells Cij, C *ij are convex, we consider H ij = Cω + . If for Cω + we have * self-intersecting cells, and if we have other cases for Cij , C ij in the definition of a node H n+1 ij , we carry out a global search for the minimum on the selected set of points. In this method in the whole domain only one criterion of adaptation is taken into account. The method of organization of calculations is given in [37]. 36.3.3.3 About the Choice of Control Parameters The methods of the construction of an initial approximation are described in detail in Section 36.3.2. The grid constructed by one of those methods is represented in Figure 36.10. Note that for the algorithm it is important that the initial approximation is not a self-crossing grid. As an initial approximation for generation of adaptive grids, the optimal grids constructed by the given algorithm without criterion of the adaptation (λA = 0) can be used. Such initial approximation is represented in Figure 36.11a. The constants λU , α are selected equal to 1. The parameters λO, λA are chosen from the requirements on the quality of a grid, estimated with the help of Eqs. 36.16 and 36.17, and according to criteria offered

©1999 CRC Press LLC

FIGURE 36.11

in [22]. The most frequently used values λO = 10k, k = –2, –1, 0, 1, 2, λA depend on values of the function Φ. In the local search for the minimum, λA is supposed equal to zero. For the initial approximation in Figure 36.10, JU = 49.71, JO = 23925.92. For the optimal grid in Figure 36.11a, λO = 0.05, λA = 0, JU = 11.08, JO = 22776.7. For an adaptive grid in Figure 36.11b, λO = 0.05, λA = 106, JU = 236.2801, JO = 22770.05, JA = 2.57. The choice of the methods of calculations and the set of points for minimization of the functional Eq. 36.19 is made from the requirements on the quality of a grid and effectiveness of algorithm. Global searching for the minimum and set of points 1 are more effective. An example of a grid for function 3

z = Φ(x, y) =

∑ Φ (x, y) where i

i=1

[

]

 1 2 2  Φ1 ( x, y) = exp− x − a11 ) + ( y − a12 ) , (  ε1 

[ [

] ]

  1  ( x − ai1 )2 + ( y − ai 2 )2 − ri2 , exp −   εi  Φ i ( x, y) =  1 2 2  2 ( x − ai1 ) + ( y − ai 2 ) − ri ,  εi 

( x − ai1 )2 + ( y − ai 2 )2 > ri2 , ( x − ai1 )2 + ( y − ai 2 )2 ≤ ri2 ,

i = 2, 3 is demonstrated in Figure 36.11b. Here εi = 0.001, i = 1, 2, 3, r2 = 0.15, r3 = 0.1, a11 = 0.3, a12 = 0.7, a21 = 0.7, a22 = 0.4, a31 = 0.9, a32 = 0.8.

36.4 Simulation of Rotational Flows of Gas in Channels of Complex Geometries by Means of Optimal Grids Frequently in technological installations there are axisymmetrical channels of complex configurations in which complicated nonstationary hydro- and gasdynamic flows occur. In constructions of such installations, one of the important points is knowledge both of the structures of the flows and the parameters describing them. For the purpose of reducing field tests, effective numerical methods that permit calculations that can rather quickly and reliably predict parameters of flows are necessary. The development of numerical methods for calculation of gas flows in channels with complicated geometries is connected with large difficulties. These are complex geometries of calculated domains, large range of flow velocities,

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formation of many rotational zones with closed streamlines caused by interaction of counter streams. As a rule, the calculations described in the publications (for example, [9, 24, 31]) are connected with serious restrictions on geometry of channels or on structure of flows. The application of optimal smooth block-structured curvilinear grids, described in Section 36.3.2, has appeared as the rather essential factor in solving the problems [1, 10, 11, 16]. Good approximating qualities of used grids and mappings [28, 40, 41] has become the basis of attained results. So, axisymmetrical simply connected channels of complicated configurations are considered. The surfaces of channels consist of parts of a porous surface through which gas is blown in solid walls, and parts for exit of gas. For modeling the gas stream in the channel, some simplifications [23] are introduced. We consider that the sizes of boundary layers, increasing along walls, are small in comparison with transversal sizes of channels; boundary layers do not interact with each other; gas that is blown in is homogeneous; and gas flow is stationary. Then for the numerical simulation of gasdynamic processes in channels it is possible to use the model of perfect gas, the flow of which satisfies the Euler equations. For numerical simulation the Euler equations are written in the stream functions ϕ – vortex function ω [9] in integral form in curvilinear coordinates (ξ, η ):

 1

∫  ρr∆ ( A ϕ

1 η

)

− A3ϕ ξ dξ −

C

ω

 1 A2ϕ ξ − A3ϕη dη  = ∫∫ ω∆dξdη, ρr∆  GC

(

)

ρ

∫ r (ϕ dξ + ϕ dη) = − ∫ 2 (V dξ + V dη), η

ξ

C

2 ξ

2 η

C

∫ H(ϕ dξ + ϕ dη) = 0, ξ

η

C

where

A1 = xξ2 + rξ2 , A2 = xη2 + rη2 , A3 = xξ xη + rξ rη , ∆ = xξ rη − xη rξ . Velocity vectors V1, V2, stream function ϕ , vortex function ω , enthalpy H, pressure P, and density ρ must satisfy the relations

V1 = −

(

)

(

)

1 1 ϕ ξ xη − ϕη xξ , V2 = − ϕ rη − ϕη rξ , ρr∆ ρr∆ ξ

(

)

1 V xη − V1η xξ + V2ξ rη − V2η rξ , ∆ 1ξ ρ ω ρ ω P = P0 − ∫  Vξ2 + ϕ ξ  dξ − ∫  Vη2 + ϕη  dη, 2   2 r r  L ( M0 , M ) L ( M0 , M )

ω=

ξ

[

]

ϕ = ϕ 0 + ∫ r( µ, η)ρ( µ, η) V1 ( µ, η)rξ ( µ, η) − V2 ( µ, η) xξ ( µ, η) dµ ξ0

η

[

]

+ ∫ r(ξ, v)ρ(ξ, v) V1 (ξ, v)rη (ξ, v) − V2 (ξ, v) xη (ξ, v) dv, η0

where x, r are cylindrical coordinates, GC is the arbitrary domain with the smooth boundary C from a given domain G, L(M0, M) being the arbitrary curve, connecting the point M ∈ G with the point M0, in

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which pressure P0 is given, ϕ 0 is an arbitrary constant, ξ 0, η0 is the coordinate of a point of the beginning of going around the boundary at calculation of stream function. To calculate the subsonic flow we specify at the exit the mass flow of gas, at the entrance parts the density and velocity in the direction of the normal, on solid walls the condition of nonpenetration, and on the axis of a symmetry the condition of symmetry. The boundary conditions should satisfy the relation of mass balance. The flow in the subsonic region is calculated by a finite difference iterative method, being the modification of the approach [9], in which it is supposed that there are no closed streamlines. In the proposed method, the special approximations of integral equations in the subsonic zone, taking into account the peculiarities of curvilinear grids and also the direction of a stream turns out to be successful. The pressure is calculated by the method of coordinated approximations [8] permitting to avoid the origin of parasitic fluctuations. To solve the algebraic linear system of equations obtained during the approximation, the matrix of which at formation of closed rotational streams is stiff, we use on each iteration a direct economic method with a regularization essentially taking into account block-diagonal structure of matrices. The offered method is realized in the programs SOKOL [1, 10, 11, 16]. The following results are obtained: • The use of optimal curvilinear grids removes restrictions on class of considered

configurations of channels. • The offered method allows calculation of effective both compressible and incompressible streams

with numerous rotational zones. • It is necessary to take into account a compressibility of a medium. • In channels with a nozzle part, taking account of parameters of a stream in the transonic part allows input data, obtained with some error from experiments, to be corrected. • The calculations can be carried out for different types of boundary conditions. Taking into account compressibility of gas and its parameters in the transonic regime, the correct boundary conditions in a series of cases of the domains lead to completely different structure of the flow in channels, namely the formation of closed rotational streams of gas. Figure 36.12b demonstrates the streamlines of gas flow obtained in calculation of rotational flow of compressible fluid in the model channel, when gas moved on lateral surface CD with constant velocity. On the surface EF the velocity was set piecewise constant, at end-wall of the channel AB — under the cosine law [18]. The density of gas on sides where it is blown in is constant. On the exit KL massflow of gas was prescribed from the relation of the mass balance. With this input data three closed vortices have been obtained as the result of calculation. In Figure 36.12 there is a calculated grid that has been cut through one grid line for visualization.

36.5 Conclusion The iterative algorithms for the calculation of three-dimensional grids can be constructed on the same approaches used in Section 36.3, ideas of a combination of explicit iterative methods of the solution of the system of Eq. 36.32 and direct local minimization of the functional Eq. 36.27. Though we do not have effective automated programs in the three-dimensional case, the first positive experience in this direction was described in [27]. For three-dimensional star-shaped domains (they can also evolve in time), a direct transferring of algorithms, used in MOPS-2a and LADA, is possible. More complicated is the question about dividing the complex three-dimensional domain into star-shaped blocks which now is practically not automated. At present, the problem of implementing the algorithms for parallel computation of grids of large dimension with number of cells greater than 106 (for some problems of continuum mechanics requiring large volume of calculations, simulation could be realized only by utilizing the parallel processors) is

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FIGURE 36.12

critical. Such problems include, in particular, the problems of gas dynamics with large deformations that need to be calculated both on moving and on stationary grids. Algorithms in Section 36.3 describe a few ways of parallelizations. These are parallelizing according to blocks for the computation of block-structured grids; parallelizing explicit iterative processes according to groups of neighboring cells [38]; and use of decomposition methods in the solution of E-O equations by iterative methods.

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