Material

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)[source]#

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”. The energy density function is:

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

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 first Lamé parameter (shear modulus)

  • \(\lambda = \frac{E\nu}{(1+\nu)(1-2\nu)}\) is the second Lamé parameter

The formulation ensures numerical stability even under large deformations and near-inversion by using logarithmic terms instead of polynomial terms for the volumetric component.

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#

First Lamé parameter (shear modulus)

Type:

float

lam#

Second Lamé 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)[source]#

Initialize material with elastic constants

Parameters:

  • youngs_modulus – Young’s modulus (E)

  • poissons_ratio – Poisson’s ratio (ν)

energy_density(F)[source]#

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)[source]#

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)[source]#

Compute first Piola-Kirchhoff stress tensor analytically with stability

P = ∂Ψ/∂F = μ*F - μ*F^{-T} + λ*(J-1)*J*F^{-T}

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)[source]#

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

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