bilinear interpolationgtryggva/cfd-course/2013-lecture-24.pdf · bilinear interpolation!...

Computational Fluid Dynamics

Elementary!Grid Generation!

Grétar Tryggvason !Spring 2013!

http://www.nd.edu/~gtryggva/CFD-Course/!Computational Fluid Dynamics

Bilinear Interpolation!

Results for the Vorticity-Streamfunction formulation!

Elliptic grid generation!

Outline!




Simples grid generation is to break the domain into blocks and use bilinear interpolation within each block!

As an example, we will write a simple code to grid the domain to the right!

(x1,y1)!

(x2,y2)!(x3,y3)!

(x4,y4)!(x5,y5)!

(x6,y6)!(x7,y7)!

(x8,y8)!



Start by breaking the domain into blocks!

Then grid each block separately!

(x1,y1)!

(x2,y2)!(x3,y3)!

(x4,y4)!(x5,y5)!

(x6,y6)!(x7,y7)!

(x8,y8)!

Bilinear Interpolation!Computational Fluid Dynamics

P1,0!1!

2!

3!

P1,3!

P1,2=P2,3!

P1,1=P2,0! P2,1=P3,0!

P3,2! P3,1!P2,2=P3,3!





2!

P1,0!1!

P1,3!

P1,2!

P1,1!

3!

P3,0!

P3,2! P3,1!

P2,2!

P3,3!

P2,1!P2,0!

P2,3!




Consider an arbitrary shaped quadrilateral block!1. Select the ξ and ξ direction.!

(x0,y0)!

(x1,y1)!

(x2,y2)!(x3,y3)!

�

ξ

�

η

N!

M!2. Divide the

opposite sides evenly with N points in the ξ direction and !

M in the ξ direction and draw

straights line between the points

on the opposite sides!



�

x ξ ,1( ) =N −ξN −1

⎛ ⎝

⎞ ⎠ x0 +

ξ −1N −1

⎛ ⎝

⎞ ⎠ x1

Along the edge between points 0 and 1!

Along the edge between points 3 and 2!

�

x ξ ,M( ) =N −ξN −1

⎛ ⎝

⎞ ⎠ x3 +

ξ −1N −1

⎛ ⎝

⎞ ⎠ x2

Then interpolate again for points between the edges!

�

x ξ ,η( ) =M −ηM −1

⎛ ⎝

⎞ ⎠ N −ξN −1

⎛ ⎝

⎞ ⎠ x0 +

ξ −1N −1

⎛ ⎝

⎞ ⎠ x1

⎛ ⎝ ⎜ ⎞

⎠ +

η−1M −1

⎛ ⎝

⎞ ⎠

N −ξN −1

⎛ ⎝

⎞ ⎠ x3 +

ξ −1N −1⎛ ⎝

⎞ ⎠ x2

⎛ ⎝ ⎜ ⎞

⎠

The y-coordinate is found in the same way!


�

x ξ ,η( ) =M −ηM −1

⎛ ⎝

⎞ ⎠ N −ξN −1

⎛ ⎝

⎞ ⎠ x0 +

ξ −1N −1

⎛ ⎝

⎞ ⎠ x1

⎛ ⎝ ⎜ ⎞

⎠ +

η−1M −1

⎛ ⎝

⎞ ⎠

N −ξN −1

⎛ ⎝

⎞ ⎠ x3 +

ξ −1N −1⎛ ⎝

⎞ ⎠ x2

⎛ ⎝ ⎜ ⎞

⎠

�

y ξ ,η( ) =M −ηM −1

⎛ ⎝

⎞ ⎠ N −ξN −1

⎛ ⎝

⎞ ⎠ y0 +

ξ −1N −1

⎛ ⎝

⎞ ⎠ y1

⎛ ⎝ ⎜ ⎞

⎠ +

η−1M −1

⎛ ⎝

⎞ ⎠

N −ξN −1

⎛ ⎝

⎞ ⎠ y3 +

ξ −1N −1

⎛ ⎝

⎞ ⎠ y2

⎛ ⎝ ⎜ ⎞

⎠

(x0,y0)!

(x1,y1)!

(x2,y2)!(x3,y3)!

�

ξ

�

η

N!

M!

For a single block, we therefore have:!



For many blocks:!1. we must interpolate for each block and !2. ensure that their boundaries are correct!

P1,0!1!

2!

3!

P1,3!

P1,2=P1,2!

P1,1=P2,0! P2,1=P3,0!

P3,2! P3,1!P2,2=P3,3!


Short MATLAB program to generate the grid!

xp(1,1:4)=[0,0,1,2];xp(2,1:4)=[0.5,0.5,1,1.5];!yp(1,1:4)=[1,0,0,0.5];yp(2,1:4)=[1,0.5,0.5,1];!

(0.0,1.0)!

(0.0,0.0)!

(0.5,1.0)!

(1.0,0.5)!

(1.0,0.0)!

(2.0,0.5)!(0.5,0.5)!

(1.5,1.0)!



xp(1,1:4)=[0,0,1,2];xp(2,1:4)=[0.5,0.5,1,1.5];!yp(1,1:4)=[1,0,0,0.5];yp(2,1:4)=[1,0.5,0.5,1];!

NumBlocks=3;!Nblock(1)=8;Nblock(2)=5;Nblock(3)=12; NTot=0;!Mblock(1)=6;Mblock(2)=6;Mblock(3)=6; MTot=6;!

for l=1:NumBlocks! N=Nblock(l);M=Mblock(l);! x0=xp(1,l);x1=xp(1,l+1);x3=xp(2,l);x2=xp(2,l+1);! y0=yp(1,l);y1=yp(1,l+1);y3=yp(2,l);y2=yp(2,l+1);! Nf=2;if(l == 1),Nf=1;end! for i=Nf:N, for j=1:M !! ii=i+NTot;! x(ii,j)=((M-j)/(M-1))*(((N-i)/(N-1))*x0+((i-1)/(N-1))*x1)+...! ((j-1)/(M-1))*(((N-i)/(N-1))*x3+((i-1)/(N-1))*x2);! y(ii,j)=((M-j)/(M-1))*(((N-i)/(N-1))*y0+((i-1)/(N-1))*y1)+...! ((j-1)/(M-1))*(((N-i)/(N-1))*y3+((i-1)/(N-1))*y2);! end, end;! NTot=NTot+Nblock(l)-1;!end!

axis(’equal'), hold on!for j=1:MTot;plot(x(1:NTot,j),y(1:NTot,j));end;!for i=1:NTot;plot(x(i,1:MTot),y(i,1:MTot));end;!


The grid!



Sometimes the grid can be improved by smoothing. The simples smoothing is to

replace the coordinate of each grid point by the average of the coordinates around it. This

process can be repeated several times to improve the smoothness.!


�

x i, j( ) = 0.25* x i +1, j( ) + x i −1, j( ) + x i, j +1( ) + x i, j −1( )( )y i, j( ) = 0.25* y i +1, j( ) + y i −1, j( ) + y i, j +1( ) + y i, j −1( )( )


The grid after two smoothing iterations!



Bilinear Interpolation can also be used for curved boundaries, if the points on the

boundaries are given.!

Higher order Interpolation functions can also be used to generate stretched grids for

complex boundaries.!


The grid generation results in an array of x and y coordinates for each grid point:!

For simple problems we can include the grid generator in the fluid solver and generate the coordinates before we solve for the fluid motion!

For more complex problems the grid generation step is usually separated from the flow solver and the grid points are read from a file!

For commercial codes you can generally use many different grid generators—as long as the data format is consistent !


�

x ξ,η( ), η ξ,η( )


%Function gridgen_elbow.m!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%!function grid=gridgen_elbow(x,y)!global x y!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%!xp(1,1:3)=[0,0,1.5];!xp(2,1:3)=[0.5,0.5,1.5];!yp(1,1:3)=[1,0,0];!yp(2,1:3)=[1,0.5,0.5];!

NumBlocks=2;%3!Nblock(1)=8;!Nblock(2)=14;%5!

NTot=0;!Mblock(1)=8;!Mblock(2)=Mblock(1);!MTot=Mblock(1);!

for l=1:NumBlocks! N=Nblock(l);! M=Mblock(l);! x0=xp(1,l);! x1=xp(1,l+1);! x2=xp(2,l+1);! x3=xp(2,l);! y0=yp(1,l);! y1=yp(1,l+1);! y2=yp(2,l+1);! y3=yp(2,l);! Nf=2;! if(l == 1)! Nf=1;! end!

for i=Nf:N, ! for j=1:M !! ii=i+NTot;! x(ii,j)=((M-j)/(M-1))*(((N-i)/(N-1))*x0+((i-1)/(N-1))*x1)+...! ((j-1)/(M-1))*(((N-i)/(N-1))*x3+((i-1)/(N-1))*x2);! y(ii,j)=((M-j)/(M-1))*(((N-i)/(N-1))*y0+((i-1)/(N-1))*y1)+...! ((j-1)/(M-1))*(((N-i)/(N-1))*y3+((i-1)/(N-1))*y2);! end, ! end;! NTot=NTot+Nblock(l)-1;! %NTot=NTot+Nblock(l);!end!

axis('equal'), hold on!for j=1:MTot;! % plot(x(1:NTot,j),y(1:NTot,j),'linewidth',1);!end;!for i=1:NTot;! %plot(x(i,1:MTot),y(i,1:MTot),'linewidth',1);!end;!axis('square'),axis([-0.10, 2, 0, 2]); hold on!axis off!disp('un-smoothed grid')!

for i=1:3!!for i=2:NTot-1!! for j=2:MTot-1!! x(i,j)=0.25*(x(i+1,j)+x(i-1,j)+x(i,j+1)+x(i,j-1));!

y(i,j)=0.25*(y(i+1,j)+y(i-1,j)+y(i,j+1)+y(i,j-1));!! end;!!end;!

end;!axis('square'),axis([-0.10, 2, 0, 2]); hold on!for j=1:MTot;! plot(x(1:NTot+1,j),y(1:NTot+1,j),'r','linewidth',1);!end;!for i=1:NTot+1;! plot(x(i,1:MTot),y(i,1:MTot),'r','linewidth',1);!end;!axis off!

Function gridgen_elbow.m!use with NS_Vorticity_Stream_Elbow.m!

Computational Fluid Dynamics Bilinear Interpolation!

Flow in an elbow, computed using a body fitted grid, using the streamfunction-vorticity formulation of the Navier-

Stokes equations!

Grid—bilinear interpolation with smoothing!

Streamfunction!

Computational Fluid Dynamics Bilinear Interpolation!

While bilinear interpolation is often the simplest approach for relatively simple domains, it usually

requires fairly large amount of human input.!

Thus, there have been major attempts to make the grid generation more automatic. !


Elliptic Grid Generation!


Elliptic Scheme!

“Isotherms” of the conduction equation!

02 =∇ Thave nice properties of smoothness and concentrated !contour spacing where solution has a large spatial gradient.!

è Why not use the solution to the Laplace equation as the new coordinate?!

0,0 22 =∇=∇ ηξ

Elliptic Grid Generation!Computational Fluid Dynamics


Elliptic Scheme! 0,0 22 =∇=∇ ηξ

�

ξ = 0, ∂η /∂ξ = 0

�

ξ = M, ∂η /∂ξ = 0

�

η = 0∂ξ∂η

= 0

�

η = N∂ξ∂η

= 0


Few points in regions of interest!

Need more control over the point location!

0,0 22 =∇=∇ ηξ


Further control can be applied by adding a source term:!

Proper choice of provides the shape of the mesh.!

In practice, instead of solving for in terms of!

, we want to solve for in terms of .!

),(),,( 22 ηξηηξξ QP =∇=∇

QP,

ηξ ,yx, yx, ηξ ,

Transform the Poisson equation into space.!( )ηξ ,



Adding the two equations:!

we obtain !�

xξ ∇2ξ + xη∇

2η = xξ P(ξ ,η) + xηQ(ξ ,η)

yξ ∇2ξ + yη∇

2η = yξ P(ξ ,η) + yηQ(ξ ,η)

( )ηξηηξηξξ QxPxJxqxqxq +−=+− 2123 2

( )ηξηηξηξξ QyPyJyqyqyq +−=+− 2123 2

where !

223

2

221

ξξ

ηξηξ

ηη

yxq

yyxxq

yxq

+=

+=

+=


In discretized form (x-equation):!

�

α xi+1, j − 2xi, j + xi−1, j( ) − 0.5β xi+1, j+1 − xi+1, j−1 − xi−1, j+1 + xi−1, j−1( )

where !

�

γ xi, j+1 − 2xi, j + xi, j−1( ) + 0.5δ P xi+1, j − xi−1, j( ) + Q xi, j+1 − xi, j−1( )[ ] = 0

�

α = 0.25 xi, j+1 − xi, j−1( )2 + yi, j+1 − yi, j−1( )2[ ]

�

β = 0.25 xi+1, j − xi−1, j( ) xi, j+1 − xi, j−1( ) + yi+1, j − yi−1, j( ) yi, j+1 − yi, j−1( )[ ]

�

γ = 0.25 xi+1, j − xi−1, j( )2 + yi+1, j − yi−1, j( )2[ ]

�

δ = 116

xi+1, j − xi−1, j( ) yi, j+1 − yi, j−1( ) − yi+1, j − yi−1, j( ) xi, j+1 − xi, j−1( )[ ]2



suggested by Thompson (1977)!

�

P ξ,η( ) = − al sgn ξ −ξ l( )exp −cl ξ −ξ l( )l=1

L

∑

QP,

�

− bm sgn ξ −ξm( )exp −dm ξ −ξm( )2 + η −ηm( )2[ ]1/ 2⎛ ⎝ ⎜

⎞ ⎠ ⎟

m=1

M

∑

�

Q ξ,η( ) = − al sgn η −ηl( )exp −cl η −ηl( )l=1

L

∑

�

− bm sgn η −ηm( )exp −dm ξ −ξm( )2 + η −ηm( )2[ ]1/ 2⎛ ⎝ ⎜

⎞ ⎠ ⎟

m=1

M

∑where are chosen to generate appropriate !

grid clustering.!mlml dcba ,,,

( )⎪⎩

⎪⎨⎧

<−=>

=0if10if00if1

sgnxxx

x


Selection of!

�

P ξ,η( ) = −al sgn ξ −ξ l( )exp −cl ξ −ξ l( )QP,

Attracts grid lines to the line ξ = ξl.!

al determines how strongly and !

cl determines the “reach” of the attraction!

�

ξ l



Selection of! QP,

�

P(ξ,η) = −bm sgn ξ −ξm( )exp −dm ξ −ξm( )2 + η −ηm( )2[ ]1/ 2⎛ ⎝ ⎜

⎞ ⎠ ⎟

Attracts grid lines to the line ξ = ξm, near the point (ξ,η) = ξm, ηm).!bm determines how

strongly and !dm determines the

“reach” of the attraction!

�

ξm�

ηm


Although mapped grids have severe limitations in terms of the geometric complexity that they can handle, it is nevertheless possible to deal with reasonable complex geometries through the use of branch cuts and blocks.!

This does, however, require significant user input and can be challenging, particularly for three-dimensional geometries.!


Here the object is introduced in two ways: on the left we bend the grid around the cylinder, turning the corner in front and back into a point on the xi=0 axis. On the right we introduce a branch-cut that preserves the corner but requires us to treat an internal boundary!

From: http://www.hpc.msstate.edu/publications/gridbook/!

Mapped grids for Complex Domains!Computational Fluid Dynamics

To accommodate this complex geometry the regular mapped grid is not a rectangle. However, all boundary conditions are imposed on a grid line !


Mapped grids for Complex Domains!


More examples of complex geometries handles by either branch cuts or mapping internal boundaries to the outer boundaries of the mapped grid!




A complex geometry handled by the introduction of more than one mapped grid (multi-block grid). The values on the boundaries must be the same to maintain a smooth solution!




And yet another example!!


When corner points are mapped into a straight boundary, we usually need to treat these points in a special way!




Accuracy of Generalized Coordinates!

Computational Fluid Dynamics Accuracy of Generalized Coordinates!

Physical Domain !

xΔ xrxΔ

1−if if 1+ifMapped Domain !

ξΔ ξΔ

1−if if 1+if

11

11

−+

−+

−−≈=

∂∂

ii

ii

xxff

xf

xf

ξ

ξ

Considering 1-D Stretching !

Taylor expansion !

fi+1 − fi−1

xi+1 − xi−1

=

fξ 1+ Δξ2

6fξξξfξ

+⎛

⎝⎜

⎞

⎠⎟

xξ 1+ Δξ2

6xξξξ

xξ

+⎛

⎝⎜

⎞

⎠⎟

rx =xi+1 − xixi − xi−1


fi+1 − fi−1

xi+1 − xi−1

=

fξ 1+ Δξ2

6fξξξfξ

+⎛

⎝⎜

⎞

⎠⎟

xξ 1+ Δξ2

6xξξξ

xξ

+⎛

⎝⎜

⎞

⎠⎟

=

fξxξ

1+Δξ2

6fξξξfξ

+⎛

⎝⎜

⎞

⎠⎟ 1−

Δξ2

6xξξξ

xξ

+⎛

⎝⎜

⎞

⎠⎟ Δξ << 1( )

=

fξxξ

1+Δξ2

6fξξξfξ

−xξξξ

xξ

⎛

⎝⎜

⎞

⎠⎟ +

⎡

⎣⎢⎢

⎤

⎦⎥⎥

It may appear 2nd order accuracy, but…!

Accuracy of Generalized Coordinates!Computational Fluid Dynamics

�

f i+1 − fi−1xi+1 − xi−1

= fx + Δξ 2

6fξξξ xξ( )2 + 3 fxx xξξ[ ] +

Further algebraic expansion gives!

xξ =

x j+1 − x j−1

2Δξ+ O(Δξ2 ) = 0.5 1+ rx( ) Δx

Δξ+ O(Δξ2 )

xξξ =

x j+1 − 2x j + x j−1

Δξ2 + O(Δξ2 ) = rx −1( ) ΔxΔξ2 + O(Δξ2 )

fi+1 − fi−1

xi+1 − xi−1

= fx +16

16

1+ rx

2⎛

⎝⎜⎞

⎠⎟

2

Δx2 fxxx +rx −1

2⎛

⎝⎜⎞

⎠⎟Δxfxx

⎡

⎣⎢⎢

⎤

⎦⎥⎥+

Moral: avoid large changes in ! )1( ≈xi rx



�

∂f∂x

≈ f i+1 − fi−12Δξxξ

= fx + Δξ 2

6fξξξfξ

fx + 3xξξ fxx + xξ( )2 fxxx⎡

⎣ ⎢

⎤

⎦ ⎥ +

If is evaluated exactly!

The use of exact evaluation of introduces an additional!(and dominant) term in the truncation error. !Therefore, numerical evaluation is preferred.!

ξx

ξx


How NOT to do uneven grids!

i i +1 i −1

fi+1 = fi +

∂f∂x

Δxi+1/ 2 +∂2 f∂x2

Δxi+1/ 22

2+∂3 f∂x3

Δxi+1/ 23

6+…

fi−1 = fi −

∂f∂x

Δxi−1/ 2 +∂2 f∂x2

Δxi−1/ 22

2−∂3 f∂x3

Δxi−1/ 23

6+…

∂f∂x

=Δxi−1/ 2

2 fi+1 + Δxi+1/ 22 + Δxi−1/ 2

2( ) fi − Δxi+1/ 22 fi−1

Δxi−1/ 2Δxi+1/ 2 Δxi−1/ 2 + Δxi+1/ 2( )



This approximation generally does not have the same symmetry properties as discretizing the mapped derivative and should generally be avoided!

From:!R.W.C.P. Verstappen and A.E.P. Veldman!Symmetry-preserving discretization of turbulent flow JCP187 (2003) 343–368!


Working with the mapped derivative also makes it straight forward to obtain higher order approximations. For example:!

∂f∂x

=

∂f∂ξ

∂x∂ξ

∂f∂ξ

=− fi+2 + 8 fi+1 − 8 fi−1 + fi−2

12Δξ+ O Δξ4( )

yields:!

∂f∂x

=− fi+2 + 8 fi+1 − 8 fi−1 + fi−2

−xi+2 + 8xi+1 − 8xi−1 + xi−2

and!

From:!R.W.C.P. Verstappen and A.E.P. Veldman!Symmetry-preserving discretization of turbulent flow JCP187 (2003) 343–368!



By mapping a complex domain can be mapped into a rectangular one, where the techniques that we have covered can be used directly!

The governing equations, however, become more complex!

Although the geometry of the domain can be fairly complex, it must be possible to map it onto a rectangular domain. For more complex domains it is possible to use block-structured grids, but very complex shapes are more easily done by unstructured grids!

Mapped Grids!Computational Fluid Dynamics

http://www.hpc.msstate.edu/publications/gridbook/!

The “classic”!

Here we have examined the elementary aspects of representing the governing equations in a mapped coordinate system. This is a complex topics with its own “language” that needs to be mastered for further studies. !

bilinear interpolationgtryggva/cfd-course/2013-lecture-24.pdf · bilinear interpolation!...

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