# An energy-stable mixed formulation for isogeometric analysis of incompressible hyper-elastodynamics

## TL;DR

This paper introduces an energy-stable mixed formulation for isogeometric analysis of incompressible hyper-elastodynamics, linking fluid and solid dynamics, with demonstrated stability, convergence, and applicability to dynamic problems.

## Contribution

It presents a novel energy-stable mixed formulation using NURBS-based elements for incompressible hyper-elastodynamics, improving stability and convergence over traditional methods.

## Key findings

- Energy stability estimate established for the discretization.
- Numerical assessment confirms inf-sup stability for various NURBS pairs.
- The formulation demonstrates accurate results under different loading conditions.

## Abstract

We develop a mixed formulation for incompressible hyper-elastodynamics based on a continuum modeling framework recently developed and smooth generalizations of the Taylor-Hood element based on non-uniform rational B-splines (NURBS). This continuum formulation draws a link between computational fluid dynamics and computational solid dynamics. This link inspires an energy stability estimate for the spatial discretization, which favorably distinguishes the formulation from the conventional mixed formulations for finite elasticity. The inf-sup condition is utilized to provide a bound for the pressure field. The generalized-$\alpha$ method is applied for temporal discretization, and a nested block preconditioner is invoked for the solution procedure. The inf-sup stability for different pairs of NURBS elements is elucidated through numerical assessment. The convergence rate of the proposed formulation with various combinations of mixed elements is examined by the manufactured solution method. The numerical scheme is also examined under compressive and tensile loads for isotropic and anisotropic hyperelastic materials. Finally, a suite of dynamic problems is numerically studied to corroborate the stability and conservation properties.