A domain decomposition strategy for natural imposition of mixed boundary conditions in port-Hamiltonian systems

TL;DR

This work develops a structure-preserving domain-decomposition strategy for port-Hamiltonian systems by exploiting finite element exterior calculus and Hilbert complexes to weakly impose mixed boundary conditions without Lagrange multipliers. Each subdomain is discretized with a mixed FE formulation drawn from a Hilbert subcomplex, and the two subproblems are interconnected through a gyrator-like interface to enforce continuity and boundary coupling. Time integration options include the implicit midpoint scheme, which preserves the Poisson structure in linear cases, and Störmer–Verlet, which decouples subdomains for efficiency while still respecting energy considerations. The approach is demonstrated on a suite of examples (geometrically exact intrinsic beams, 2D wave, elastodynamics, Mindlin plate) showing accurate convergence, energy conservation, and elimination of shear locking, with extensions to semi-linear problems and potential for fully DG coupling and model reduction. Overall, the paper provides a conceptually clear, numerically robust framework for mixed boundary conditions in pH systems with broad applicability to structural dynamics and wave propagation problems.

Abstract

In this contribution, a finite element scheme to impose mixed boundary conditions without introducing Lagrange multipliers is presented for hyperbolic systems described as port-Hamiltonian systems. The strategy relies on finite element exterior calculus and domain decomposition to interconnect two systems with dual input-output behavior. The spatial domain is split into two parts by introducing an arbitrary interface. Each subdomain is discretized with a mixed finite element formulation that introduces a uniform boundary condition in a natural way as the input. In each subdomain the finite element spaces are selected from a finite element subcomplex to obtain a stable discretization. The two systems are then interconnected together by making use of a feedback interconnection. This is achieved by discretizing the boundary inputs using appropriate spaces that couple the two formulations. The final systems include all boundary conditions explicitly and do not contain any Lagrange multiplier. Time integration is performed using the implicit midpoint or Störmer-Verlet scheme. The method can also be applied to semilinear systems containing algebraic nonlinearities. The proposed strategy is tested on different examples: geometrically exact intrinsic beam model, the wave equation, membrane elastodynamics and the Mindlin plate. Numerical tests assess the conservation properties of the scheme, the effectiveness of the methodology and its robustness against shear locking phenomena.

Figures (13)

• Figure 1: Splitting of the domain.
• Figure 2: Interconnection at the interface $\Gamma_{12}$.
• Figure 3: Feedback interconnection of the two systems arising from the domain decomposition.
• Figure 4: The decomposed beam with mixed boundary conditions.
• Figure 5: Configurations for the quasi static roll up using the free-free, clamped-clamped and interconnected model. System \ref{['eq:discrete_clamped_intrinsic']} is used for Fig. \ref{['fig:clamped_nonlinear_beam']}. System \ref{['eq:discrete_free_intrinsic']} is used for Fig. \ref{['fig:free_nonlinear_beam']}. Fig. \ref{['fig:interonnected_nonlinear_beam']} shows the results when using the interconnection of the two. The first two cases require a Lagrange multiplier to impose the boundary conditions while the last does not.

Theorems & Definitions (11)

• Definition 1: Formal Adjoint
• Example 1: Gradient and divergence operators
• Remark 1: Equivalence with Lagrangian dynamics
• Definition 2: Hilbert Complex
• Example 2: de-Rham Complex
• Definition 3: Hilbert Subcomplex
• Remark 2: Equivalence with the second order formulation
• Remark 3
• Proposition 1
• Proof 1