Optimization and Computational Fluid Dynamics - Department of ...

116 Nicolas R. Gauger
and the four additional conditions
n�
βi = 0
i=1
n�
i=1
n�
i=1
n�
i=1
βi xi =0
βi yi =0
βi zi =0
which can be physically interpreted as equilibrium equations.
This results in solving the following linear system of equations
⎛ ⎞ ⎛
⎞ ⎛ ⎞
0 00 0 0 1 1... 1 α1
⎜ 0 ⎟ ⎜
⎟ ⎜00
0 0x1 x2 ... xn⎟
⎜α2⎟
⎟ ⎜ ⎟
⎜ 0 ⎟ ⎜
⎟ ⎜00
0 0y1 y2 ... yn⎟
⎜α3⎟
⎟ ⎜ ⎟
⎜ 0 ⎟ ⎜
⎟ ⎜00
0 0z1 z2 ... zn⎟
⎜α4⎟
⎟ ⎜ ⎟
⎜f1⎟
= ⎜
⎜ ⎟ ⎜1
x1 y1 z1 0 ǫ12 ...ǫ1n⎟
· ⎜β1⎟
⎟ ⎜ ⎟
⎜f2⎟
⎜
⎜ ⎟ ⎜1
x2 y2 z2 ǫ21 0 ...ǫ2n⎟
⎜β2⎟
⎟ ⎜ ⎟
⎜ ⎟ ⎜
⎟ ⎜ ⎟
⎝ . ⎠ ⎝.
. . . . . ... . ⎠ ⎝ . ⎠
1 xn yn zn ǫn2 ǫn2 ... 0
fn
where ǫij = � (xi − xj) 2 +(yi − yj) 2 +(zi − zj) 2 is the Euclidean distance
between the interpolation points (xi,yi,zi) and(xj,yj,zj).
After solving this system of equations the interpolation is ready to be used
with the given formula (5.1) for arbitrary points (x, y, z).
This general interpolation method is now applied to the differences of the
original and deformed surface dx, dy, dz. These functions are each interpolated
with the differences of the surfaces as interpolation points. Afterwards,
dx, dy, dz are applied to the computational grid and therefore yield a grid
deformation.
Therefore, let (xold,i,yold,i,zold,i) be the old and (xnew,i,ynew,i,znew,i) be
the new surface points (1 ≤ i ≤ n). Then the functions dx, dy, dz can be
interpolated with the interpolation point values
dxi = xnew,i − xold,i ,
dyi = ynew,i − yold,i ,
dzi = znew,i − zold,i
at (xold,i,yold,i,zold,i). These obtained functions dx, dy, dz can then be computed
at arbitrary points. Now let (aold,j,bold,j,cold,j) be the old computa-
βn