Materials

SuperDex Physics supports several material models for soft body simulation. Each model is defined by a strain energy density $\Psi(\mathbf{F})$, where $\mathbf{F}$ is the deformation gradient. The first Piola–Kirchhoff stress $\mathbf{P} = \partial\Psi/\partial\mathbf{F}$ and material tangent $\partial\mathbf{P}/\partial\mathbf{F}$ follow from this energy.

Common Parameters

Most isotropic material models use the following constitutive parameters:

Parameter	Symbol	Units	Description
Young's modulus	$E$	Pa	Overall stiffness. Higher values produce stiffer objects. Must be finite and positive.
Poisson's ratio	$\nu$	--	Compressibility. Must be finite and satisfy $-1 < \nu < 0.5$.

SoftMaterialParams::density is an actor-wide parameter shared by every material model, rather than a field of a model-specific *MaterialParams struct. It specifies mass per unit reference volume in kg/m$^3$ and defaults to 1000.

Young's modulus and Poisson's ratio define the Lamé parameters

$$\lambda = \frac{E \nu}{(1 + \nu)(1 - 2\nu)}~, \qquad \mu = \frac{E}{2(1 + \nu)}~,$$

which are used in the definitions of several different material formulations.

Nearly incompressible materials

Notice that the first Lamé parameter, $\lambda$, diverges when $\nu$ approaches the ends of its allowed range. The limit of $\nu \to -1$ is of little practical interest, but the limit of $\nu\to 0.5$ corresponds to near-incompressibility, and is common in soft materials like rubber and biological tissues. The deformation of these materials occurs almost entirely in shearing modes, whose stiffness is controlled by $\mu$, also known as the shear modulus. The stiffness of volume changes is controlled by the bulk modulus, $\kappa = \lambda + 2\mu/3$, which diverges alongside $\lambda$. Approximating non-trivial shearing deformations on a discrete mesh often still requires some small volume changes of elements, which are penalized by $\kappa$. This causes artificial stiffening known as volumetric locking in the finite element literature. For general simulations of human-scale manipulation problems, it is recommended to avoid this limit and regularize handbook bulk modulus values of soft materials to keep $\nu\lesssim 0.49$.

Available Models

Model	Use Case	Large Deformation	Inversion-Safe	Cost
Linear Elastic	Small displacement gradients only	No	N/A	Lowest
Neo-Hookean	General-purpose (recommended)	Yes	Yes	Medium
Saint Venant–Kirchhoff	Large rotation, moderate strain	Yes	No	Medium
ARAP	Shape-preserving deformation	Yes	Yes	Medium
Active Neo-Hookean	Muscle tissue with fiber actuation	Yes	Yes	High
Active Shape Targeting ARAP	Programmable local shape targets	Yes	Yes	High

For most simulations, Neo-Hookean is the recommended default.

PSD Enforcement

The default choices of algebraic Solvers used to solve implicit stages during time integration assume that energy contributions from soft materials have positive-semidefinite (PSD) Hessians. The exact nonlinear material tangents can become indefinite and break this assumption. The default behavior is therefore to replace them with PSD approximations, to improve solver performance.

A full-eigensystem MaterialPsdStrategy::Projection writes a symmetric material tangent as

$$\frac{\partial\mathbf{P}}{\partial\mathbf{F}} = \mathbf{Q}\boldsymbol{\Lambda}\mathbf{Q}^T~,$$

where the columns of $\mathbf{Q}$ are orthonormal eigenvectors and $\boldsymbol{\Lambda}$ is the diagonal matrix of eigenvalues, then replaces it with

$$\mathbf{Q}\max(\boldsymbol{\Lambda}, \varepsilon)\mathbf{Q}^T~.$$

Here $\varepsilon > 0$ is a small eigenvalue floor, and each diagonal entry $\lambda_i$ is replaced by $\max(\lambda_i, \varepsilon)$. Some materials use equivalent analytic projections of only their indefinite modes and may preserve structural zero modes.

PSD projection is controlled by two parameters:

• NonLinearSolverParams::psdProjMode decides when the nonlinear solve requests PSD projection. Its default is PsdProjectionMode::Always; retry modes request projection only after an iteration of Newton's method fails, while PsdProjectionMode::Never never requests it.
• Each material's MaterialPsdStrategy decides how that material responds to a request. MaterialPsdStrategy::None ignores the request and leaves the material tangent unmodified.

The available material-level PSD strategies are summarized in the following table:

Strategy	Description	Availability
MaterialDefault	Resolve to the selected material's concrete default strategy.	All nonlinear models
Projection	Project the tangent to be PSD, commonly by replacing $\lambda_i$ with $\max(\lambda_i, \varepsilon)$.	All nonlinear models
Fast	Drop or filter problematic terms. Faster, but may degrade nonlinear convergence.	StVK, Neo-Hookean
AbsEigenProjection	Use absolute eigenvalue filtering, commonly replacing $\lambda_i$ with $\max(\lvert\lambda_i\rvert, \varepsilon)$.	All nonlinear models
PerTermProjection	Project tangent summands independently.	Active Shape Targeting ARAP only
None	Do not modify the material tangent.	All nonlinear models

The Linear Elastic model has no material-level psdStrategy field because its tangent is unconditionally PSD.