On the velocity-stress formulation for geometrically nonlinear elastodynamics and its structure-preserving discretization

TL;DR

The paper addresses nonlinear elastodynamics under large deformation with small strain by formulating a velocity-stress system that reveals a port-Hamiltonian structure modulated by the displacement field $u$. It develops a weak variational formulation with Dirichlet constraints enforced via Lagrange multipliers, establishing a global power balance and enabling a structure-preserving discretization via mixed finite elements in space and an implicit midpoint time integrator. The resulting finite-dimensional port-Hamiltonian DAEs preserve energy exchange both inside the domain and at the boundary, and the fully discrete scheme maintains the discrete power balance. Numerical tests on a planar compliant mechanism demonstrate energy preservation and dynamics close to a rigid-body model, validating the approach and illustrating practical applicability to nonlinear elasticity with efficient computation. The work lays groundwork for extending to damping and multiphysics while maintaining energy-consistent simulations akin to linear cases.

Abstract

We consider the dynamics of an elastic continuum under large deformation but small strain. Such systems can be described by the equations of geometrically nonlinear elastodynamics in combination with the St. Venant-Kirchhoff material law. The velocity-stress formulation of the problem turns out to have a formal port-Hamiltonian structure. In contrast to the linear case, the operators of the problem are modulated by the displacement field which can be handled as a passive variable and integrated along with the velocities. A weak formulation of the problem is derived and essential boundary conditions are incorporated via Lagrange multipliers. This variational formulation explicitly encodes the transfer between kinetic and potential energy in the interior as well as across the boundary, thus leading to a global power balance and ensuring passivity of the system. The particular geometric structure of the weak formulation can be preserved under Galerkin approximation via appropriate mixed finite elements. In addition, a fully discrete power balance can be obtained by appropriate time discretization. The main properties of the system and its discretization are shown theoretically and demonstrated by numerical tests.

Figures (5)

• Figure 1: Mapping of an infinitesimal line element $dX$ from the material (undeformed) configuration ($\Omega\subset\mathbb{R}^2$) to the spatial (deformed) one ($\Omega_t\subset\mathbb{R}^2$) via the deformation gradient $F = \operatorname{Grad}(x)$, $dx=F\cdot dX$.
• Figure 2: Schematic sketch of the geometry. Red lines mark the Dirichlet boundary $\partial\Omega_D$ for which the displacement velocity $v_D$ is prescribed. The Lagrange multiplier corresponds to the reaction forces.
• Figure 3: Snapshots at $t=0\;\text{s}$ (green) and $t=0.722\;\text{s}$ (blue), dots represent the reference configuration $\Omega$, and the red dashed lines demonstrate the rigid body configuration. Images generated with ParaView Ayachit2015.
• Figure 4: Absolut velocities $|v|$ at the point $Q=(L_x,0)$ for a rigid body motion (blue) compared to the true elastic motion (orange). Elastic vibrations pertain after the motion of the pivot has been stopped.
• Figure 5: Approximation of the total energy $H_{tot}$ obtained for the rigid body model and the structure-preserving fully discrete approximation of the elasticity system proposed in this paper.

Theorems & Definitions (10)

• Remark 1
• Remark 2
• Lemma 2.1
• Proposition 2.2
• Remark 3
• Proposition 3.1
• Lemma 3.2
• Remark 4
• Proposition 3.3
• Remark 5