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

TL;DR

This paper introduces a finite element domain decomposition method for port-Hamiltonian systems that naturally imposes mixed boundary conditions without Lagrange multipliers, ensuring stability and conservation in complex hyperbolic PDEs.

Contribution

It presents a novel finite element scheme combining exterior calculus and domain decomposition to naturally enforce mixed boundary conditions in port-Hamiltonian systems without Lagrange multipliers.

Findings

Accurately conserves physical quantities in numerical simulations.

Effectively handles complex boundary conditions in hyperbolic systems.

Demonstrates robustness against shear locking phenomena.

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 St\"ormer-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.