Comfusion about implementation of XPBD FEM constraint

[blacksino]

Hi, I have been reading the code of the implementation of the XPBD FEM constraint, however, it seems to be a bit different from the formula from the XPBD paper.

Real sum_normGradC =
        invMass0 * gradC[0].squaredNorm() +
        invMass1 * gradC[1].squaredNorm() +
        invMass2 * gradC[2].squaredNorm() +
        invMass3 * gradC[3].squaredNorm();

C = sqrt(2.0 * C); // why?

Real alpha = static_cast<Real>(1.0) / (youngsModulus * dt * dt);
sum_normGradC += C * C * alpha; // why multiply alpha with C**2?

if (sum_normGradC < eps)
    return false;

// compute scaling factor
//const Real s = C / sum_normGradC;
const Real s = (C * C + C * alpha * multiplier) / sum_normGradC; // Why is there an extra C multiplied on numerator?

//update mulitiplier,aka lamda?

corr0 = -s * invMass0 * gradC[0];
corr1 = -s * invMass1 * gradC[1];
corr2 = -s * invMass2 * gradC[2];
corr3 = -s * invMass3 * gradC[3];`

[janbender]

Hm, the names of the variables are not self-explaining due to copy & paste. I have to fix that. The explanation is that we use the potential energy:

U = 0.5 C^T alpha^-1 C (Eq. 5 in XPBD paper)

In our original paper we used C= U in the PBD framework. To adapt this to XPBD we now use:

C = sqrt(2 U')

with U = alpha^-1 U' and alpha = 1/youngsModulus.
This yields:

grad C = 1/C grad U'

This is where the additional C's come from. In the code grad U' has the name "gradC" which is quite misleading which comes from lazy copy & pasting stuff.

I hope that clarifies this a bit.

[blacksino]

thank you, that helps a lot!