Material

Material models for finite element simulation

This module implements hyperelastic material models for FEM simulation. The primary model is the Stable Neo-Hookean formulation, which provides numerical stability even under large deformations and element inversion.

The strain energy density is derived from the deformation gradient \(\mathbf{F}\), and forces are computed as the negative gradient of the total elastic energy.

class diffsim.material.StableNeoHookean(youngs_modulus=1000000.0, poissons_ratio=0.45)

Bases: object

Stable Neo-Hookean hyperelastic material model

This class implements the stable Neo-Hookean formulation from Smith et al. (2018), “Stable Neo-Hookean Flesh Simulation”. Up to an additive constant, its energy density is:

\[\Psi(\mathbf{F}) = \frac{\bar\mu}{2}\left[(I_C-3)-\log\left(1+\frac{I_C-3}{4}\right)\right] + \frac{\bar\lambda}{2}(J-1)^2 - \frac{3\bar\mu}{4}(J-1)\]

where:

• \(\mathbf{F}\) is the deformation gradient tensor (3×3)

• \(I_C = \text{tr}(\mathbf{F}^T \mathbf{F}) = \|\mathbf{F}\|_F^2\) is the first invariant

• \(J = \det(\mathbf{F})\) is the Jacobian determinant (volume ratio)

• \(\mu = \frac{E}{2(1+\nu)}\) is the shear modulus

• \(\lambda = \frac{E\nu}{(1+\nu)(1-2\nu)}\) is Lamé’s first parameter

Here \(\bar\mu=4\mu/3\) and \(\bar\lambda=\lambda+5\mu/6\). This formulation remains finite for singular and inverted elements; explicit time integration still requires a sufficiently small time step.

Parameters:

• youngs_modulus (float) – Young’s modulus \(E\) in Pascals (default: 1e6)

• poissons_ratio (float) – Poisson’s ratio \(\nu\) (default: 0.45)

E

Young’s modulus

Type:

float

nu

Poisson’s ratio

Type:

float

mu

Shear modulus (Lamé’s second parameter)

Type:

float

lam

Lamé’s first parameter

Type:

float

Reference:

Smith, B., De Goes, F., & Kim, T. (2018). Stable neo-hookean flesh simulation. ACM Transactions on Graphics (TOG), 37(2), 1-15.

__init__(youngs_modulus=1000000.0, poissons_ratio=0.45)

Initialize material with elastic constants

Parameters:

• youngs_modulus – Young’s modulus (E)

• poissons_ratio – Poisson’s ratio (ν)

energy_density(F)

Compute strain energy density for deformation gradient F

Parameters:

F – \((M, 3, 3)\) deformation gradient tensor

Returns:

\((M,)\) energy density for each element

Return type:

psi

first_piola_kirchhoff_stress(F)

Compute first Piola-Kirchhoff stress tensor P = ∂Ψ/∂F

Parameters:

F – \((M, 3, 3)\) deformation gradient tensor

Returns:

\((M, 3, 3)\) first Piola-Kirchhoff stress tensor

Return type:

P

first_piola_kirchhoff_stress_analytic(F)

Compute the exact first Piola-Kirchhoff stress tensor analytically.

Parameters:

F – \((M, 3, 3)\) deformation gradient tensor

Returns:

\((M, 3, 3)\) first Piola-Kirchhoff stress tensor

Return type:

P

compute_elastic_forces(F, Dm_inv, volume)

Compute elastic forces from stress tensor

Parameters:

• F – \((M, 3, 3)\) deformation gradient

• Dm_inv – \((M, 3, 3)\) inverse rest shape matrix

• volume – \((M,)\) rest volume of each element

Returns:

\((M, 4, 3)\) forces on vertices of each element

Return type:

forces