Extended body dynamics in general relativity: hyperelastic models

TL;DR

This paper introduces a numerical framework for simulating the dynamics of extended hyperelastic bodies in general relativity using finite element methods and Lagrangian formulation, validated against known oscillation modes.

Contribution

It develops a novel finite element-based numerical approach for modeling relativistic hyperelastic bodies, capable of handling arbitrary shapes and metrics, with validated convergence properties.

Findings

Algorithm displays second order convergence.

Framework accurately models oscillation modes of hyperelastic spheres.

Applicable to diverse shapes and spacetime metrics.

Abstract

We present a numerical framework for modeling extended hyperelastic bodies based on a Lagrangian formulation of general relativistic elasticity theory. We use finite element methods to discretize the body, then use the semi--discrete action to derive ordinary differential equations of motion for the discrete nodes. The nodes are evolved in time using fourth--order Runge--Kutta. We validate our code against the normal modes of oscillation of a hyperelastic sphere, which are known analytically in the limit of small (linear), slow (Newtonian) oscillations. The algorithm displays second order convergence. This numerical framework can be used to obtain the orbital motion and internal dynamics of a hyperelastic body of any shape, for any spacetime metric, and for varying hyperelastic energy models.