[Insight-users] Implementation Finite Element Method (FEM) in ITK

[brian avants]

hi jeremy , cristina

here is a brief latex sketch of the method used in the
FEMCrankNicolsonSolver --

\text{matrix form parabolic PDE, heat equation type} \\
M\frac{ \partial U}{\partial t} +  K U  = f \\
\frac{ M U_t - M  U_{t-1}}{\delta_t} +  K U  = f \\
\frac{ M U_t - M  U_{t-1} +  \delta_t K U }{\delta_t} = f \\
\text{ here i make the decision to represent $U$ and $f$ at $t$ rather
than $t-1$ i believe this is backward Euler}\\
( M  + \delta_t K ) U_t  =  M  U_{t-1} + \delta_t f_t \\
\\
\text{other choices are possible, e.g. } \\
\text{ Crank-Nicolson discretization is based on averaging forward and
backward Euler } \\
( M + \alpha \delta_t K ) U_t = (  M - ( 1- \alpha ) \delta_t  K )
U_{t-1} + f \\
 f  =  \delta_t ( \alpha f_t + ( 1 - \alpha ) f_{t-1} ) \\
\\
\text{if}~ \alpha = 1 \text{ backward Euler }\\
( M +  \delta_t K ) U_t = M  U_{t-1} + \delta_t f \\
\\
\text{if}~ \alpha = 0 \text{ forward Euler }\\
 M U_t =(  M - \delta_t  K ) U_{t-1} + \delta_t f_{t-1}

hopefully this is helpful.

brian




On Tue, Feb 1, 2011 at 4:01 PM, Jeremy Bournesel
<jeremy.bournesel at gmail.com> wrote:
> Hi,
> I have a decent background regarding FEM in the engineering domain and have
> started looking at it for registering medical images as well.
> I ran one of the ITK deformable registration demos, read the Software Guide
> and the ITK Powerpoint Presentations, however there is still some magic
> going on that I don't completely get.
> From what I understand ITK is using a uniform grid/mesh (if a custom one
> wasn't supplied) and then calculates the deformation for every element at
> the nodal points (iteratively).
> I couldn't find any information on how the exact process is working (besides
> some high level slides of the type K U = F).
> It'd be awesome if someone can point me to the right document or explain it
> to me.
> Thanks,
> Jeremy
>
>
>
>
>
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