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Choose of O-grid and C-grid
sharp edge
depend on the flow problem
the principle is more grid points in the important region
Avoid skewed or distorted grid elements
Overset/overlapping grids(moving objects)
V1
V2
-5 -4 -3 -2 -1 0 1 2
-3
-2
-1
0
1
2
3
Frame 001 12 Dec 2006 | |Frame 001 12 Dec 2006 | |
Unstructured grid ------ in FVM(triangular, quadrilateral, prism, tetrahedra, hexahedra)
(more easier to generate, automatic, adaptive refinement)
Transformationcomplex geometry body-fitted grid----the generalized curvilinear coordinate lines are used to describe this grid.
finite difference are difficult to use for curvilinear grid in x-y plane.it is convenient to set finite difference in - plane . PDE should be transformed into - plane and the FDE in - plane is calculated.
Structured grid----- in FDM
Jacobian and metrics of transformation
x=x(,), y=y(,) or =(x,y), =(x,y)
dd
yyxx
dydx
dydx
dd
yx
yx
,
dydx
xyxy
J1
dd
J=xy-xy J is called jacobian.
x,y,x,y etc. are called metrics.
PDEs transformed into -- space
NS eqs transformation
In x-y-t space, the conservative NS eqs is Ut+Fx+Gy=0 In -- space, a conservative form of NS eqs is also needed.
In -- space, U , F, G are functions of (,,), using chain rule in differential calculus,
U
+F+G
=0
where U=JU, F=JUt+JFx+JGy, G=JUt+JFx+JGy
Grid quality
several requirements1 One-to-one correspondence. if J0, it is ok, if J=0, some errors exist in the grid.2 Smooth. the grid spacing step change slowly3 Orthogonal or near-orthogonal. the grid lines of two families (,) are perpendicular or nearly perpendicular to each other
4 Enough grid points in the important region
Example . Considering the Ux at point (i),
Ui+1=Ui+(Ux)i∆x1+0.5(Uxx)i(∆x1)2+
Ui-1= Ui(Ux)i∆x2+0.5(Uxx)i(∆x2)2+
1 12 1
2 1
( ) 0.5( ) ( )i ix i xx i
U UU U x x
x x
If ∆x2=∆x1 2nd order
If ∆x2 «∆x1
)xx()U(5.0 12ixx 1ixx x)U(5.0
1st order
1 12 1
2 1
( ) 0.5( ) ( )i ix i xx i
U UU U x x
x x
Grid generation---Possion equation method
Considering the solution of a steady heat conduction problem in 2D with Dirichlet B.C., the solution of this problem produces isotherms which are smooth and are nonintersecting. The number of isotherms in a given region can be increased by adding a source term. Hence, isotherms can be viewed as grid lines.
So we consider Poisson equations:
),( Pyyxx
),( Qyyxx
0)(2)( qxxxpxx
0)(2)( qyyypyy
22 yx yyxx 22
yx
/2PJp /2QJq yxyxyxJ
),(),(1
After the source terms p and q are specified with some methods on the boundary and interpolated into the inner domain, we can solve the equations,then the grid is obtained.
How to solve the above equations ?One solution: Five-point formula and GS.