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

T0 review · 2 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read A kinematic lifting framework strongly imposes Dirichlet boundary velocities in port-Hamiltonian elastodynamics while preserving an ordinary differential equation structure, avoiding Lagrange multipliers and penalty parameters.

desk verdict A careful, genuinely useful PHS-FEM construction that achieves strong Dirichlet velocity imposition as an ODE — but only under an equal-interpolation compatibility assumption that the paper neither relaxes nor tests. read the letter →

arxiv 2607.26248 v1 pith:KBBVZVH7 submitted 2026-07-28 cs.CE cs.NAmath.NAphysics.comp-ph

Cristobal Ponce , Hector Ramirez , Yongxin Wu , Ning Liu , Yann Le Gorrec This is my paper · ORCID
classification cs.CEcs.NAmath.NAphysics.comp-ph MSC 65M6070H0574S05
keywords port-HamiltoniansystemsDirichletboundaryconditionskinematicliftingstructure-preservingdiscretizationfiniteelementmethodelastodynamicsDAEavoidancemassmatrixpartitioning
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish that Dirichlet boundary velocities in elastodynamics can be imposed exactly at the discrete level without turning the semi-discrete model into a differential-algebraic equation or degrading accuracy at the boundary. The authors achieve this by splitting displacement and velocity into a relative component that vanishes on the boundary and a prescribed lifting function, and by embedding this split into virtual power principles. The resulting finite-dimensional systems are port-Hamiltonian, have ODE topology, and reduce to classical algebraic mass-matrix partitioning under compatible shape functions. If true, this gives a systematic structure-preserving route to strong boundary-velocity imposition in flexible mechanical systems.

What carries the argument

The load-bearing mechanism is the kinematic lifting decomposition combined with a distributed power-neutral port. The lifting function vL extends the Dirichlet velocity into the interior and enters the system through a collocated input/output pair (uΩ, yΩ) with yΩ = 0, so no energy is injected or extracted by the arbitrary interior extension. The construction works because the shape functions for momentum and displacement are required to satisfy Np = Ap Nr with Ap symmetric positive definite, and the interior lifting shape functions equal the displacement shape functions, which guarantees invertibility of the coupling matrix Mpr and the symmetric PHS structure.

What would settle it

Discretize a 2D elastodynamics problem with momentum shape functions one polynomial degree lower than displacement (e.g., P0 momentum, P1 displacement) so that Np = Ap Nr fails, then check whether the semi-discrete system is an ODE with strongly imposed Dirichlet velocities or degenerates into a DAE or loses passivity. A simpler test: on a single element with a node at the Dirichlet boundary, verify that the coupling matrix Mpr remains invertible and that the boundary velocity appears only through the proposed input channel without algebraic constraints.

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Extended reading notes

Core claim

The paper claims the first structure-preserving finite element framework that strongly imposes Dirichlet boundary velocities while yielding an ODE topology. The key step is a continuous additive kinematic decomposition r = rr + rL and ṙ = ṙr + vL, where rr and ṙr vanish on the Dirichlet boundary and vL carries the prescribed velocity. Inserting this decomposition into rate-form virtual power principles (inspired by Hamilton-Pontryagin and Hu-Washizu) produces lifted port-Hamiltonian systems with a distributed port whose output is identically zero, preserving the physical power balance. Upon finite element discretization with compatible shape functions Np = Ap Nr and Nr = NrΩ, the discrete sy

Load-bearing premise

The central claim collapses if the shape functions cannot be chosen so that momentum and displacement interpolations are linked by a symmetric positive-definite matrix and the interior lifting uses the same shape functions as the relative displacement; the paper states this compatibility as mandatory but does not relax it.

Editorial extensions

If this is right

  • Engineers can impose boundary velocities exactly in port-Hamiltonian elastodynamics without solving DAEs or tuning penalty parameters, simplifying simulation and control design.
  • The discrete equations reduce to the classical algebraic partitioning of the consistent mass matrix, meaning existing FEM codes can be reinterpreted as structure-preserving PHS discretizations with strong Dirichlet velocities.
  • Both jet-bundle (2-field) and Stokes-Dirac (4-field) formulations remain energy-conserving at the semi-discrete level, as verified numerically for strings and a 2D Neo-Hookean frame.
  • The distributed interior port, though energy-neutral in the output, expands the state space and may be used for observer design or trajectory planning without corrupting the power balance.
  • The Hu-Washizu-based lifting preserves the locking-free character of mixed formulations, as shown for a Timoshenko beam.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The framework may extend beyond elastodynamics to any port-Hamiltonian system with Dirichlet boundary ports (e.g., fluids, electromagnetics) as long as the base variational principle has a PHS structure and the shape-function compatibility condition holds.
  • The equivalence to algebraic mass-matrix partitioning suggests that the lifting framework is not a new numerical method but a systematic way to retrofit existing FEM discretizations with a PHS structure and strong boundary velocity inputs, potentially enabling straightforward reuse of mature codebases.
  • A testable extension would be to relax the Np = Ap Nr condition and experiment with mixed-order spaces (e.g., P0 momentum, P1 displacement) to see whether an ODE structure can still be recovered via different lumping or projection strategies.
  • The energy-neutral distributed port could be exploited as a 'virtual actuation' channel for control design, since it does not alter the physical power balance yet can shape the relative-state dynamics.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper introduces a continuous kinematic lifting framework for port-Hamiltonian elastodynamics: displacement and velocity are split into a relative dynamic field vanishing on the Dirichlet boundary and a prescribed lifting field, inducing a distributed port. Two continuous lifted PHS models are derived from virtual-power principles, and their Galerkin discretizations yield finite-dimensional PHS systems with an ODE topology in which Dirichlet boundary velocities are imposed as inputs. Corollaries connect the discrete lifted models to classical algebraic partitioning of the consistent mass matrix. Numerical benchmarks treat a Timoshenko beam (static shear locking), a geometrically exact string (energy conservation), and a 2D Neo-Hookean frame under prescribed boundary velocity.

Significance. If the claims hold, the paper is a useful contribution to structure-preserving FEM for PHS. It offers a route to exact imposition of Dirichlet velocities without Lagrange multipliers or DAEs, while retaining a discrete PHS structure and explicit equivalence to standard mass-matrix partitioning. The appendices contain detailed algebraic proofs of the discrete PHS structure and energy balance, and the paper provides reproducible code and data. Numerical examples reproduce analytical static solutions, conserve energy for homogeneous boundary conditions, and demonstrate stable dynamic responses. No fitted parameters appear. The main caveat, discussed below, is that the central ODE result is proven under a specific shape-function compatibility condition, so the generality of the headline claim is narrower than stated.

major comments (2)
  1. [§4.1, Remark 3; Theorems 1 and 2] The ODE/strong-imposition result is conditional on the shape-function compatibility N_p = A_p N_r with A_p = A_p^T > 0 and N_r = N_rΩ. These conditions are what make the coupling matrix \hat M_pr invertible and make the distributed output \hat y_Ω vanish. The paper states these as mandatory but does not investigate or test non-compliant element choices. Independent interpolation orders for momentum and displacement are common in Stokes-Dirac and mixed FEM, and for such spaces the central theorem is not established. This is load-bearing because the headline claim—strong imposition without DAEs—is proven only for equal-order momentum/displacement spaces. Please either state this restriction prominently in the abstract and introduction, or extend the analysis to see whether the result can be relaxed, and add a numerical experiment with a non-compliant pair to delimit the failure mode.
  2. [§4.3.1, Corollary 1] The PHS obtained after algebraic reduction uses the momentum p1 = M11\dot q1 + M12 vD and the storage function H = (1/2) p1^T M11^{-1} p1 + V. When vD is nonzero, this H differs from the physical total mechanical energy of the constrained system by terms quadratic in vD (and the cross term is removed from the physical kinetic energy). The paper motivates the lifting framework by stating that the Hamiltonian should not degenerate and that the physical power balance should be preserved. Please clarify that, after reduction to absolute displacements with prescribed boundary velocities, H is a valid PHS storage function but not necessarily the physical energy, and state explicitly what the discrete energy balance means in this case. This distinction matters for the interpretation of the energy plots in §5.3 during the actuation phase.
minor comments (5)
  1. [§4.1, after Eq. (43)] The proof of Theorem 1 uses L_rΩ^e = L_r^e when forming \hat M_pr with \hat v_Ω. Remark 3 states N_r = N_rΩ but does not explicitly state that the corresponding assembly operators are identical. Please add this to the assumptions.
  2. [Appendix B.1, Step 2] The invertibility of \hat M_pr is justified locally from A_p>0, but the global statement also needs that the momentum and displacement assembly operators L_p and L_r select the same free DOFs. Since this is not stated, readers cannot tell whether the proof covers, e.g., discontinuous momentum with continuous displacement.
  3. [Abstract and §1] The motivation that weak imposition methods 'often exhibit poor accuracy at Dirichlet boundaries' is supported only by a reference. Since the paper's title includes 'strong imposition,' a small numerical comparison against a weak-imposition method (or at least an explicit statement of the reported accuracy deficiency) would strengthen the motivation.
  4. [§3, Propositions 2 and 4] Calling (u_Ω, y_Ω) a 'distributed port' when y_Ω is identically zero is unconventional. Consider using 'auxiliary input' or 'energy-neutral input' to avoid implying an actual energy-exchange port.
  5. [§5.3] The computational times are hardware-dependent and obtained with a 'sub-optimal custom MATLAB code.' They are not essential and may be omitted or clearly labeled as indicative only.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; the strong-imposition ODE result is a genuine derivation under the explicitly stated shape-function compatibility conditions.

full rationale

The paper contains no fitted parameters: material, geometric, load, and boundary-velocity inputs are prescribed benchmark data, and the discrete lifted PHS models are verified against analytical solutions and exact energy conservation. The claimed equivalence to standard FEM algebraic partitioning (Section 4.3) is an explicit algebraic demonstration, not a renaming or a fitted prediction. The zero-power distributed port yOmega = 0 is a designed consequence of writing the Hamiltonian in absolute fields (so that err = erOmega), and the paper presents it as a structural property rather than as an independent empirical prediction; it does not carry the central ODE claim. Self-citations [52], [53], and [58] supply background formulations and motivating comparisons, but the main derivation (Propositions 2 and 4, Theorems 1 and 2) is proved in the appendices from the stated variational principles and does not reduce to those citations. The equal-interpolation requirement in Remark 3 (N_p = A_p N_r with A_p = A_p^T > 0 and N_r = N_rOmega) is an explicit, load-bearing assumption that limits the framework's scope; this is a correctness/scope risk, not a circularity. Overall, the derivation chain is self-contained, with only minor and non-load-bearing reliance on prior work by the same authors.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

The central derivation uses no fitted parameters; all material and geometric constants are prescribed benchmark definitions. The main auxiliary construct is the distributed lifting port, a mathematical device whose power output is zero by construction. The remaining axioms are standard Sobolev-lifting and FEM compatibility conditions.

assumptions (4)
  • standard math Existence of a smooth lifting function rL with prescribed boundary values, via Sobolev trace theorem (Definition 5, Section 2.4).
    Used to define the kinematic decomposition and to extend Dirichlet boundary data into the interior domain.
  • domain assumption Shape-function compatibility: N_p = A_p N_r with A_p = A_p^T > 0, and N_r = N_rΩ (Remark 3, Section 4.1).
    Required for the discrete PHS structure, invertibility of M_pr, and the reduction to absolute displacements in Corollary 3.
  • domain assumption Mass and constitutive mapping matrices are symmetric positive definite (M, K_epsilon, A_p, A_epsilon).
    Standard in FEM; needed for invertibility and passivity of the discrete port-Hamiltonian system.
  • domain assumption Dirichlet and Neumann boundaries are disjoint, and boundary data are sufficiently smooth (H^1 fields).
    Definition 5 requires the lifting fields rL and vL to be smooth on the closure of the domain.
invented entities (1)
  • Distributed interior port (uΩ = vΩ, yΩ = 0)
    purpose: Maps boundary velocity actuation into the interior of the domain as a lifting velocity while maintaining power neutrality.
    Introduced as an auxiliary input in the continuous and discrete models (Propositions 2 and 4; Theorems 1 and 2). It is a mathematical construct, not a physical entity, and has no falsifiable handle outside the paper; its output is zero by construction.

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Cite this review

Pith. "Pith review of Strong imposition of Dirichlet boundary velocities in structure-preserving discretizations of elastodynamics." pith.science (2026). https://pith.science/paper/KBBVZVH7

@misc{pith2026260726248,
  author       = {Pith},
  title        = {Pith review of: Strong imposition of Dirichlet boundary velocities in structure-preserving discretizations of elastodynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KBBVZVH7}},
  note         = {Machine review of arXiv:2607.26248}
}
read the original 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.

Figures

Figures reproduced from arXiv: 2607.26248 by the authors.

Figure 1
Figure 1. Static deflections computed via the discrete lifted jet-bundle formulation (Theorem [PITH_FULL_IMAGE:figures/full_fig_p019_1.png] view at source ↗
Figure 2
Figure 2. Static deflections computed via the discrete lifted Stokes-Dirac formulation (Theorem [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗
Figure 3
Figure 3. Spatial configuration of the string computed via the discrete lifted jet-bundle formulation (Theorem [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Energy evolution of the string computed via the discrete lifted jet-bundle formulation (Theorem [PITH_FULL_IMAGE:figures/full_fig_p022_4.png]
Figure 5
Figure 5. Figure 5: Spatial configuration of the string computed via the discrete lifted Stokes-Dirac formulation (Theorem [PITH_FULL_IMAGE:figures/full_fig_p022_5.png]
Figure 6
Figure 6. Figure 6: Energy evolution of the string computed via the discrete lifted Stokes-Dirac formulation (Theorem [PITH_FULL_IMAGE:figures/full_fig_p022_6.png]
Figure 7
Figure 7. Figure 7: Two dimensional frame. 5.3 Dynamic response under Dirichlet boundary velocities The third numerical example evaluates the proposed formulations under imposed Dirichlet boundary velocities. For this purpose, a geometrically nonlinear two-dimensional frame with compressi…
Figure 8
Figure 8. Figure 8: Dynamic configuration of the 2D Neo-Hookean frame. [PITH_FULL_IMAGE:figures/full_fig_p024_8.png]
Figure 9
Figure 9. Figure 9: Energy evolution of the 2D frame comparing the explicit Störmer-Verlet and implicit midpoint rule integrators. [PITH_FULL_IMAGE:figures/full_fig_p024_9.png]

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Reference graph

Works this paper leans on

60 extracted references · 2 linked inside Pith

  1. [1]

    Elastodynamic equations and wave propagation for multiscale wave- induced fluid flow in partially saturated media from energy conservation,

    L. Liu, X. Zhang, X. Wang, Y. Qi, and Z. Shi, “Elastodynamic equations and wave propagation for multiscale wave- induced fluid flow in partially saturated media from energy conservation,” The Journal of the Acoustical Society of America, vol. 159, no. 6, pp. 5083–5094, 2026

    2026

  2. [2]

    Large scale parameter estimation problems in frequency-domain elastodynamics using an error in constitutive equation functional,

    B. Banerjee, T. F. Walsh, W. Aquino, and M. Bonnet, “Large scale parameter estimation problems in frequency-domain elastodynamics using an error in constitutive equation functional,” Computer methods in applied mechanics and engi- neering, vol. 253, pp. 60–72, 2013

    2013

  3. [3]

    A review on flexible robotic systems for minimally invasive surgery,

    O. M. Omisore, S. Han, J. Xiong, H. Li, Z. Li, and L. Wang, “A review on flexible robotic systems for minimally invasive surgery,” IEEE Transactions on Systems, Man, and Cybernetics: Systems , vol. 52, no. 1, pp. 631–644, 2020

    2020

  4. [4]

    Bathe, Finite element procedures

    K.-J. Bathe, Finite element procedures. Klaus-Jurgen Bathe, 2006

    2006

  5. [5]

    Belytschko, W

    T. Belytschko, W. K. Liu, B. Moran, and K. Elkhodary, Nonlinear finite elements for continua and structures . John wiley & sons, 2014

    2014

  6. [6]

    J. N. Reddy, Energy principles and variational methods in applied mechanics . John Wiley & Sons, 2017

    2017

  7. [7]

    The hamiltonian structure of nonlinear elasticity: the material and convective representations of solids, rods, and plates,

    J. C. Simo, J. E. Marsden, and P. Krishnaprasad, “The hamiltonian structure of nonlinear elasticity: the material and convective representations of solids, rods, and plates,” Arch. Ration. Mech. Anal , vol. 104, no. 2, pp. 125–183, 1988

    1988

  8. [8]

    Classical mechanics,

    J. Safko, H. Goldstein, and C. Poole, “Classical mechanics,” 2002

    2002

Show all 60 references
  1. [9]

    W. Yao, W. Zhong, and C. W. Lim, Symplectic elasticity . World Scientific, 2009

    2009

  2. [10]

    Symplectic Hamiltonian finite element methods for linear elastodynamics,

    M. A. Sánchez, B. Cockburn, N.-C. Nguyen, and J. Peraire, “Symplectic Hamiltonian finite element methods for linear elastodynamics,” Computer Methods in Applied Mechanics and Engineering , vol. 381, p. 113843, 2021

    2021

  3. [11]

    Hamiltonian formulation of distributed-parameter systems with boundary energy flow,

    A. van der Schaft and B. Maschke, “Hamiltonian formulation of distributed-parameter systems with boundary energy flow,” Journal of Geometry and physics , vol. 42, no. 1-2, pp. 166–194, 2002

    2002

  4. [12]

    Port-controlled Hamiltonian systems: modelling origins and systemtheoretic prop- erties,

    B. Maschke and A. van der Schaft, “Port-controlled Hamiltonian systems: modelling origins and systemtheoretic prop- erties,” in Nonlinear control systems design 1992 , pp. 359–365, Elsevier, 1993

    1992

  5. [13]

    Duindam, A

    V. Duindam, A. Macchelli, S. Stramigioli, and H. Bruyninckx, Modeling and control of complex physical systems: the port-Hamiltonian approach. Springer Science & Business Media, 2009

    2009

  6. [14]

    Port-Hamiltonian systems modelling in electrical engineer- ing,

    A. Bartel, M. Clemens, M. Günther, B. Jacob, and T. Reis, “Port-Hamiltonian systems modelling in electrical engineer- ing,” in International Conference on Scientific Computing in Electrical Engineering , pp. 133–143, Springer, 2022

    2022

  7. [15]

    Distributed control and optimization of dc microgrids: A port-Hamiltonian ap- proach,

    B. Abdolmaleki and G. Bergna-Diaz, “Distributed control and optimization of dc microgrids: A port-Hamiltonian ap- proach,” IEEE Access, vol. 10, pp. 64222–64233, 2022

    2022

  8. [16]

    Port-Hamiltonian modeling and control of electric vehicle charging stations,

    H. Gernandt, B. Severino, X. Zhang, V. Mehrmann, and K. Strunz, “Port-Hamiltonian modeling and control of electric vehicle charging stations,” IEEE Transactions on Transportation Electrification , vol. 11, no. 1, pp. 2897–2907, 2024

    2024

  9. [17]

    Port-based modeling and simulation of mechanical systems with rigid and flexible links,

    A. Macchelli, C. Melchiorri, and S. Stramigioli, “Port-based modeling and simulation of mechanical systems with rigid and flexible links,” IEEE transactions on robotics , vol. 25, no. 5, pp. 1016–1029, 2009

    2009

  10. [18]

    Port-Hamiltonian representation of mechanical systems with velocity inputs,

    J. Ferguson and C. Renton, “Port-Hamiltonian representation of mechanical systems with velocity inputs,” IEEE Control Systems Letters , vol. 8, pp. 1367–1372, 2024

    2024

  11. [19]

    Port-Hamiltonian flexible multibody dynamics,

    A. Brugnoli, D. Alazard, V. Pommier-Budinger, and D. Matignon, “Port-Hamiltonian flexible multibody dynamics,” Multibody System Dynamics , vol. 51, no. 3, pp. 343–375, 2021

    2021

  12. [20]

    Port-Hamiltonian multibody dynamics: Lagrangian formulation, consistent interconnection, structure-preserving simulation and index-reduction,

    L. Latussek, P. L. Kinon, and P. Betsch, “Port-Hamiltonian multibody dynamics: Lagrangian formulation, consistent interconnection, structure-preserving simulation and index-reduction,” arXiv preprint arXiv:2603.12841 , 2026

    2026

  13. [21]

    Port-Hamiltonian modeling and control of a micro-channel experimental plant,

    N. Cisneros, A. J. Rojas, and H. Ramirez, “Port-Hamiltonian modeling and control of a micro-channel experimental plant,” IEEE Access, vol. 8, pp. 176935–176946, 2020

    2020

  14. [22]

    On port-Hamiltonian formulations of 3-dimensional compressible Newtonian fluids,

    L. Mora, Y. Le Gorrec, D. Matignon, H. Ramirez, and J. Yuz, “On port-Hamiltonian formulations of 3-dimensional compressible Newtonian fluids,” Physics of Fluids , vol. 33, no. 11, p. 117117, 2021. 34

    2021

  15. [23]

    Geometric and energy-aware decomposition of the Navier– Stokes equations: A port-Hamiltonian approach,

    F. Califano, R. Rashad, F. P. Schuller, and S. Stramigioli, “Geometric and energy-aware decomposition of the Navier– Stokes equations: A port-Hamiltonian approach,” Physics of fluids , vol. 33, no. 4, 2021

    2021

  16. [24]

    Port-Hamiltonian formulations for the modeling, simulation and control of fluids,

    F. L. Cardoso-Ribeiro, G. Haine, Y. Le Gorrec, D. Matignon, and H. Ramirez, “Port-Hamiltonian formulations for the modeling, simulation and control of fluids,” Computers & Fluids , vol. 283, p. 106407, 2024

    2024

  17. [25]

    Port Hamiltonian systems extended to irreversible systems: the example of the heat conduction,

    D. Eberard and B. Maschke, “Port Hamiltonian systems extended to irreversible systems: the example of the heat conduction,” IF AC Proceedings Volumes, vol. 37, no. 13, pp. 243–248, 2004

    2004

  18. [26]

    A two-dimensional port-Hamiltonian model for coupled heat transfer,

    J. Jäschke, M. Ehrhardt, M. Günther, and B. Jacob, “A two-dimensional port-Hamiltonian model for coupled heat transfer,” Mathematics, vol. 10, no. 24, p. 4635, 2022

    2022

  19. [27]

    An overview on irreversible port-Hamiltonian systems,

    H. Ramirez and Y. Le Gorrec, “An overview on irreversible port-Hamiltonian systems,” Entropy, vol. 24, no. 10, p. 1478, 2022

    2022

  20. [28]

    Modeling of a fluid-structure coupled system using port-Hamiltonian formulation,

    F. L. Cardoso-Ribeiro, D. Matignon, and V. Pommier-Budinger, “Modeling of a fluid-structure coupled system using port-Hamiltonian formulation,” IF AC-PapersOnLine, vol. 48, no. 13, pp. 217–222, 2015

    2015

  21. [29]

    A thermodynamically consistent port-Hamiltonian model for dielectric elastomer membrane actuators and generators,

    G. Rizzello, D. Naso, and S. Seelecke, “A thermodynamically consistent port-Hamiltonian model for dielectric elastomer membrane actuators and generators,” IF AC-PapersOnLine, vol. 50, no. 1, pp. 4855–4862, 2017

    2017

  22. [30]

    A port-Hamiltonian formulation of linear thermoe- lasticity and its mixed finite element discretization,

    A. Brugnoli, D. Alazard, V. Pommier-Budinger, and D. Matignon, “A port-Hamiltonian formulation of linear thermoe- lasticity and its mixed finite element discretization,” Journal of Thermal Stresses , vol. 44, no. 6, pp. 643–661, 2021

    2021

  23. [31]

    Dynamic modeling of a curling HASEL actuator using the port Hamiltonian framework with experimental validation,

    N. Cisneros, Y. Wu, K. Rabenorosoa, and Y. Le Gorrec, “Dynamic modeling of a curling HASEL actuator using the port Hamiltonian framework with experimental validation,” Mechatronics, vol. 109, p. 103342, 2025

    2025

  24. [32]

    The port-Hamiltonian structure of continuum mechanics,

    R. Rashad and S. Stramigioli, “The port-Hamiltonian structure of continuum mechanics,” Journal of nonlinear science , vol. 35, no. 2, p. 35, 2025

    2025

  25. [33]

    Formal distributed port-Hamiltonian representation of field equations,

    G. Nishida and M. Yamakita, “Formal distributed port-Hamiltonian representation of field equations,” in Proceedings of the 44th IEEE Conference on Decision and Control , pp. 6009–6015, IEEE, 2005

    2005

  26. [34]

    Jet bundle formulation of infinite-dimensional port-Hamiltonian systems using differential operators,

    M. Schöberl and A. Siuka, “Jet bundle formulation of infinite-dimensional port-Hamiltonian systems using differential operators,” Automatica, vol. 50, no. 2, pp. 607–613, 2014

    2014

  27. [35]

    Port-Hamiltonian formulation for higher-order PDEs,

    M. Schöberl and K. Schlacher, “Port-Hamiltonian formulation for higher-order PDEs,” IF AC-PapersOnLine, vol. 48, no. 13, pp. 244–249, 2015

    2015

  28. [36]

    Port-Hamiltonian differential-algebraic systems,

    A. van der Schaft, “Port-Hamiltonian differential-algebraic systems,” Surveys in Differential-Algebraic Equations I , pp. 173–226, 2013

    2013

  29. [37]

    Linear port-Hamiltonian descriptor systems,

    C. Beattie, V. Mehrmann, H. Xu, and H. Zwart, “Linear port-Hamiltonian descriptor systems,” Mathematics of Control, Signals, and Systems , vol. 30, no. 4, p. 17, 2018

    2018

  30. [38]

    Generalized port-Hamiltonian DAE systems,

    A. van der Schaft and B. Maschke, “Generalized port-Hamiltonian DAE systems,” Systems & Control Letters , vol. 121, pp. 31–37, 2018

    2018

  31. [39]

    Linear boundary port Hamiltonian systems defined on Lagrangian submanifolds,

    B. Maschke and A. van der Schaft, “Linear boundary port Hamiltonian systems defined on Lagrangian submanifolds,” IF AC-PapersOnLine, vol. 53, no. 2, pp. 7734–7739, 2020

    2020

  32. [40]

    Linear boundary port-Hamiltonian systems with implicitly defined energy,

    B. Maschke and A. van der Schaft, “Linear boundary port-Hamiltonian systems with implicitly defined energy,” arXiv preprint arXiv:2305.13772 , 2023

    2023 arXiv

  33. [41]

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

    A. Bendimerad-Hohl, G. Haine, L. Lefèvre, and D. Matignon, “Structure-preserving space discretization of differential and nonlocal constitutive relations for port-Hamiltonian systems,” Journal of Computational Physics , p. 114951, 2026

    2026

  34. [42]

    Dirac and Lagrange algebraic constraints in nonlinear port-Hamiltonian systems,

    A. van der Schaft and B. Maschke, “Dirac and Lagrange algebraic constraints in nonlinear port-Hamiltonian systems,” Vietnam Journal of Mathematics , vol. 48, no. 4, pp. 929–939, 2020

    2020

  35. [43]

    Stokes-Lagrange and Stokes-Dirac representations of N - dimensional port-Hamiltonian systems for modeling and control,

    A. Bendimerad-Hohl, G. Haine, L. Lefèvre, and D. Matignon, “Stokes-Lagrange and Stokes-Dirac representations of N - dimensional port-Hamiltonian systems for modeling and control,” Communications in Analysis and Mechanics , vol. 17, no. 2, pp. 474–519, 2025

    2025

  36. [44]

    Port-Hamiltonian systems theory: An introductory overview,

    A. van der Schaft and D. Jeltsema, “Port-Hamiltonian systems theory: An introductory overview,” Foundations and Trends® in Systems and Control , vol. 1, no. 2-3, pp. 173–378, 2014

    2014

  37. [45]

    Twenty years of distributed port-Hamiltonian systems: a literature review,

    R. Rashad, F. Califano, A. van der Schaft, and S. Stramigioli, “Twenty years of distributed port-Hamiltonian systems: a literature review,” IMA Journal of Mathematical Control and Information , vol. 37, no. 4, pp. 1400–1422, 2020

    2020

  38. [46]

    Partitioned finite element method for structured discretization with mixed boundary conditions,

    A. Brugnoli, F. L. Cardoso-Ribeiro, G. Haine, and P. Kotyczka, “Partitioned finite element method for structured discretization with mixed boundary conditions,” IF AC-PapersOnLine, vol. 53, no. 2, pp. 7557–7562, 2020

    2020

  39. [47]

    Explicit port-Hamiltonian FEM-models for linear mechanical systems with non-uniform boundary conditions,

    T. Thoma and P. Kotyczka, “Explicit port-Hamiltonian FEM-models for linear mechanical systems with non-uniform boundary conditions,” IF AC-PapersOnLine, vol. 55, no. 20, pp. 499–504, 2022

    2022

  40. [48]

    On the velocity-stress formulation for geometrically nonlinear elastodynamics and its structure-preserving discretization,

    T. Thoma, P. Kotyczka, and H. Egger, “On the velocity-stress formulation for geometrically nonlinear elastodynamics and its structure-preserving discretization,” Mathematical and Computer Modelling of Dynamical Systems , vol. 30, no. 1, pp. 701–720, 2024

    2024

  41. [49]

    Generalized Maxwell viscoelasticity for geometrically exact strings: Nonlinear port-Hamiltonian formulation and structure-preserving discretization,

    P. Kinon, T. Thoma, P. Betsch, and P. Kotyczka, “Generalized Maxwell viscoelasticity for geometrically exact strings: Nonlinear port-Hamiltonian formulation and structure-preserving discretization,” IF AC-PapersOnLine, vol. 58, no. 6, 35 pp. 101–106, 2024

    2024

  42. [50]

    Structure-preserving discretization of multidimensional linear port- Hamiltonian systems using FEM approaches,

    C. Ponce, Y. Wu, Y. Le Gorrec, and H. Ramirez, “Structure-preserving discretization of multidimensional linear port- Hamiltonian systems using FEM approaches,” in 2024 IEEE 63rd Conference on Decision and Control (CDC) , pp. 2676– 2681, IEEE, 2024

    2024

  43. [51]

    Quarteroni, Numerical models for differential problems

    A. Quarteroni, Numerical models for differential problems . Springer, 2009

    2009

  44. [52]

    Ponce, Port-Hamiltonian modeling, discretization, and shape control of multidimensional flexible mechanical systems

    C. Ponce, Port-Hamiltonian modeling, discretization, and shape control of multidimensional flexible mechanical systems . PhD thesis, Université Bourgogne Franche-Comté, Universidad Técnica Federico Santa María, 2024

    2024

  45. [53]

    A port-Hamiltonian framework for the modeling and FEM discretization of hyperelastic systems,

    C. Ponce, Y. Wu, Y. Le Gorrec, and H. Ramirez, “A port-Hamiltonian framework for the modeling and FEM discretization of hyperelastic systems,” Applied Mathematical Modelling , p. 116403, 2025

    2025

  46. [54]

    Explicit structure-preserving discretization of port-Hamiltonian systems with mixed boundary control,

    A. Brugnoli, G. Haine, and D. Matignon, “Explicit structure-preserving discretization of port-Hamiltonian systems with mixed boundary control,” IF AC-papersonline, vol. 55, no. 30, pp. 418–423, 2022

    2022

  47. [55]

    Port-Hamiltonian formulation and structure-preserving discretization of hyperelastic strings,

    P. Kinon, T. Thoma, P. Betsch, and P. Kotyczka, “Port-Hamiltonian formulation and structure-preserving discretization of hyperelastic strings,” arXiv preprint arXiv:2304.10957 , 2023

    2023 arXiv

  48. [56]

    Port-Hamiltonian formulation and structure-preserving discretization of finite elasticity based on a mixed Hu-Washizu-type formulation,

    M. Hille, P. Betsch, and M. Franke, “Port-Hamiltonian formulation and structure-preserving discretization of finite elasticity based on a mixed Hu-Washizu-type formulation,” Computer Methods in Applied Mechanics and Engineering , vol. 458, p. 118790, 2026

    2026

  49. [57]

    Mixed formulation and structure-preserving discretization of Cosserat rod dynamics in a port-Hamiltonian framework,

    P. L. Kinon, S. R. Eugster, and P. Betsch, “Mixed formulation and structure-preserving discretization of Cosserat rod dynamics in a port-Hamiltonian framework,” Computer Methods in Applied Mechanics and Engineering , vol. 458, p. 118966, 2026

    2026

  50. [58]

    Port-Hamiltonian modeling of a geometrically nonlinear hyperelastic beam,

    C. Ponce, Y. Wu, Y. Le Gorrec, and H. Ramirez, “Port-Hamiltonian modeling of a geometrically nonlinear hyperelastic beam,” IF AC-PapersOnLine, vol. 58, no. 6, pp. 309–314, 2024

    2024

  51. [59]

    Field port-Lagrangian representation of conservation laws for variational symmetries,

    G. Nishida, M. Yamakita, and Z.-w. Luo, “Field port-Lagrangian representation of conservation laws for variational symmetries,” in Proceedings of the 45th IEEE Conference on Decision and Control , pp. 5875–5881, IEEE, 2006

    2006

  52. [60]

    Dirac structures in Lagrangian mechanics Part II: Variational structures,

    H. Yoshimura and J. Marsden, “Dirac structures in Lagrangian mechanics Part II: Variational structures,” Journal of Geometry and Physics , vol. 57, no. 1, pp. 209–250, 2006. 36

    2006

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.