Corotational FEM cloth

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Triangle membrane elements with corotational linear elastic energy^[1]. Implicit Euler with CG, same outer loop as the BW98-like mass-spring sibling.

50×50 grid pinned at the two top corners, 4802 triangles. Material constants: $\mu = 48000$ (shear modulus), $\lambda = 16000$ (first Lamé), thickness $= 0.01$. Edit the scene above; changes auto-apply.

Per triangle, build the deformation gradient $F = D_s D_m^{-1}$ from the rest pose, extract rotation via SVD ($F = U\Sigma V^T,\; R = UV^T$ with reflection fix), and evaluate

$$\Psi = \mu\,\lVert F - R \rVert_F^2 + \tfrac{\lambda}{2}\bigl(\mathrm{tr}(R^T F) - 2\bigr)^2$$

The Hessian here is the Gauss-Newton approximation (drops $\partial R / \partial F$ terms). SPD by construction, so CG converges; recovering those dropped terms via operator splitting (cf. Kugelstadt 2018^[2]) gives faster convergence on stiff materials.

Color encodes per-triangle area strain (green = rest, red = stretched, blue = compressed). Click and drag any vertex.

What you trade for going FEM over mass-spring: shear behaviour falls out of $\mu, \lambda$ instead of needing diagonal springs, and resolution refinement no longer changes the effective stiffness.

References

1. Müller & Gross 2004 -- Interactive Virtual Materials
2. Kugelstadt et al. 2018 -- Fast Corotated FEM