An interaction potential method for passive and active dynamics of hyperelastic materials

Abstract

Simulating active biological tissues, such as the myocardium, requires constitutive models that are both physically faithful and computationally efficient. The most common approach relies on finite element methods that accurately discretize the underlying continuum hyperelastic problem which, in turn, require global nonlinear solve at each time step. On the other hand, fast, interaction-potential methods replace the continuum with a network of independent links approximating the mechanical response. We propose an interaction potential formulation for simulating active biological tissues that bridges this gap. The method recasts continuum hyperelastic constitutive laws in terms of tetrahedral edge strain. Unlike classical mass-spring models, the proposed formulation does not approximate the tissue as independent spring elements but preserves the energetic coupling between adjacent edges. Passive tissue mechanics is described by hyperelastic constitutive laws, while active contraction is incorporated through the active-strain multiplicative decomposition. Within the edge-based formulation, the active strain is incorporated through a time-dependent activated reference configuration. We further introduce a strategy for enforcing the material incompressibility constraint while avoiding volumetric locking. The resulting method can be interpreted as an edge-strain representation of a constant-strain tetrahedral continuum element, providing a bridge between continuum mechanics and discrete interaction potential solvers. Numerical tests demonstrate the capability of the method to effectively simulate different hyperelastic constitutive laws and to properly preserve the energy balance equation. Finally, the proposed method is applied to the active deformation of a realistic ventricle, reproducing longitudinal shortening and wall-thickening values consistent with the literature.