Real-Time Physics Simulation Forum

Question about deformable solid paper

2007-03-01T23:57:06+00:00

Ah I see, I wasnt quite sure what exactly a Jacobi was, but I get it now

I got the damping working correctly now, thanks for the help

Statistics: Posted by coderchris — Thu Mar 01, 2007 11:57 pm

Question about deformable solid paper

2007-03-01T23:37:42+00:00

dC/dp is the Jacobi matrix and in this particlur case it is the gradient of the constraint function, so you better think of it in this case as a row vector. In our example it is.

J = ( Transpose(n) -Transpose(n) )

So in this case you have a 1 x 6 matrix.

Basically the Jacobi matrix stores all gradient vectors. by definition (there are some reasons for this) the gradients are stored in the rows. So this is always a bit confusing the 1D case where you can either interpret it as a vector (the gradient) or as matrix.

Note that this yields the same results:

a * b = Transpose( a ) * b // where a and b are vectors

Does this help?

Statistics: Posted by Dirk Gregorius — Thu Mar 01, 2007 11:37 pm

Question about deformable solid paper

2007-03-01T23:29:57+00:00

Ahh, I understand completely now
seems to be working out right for me now

I just have one more question regarding the paper; In equation (3) they describe how to dampen the forces. Inside the sumnation, they have

dC
--- v
dp

As I understand it, dC/dp is a vector but so is velocity (v).
Are they implying a dot product here (x1*x2 + y1*y2 + z1*z2)?
Or, do they imply a component-wise multiplication? (x1*x2, y1*y2, z1*z2)

Statistics: Posted by coderchris — Thu Mar 01, 2007 11:29 pm

Question about deformable solid paper

2007-03-01T21:42:48+00:00

if x1 is the particle position then the change of the position in time is the velocity v1 = dx1/dt

(v1 - v2) is the inner derivative (I hope you call it like this in english).
Take the simple example C( x(t) ) = | x(t) |

Then dC/dt = d/dx * | x(t) | * dx(t) / dt (Chain rule)

So,

dC/dt = x(t) / |x(t)| * v

the dC/dz was a typo and I fixed it. It should read dC/dt of course

Statistics: Posted by Dirk Gregorius — Thu Mar 01, 2007 9:42 pm

Question about deformable solid paper

2007-03-01T21:20:03+00:00

I took a look at the livemath maker; i cant seem to get it to derive anything, but I probably just havent spent enough time with it

I wouldn't have figured it out, except that there are help files/videos:

video: http://support.livemath.com/lmhelp/main ... =954790820

you can download a livemath workbook from the bottom of this page that has working partial derivative calculation:
http://support.livemath.com/lmhelp/main ... =953485362

Statistics: Posted by raigan2 — Thu Mar 01, 2007 9:20 pm

Question about deformable solid paper

2007-03-01T22:08:55+00:00

It almost makes sense now;

Why do we find derive it with respect to time?
Shouldnt dC/dx be enough to solve the constraints?

I also noticed that L0 gets lost in the derivation. If L0 has no effect on the end result, how is the constraint able to get back to rest length?

Also, does dC/dt represent the force?
or do I have to multiply dC/dt by -kC

Thanks for the help

Statistics: Posted by coderchris — Thu Mar 01, 2007 8:59 pm

Question about deformable solid paper

2007-03-01T21:38:37+00:00

The constraint is C(x1, x2) = |x1 - x2| - L0

Since both x1 and x2 are functions of the time I recommend building the time derivative and then find the Jacobian by inspection. You use

dC/dt = dC/dx * dx/dt = J * v

This is the same as the chain rule (e.g. f'( h(x) ) = f' ( h(x) ) * h'(x) )

We use the this identity |x|' = x/|x| (Try to prove this yourself for practise. Hint |x| = Sqrt( x^2 + y^2 + z^2 )

dC/dt = (x1 - x2) / |x1 - x2| * ( v1 - v2 )

Now we sort this and define for readability n = (x1 - x2) / |x1 - x2|

dC/dt = n * v1 - n * v2

We can bring this in matrix form

dC/dt = ( n -n ) * Transpose( v1 v2 )

dC/dt = J * v

Does this make any sense to you?

Statistics: Posted by Dirk Gregorius — Thu Mar 01, 2007 8:21 pm

Question about deformable solid paper

2007-03-01T20:10:51+00:00

I took a look at the livemath maker; i cant seem to get it to derive anything, but I probably just havent spent enough time with it

Anyway I have a math related question concerning the formulas in the paper; Does anyone think they could show me the steps for deriving the distance constraint outlined in the paper?

I took a look at partial derivatives and I understand them now, but im unsure where to go with them...

Statistics: Posted by coderchris — Thu Mar 01, 2007 8:10 pm

Question about deformable solid paper

2007-02-24T14:11:26+00:00

Again, I must recommend LiveMath Maker: you just write your constraint function in scalar form and it calculates the partial derivatives wrt each scalar: http://www.livemath.com/download/

Statistics: Posted by raigan2 — Sat Feb 24, 2007 2:11 pm

Question about deformable solid paper

2007-02-24T11:32:26+00:00

I tryed to derive volume constraint gradients by myself and got following:

constraint C(p1,p2,p3,p4) = 1/6 * dot(p2 - p1, (p3 - p1) x (p4 - p1)) - V0

gradients:
p1: 1/6 * ((p4 - p3) x (p2 - p1) - (p3 - p1) x (p4 - p1))
p2: 1/6 * (p3 - p1) x (p4 - p1)
p3: -1/6 * (p4 - p1) x (p2 - p1)
p4 1/6 * (p3 - p1) x (p2 - p1)
but it seems that they are wrong
(I've already implemented it)

Statistics: Posted by stbuzer — Sat Feb 24, 2007 11:32 am