Strong imposition of Dirichlet boundary velocities in structure-preserving discretizations of elastodynamics

2026-07-28

AI Summary

This work addresses the challenge of enforcing Dirichlet boundary velocities in structure-preserving discretizations, which traditionally either introduce Lagrange multipliers—yielding differential-algebraic equations—or weakly impose boundary conditions at the expense of accuracy. The authors propose a continuous-level additive kinematic decomposition that splits the displacement and velocity fields into a dynamic component satisfying homogeneous boundary conditions and a prescribed lifting function. Building on the principle of virtual power, they formulate a lifted port-Hamiltonian system. Upon finite element discretization, the resulting system is an ordinary differential equation that strongly enforces Dirichlet velocity boundary conditions—a first within the port-Hamiltonian framework—while preserving energy conservation and the ODE structure. The approach also unifies classical finite element matrix partitioning techniques. Numerical experiments confirm its energy balance, computational efficiency, and equivalence to standard formulations.

Abstract

The imposition of boundary velocities in finite element models of port-Hamiltonian elastodynamics typically relies on Lagrange multipliers, yielding Differential-Algebraic Equations (DAEs). Alternatively, weak imposition methods that maintain an Ordinary Differential Equation (ODE) structure often exhibit poor accuracy at Dirichlet boundaries. To address these limitations, this paper introduces an additive kinematic decomposition at the continuous level, splitting the displacement and velocity fields into a relative dynamic component that vanishes on the boundary and a prescribed lifting function extending into the interior domain. This decomposition induces a distributed port that maps the effects of the boundary actuation inside the domain. By incorporating this mapping into suitable virtual power principles, we derive lifted port-Hamiltonian system (PHS) models that, upon finite element discretization, reduce to ODE systems in which Dirichlet boundary velocities are strongly imposed. The framework is applied to derive 2-field and 4-field formulations suited to distinct PHS geometric representations. Furthermore, we show that under specific shape functions, standard FEM schemes are recovered, demonstrating that the lifting framework in the discrete models is equivalent to the classic algebraic matrix partitioning in computational mechanics practice. The energy-balance properties and computational performance of the proposed methodology are verified through numerical simulations.

Problem

• Dirichlet boundary conditions
• elastodynamics
• structure-preserving discretization
• finite element method
• boundary velocity imposition

Innovation

• port-Hamiltonian systems
• Dirichlet boundary conditions
• kinematic decomposition
• structure-preserving discretization
• finite element method