arXiv subjects

Denis Matignon

Publications and source records attributed to Denis Matignon.

At least 19 recordsLinked to original sources

arXiv2026

Conduction-Diffusion in N-Dimensional settings as irreversible port-Hamiltonian systems

This work extends previous 1D irreversible port-Hamiltonian system (IPHS) formulations to boundary-controlled ND distributed parameter systems describing conduction-diffusion fluid phenomena. Within a unified and thermodynamically consistent framework, we show that conduction and diffusion can be represented through a single coherent structure that preserves global energy balance and ensures a correct characterization of entropy production. The resulting formulation provides a foundation for the systematic modeling and control of complex multi-physical processes governed by coupled transport mechanisms in N dimensions. In the longer term, this framework opens the door to structure-preserving numerical schemes capable of enforcing thermodynamic principles directly at the discretized level.

eess.SY↗

arXiv2025

PFEM-GP-dPHS : a finite element framework for combining Gaussian processes and infinite-dimensional port-Hamiltonian systems

In order to learn distributed port-Hamiltonian systems (dPHS) using Gaussian processes (GPs), the partitioned finite element method (PFEM) is combined with the Gp-dPHS method. By following a late lumping approach, the discretization of the functional hyperparameters of the GP prior over the Hamiltonian functional is chosen independently from the discretization of the dPHS, thus reducing the numerical complexity of our method. We next model the mean of the GP prior of the Hamiltonian as a quadratic form, enabling the GP kernel to focus on the nonlinear part of a given dPHS. We illustrate our method on a nonlinear one dimensional wave equation with unknown physical parameters (tension and linear mass).

math.AP↗

arXiv2025

Structure-preserving space discretization of differential and nonlocal constitutive relations for port-Hamiltonian systems

We study the structure-preserving space discretization of port-Hamiltonian (pH) systems defined with differential constitutive relations. Using the concept of Stokes-Lagrange structure to describe these relations, these are reduced to a finite-dimensional Lagrange subspace of a pH system thanks to a structure-preserving Finite Element Method. To illustrate our results, the 1D nanorod case and the shear beam model are considered, which are given by differential and implicit constitutive relations for which a Stokes-Lagrange structure along with boundary energy ports naturally occur. Then, these results are extended to the nonlinear 2D incompressible Navier-Stokes equations written in a vorticity-stream function formulation. It is first recast as a pH system defined with a Stokes-Lagrange structure along with a modulated Stokes-Dirac structure. A careful structure-preserving space discretization is then performed, leading to a finite-dimensional pH system. Theoretical and numerical results show that both enstrophy and kinetic energy evolutions are preserved both at the semi-discrete and fully-discrete levels.

math.NA↗

arXiv2025

A linearly-implicit energy-momentum preserving scheme for geometrically nonlinear mechanics based on non-canonical Hamiltonian formulations

This work presents a novel formulation and numerical strategy for the simulation of geometrically nonlinear structures. First, a non-canonical Hamiltonian (Poisson) formulation is introduced by including the dynamics of the stress tensor. This framework is developed for von-K\'arm\'an nonlinearities in beams and plates, as well as geometrically nonlinear elasticity with Saint-Venant material behavior. In the case of plates, both negligible and non-negligible membrane inertia are considered. For the former case the two-dimensional elasticity complex is leveraged to express the dynamics in terms of the Airy stress function. The finite element discretization employs a mixed approach, combining a conforming approximation for displacement and velocity fields with a discontinuous stress tensor representation. A staggered, linear implicit time integration scheme is proposed, establishing connections with existing explicit-implicit energy-preserving methods. The stress degrees of freedom are statically condensed, reducing the computational complexity to solving a system with a positive definite matrix. The integration strategy preserves energy and angular momentum exactly. The methodology is validated through numerical experiments on the Duffing oscillator, a von-K\'arm\'an beam, and a column undergoing finite deformations. Comparisons with fully implicit energy-preserving method and the leapfrog scheme demonstrate that the proposed approach achieves superior accuracy while maintaining energy stability. Additionally, it enables larger time steps compared to explicit schemes and exhibits computational efficiency comparable to the leapfrog method.

math.NA↗

arXiv2024

Stokes-Lagrange and Stokes-Dirac representations of $N$-dimensional port-Hamiltonian systems for modelling and control

In this paper, we extend the port-Hamiltonian framework by introducing the concept of Stokes-Lagrange structure, which enables the implicit definition of a Hamiltonian over an $N$-dimensional domain and incorporates energy ports into the system. This new framework parallels the existing Dirac and Stokes-Dirac structures. We propose the Stokes-Lagrange structure as a specific case where the subspace is explicitly defined via differential operators that satisfy an integration by parts formula. By examining various examples through the lens of the Stokes-Lagrange structure, we demonstrate the existence of multiple equivalent system representations. These representations provide significant advantages for both numerical simulation and control design, offering additional tools for the modelling and control of port-Hamiltonian systems.

math.OC↗

arXiv2024

On implicit and explicit representations for 1D distributed port-Hamiltonian systems

First, two examples of 1D distributed port-Hamiltonian systems with dissipation, given in explicit (descriptor) form, are considered: the Dzekster model for the seepage of underground water and a nanorod model with non-local viscous damping. Implicit representations in Stokes-Lagrange subspaces are formulated. These formulations lead to modified Hamiltonian functions with spatial differential operators. The associated power balance equations are derived, together with the new boundary ports. Second, the port-Hamiltonian formulations for the Timoshenko and the Euler-Bernoulli beams are recalled, the latter being a flow-constrained version of the former. Implicit representations of these models in Stokes-Lagrange subspaces and corresponding power balance equations are derived. Bijective transformations are proposed between the explicit and implicit representations. It is proven these transformations commute with the flow-constraint projection operator.

math.DS↗

arXiv2024

Structure-Preserving Discretization and Model Order Reduction of Boundary-Controlled 1D Port-Hamiltonian Systems

This paper presents a systematic methodology for the discretization and reduction of a class of one-dimensional Partial Differential Equations (PDEs) with inputs and outputs collocated at the spatial boundaries. The class of system that we consider is known as Boundary-Controlled Port-Hamiltonian Systems (BC-PHSs) and covers a wide class of Hyperbolic PDEs with a large type of boundary inputs and outputs. This is, for instance, the case of waves and beams with Neumann, Dirichlet, or mixed boundary conditions. Based on a Partitioned Finite Element Method (PFEM), we develop a numerical scheme for the structure-preserving spatial discretization for the class of one-dimensional BC-PHSs. We show that if the initial PDE is passive (or impedance energy preserving), the discretized model also is. In addition and since the discretized model or Full Order Model (FOM) can be of large dimension, we recall the standard Loewner framework for the Model Order Reduction (MOR) using frequency domain interpolation. We recall the main steps to produce a Reduced Order Model (ROM) that approaches the FOM in a given range of frequencies. We summarize the steps to follow in order to obtain a ROM that preserves the passive structure as well. Finally, we provide a constructive way to build a projector that allows to recover the physical meaning of the state variables from the ROM to the FOM. We use the one-dimensional wave equation and the Timoshenko beam as examples to show the versatility of the proposed approach.

math.NA↗

arXiv2024

Scattering-Passive Structure-Preserving Finite Element Method for the Boundary Controlled Transport Equation with a Moving Mesh

A structure-preserving Finite Element Method (FEM) for the transport equation in one- and two-dimensional domains is presented. This Distributed Parameter System (DPS) has non-collocated boundary control and observation, and reveals a scattering-energy preserving structure. We show that the discretized model preserves the aforementioned structure from the original infinite-dimensional system. Moreover, we analyse the case of moving meshes for the one-dimensional case. The moving mesh requires less states than the fixed one to produce solutions with a comparable accuracy, and it can also reduce the overshoot and oscillations of Gibbs phenomenon produced when using the FEM. Numerical simulations are provided for the case of a one-dimensional transport equation with fixed and moving meshes.

math.NA↗

arXiv2023

Stokes-Dirac structures for distributed parameter port-Hamiltonian systems: an analytical viewpoint

In this paper we prove that a large class of linear evolution PDEs defines a Stokes-Dirac structure over Hilbert spaces. To do so, the theory of boundary control system is employed. This definition encompasses problems from mechanics, that cannot be handled by the seminal geometric setting given in [van der Schaft and Maschke, Hamiltonian formulation of distributed-parameter systems with boundary energy flow, 2002 ]. Many worked-out examples stemming from continuum mechanics and physics are presented in detail, and a particular focus is given on the functional spaces in duality at the boundary of the geometrical domain. For each example, the connection between the differential operators and the associated Hilbert complexes is illustrated.

math.AP↗

arXiv2022

Data-driven port-Hamiltonian structured identification for non-strictly passive systems

In this work, we detail a procedure to construct a reduced order model on the basis of frequency-domain data, that preserves the non-strictly passive property and the port-Hamiltonian structure. The proposed scheme is based on Benner et al. (2020) contribution, which has been adapted (i) to handle non-strictly passive model, and (ii) to handle numerical issues observed when applying the Loewner framework on complex configurations. We validate the proposed scheme on a very complex two-dimensional wave equation, for which the discretized version preserves the port-Hamiltoninan form.

eess.SY↗

arXiv2020

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. A recent structure-preserving mixed Galerkin method is applied, leading directly to a finite-dimensional port-Hamiltonian system: its numerical analysis is carried out in a general framework. Optimal choices of mixed finite elements are then proved to reach the best trade-off between the convergence rate and the number of degrees of freedom for the state error. Exta compatibility conditions are identified for the Hamiltonian error to be twice that of the state error, and numerical evidence is provided that some combinations of finite element families meet these conditions. Numerical simulations in 2D are performed to illustrate the main theorems among several choices of classical finite element families. Several test cases are provided, including non-convex domain, anisotropic or hetergoneous cases and absorbing boundary conditions.

math.NA↗

arXiv2020

Structure-preserving discretization of port-Hamiltonian plate models

Methods for discretizing port-Hamiltonian systems are of interest both for simulation and control purposes. Despite the large literature on mixed finite elements, no rigorous analysis of the connections between mixed elements and port-Hamiltonian systems has been carried out. In this paper we demonstrate how existing methods can be employed to discretize dynamical plate problems in a structure-preserving way. Based on convergence results of existing schemes, new error estimates are conjectured; numerical simulations confirm the expected behaviors.

math.NA↗

arXiv2020

About Dissipative and Pseudo Port-Hamiltonian Formulations of Irreversible Newtonian Compressible Flows

In this paper we consider the problem of obtaining a general port-Hamiltonian formulation of Newtonian fluids. We propose the port-Hamiltonian models to describe the energy flux of rotational three-dimensional isentropic and non-isentropic fluids, whose boundary flows and efforts can be used for control purposes or for power-preserving interconnection with other physical systems. In case of two-dimensional flows, we include the considerations about the operators associated with fluid vorticity, preserving the port-Hamiltonian structure of the models proposed.

physics.flu-dyn↗

arXiv2020

Port-Hamiltonian flexible multibody dynamics

A new formulation for the modular construction of flexible multibody systems is presented. By rearranging the equations for a flexible floating body and introducing the appropriate canonical momenta, the model is recast into a coupled system of ordinary and partial differential equations in port-Hamiltonian (pH) form. This approach relies on a floating frame description and remains valid under the assumption of small deformations. This allows including mechanical models that cannot be easily formulated in terms of differential forms. Once a pH model is established, a finite element based method is then introduced to discretize the dynamics in a structure-preserving manner. Thanks to the features of the pH framework, complex multibody systems are constructed in a modular way. Constraints are imposed at the velocity level, leading to an index 2 quasi-linear differential-algebraic system. Numerical tests are carried out to assess the validity of the proposed approach.

physics.class-ph↗

arXiv2019

A Partitioned Finite Element Method for power-preserving discretization of open systems of conservation laws

This paper presents a structure-preserving spatial discretization method for distributed parameter port-Hamiltonian systems. The class of considered systems are hyperbolic systems of two conservation laws in arbitrary spatial dimension and geometries. For these systems, a partioned finite element method is derived, based on the integration by parts of one of the two conservation laws written in weak form. The nonlinear 1D Shallow Water Equation (SWE) is first considered as a motivation example. Then the method is investigated on the example of the nonlinear 2D SWE. Complete derivation of the reduced finite-dimensional port-Hamiltonian system is provided and numerical experiments are performed. Extensions to curvilinear (polar) coordinate systems, space-varying coefficients and higher-order port-Hamiltonian systems (Euler-Bernoulli beam equation) are provided.

math.NA↗

arXiv2018

Asymptotic stability of the multidimensional wave equation coupled with classes of positive-real impedance boundary conditions

This paper proves the asymptotic stability of the multidimensional wave equation posed on a bounded open Lipschitz set, coupled with various classes of positive-real impedance boundary conditions, chosen for their physical relevance: time-delayed, standard diffusive (which includes the Riemann-Liouville fractional integral) and extended diffusive (which includes the Caputo fractional derivative). The method of proof consists in formulating an abstract Cauchy problem on an extended state space using a dissipative realization of the impedance operator, be it finite or infinite-dimensional. The asymptotic stability of the corresponding strongly continuous semigroup is then obtained by verifying the sufficient spectral conditions derived by Arendt and Batty (Trans. Amer. Math. Soc., 306 (1988)) as well as Lyubich and V\~u (Studia Math., 88 (1988)).

math.DS↗

arXiv2018

Port-Hamiltonian formulation and symplectic discretization of Plate models. Part I : Mindlin model for thick plates

The port-Hamiltonian formulation is a powerful method for modeling and interconnecting systems of different natures. In this paper, the port-Hamiltonian formulation in tensorial form of a thick plate described by the Mindlin-Reissner model is presented. Boundary control and observation are taken into account. Thanks to tensorial calculus, it can be seen that the Mindlin plate model mimics the interconnection structure of its one-dimensional counterpart, i.e. the Timoshenko beam. The Partitioned Finite Element Method (PFEM) is then extended to both the vectorial and tensorial formulations in order to obtain a suitable, i.e. structure-preserving, finite-dimensional port-Hamiltonian system (PHs), which preserves the structure and properties of the original distributed parameter system. Mixed boundary conditions are finally handled by introducing some algebraic constraints. Numerical examples are finally presented to validate this approach.

math.AP↗

arXiv2018

Port-Hamiltonian formulation and symplectic discretization of plate models. Part II : Kirchhoff model for thin plates

The mechanical model of a thin plate with boundary control and observation is presented as a port-Hamiltonian system (pHs), both in vectorial and tensorial forms: the Kirchhoff-Love model of a plate is described by using a Stokes-Dirac structure and this represents a novelty with respect to the existing literature. This formulation is carried out both in vectorial and tensorial forms. Thanks to tensorial calculus, this model is found to mimic the interconnection structure of its one-dimensional counterpart, i.e. the Euler-Bernoulli beam. The Partitioned Finite Element Method (PFEM) is then extended to obtain a suitable, i.e. structure-preserving, weak form. The discretization procedure, performed on the vectorial formulation, leads to a finite-dimensional port-Hamiltonian system. This part II of the companion paper extends part I, dedicated to the Mindlin model for thick plates. The thin plate model comes along with additional difficulties, because of the higher order of the differential operator under consideration.

math.AP↗