As-Rigid-As-Possible (ARAP)

ARAP penalizes deformation that differs from a rigid rotation, preserving local shape rather than matching a conventional isotropic elastic law. SuperDex Physics follows the rotation-variant SVD treatment described by Kim and Eberle (2022).

Enum value: SoftMaterialType::Arap

Formulation

SuperDex Physics computes a rotation-variant singular value decomposition

$$\mathbf{F} = \mathbf{U}\boldsymbol{\Sigma}\mathbf{V}^T~,$$

in which $\mathbf{U}$ and $\mathbf{V}$ are proper rotations, so an inversion appears as a negative final signed principal stretch. The matrix

$$\mathbf{R} = \mathbf{U}\mathbf{V}^T$$

is therefore the closest proper rotation to $\mathbf{F}$, including for inverted elements. Let $\boldsymbol{\sigma} = (\sigma_1, \sigma_2, \sigma_3)$ contain the diagonal entries of $\boldsymbol{\Sigma}$ and let $\mathbf{1}_3 = (1, 1, 1)$. The strain energy density is

$$\Psi(\mathbf{F}) = \frac{\mu}{2}\Vert\boldsymbol{\sigma} - \mathbf{1}_3\Vert_2^2 = \frac{\mu}{2}\|\mathbf{F} - \mathbf{R}\|^2~,$$

and the first Piola–Kirchhoff stress is

$$\mathbf{P} = \mu(\mathbf{F} - \mathbf{R})~.$$

Compression and inversion can make the material tangent indefinite. The public PSD strategies regularize this tangent when requested by the nonlinear solver.

Parameters

The public parameter type is ArapMaterialParams.

Parameter	Default	Description
stiffness	1,000 Pa	Resistance to non-rigid deformation
psdStrategy	Projection	PSD enforcement strategy

The default stiffness is numerically 100 times smaller than the default Young's modulus $E=100{,}000$ Pa used by the isotropic models. These parameters belong to different constitutive models and are not directly equivalent.

Supported concrete PSD strategies: None, Projection, AbsEigenProjection. MaterialDefault resolves to the model's default, Projection.

When to Use

• Shape-preserving deformation where local rigidity matters more than conventional material calibration.
• As the passive basis for Active Shape Targeting ARAP.

References

• T. Kim and D. Eberle, Dynamic Deformables, 2022.