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Discontinuous Galerkin Finite Element methods for Linear Port-Hamiltonian Dynamical systems

Journal of Scientific Computing 2025年第1期104卷 1-47页

作者： Cheng, Xiaoyu van der Vegt, J.J.W. Xu, Yan Zwart, H.J. School of Mathematics University of Science and Technology of China Anhui Hefei 230026 China Department of Applied Mathematics Mathematics of Computational Science Group University of Twente P.O. Box 217 Enschede 7500 AE Netherlands Department of Applied Mathematics Mathematics of Systems Theory Group University of Twente P.O. Box 217 Enschede 7500 AE Netherlands Department of Mechanical Engineering Dynamics and Control Group Eindhoven University of Technology P.O. Box 513 Eindhoven 5600 MB Netherlands

In this paper, we present discontinuous Galerkin (DG) finite element discretizations for a class of linear hyperbolic port-Hamiltonian dynamical systems. The key point in constructing a port-Hamiltonian system is a Stokes-Dirac structure. Instead of following the traditional approach of defining the strong form of the Dirac structure, we define a Dirac structure in weak form, specifically in the input-state-output form. This is implemented within broken Sobolev spaces on a tessellation with polyhedral elements. After that, we state the weak port-Hamiltonian formulation and prove that it relates to a Poisson bracket. In our work, a crucial aspect of constructing the above-mentioned Dirac structure is that we provide a conservative relation between the boundary ports. Next, we state DG discretizations of the port-Hamiltonian system by using the weak form of the Dirac structure and broken polynomial spaces of differential forms, and we provide a priori error estimates for the structure-preserving port-Hamiltonian discontinuous Galerkin (PHDG) discretizations. The accuracy and capability of the methods developed in this paper are demonstrated by presenting several numerical experiments. © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2025.

关键词： Dirac structure Discontinuous Galerkin methods Exterior calculus Port-Hamiltonian systems

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Portable Electronic Devices in Cabin and Cockpit – Analysis of Hazards

Portable Electronic Devices in Cabin and Cockpit – Analysis...

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AIAA Science and Technology Forum and Exposition, AIAA SciTech Forum 2025

作者： Holz, Simon Pfaff, Jonas Kuder, Jürgen Schopferer, Sebastian May, Michael Norrefeldt, Victor Pathak, Arnav Pschirer, Marie Kienberger, Maximilian Fraunhofer Institute for High-Speed Dynamics Ernst-Mach-Institut Research Group Computational Physics EMI Dpt. Materials and Simulation Methods Freiburg79104 Germany Fraunhofer Institute for High-Speed Dynamics Dpt. Measurement and Sensor Technology Ernst-Mach-Institut EMI Freiburg79104 Germany Fraunhofer Institute for High-Speed Dynamics Department Resilience Safety and Security Ernst-Mach-Institut EMI Freiburg79104 Germany Fraunhofer Institute for Building Physics IBP Research Group Vehicle Climate Control Systems Dpt. Indoor Climate Valley83626 Germany Fraunhofer Institute for Building Physics IBP Valley83626 Germany

ISBN: (数字)9781624107238

ISBN: (纸本)9781624107238

Multiple portable electronic devices (PED) such as smartphones and laptops are carried during flight in cabin and cockpit by passenger and crew. When the Lithium-Ion batteries of PEDs go into thermal runaway flammable and toxic gases are released. In the first part of this study a total of 26 laptops, tablets, smartphones, and power tool batteries have been forced into thermal runaway in a ventilated test box (V=350 l, V=13 l/s) observing smoke and fire as the main hazards. Correlations of the venting volume depending on the energy content of the PED are presented. The volume released during fire hazard is seven times the volume released during smoke hazard. Up to 3 l of electrolyte and up to 0.4 l hydrogen fluoride, hydrogen chloride and formaldehyde as well as up 20 l of combustion products of plastics are released during the full test time. For tablets and smartphones, the smoldering of plastics prior to the thermal runaway could be noted visually and due to a rise in propane emissions. In the second part of this study thermal runaway tests with laptops in a A320 mockup (V=13 m³) under realistic ventilation (V=60 l/s) are presented. The nine tested laptops were used to generate thermal runaway events on the floor, on seats and in the overhead bin. The increase of air temperature in the cabin depends on the distance from the floor, and may be as high as 60 K. The CO2 concentration level in the cabin increases and it takes 16 to 26 min to restore 105 % of the level measured prior to thermal runaway. Finally, the absolute amounts of vent gases are presented and clustered for e.g. toxic and flammable gases or evaporated electrolyte. The presented data will enable the prediction of local concentrations inside occupied areas of the cabin during thermal runaway by CFD simulations. © 2025, American Institute of Aeronautics and Astronautics Inc, AIAA. All rights reserved.

关键词： Fire hazards

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Frequency scaling and energy efficiency regarding the Gauss-Jordan elimination scheme with application to the matrix-sign-function on OpenPOWER 8

Frequency scaling and energy efficiency regarding the Gauss-...

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作者： Köhler, Martin Saak, Jens Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems MagdeburgD-39104 Germany

The Gauss-Jordan Elimination scheme is an alternative to the LU decomposition for solving linear systems or computing the inverse of a matrix. We develop a multi-GPU aware implementation of this algorithm on an OpenPOWER 8 system with application to the Matrix Sign Function. Thereby, we analyze the influence of the CPU clock frequency scaling on the overall energy consumption. The results show possible energy saving of 14.2% without a noteworthy increase of the runtime. © 2018 John Wiley & Sons, Ltd.

关键词： Energy efficiency

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On an invariance principle for the solution space of the differential riccati equation

On an invariance principle for the solution space of the dif...

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89th Annual Meeting on the International Association of Applied Mathematics and Mechanics, GAMM 2018

作者： Behr, Maximilian Benner, Peter Heiland, Jan Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 Magdeburg39106 Germany

The differential Riccati equation appears in different fields of applied mathematics like control theory and systems theory. For large-scale systems the numerical solution comes with a large amount of storage requirements. This motivates the use of Krylov subspace and projection based methods [1–3]. In the present paper we apply an invariance theorem for ODEs to the differential Riccati Equation. We show that the solution is contained in a Krylov like subspace and extend our results to certain time-varying cases. © 2018 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim.

关键词： Riccati equations

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Numerical solution to low rank perturbed lyapunov equations by the sign function method

Numerical solution to low rank perturbed lyapunov equations ...

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Joint 87th Annual Meeting on the International Association of Applied Mathematics and Mechanics, GAMM 2016 and Deutsche Mathematiker‐Vereinigung, DMV 2016

作者： Benner, Peter Denißen, Jonas Max Planck Institute for Dynamics of Complex Technical Systems Computational Methods in Systems and Control Theory Sandtorstr. 1 Magdeburg39106 Germany

We investigate the numerical solution to a low rank perturbed Lyapunov equation AT X + XA = W via the sign function method (SFM). The sign function method has been proposed to solve Lyapunov equations, see e.g. [1], but here we focus on a framework where the matrix A has a special structure, i.e. A = B + UCVT, where B is a blockdiagonal matrix and UCVT is a low rank perturbation. We show that this structure can be kept throughout the sign function iteration but the rank of the perturbation doubles per iteration. Therefore, we apply a low rank approximation to the perturbation in order to keep its numerical rank small. We compare the standard SFM with its structure preserving variant presented in this paper by means of numerical examples from viscously damped mechanical systems. © 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim.

关键词： Matrix algebra

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Towards riccati-feedback control of complex flows with moving interfaces

Towards riccati-feedback control of complex flows with movin...

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88th Annual Meeting on the International Association of Applied Mathematics and Mechanics, GAMM 2017

作者： Baran, Björn Benner, Peter Heiland, Jan Saak, Jens Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 Magdeburg39106 Germany

Problems featuring moving interfaces appear in many applications. They can model solidification and melting of pure materials, crystal growth and other multi-phase problems. The control of the moving interface enables to, for example, influence production processes and, thus, the product material quality. We consider the two-phase Stefan problem that models a solid and a liquid phase separated by the moving interface. In the liquid phase, the heat distribution is characterized by a convection-diffusion equation. The fluid flow in the liquid phase is described by the Navier–Stokes equations which introduces a differential algebraic structure to the system. The interface movement is coupled with the temperature through the Stefan condition, which adds additional algebraic constraints. Our formulation uses a sharp interface representation and we define a quadratic tracking-type cost functional as a target of a control input. We compute an open loop optimal control for the Stefan problem using an adjoint system. For a feedback representation, we linearize the system about the trajectory defined by the open loop control. This results in a linear-quadratic regulator problem, for which we formulate the differential Riccati equation with time varying coefficients. This Riccati equation defines the corresponding feedback gain. Further, we present the feedback formulation that takes into account the structure and the differential algebraic components of the problem. Also, we discuss how the complexities that come, for example, with mesh movements, can be handled in a feedback setting. © 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim.

关键词： Feedback

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Adjoint-based optimal boundary control of a stefan problem system fully coupled with navier-stokes equations

Adjoint-based optimal boundary control of a stefan problem s...

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Joint 87th Annual Meeting on the International Association of Applied Mathematics and Mechanics, GAMM 2016 and Deutsche Mathematiker‐Vereinigung, DMV 2016

作者： Baran, Björn Heiland, Jan Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 Magdeburg39106 Germany

Free boundary and moving boundary problems, that can be used to model crystal growth or the solidification and melting of pure materials, receive growing attention in science and technology. The optimal control of these problems appear even more interesting since certain desired shapes of the boundaries improve, e.g., the material quality in the case of crystal growth. We consider the so called two-phase Stefan problem that models a solid and a liquid phase separated by a moving interface. In the work presented, we take a sharp interface model approach and define a quadratic tracking-type cost functional that penalizes the deviation of the interface from the desired state at a final time as well as the control costs. Following the "optimize-then-discretize" paradigm, we formulate a first order optimality system using the formal Lagrange approach and derive the adjoint PDE system that provides the needed gradient of the cost functional. By means of an example setup of a container with an in-and outflow of water and a cooling unit at the bottom, we illustrate how the derived formulations can be used to achieve a desired interface between the solid and the fluid phase by controlling the flow at the inlet. © 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim.

关键词： Crystal growth

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An artificial neural network for surrogate modeling of stress fields in viscoplastic polycrystalline materials

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npj computational Materials 2023年第1期9卷 1981-1990页

作者： Mohammad S.Khorrami Jaber R.Mianroodi Nima H.Siboni Pawan Goyal Bob Svendsen Peter Benner Dierk Raabe Microstructure Physics and Alloy Design Max-Planck-Institut für EisenforschungDüsseldorfGermany Ergodic Labs Lohmühlenstraße 6512435 BerlinGermany Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical SystemsMagdeburgGermany Material Mechanics RWTH Aachen UniversityAachenGermany

The purpose of this work is the development of a trained artificial neural network for surrogate modeling of the mechanical response of elasto-viscoplastic grain *** this end,a U-Net-based convolutional neural network(CNN)is trained using results for the von Mises stress field from the numerical solution of initial-boundary-value problems(IBVPs)for mechanical equilibrium in such microstructures subject to quasi-static uniaxial *** resulting trained CNN(tCNN)accurately reproduces the von Mises stress field about 500 times faster than numerical solutions of the corresponding IBVP based on spectral *** of the tCNN to test cases based on microstructure morphologies and boundary conditions not contained in the training dataset is also investigated and discussed.

关键词： methods boundary faster

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Solving differential matrix equations using parareal

Solving differential matrix equations using parareal

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Joint 87th Annual Meeting on the International Association of Applied Mathematics and Mechanics, GAMM 2016 and Deutsche Mathematiker‐Vereinigung, DMV 2016

作者： Köhler, Martin Lang, Norman Saak, Jens Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg Computational Methods in Systems and Control Theory Germany Technische Universität Chemnitz Faculty of Mathematics Germany

Differential matrix equations appear in many applications like optimal control of partial differential equations, balanced truncation model order reduction of linear time-varying systems and many more. Here, we will focus on differential Riccati equations (DRE). Solving such matrix-valued ordinary differential equations (ODE) is a highly time consuming process. We present a Parareal based algorithm applied to Rosenbrock methods for the solution of the matrix-valued differential Riccati equations. Considering problems of moderate size, direct matrix equation solvers for the solution of the algebraic Lyapunov equations arising inside the time intgration methods are used. © 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim.

关键词： Ordinary differential equations

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AFast Solver for an H_(1) Regularized PDE-Constrained Optimization Problem

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Communications in computational Physics 2016年第1期19卷 143-167页

作者： Andrew T.Barker Tyrone Rees Martin Stoll Center for Applied Scientific Computing Lawrence Livermore National LaboratoryMail Stop L-561LivermoreCA 94551USA Numerical Analysis Group Scientific Computing DepartmentRutherford Appleton LaboratoryChiltonDidcotOxfordshireOX110QXUnited Kingdom Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical SystemsSandtorstr.139106MagdeburgGermany

In this paperwe consider PDE-constrained optimization problemswhich incorporate an H_(1)regularization control *** focus on a time-dependent PDE,and consider both distributed and boundary *** problems we consider include bound constraints on the state,and we use a Moreau-Yosida penalty function to handle *** propose Krylov solvers and Schur complement preconditioning strategies for the different problems and illustrate their performance with numerical examples.

关键词： Preconditioning Krylov methods PDE-constrained optimization optimal control of PDEs

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