Behaviour integrators

A Behaviour integrator computes residual and jacobian terms for a material. Behaviour integrators are associated with:

• a physical phenomenon (mechanics, heat transfer, etc.),

• a modeling hypothesis (plane strain, plane stress, etc.),

• a material symmetry (isotropy, geometry).

Note

Behaviour integrators are generally declared using the addBehaviourIntegrator method of the NonLinearEvolutionProblem class, as in the following example:

mechanics.addBehaviourIntegrator(ctx, "Mechanics", "beam",
                                 "src/libBehaviour.so",
                                 "MicromorphicDamageI_SpectralSplit");

Based on the generic category Mechanics, this declaration automatically handles the selection of a behaviour integrator consistent with the modeling hypothesis declared by the nonlinear evolution problem and the material symmetry declared by the behaviour.

Behaviour integrators based on MFront behaviours

Most behaviour integrators rely on external behaviours that shall be generated by MFront. In practice, those external behaviours are compiled by MFront into shared libraries that are loaded dynamically when an behaviour integrator is loaded.

Mechanical behaviour integrator

A mechanical behaviour integrator computes the residual and jacobian associated with one of the following variational operators:

\[\begin{split}\begin{aligned} \int_{\Omega_r} \underline{\sigma}\,\colon\,\underline{\varepsilon}^{\star}\,\mathrm{d}\,V & \quad\quad &\text{(small strain)} \\ \int_{\Omega_r} \underline{P}\,\colon\,\underline{F}^{\star}\,\mathrm{d}\,V & \quad\quad & \text{(finite strain)} \end{aligned}\end{split}\]

where:

• \(\Omega_r\) is the domain occupied by the material of interest in the reference configuration,

• \(\underline{\sigma}\) is the Cauchy stress (expressed in the reference configuration for a small strain analysis,

• \(\underline{\varepsilon}^{\star}\) is the virtual deformation,

• \(\underline{P}\) is the first Piola-Kirchhoff stress,

• \(\underline{F}^{\star}\) is the virtual deformation gradient.

The mechanical behaviour integrators available in MFEM/MGIS support:

• small and finite strain behaviours,

• isotropic and orthotropic behaviours,

• plane strain, plane stress and tridimensional modeling hypotheses.

Faltus 2026 regularization

The regularization proposed by Faltus et al. in the context of contact mechanics using a third medium [FAH]. This regularization only applies to finite strain behaviours. Currently, only this regularization is only available for isotropic behaviours.

This regularization adds a contribution to the previous variational operator in finite strain to can be derived from an energy \(W\) which penalizes the difference between the deformation gradient \(\underline{F}\) at a given quadrature point and its value \(\bar{\underline{F}}\) at the centroid of the element:

\[W\left(\underline{F}, \bar{\underline{F}}\right) = \alpha\,\left(\underline{F}-\bar{\underline{F}}\right)\,\colon\, \left(\underline{F}-\bar{\underline{F}}\right)\]

where \(\alpha\) is a penalization coefficient.

This regularization is enabled by passing an additional parameter to the the Mechanics behaviour integrator, as follows:

const auto faltus_parameters = mfem_mgis::Parameters{
    {"Regularization",
     mfem_mgis::Parameters{
         {"Faltus2026",
          mfem_mgis::Parameters{{"PenalizationCoefficient", 1e11}}}}}};
mechanics.addBehaviourIntegrator(ctx, "Mechanics", "ThirdMedium", library,
                                 behaviour2, faltus_parameters) | or_die;

Heat transfer behaviour integrators

Nonlinear heat transfer behaviour integrators compute the residual and jacobian associated with the following variational operator:

\[-\int_{\Omega_r} \vec{j}\,\cdot\,\vec{\nabla} T^{\star}\,\mathrm{d}\,V\]

where:

• \(\Omega_r\) is the domain occupied by the material of interest in the reference configuration,

• \(\vec{j}\) is the heat flux,

• \(\vec{\nabla} T^{\star}\) is the virtual temperature.

The heat transfer behaviour integrators available in MFEM/MGIS support:

• small and finite strain behaviours,

• isotropic and orthotropic behaviours,

• plane strain, plane stress and tridimensional modeling hypotheses.

Micromorphic damage behaviour integrators

Micromorphic damage behaviour integrators compute the residual and jacobian associated with the following variational operator:

\[\int_{\Omega_r} \vec{b}_{\chi}\,\cdot\,\vec{\nabla} d_{\chi}^{\star}+a_{\chi}\,d_{\chi}^{\star}\,\mathrm{d}\,V\]

where:

• \(\Omega_r\) is the domain occupied by the material of interest in the reference configuration,

• \(b_{\chi}\) is the thermodynamic force associated with the gradient of the micromorphic damage \(\vec{\nabla} d_{\chi}\),

• \(a_{\chi}\) is the thermodynamic force associated with the micromorphic damage \(d_{\chi}\),

• \(d_{\chi}^{\star}\) is the virtual micromorphic damage.

The micromorphic damage integrators available in MFEM/MGIS support:

• isotropic behaviour in \(2D\) (plane strain and plain stress) and \(3D\) computations,

• orhotropic behaviours in \(2D\).