Symplectic Model Reduction of Hamiltonian Systems on Nonlinear Manifolds

Buchfink, Patrick; Glas, Silke; Haasdonk, Bernard

doi:10.48550/arxiv.2112.10815

Cited by 3 publications

(3 citation statements)

References 39 publications

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“…Remark 8.2. MOR for pHODEs is discussed for instance in Hesthaven (2017, 2019), Gugercin (2011), Borja, Scherpen andFujimoto (2023), Breiten and Unger (2022), Breiten, Morandin and Schulze (2022b), Buchfink, Glas and Haasdonk (2021), Chaturantabut, Beattie andGugercin (2016), Egger et al (2018), Fujimoto and Kajiura (2007), van der Schaft (2009, 2012), Ionescu and Astolfi (2013), Kawano and Scherpen (2018)…”

Section: Model-order Reductionmentioning

confidence: 99%

Control of port-Hamiltonian differential-algebraic systems and applications

Mehrmann¹,

Unger²

2023

Acta Numerica

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We discuss the modelling framework of port-Hamiltonian descriptor systems and their use in numerical simulation and control. The structure is ideal for automated network-based modelling since it is invariant under power-conserving interconnection, congruence transformations and Galerkin projection. Moreover, stability and passivity properties are easily shown. Condensed forms under orthogonal transformations present easy analysis tools for existence, uniqueness, regularity and numerical methods to check these properties.After recalling the concepts for general linear and nonlinear descriptor systems, we demonstrate that many difficulties that arise in general descriptor systems can be easily overcome within the port-Hamiltonian framework. The properties of port-Hamiltonian descriptor systems are analysed, and time discretization and numerical linear algebra techniques are discussed. Structure-preserving regularization procedures for descriptor systems are presented to make them suitable for simulation and control. Model reduction techniques that preserve the structure and stabilization and optimal control techniques are discussed.The properties of port-Hamiltonian descriptor systems and their use in modelling simulation and control methods are illustrated with several examples from different physical domains. The survey concludes with open problems and research topics that deserve further attention.

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Section: Model-order Reductionmentioning

confidence: 99%

Control of port-Hamiltonian differential-algebraic systems and applications

Mehrmann¹,

Unger²

2023

Acta Numerica

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“…To this end, such methods rely on nonlinear and/or hybrid space-time approaches. For more details, we refer to [8,10,18,21,26,29,31].…”

Section: Introductionmentioning

confidence: 99%

Geometry-based approximation of waves in complex domains

Pradovera¹,

Nonino²,

Perugia³

2023

Preprint

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We consider wave propagation problems over 2-dimensional domains with piecewise-linear boundaries, possibly including scatterers. Under the assumption that the initial conditions and forcing terms are radially symmetric and compactly supported (which is common in applications), we propose an approximation of the propagating wave as the sum of some special nonlinear space-time functions: each term in this sum identifies a particular ray, modeling the result of a single reflection or diffraction effect. We describe an algorithm for identifying such rays automatically, based on the domain geometry. To showcase our proposed method, we present several numerical examples, such as waves scattering off wedges and waves propagating through a room in presence of obstacles.

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“…While the field of symplectic model reduction of Hamiltonian systems has grown considerably in the recent years [35,1,15,33,6], progress on structure-preserving model reduction of Lagrangian systems has been less rapid. The intrusive model reduction for Lagrangian systems was introduced in [26] where the authors showed that performing a Galerkin projection on the Euler-Lagrange equations preserves the Lagrangian structure.…”

Section: Introductionmentioning

confidence: 99%

Preserving Lagrangian structure in data-driven reduced-order modeling of large-scale mechanical systems

Sharma¹,

Kramer²

2022

Preprint

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This work presents a nonintrusive physics-preserving method to learn reduced-order models (ROMs) of Lagrangian mechanical systems. Existing intrusive projection-based model reduction approaches construct structure-preserving Lagrangian ROMs by projecting the Euler-Lagrange equations of the full-order model (FOM) onto a linear subspace. This Galerkin projection step requires complete knowledge about the Lagrangian operators in the FOM and full access to manipulate the computer code. In contrast, the proposed Lagrangian operator inference approach embeds the mechanics into the operator inference framework to develop a data-driven model reduction method that preserves the underlying Lagrangian structure. The proposed approach exploits knowledge of the governing equations (but not their discretization) to define the form and parametrization of a Lagrangian ROM which can then be learned from projected snapshot data. The method does not require access to FOM operators or computer code. The numerical results demonstrate Lagrangian operator inference on an Euler-Bernoulli beam model and a large-scale discretization of a soft robot fishtail with 779,232 degrees of freedom. Accurate long-time predictions of the learned Lagrangian ROMs far outside the training time interval illustrate their generalizability.

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Symplectic Model Reduction of Hamiltonian Systems on Nonlinear Manifolds

Cited by 3 publications

References 39 publications

Control of port-Hamiltonian differential-algebraic systems and applications

Control of port-Hamiltonian differential-algebraic systems and applications

Geometry-based approximation of waves in complex domains

Preserving Lagrangian structure in data-driven reduced-order modeling of large-scale mechanical systems

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