结构与非结构网格之间的转换及应用

Applications Of Transformation Of Structured To
Unstructured Meshes
Liu Jing1, 2，Zhang Min1，John C. Chai2，Xu Bin1
1School of Power Eng.，Nanjing University of Science & Technology，Nanjing (210094)
2School of Mechanical and Aero spacing Eng.，Nanyang Tech. University，Singapore (639798)
E-mail：mz2455@
Abstract
The transformation of structured meshes to unstructured meshes is a branch of mesh generation technology. We can obtain the advantages of both grids that structure grids have the characteristics of convergence quickly and unstructured grids have the characteristics of matching sophisticated calculating domains well from this conversion. Meanwhile, it is expanding the widespread useful application of unstructured mesh codes. This paper gave the models of the transformations of the orthogonal meshes and body-fitted meshes. And, the heat conduction equation was solved using the based cell finite volume method and the secondary order accuracy. Finally, a couple of three dimension examples of heat transfer that included different geometries and boundary conditions were given. Therefore, the procedure was validated exactly and actually.
Keywords：structured grids/meshes，unstructured grids/meshes，heat conduction
1.Introduction
The first step of numerical simulation is mesh generation that is cutting the continuous computational space into subdomains and identifying each node. The accuracy and efficiency of engineering numerical simulation mainly defend on the meshes and algorisms. In generally, all kind of mesh has its advantages and disadvantages; also the every numerical method has its constraints. Therefore, successful numerical simulation can only be done on the conditions that meshes and algorisms match perfectly [1].
Two commonly kinds of mesh are structured and unstructured mesh/grid. The former characteristic is that the relationship between nodes is fixed and implied in the mesh. Thus, no special action is needed to ensure the relationship. But there don’t exists the property in unstructured mesh, so we must store the information about nodes such as volume nodes number, interfaces nodes number, and neighbor volume number[2-4] .
It is stubborn to compare structured grid and unstructured grid exactly, besides considering the numerical algorism. In the brief, structured mesh has the good feature, simplex in generating, converging fast, and steady etc, while unstructured mesh can be more applicable for irregular domain, decomposing and encrypting in whole or part domain and used widely in later computation[4] . The paper takes advantage of two kinds of mesh to get fine results by the transformation between them.
2.Transformation Between Both Meshes
Regular structured mesh in orthogonal coordination is the oldest, most basic and simplex generation technique, including rectangle mesh of Cartesian coordinates and curve mesh in cylindrical coordinates or spherical coordinates. No detail about this kind of mesh, but the paper based on orthogonal mesh and body-fitted grid.
First, we have to get the grid nodes of coordination in three dimensions, and then transform them to unstructured grid nodes number. Finally, numerical simulation will be done based on the unstructured mesh. For the transformation, at first, select cells shape and nodes NCTYPE(I) and NCNODE(J,I), here they are vertex number and coordination value (X(I),Y(J),Z(K)) of cell, respectively. Secondly, get the surface information NFTYPE (I) and NFNODE (J, I) of the cells. Where, the node order conform right hand rule, which is, ensuring the direction of surface normal is outside the cells.
At the end, storing all neighbor cells information and their boundary property by KBCC (I).
Ultimately, we can obtain the six data files. It is exactly these files comprise surfaces and nodes number for every cell and surface. The key of transformation is rearranging the I/J/K order of structured grid nodes to cell series data structure. Although the program is easy to do, the technique proved to be a handicap. Next part program is given in two dimensions.
C**************************************************
COME HERE FOR THE NODES OF CELL (cell_node.dat)
LM=L2*M2 I0=0 J0=0
DO 30 I=1,NCV NCTYPE(I)=8 NCNODE(1,I)=I
+I0+J0
NCNODE(2,I)=I+1 +I0+J0 NCNODE(3,I)=I+L1+I0+J0 NCNODE(4,I)=I+L2+I0+J0 NCNODE(5,I)=I
+I0+J0+LM
NCNODE(6,I)=I+1 +I0+J0+LM NCNODE(7,I)=I+L1+I0+J0+LM NCNODE(8,I)=I+L2+I0+J0+LM IF(MOD(I,L3).EQ.0) I0=I0+1
IF(MOD(I,L3*M3).EQ.0) J0=J0+L2
30 CONTINUE
C**************************************************
The particular examples and their results analysis are provided in following paragraphs.
3. Heat Conduction Exampls
Problem 1: We have heat transfer conduction without heat source in cubic region. Geometry and computational grids are showed in figure1, and governing equation is heat conduct equation with constant property in three Cartesian coordinates. The left surface has higher temperature T 2, and the left five ones have lower temperature T 1. Arithmetic formula of governing function and boundary conditions are:
0=⎟⎠⎞
⎜⎝⎛∂∂∂∂+⎟⎟⎠⎞⎜⎜⎝⎛∂∂∂∂+⎟⎠⎞⎜⎝⎛∂∂∂∂z T k z y T k y x T k x
(1.1) 0.1,0.1,0.0,0.121======k T T c b a
(1.2)
(a) Cubic V olume (b) Orthogonal meshes (c) Body-fitted meshes
Figure 1 Geometry and structured/unstructured meshes
We can obtain the exact solution of (1.1) and (1.2) (Kakac and Yener, 1993)[5]，
[][]∑∑∞
=∞
=−−−−−=−−=
111
21
sinh )(sinh )sin()sin(])1(1[])1(1[4),(),(m n mn mn m n m m
n n
b y b z x a
c T T T y x T y x ααβλβλθ
(1.3)
Where,
n λa n π
=
(n = 1, 2,…,i ) =m βc
m π (m = 1, 2, …,i )
2
2m
n mn βλα+=
(1.4)
In Figure 2, the results of temperature distributions were from the transformation of orthogonal
meshes to unstructured grids. The same one was from the transformation of body-fitted meshes to unstructured grids in Figure 3. The solid lines stand for the exact solution. The dashed lines represent numerical solution. The numbers of grid are 10*10*10. There are agreements of temperature fields in both meshes.
(a) X =0.5 (b) Y =0.2 (c) Z =0.5
Figure 2. The temperature field of orthogonal meshes
(————Exact Solution - - - - - -Numerical Solution)
(a) X =0.5 (b) Y =0.2 (c) Z =0.5
Figure 3. The temperature field of body-fitted meshes
Problem 2: We have heat transfer conduction within heat source in cubic region. Geometry and computational grids are showed in figure1, and governing equation is heat conduct equation with constant property in three Cartesian coordinates as following. The all surfaces maintain the constant temperature (T 1 =0) same as the first kind of boundary condition. Mathematical formula of governing function and boundary conditions are:
=∂∂+∂∂+∂∂2
22222z
T y T x T )sin()sin()(1
c z a x b y y k ππ−− (2.1)
The exact solution of this problem is [6]，
)sin()sin()sin(]
1()()1[(1
π8),,,5,3,12223
5
2
c z b y n a x c
b n a n k
b z y x T n πππ⋅++−=∑∞
=L （ (2.2)
The results of temperature distributions were from the transformation of body-fitted meshes to
unstructured grids in Figure 4. The solid lines stand for the exact solution. The dashed lines represent numerical solution. The numbers of grid are 10*10*10. There are agreements of temperature fields in both meshes. There are the symmetrical temperature distribution basic of the boundary conditions and geometry.
(a) X =0.5 (b) Y =0.5 (c) Z =0.5
Figure 4. The temperature field of body-fitted meshes
Problem 3: We have heat transfer conduction without heat source in cylindrical region. Geometry and computational grids are showed in figure 5a, and governing equation is heat conduct equation with constant property in three cylindrical coordinates as following. The outside surface has higher temperature T 2, and the inside surface has lower temperature T 1. Mathematical formula of governing function and boundary conditions are:
0112=⎟⎠⎞
⎜⎝⎛∂∂∂∂+⎟⎟⎠⎞⎜⎜⎝⎛∂∂∂∂+⎟⎠⎞⎜⎝⎛∂∂∂∂z T k z T k r r T kr r r ϕϕ
(3.1) 0.1,0.8,0.4,0.2,0.12121=====k T T r r r r
(3.2)
The exact solution of this problem is [7-8]，
)/ln()
/ln()/ln()(122211r r r r T r r T r T r r −=
(3.3)
In Figure 5, the results of temperature distributions were from the transformation of body-fitted
meshes to unstructured grids. The solid lines stand for the exact solution. The dashed lines represent numerical solution. The numbers of grid are 10*10*10. There are agreements of temperature fields in both meshes. There are the symmetrical temperature distribution basic of the boundary conditions and geometry.
(a) Meshes (b)Temperature fields in Z=0.5 (c) The flood picture of temperature
Figure 5. The temperature field of cylindrical coordinates
4.Closure Remark
The produces, which the heat conduction equations were solved, was presente d using the unstructured meshes that were transformed from structured grids. There are two kinds of meshes including orthogonal and body-fitted meshes. We show three examples for evaluating and proving this processor accruable and reasonable. The problem one and two are in the Cantinas coordinate and the problem three is in cylindrical coordinate. All results of numerical simulation were compared with the exact solutions. As a result, there is a perfect agreement between them.
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