Unified discrete multisymplectic Lagrangian formulation for hyperelastic solids and barotropic fluids

TL;DR

This paper develops a unified geometric variational discretization framework for nonlinear elasticity and fluids, enabling accurate simulation of fluid-structure interactions with constraints.

Contribution

It introduces a novel discrete geometric setting for hyperelastic solids and fluids, unifying their variational discretizations and allowing coupled simulations with constraints.

Findings

Effective discretization of nonlinear elasticity in 2D and 3D.

Unified framework compatible with multisymplectic fluid discretization.

Successful tests with fluid flow on incompressible hyperelastic bodies.

Abstract

We present a geometric variational discretization of nonlinear elasticity in 2D and 3D in the Lagrangian description. A main step in our construction is the definition of discrete deformation gradients and discrete Cauchy-Green deformation tensors, which allows for the development of a general discrete geometric setting for frame indifferent isotropic hyperelastic models. The resulting discrete framework is in perfect adequacy with the multisymplectic discretization of fluids proposed earlier by the authors. Thanks to the unified discrete setting, a geometric variational discretization can be developed for the coupled dynamics of a fluid impacting and flowing on the surface of an hyperelastic body. The variational treatment allows for a natural inclusion of incompressibility and impenetrability constraints via appropriate penalty terms. We test the resulting integrators in 2D and 3D with the case of a barotropic fluid flowing on incompressible rubber-like nonlinear models.