Regarding implicit Euler integration

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mobeen
Posts: 122
Joined: Thu May 05, 2011 11:47 am

Regarding implicit Euler integration

Post by mobeen »

Hi all,
I am trying to implement implicit Euler integration and I have run into an issue which hopefully someone here would know how to solve. In the general case, the implicit Euler integration is based on the following equation (ref: http://www.matthiasmueller.info/realtim ... enotes.pdf)

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[M - dt^2.K]V^(t+1) = MV^t + dt.f(X^t) which gives 
A = M - dt^2.K.
b = MV^t + dt.f(X^t)
and this system is solved for V^(t+1). Here, K is the stiffness matrix that is Jacobian of force (f). Now this matrix (K) is a 3n x 3n matrix, the mass matrix (M) is also 3n x 3n.
K is composed of 4 sub matrices K = [K_ii K_ij K_ji K_jj] each individual matrix K_nn is a 3x3 matrix which makes K a 6x6 matrix. Now if I am correct, to solve this, the mass matrix will also be 6x6 diagonal matrix containing the masses in the diagonal.

Now my question is when i am trying to solve this system using CG solver, how do i bring the large vectors back to 3 components to ultimately get the new velocity?
Currently, I am only taking the first sub matrix of K i.e K_ii however, there are two issues
1) the simulation sort of drifts to one side.
2) the simulation progresses slower and the springs appear stiffer as compared to explicit integration. Is this expected?
Any ideas?
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