Twenty years of distributed port-Hamiltonian systems:A literature review

The port-Hamiltonian (pH) theory for distributed parameter systems has developed greatly in the past two decades. The theory has been successfully extended from finite-dimensional to infinite-dimensional systems through a lot of research efforts. This article collects the different research studies carried out for distributed pH systems. We classify over a hundred and fifty studies based on different research focuses ranging from modeling, discretization, control and theoretical foundations. This literature review highlights the wide applicability of the pH systems theory to complex systems with multi-physical domains using the same tools and language. We also supplement this article with a bibliographical database including all papers reviewed in this paper classified in their respective groups

Numerics for Physics-Based PDEs with Boundary Control: the Partitioned Finite Element Method for Port-Hamiltonian Systems

The numerical simulation of complex open multiphysics systems in Computational Science and Engineering is a challenging topic. Based on energy exchanges, the port-Hamiltonian formalism aims at describing physics in a structured manner. One of the major interests of this approach is its versatility, allowing for coupling and interconnection that preserve this structure. We propose a Finite Element based technique for the structure-preserving discretization of a large class of port-Hamiltonian systems. Assuming a partitioned structure of the system associated to an integration-by-parts formula, it is possible to derive a consistent weak-formulation sharing the main features of the original boundary-controlled PDE. This allows using Galerkin approximations to obtain finite-dimensional systems that mimic the properties of the original distributed ones: the Partitioned Finite Element Method producing sparse matrices enables the use of dedicated algorithms in scientific computing. Indeed, this method can be easily implemented using well-established and robust libraries. This strategy is illustrated by means of physically motivated PDEs: acoustic waves, Mindlin and Kirchhoff plates, heat equation, Maxwell's equation. Interactive Jupyter notebooks are available, relying on the FEniCS open-source software. Advanced applications include multiphysics problems, e.g. fluid-structure interactions, thermoelasticity, and modular modelling of complex systems, e.g. multibody dynamics

Partitioned finite element method for structured discretization with mixed boundary conditions

The propagation of acoustic waves in a 2D geometrical domain under mixed boundary control is here described by means of the port-Hamiltonian (pH) formalism. A finite element based method is employed to obtain a consistently discretized model. To construct a model with mixed boundary control, two different methodologies are detailed: one employs Lagrange multipliers, the other relies on a virtual domain decomposition to interconnect models with different causalities. The two approaches are assessed numerically, by comparing the Hamiltonian and the state variables norm for progressively refined meshes

Anisotropic heterogeneous n-D heat equation with boundary control and observation : II. Structure-preserving discretization

The heat equation with boundary control and observation can be described by means of three different Hamiltonians, the internal energy, the entropy, or a classical Lyapunov functional, as shown in the companion paper (Serhani et al. (2019a)). The aim of this work is to apply the partitioned finite element method (PFEM) proposed in Cardoso-Ribeiro et al. (2018) to the three associated port-Hamiltonian systems. Differential Algebraic Equations are obtained. The strategy proves very efficient to mimic the continuous Stokes-Dirac structure at the discrete level, and especially preserving the associated power balance. Anisotropic and heterogeneous 2D simulations are finally performed on the Lyapunov formulation to provide numerical evidence that this strategy proves very efficient for the accurate simulation of a boundary controlled and observed infinite-dimensional system

Numerical analysis of a structure-preserving space-discretization for an anisotropic and heterogeneous boundary controlled N-dimensional wave equation as port-Hamiltonian system

The anisotropic and heterogeneous N-dimensional wave equation, controlled and observed at the boundary, is considered as a port-Hamiltonian system. The recent structure-preserving Partitioned Finite Element Method is applied, leading directly to a finite-dimensional port-Hamiltonian system, and its numerical analysis is done in a general framework, under usual assumptions for finite element. Compatibility conditions are then exhibited to reach the best trade off between the convergence rate and the number of degrees of freedom for both the state error and the Hamiltonian error. Numerical simulations in 2D are performed to illustrate the optimality of the main theorems among several choices of classical finite element families.Comment: 36 pages, 1 figure, submitte

A port-Hamiltonian formulation of flexible structures. Modelling and structure-preserving finite element discretization

Despite the large literature on port-Hamiltonian (pH) formalism, elasticity problems in higher geometrical dimensions have almost never been considered. This work establishes the connection between port-Hamiltonian distributed systems and elasticity problems. The originality resides in three major contributions. First, the novel pH formulation of plate models and coupled thermoelastic phenomena is presented. The use of tensor calculus is mandatory for continuum mechanical models and the inclusion of tensor variables is necessary to obtain an equivalent and intrinsic, i.e. coordinate free, pH description. Second, a finite element based discretization technique, capable of preserving the structure of the infinite-dimensional problem at a discrete level, is developed and validated. The discretization of elasticity problems requires the use of non-standard finite elements. Nevertheless, the numerical implementation is performed thanks to well-established open-source libraries, providing external users with an easy to use tool for simulating flexible systems in pH form. Third, flexible multibody systems are recast in pH form by making use of a floating frame description valid under small deformations assumptions. This reformulation include all kinds of linear elastic models and exploits the intrinsic modularity of pH systems

Advanced Fluid Dynamics

This book provides a broad range of topics on fluid dynamics for advanced scientists and professional researchers. The text helps readers develop their own skills to analyze fluid dynamics phenomena encountered in professional engineering by reviewing diverse informative chapters herein