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Port-Hamiltonian formulations of the incompressible Euler equations with a free surface

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arXiv 2023年

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 Hefei230026 China Department of Applied Mathematics Mathematics of Computational Science Group University of Twente P.O. Box 217 Enschede7500 AE Netherlands Department of Applied Mathematics Mathematics of Systems Theory Group University of Twente P.O. Box 217 Enschede7500 AE Netherlands Department of Mechanical Engineering Dynamics and Control Group Eindhoven University of Technology P.O. Box 513 Eindhoven5600 MB Netherlands

In this paper, we present port-Hamiltonian formulations of the incompressible Euler equations with a free surface governed by surface tension and gravity forces, modelling e.g. capillary and gravity waves and the evolution of droplets in air. Three sets of variables are considered, namely (v, Σ), (η, ∂, Σ) and (ω, ∂, Σ), with v the velocity, η the solenoidal velocity, ∂ a potential, ω the vorticity, and Σ the free surface, resulting in the incompressible Euler equations in primitive variables and the vorticity equation. First, the Hamiltonian formulation for the incompressible Euler equations in a domain with a free surface combined with a fixed boundary surface with a homogeneous boundary condition will be derived in the proper Sobolev spaces of differential forms. Next, these results will be extended to port-Hamiltonian formulations allowing inhomogeneous boundary conditions and a non-zero energy flow through the boundaries. Our main results are the construction and proof of Dirac structures in suitable Sobolev spaces of differential forms for each variable set, which provides the core of any port-Hamiltonian formulation. Finally, it is proven that the state dependent Dirac structures are related to Poisson brackets that are linear, skew-symmetric and satisfy the Jacobi *** Codes 35Q35, 35Q31 Copyright © 2023, The Authors. All rights reserved.

关键词： Vorticity

Inf-Sup-Constant-Free State Error Estimator for Model Order Reduction of Parametric systems in Electromagnetics

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arXiv 2021年

作者： Chellappa, Sridhar Feng, Lihong de la Rubia, Valentín Benner, Peter Department of Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg39106 Germany Departamento de Matemática Aplicada a las TIC ETSI de Telecomunicación Universidad Politécnica de Madrid Madrid28040 Spain

A reliable model order reduction process for parametric analysis in electromagnetics is detailed. Special emphasis is placed on certifying the accuracy of the reduced-order model. For this purpose, a sharp state error estimator is proposed. Standard a posteriori state error estimation for model order reduction relies on the inf-sup constant. For parametric systems, the inf-sup constant is parameter-dependent. The a posteriori error estimation for systems with very small or vanishing inf-sup constant poses a challenge, since it is inversely proportional to the inf-sup constant, resulting in overly pessimistic error estimation especially at and around resonance frequencies. Such systems appear in electromagnetics since the inf-sup constant values are close to zero at points close to resonant frequencies, where they eventually vanish. We propose a novel a posteriori state error estimator which avoids the calculation of the inf-sup constant. The proposed state error estimator is compared with the standard error estimator and a recently proposed one in the literature. It is shown that our proposed error estimator outperforms both existing estimators. Numerical experiments are performed on real-life microwave devices such as narrowband and wideband antennas, two types of dielectric resonator filters as well as a dual-mode waveguide filter. These examples show the capabilities and efficiency of the proposed methodology. © 2021, CC BY.

关键词： Galerkin methods

Fast a posteriori state error estimation for reliable frequency sweeping in microwave circuits via the reduced-basis method

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arXiv 2021年

作者： de la Rubia, Valentín Chellappa, Sridhar Feng, Lihong Benner, Peter Departamento de Matemática Aplicada a las TIC ETSI de Telecomunicación Universidad Politécnica de Madrid Madrid28040 Spain Department of Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg39106 Germany

We develop a compact, reliable model order reduction approach for fast frequency sweeps in microwave circuits by means of the reduced-basis method. Contrary to what has been previously done, special emphasis is placed on certifying the accuracy of the reduced-order model with respect to the original full-order model in an effective and efficient way. Previous works on model order reduction accuracy certification rely on costly a posteriori error estimators, which typically require expensive inf-sup constant evaluations of the underlying full-order model. This scenario is often too time-consuming and unaffordable in electromagnetic applications. As a result, less expensive and heuristic error estimators are commonly used instead. Very often, one is interested in knowing about the full state vector, instead of just some output quantities derived from the full state. Therefore, error estimators for the full state vector become relevant. In this work, we detail the frequency behavior of both the electric field and the state error when an approximation to the electric field solution is carried out. Both field quantities share the same frequency behavior. Based on this observation, we focus on the efficient estimation of the electric field state error and propose a fast evaluation of the reduced-order model state error in the frequency band of analysis, minimizing the number of full-order model evaluations. This methodology is of paramount importance to carry out a reliable fast frequency sweep in microwave circuits. Finally, real-life applications will illustrate the capabilities and efficiency of the proposed approach. © 2021, CC BY.

关键词： Computer aided design

Topical Issue Scientific Machine Learning (1/2)

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GAMM Mitteilungen 2021年第1期44卷 e202100010-e202100010页

作者： Benner, Peter Klawonn, Axel Stoll, Martin Computational Methods in Systems and Control Theory Max Planck institute for Dynamics of Complex Technical Systems Magdeburg Germany Fakultät für Mathematik Otto-von-Guericke Universität Magdeburg Germany Department of Mathematics and Computer Science University of Cologne Germany Center for Data and Simulation Science University of Cologne Germany Chair of Scientific Computing Department of Mathematics TU Chemnitz Germany

Parallel algorithms for tensor train arithmetic

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arXiv 2020年

作者： Al Daas, Hussam Ballard, Grey Benner, Peter Department of Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg Germany Computer Science Department Wake Forest University Winston SalemNC United States

We present efficient and scalable parallel algorithms for performing mathematical operations for low-rank tensors represented in the tensor train (TT) format. We consider algorithms for addition, elementwise multiplication, computing norms and inner products, orthonormalization, and rounding (rank truncation). These are the kernel operations for applications such as iterative Krylov solvers that exploit the TT structure. The parallel algorithms are designed for distributed- memory computation, and we propose a data distribution and strategy that parallelizes computations for individual cores within the TT format. We analyze the computation and communication costs of the proposed algorithms to show their scalability, and we present numerical experiments that demonstrate their efficiency on both shared-memory and distributed-memory parallel systems. For example, we observe better single-core performance than the existing MATLAB TT-Toolbox in rounding a 2GB TT tensor, and our implementation achieves a 34× speedup using all 40 cores of a single node. We also show nearly linear parallel scaling on larger TT tensors up to over 10,000 cores for all mathematical operations. Copyright © 2020, The Authors. All rights reserved.

关键词： Tensors

Locomotion Optimization and Manipulation Planning of a Tetrahedron-Based Mobile Mechanism with Binary control

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Chinese Journal of Mechanical Engineering 2018年第1期31卷 78-99页

作者： Ran Liu Yan-An Yao Wan Ding Xiao-Ping Liu School of Mechanical Electronic and Control EngineeringBeijing Jiaotong University Department of Mechanism Theory and Dynamics of Machines RWTH Aachen University Department of Systems and Computer Engineering Carleton University

Locomotion and manipulation optimization is essential for the performance of tetrahedron-based mobile mechanism. Most of current optimization methods are constrained to the continuous actuated system with limited degree of freedom(DOF), which is infeasible to the optimization of binary control multi-DOF system. A novel optimization method using for the locomotion and manipulation of an 18 DOFs tetrahedron-based mechanism called 5-TET is proposed. The optimization objective is to realize the required locomotion by executing the least number of *** control strategy is adopted, and forward kinematic and tipping dynamic analyses are performed, *** on a developed genetic algorithm(GA), the optimal number of alternative struts between two adjacent steps is obtained as 5. Finally, a potential manipulation function is proposed, and the energy consumption comparison between optimal 5-TET and the traditional wheeled robot is carried out. The presented locomotion optimization and manipulation planning enrich the research of tetrahedron-based mechanisms and provide the instruction to the successive locomotion and operation planning of multi-DOF mechanisms.

关键词： Tetrahedron-based mobile mechanism Binary control GA Locomotion optimization Manipulation planning

Is there a Floquet Lindbladian?

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Physical Review B 2020年第10期101卷 100301(R)-100301(R)页

作者： Alexander Schnell André Eckardt Sergey Denisov Max-Planck-Institut für Physik komplexer Systeme Nöthnitzer Straße 38 01187 Dresden Germany Center for Theoretical Physics of Complex Systems IBS Daejeon 305-732 Korea Department of Computer Science OsloMet–Oslo Metropolitan University 0130 Oslo Norway Department of Control Theory and Systems Dynamics Lobachevsky University Gagarina Avenue 23 Nizhny Novgorod 603950 Russia

The stroboscopic evolution of a time-periodically driven isolated quantum system can always be described by an effective time-independent Hamiltonian. Whether this concept can be generalized to open Floquet systems, described by a Markovian Lindblad master equation with a time-periodic generator (Lindbladian), remains an open question. By using a two-level system as a model, we explicitly show the existence of two well-defined parameter regions. In one region the stroboscopic evolution can be described by a Markovian master equation with a time-independent Floquet Lindbladian. In the other it cannot; but here the one-cycle evolution operator can be reproduced with an effective non-Markovian master equation that is homogeneous but nonlocal in time. Interestingly, we find that the boundary between the phases depends on when during the period the evolution is stroboscopically monitored. This reveals the nontrivial role played by the micromotion in the dynamics of open Floquet systems.

关键词： Nonequilibrium statistical mechanics Open quantum systems & decoherence Quantum control Quantum statistical mechanics Quantum theory Floquet systems Master equation

Efficient structure-preserving support tensor train machine

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arXiv 2020年

作者： Kour, Kirandeep Benner, Peter Dolgov, Sergey Stoll, Martin Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems MagdeburgD-39106 Germany Department of Mathematical Sciences University of Bath BathBA2 7AY United Kingdom Faculty of Mathematics Technische Universität Chemnitz ChemnitzD-09107 Germany

An increasing amount of collected data are high-dimensional multi-way arrays (tensors), and it is crucial for efficient learning algorithms to exploit this tensorial structure as much as possible. The ever present curse of dimensionality for high dimensional data and the loss of structure when vectorizing the data motivates the use of tailored low-rank tensor classification methods. In the presence of small amounts of training data, kernel methods offer an attractive choice as they provide the possibility for a nonlinear decision boundary. We develop the Tensor Train Multi-way Multi-level Kernel (TT-MMK), which combines the simplicity of the Canonical Polyadic decomposition, the classification power of the Dual Structure-preserving Support Vector Machine, and the reliability of the Tensor Train (TT) approximation. We show by experiments that the TT-MMK method is usually more reliable computationally, less sensitive to tuning parameters, and gives higher prediction accuracy in the SVM classification when benchmarked against other state-of-the-art techniques. Copyright © 2020, The Authors. All rights reserved.

关键词： Tensors

Exact Optimization: Part I

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arXiv 2019年

作者： Lin, Li-Gang Liang, Yew-Wen The Department of Mechanical Engineering National Central University Taoyuan32001 Taiwan The Department of Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg39106 Germany The Institute of Electrical Control Engineering National Yang Ming Chiao Tung University Hsinchu30010 Taiwan

Nonlinear programming is explicitly analyzed via a novel perspective/method and from a bottom-up manner. The philosophy is based on the recent findings on convex quadratic equation (CQE), which help clarify a geometric interpretation that relates CQE to convex quadratic function (CQF). More specifically, regarding the solvability of CQE, its necessary and sufficient condition as well as a unified parameterization of all the solutions has recently been analytically formulated. Moving forward, the understanding of CQE is utilized to describe the geometric structure of CQF, and the CQE-CQF relation. All these results are shown closely related to a basis in the optimization literature, namely quadratic programming (QP). For the first time from this viewpoint, the QPs subject to equality, inequality, equality-and-inequality, and extended constraints can be algebraically solved in derivative-free closed formulae, respectively. All the results are derived without knowing a feasible point, a priori and any time during the *** Codes Primary: 15A18, 52A41, 90C20, 90C25, and 90C46 Copyright © 2019, The Authors. All rights reserved.

关键词： Matrix algebra