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A formal process of hierarchical functional requirements development for Set-Based Design

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arXiv 2022年

作者： Sun, Minghui Chen, Zhaoyang Bakirtzis, Georgios Jafarzadeh, Hassan Fleming, Cody Coordinated Systems Lab Department of Mechanical Engineering Iowa State University AmesIA50011 United States Autonomous Systems Group Oden Institute for Computational Engineering & Sciences The University of Texas at Austin AustinTX78712 United States Advanced Control Methods Research and Advanced Engineering Ford Motor Company DearbornMI48126 United States

The design of complex systems is typically uncertain and ambiguous at early stages. Set-Based Design is a promising approach to complex systems design as it supports alternative exploration and gradual uncertainty reduction. When designing a complex system, functional requirements decomposition is a common and effective approach to progress the design incrementally. However, the current literature on Set-Based Design lacks formal guidance in functional requirements decomposition. To bridge the gap, we propose a formal process to hierarchically decompose the functional requirements for Set-Based Design. A four-step formal process is proposed to systematically define, reason, and narrow the sets, and eventually decompose the functional requirement into the sub-requirements. Such a process can be used by the individual suppliers working in parallel at multiple levels of abstraction and guarantee that the resulting system will eventually satisfy the top-level functional requirements. An example of designing a cruise control system is applied to demonstrate the feasibility of the proposed process. © 2022, CC BY.

关键词： Large scale systems

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

Comparing (Empirical-Gramian-Based) Model Order Reduction Algorithms

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arXiv 2020年

作者： Himpe, Christian Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 Magdeburg39106 Germany

In this work, the empirical-Gramian-based model reduction methods: Empirical poor man’s truncated balanced realization, empirical approximate balancing, empirical dominant subspaces, empirical balanced truncation, and empirical balanced gains are compared in a non-parametric and two parametric variants, via ten error measures: Approximate Lebesgue L0, L1, L2, L∞, Hardy H2, H∞, Hankel, Hilbert-Schmidt-Hankel, modified induced primal, and modified induced dual norms, for variants of the thermal block model reduction benchmark. This comparison is conducted via a new meta-measure for model reducibility called MORscore. inMSC Codes 93A15, 93B11, 65Y20 Copyright © 2020, The Authors. All rights reserved.

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GR decompositions and their relations to Cholesky-like factorizations

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arXiv 2020年

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

For a given matrix, we are interested in computing GR decompositions A = GR, where G is an isometry with respect to given scalar products. The orthogonal QR decomposition is the representative for the Euclidian scalar product. For a signature matrix, a respective factorization is given as the hyperbolic QR decomposition. Considering a skew-symmetric matrix leads to the symplectic QR decomposition. The standard approach for computing GR decompositions is based on the successive elimination of subdiagonal matrix entries. For the hyperbolic and symplectic case, this approach does in general not lead to a satisfying numerical accuracy. An alternative approach computes the QR decomposition via a Cholesky factorization, but also has bad stability. It is improved by repeating the procedure a second time. In the same way, the hyperbolic and the symplectic QR decomposition are related to the LDLT and a skew-symmetric Cholesky-like factorization. We show that methods exploiting this connection can provide better numerical stability than elimination-based *** Codes 65F99 Copyright © 2020, The Authors. All rights reserved.

关键词： Factorization

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

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

Non-intrusive Time-POD for Optimal control of a Fixed-Bed Reactor for CO 2 Methanation

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IFAC-PapersOnLine 2021年第3期54卷 122-127页

作者： Jens Bremer Jan Heiland Peter Benner Kai Sundmacher Max Planck Institute Magdeburg Dpt. Process Systems Engineering Sandtorstraße 2 39106 Magdeburg Germany and Otto von Guericke University Magdeburg Chair for Process Systems Engineering Universitätsplatz 2 39106 Magdeburg (Sundmacher) Max Planck Institute Magdeburg Dpt. Computational Methods in Systems and Control Theory and Otto von Guericke University Magdeburg Fakultät für Mathematik

The optimization of a controlled process in a simulation without access to the model itself is a common scenario and very relevant to many chemical engineering applications. A general approach is to apply a black-box optimization algorithm to a parameterized control scheme. The success then depends on the quality of the parametrization that should be low-dimensional though rich enough to express the salient features. This work proposes using solution snapshots to extract dominant modes of the temporal dynamics of a process and use them for low-dimensional parametrizations of control functions. The approach is called proper orthogonal decomposition in time (time-POD). We provide theoretical reasoning and illustrate the performance for the optimal control of a methanation reactor.

关键词： Optimal control Model order reduction Proper orthogonal decomposition Fixed-bed reactor Carbon dioxide methanation Nonlinear programming Black-box model

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

Model order reduction via substructuring for a nonlinear, differential-algebraic machine tool model with moving loads

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PAMM 2023年第1期23卷

作者： Quirin Aumann Peter Benner Jens Saak Julia Vettermann Mathematics in Industry and Technology Faculty of Mathematics Technische Universität Chemnitz Reichenhainer Str. 41 09126 Chemnitz Germany Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 39106 Magdeburg Germany

Machine tools are permanently exposed to complex static, dynamic and thermic loads. This often results in an undesired displacement of the tool center point (TCP), causing errors in the production process and thus limiting the achievable workpiece quality. Recent research efforts try to counter these effects by process-parallel solutions utilizing machine internal data. Therefore, fast simulation models are a fundamental requirement. Here, model order reduction (MOR) becomes crucial. The resulting low-dimensional models are required for various applications, e.g., in digital twins, the correction of thermally induced errors at the TCP during the production process as well as for lifetime calculations in predictive maintenance. In this contribution, we present a strategy of MOR for a coupled thermo-mechanical model with a nonlinear subsystem and moving loads, using the example of a feed axis with nonlinear machine components. Applying tailored substructuring techniques, we are able to separate the linear and nonlinear system components. This allows to apply classic linear MOR methods to the much larger linear part, whereas the small nonlinear part is kept, and thus enables drastically reduced computing times. The relative movement is considered by a switched system approach. Transient thermo-mechanical interactions of the feed axis are calculated in a final investigation, comparing the performance of the resulting reduced-order model and the original one.

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