# Unified discrete multisymplectic Lagrangian formulation for hyperelastic solids and barotropic fluids

@article{Demoures2021UnifiedDM, title={Unified discrete multisymplectic Lagrangian formulation for hyperelastic solids and barotropic fluids}, author={Franccois Demoures and Franccois Gay-Balmaz}, journal={ArXiv}, year={2021}, volume={abs/2110.00412} }

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…

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SHOWING 1-10 OF 33 REFERENCES

Multisymplectic variational integrators for barotropic and incompressible fluid models with constraints

- Computer Science, MathematicsArXiv
- 2021

It is shown how the geometric integrator can handle regular fluid motion in vacuum with free boundaries and constraints such as the impact against an obstacle of a fluid flowing on a surface.

A multisymplectic integrator for elastodynamic frictionless impact problems

- Physics
- 2017

We present a structure preserving numerical algorithm for the collision of elastic bodies. Our integrator is derived from a discrete version of the field-theoretic (multisymplectic) variational…

A geometric structure-preserving discretization scheme for incompressible linearized elasticity

- Mathematics
- 2013

Abstract In this paper, we present a geometric discretization scheme for incompressible linearized elasticity. We use ideas from discrete exterior calculus (DEC) to write the action for a discretized…

MULTISYMPLECTIC VARIATIONAL INTEGRATORS FOR NONSMOOTH LAGRANGIAN CONTINUUM MECHANICS

- MathematicsForum of Mathematics, Sigma
- 2016

This paper develops the theory of multisymplectic variational integrators for nonsmooth continuum mechanics with constraints. Typical problems are the impact of an elastic body on a rigid plate or…

EULERIAN FORMULATION AND LEVEL SET MODELS FOR INCOMPRESSIBLE FLUID-STRUCTURE INTERACTION

- Mathematics
- 2008

This paper is devoted to Eulerian models for incompressible fluid-structure systems. These models are primarily derived for computational purposes as they allow to simulate in a rather straight-…

Reduced Variational Formulations in Free Boundary Continuum Mechanics

- Mathematics, Computer ScienceJ. Nonlinear Sci.
- 2012

We present the material, spatial, and convective representations for elasticity and fluids with a free boundary from the Lagrangian reduction point of view, using the material and spatial symmetries…

Reference map technique for finite-strain elasticity and fluid-solid interaction

- Mathematics
- 2012

Abstract The reference map, defined as the inverse motion function, is utilized in an Eulerian-frame representation of continuum solid mechanics, leading to a simple, explicit finite-difference…

On geometric discretization of elasticity

- Mathematics
- 2008

This paper presents a geometric discretization of elasticity when the ambient space is Euclidean. This theory is built on ideas from algebraic topology, exterior calculus, and the recent developments…

Variational Methods, Multisymplectic Geometry and Continuum Mechanics

- Mathematics
- 2001

This paper presents a variational and multisymplectic formulation of both compressible and incompressible models of continuum mechanics on general Riemannian manifolds. A general formalism is…

A Lagrange multiplier/fictitious domain method for the numerical simulation of incompressible viscous flow around moving rigid bodies: (I) case where the rigid body motions are known a priori

- Mathematics
- 1997

Abstract We describe in this Note a method for the numerical simulation of incompressible viscous flow around moving rigid bodies; we suppose the rigid body motions a priori known. The computational…