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A solution framework for linear PDE-constrained mixed-integer problems

MATHEMATICAL PROGRAMMING 2021年第2期188卷 695-728页

作者： Gnegel, Fabian Fuegenschuh, Armin Hagel, Michael Leyffer, Sven Stiemer, Marcus Brandenburg Univ Technol Cottbus Senftenberg Pl Deutsch Einheit 1 D-03046 Cottbus Germany Helmut Schmidt Univ Univ Fed Armed Forces Hamburg Holstenhofweg 83 D-22043 Hamburg Germany Argonne Natl Lab Math & Comp Sci Div 9700 Cass Ave Lemont IL 60439 USA

We present a general numerical solution method for control problems with state variables defined by a linear PDE over a finite set of binary or continuous control variables. We show empirically that a naive approach that applies a numerical discretization scheme to the PDEs to derive constraints for a mixed-integer linear program (MILP) leads to systems that are too large to be solved with state-of-the-art solvers for MILPs, especially if we desire an accurate approximation of the state variables. Our framework comprises two techniques to mitigate the rise of computation times with increasing discretization level: First, the linear system is solved for a basis of the control space in a preprocessing step. Second, certain constraints are just imposed on demand via the IBM ILOG CPLEX feature of a lazy constraint callback. These techniques are compared with an approach where the relations obtained by the discretization of the continuous constraints are directly included in the MILP. We demonstrate our approach on two examples: modeling of the spread of wildfire and the mitigation of water contamination. In both examples the computational results demonstrate that the solution time is significantly reduced by our methods. In particular, the dependence of the computation time on the size of the spatial discretization of the PDE is significantly reduced.

关键词： Mixed-integer linear programming Partial differential equations finite-difference methods finite-element methods Convection-diffusion equation Global optimal control

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Modeling, simulation, and verification of an advanced micromirror using SUGAR

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JOURNAL OF MICROELECTROMECHANICAL SYSTEMS 2007年第6期16卷 1524-1536页

作者： Clark, Jason Vaughn Pister, Kristofer S. J. Univ Calif Berkeley Berkeley Sensor & Actuator Ctr Berkeley CA 94720 USA Univ Calif Berkeley Appl Sci & Technol Grad Grp Berkeley CA 94720 USA Univ Calif Berkeley Dept Elect Engn & Comp Sci Berkeley CA 94720 USA

We present some of the design, modeling, and simulation features of a computer-aided engineering tool for microelectromechanical systems (MEMS) called SUGAR. The features include a flexible SPICE-like netlist language for MEMS design, a simple modeling framework for computationally efficient lumped models, an extensible architecture to which users can add features, and the ability to display 3-D circuits together with deflected electromechanical structures. Since SUGAR is programmed in MATLAB, many MATLAB functions and toolboxes may be used with SUGAR. Such attributes facilitate the exploration of design spaces and feature modifications. In this paper, we describe SUGAR's extensible architecture, flexible design methodology, modeling framework, and reduced-order modeling technique. We do not present the many other advances made for SUGAR by other developers. For a test case, we choose an advanced microdevice that is difficult to simulate with conventional MEMS software. We show that the relative errors of our lumped models are less than 3% of the finite-element analysis (FEA), that the computational costs are less than 1% of the FEA, and that simulation of the test case fairly agrees with the experiment.

关键词： design methodology finite-element methods modeling simulation software SPICE system analysis and design

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A Convolution-Free Mixed finite-element Time-Domain Method for General Nonlinear Dispersive Media

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION 2019年第1期67卷 324-334页

作者： Abraham, David S. Giannacopoulos, Dennis D. McGill Univ Dept Elect & Comp Engn Montreal PQ H3A 0E9 Canada

In this paper, a mixed finite-element time-domain (FETD) method is presented for the simulation of electrically complex materials, including general combinations of linear dispersion, instantaneous nonlinearity, and dispersive nonlinearity. Using both edge and face elements, the presented method offers greater geometric flexibility than existing finite-difference time-domain (FDTD) implementations, and in contrast to existing nonlinear FETD methods, also incorporates both linear and nonlinear material dispersions. Dielectric nonlinearity is incorporated into the Crank-Nicolson mixed FETD formulation via a straightforward Newton-Raphson approach, for which the associated Jacobian is derived. Moreover, the dispersion is modeled via the Mobius z transform method, yielding a simpler more general algorithm. The method's accuracy and convergence are verified, and its capability demonstrated via the simulation of several nonlinear phenomena, including temporal and spatial solitons in two spatial dimensions.

关键词： Dispersive media finite-element methods nonlinear media time-domain analysis

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Characteristics of an electrodynamic wheel using a 2-D steady-state model

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IEEE TRANSACTIONS ON MAGNETICS 2007年第8期43卷 3395-3405页

作者： Bird, Jonathan Lipo, Thomas A. Univ Wisconsin Madison WI 53706 USA

The mechanical rotation of a radially positioned permanent-magnet Halbach array above a conducting, nonmagnetic track induces eddy currents in the track that can inductively create suspension and propulsion forces simultaneously. The parameters that affect the performance of this electrodynamic wheel are studied using a 2-D steady-state finite-element method. Tradeoffs between the lift and thrust force performance are investigated and methods to improve the thrust efficiency are proposed.

关键词： eddy currents electromagnetic analysis finite-element methods Halbach array maglev

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A fast vector-potential method using tangentially continuous vector finite elements

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES 1998年第6期46卷 863-868页

作者： Dyczij-Edlinger, R Peng, GH Lee, JF Ansoft Corp Pittsburgh PA 15219 USA Motorola CCRL Schaumburg IL 60195 USA Worcester Polytech Inst Dept Elect & Comp Engn Worcester MA 01609 USA

An efficient finite-element method for driven time-harmonic wave-propagation problems is proposed. The special properties of tangentially continuous vector finite elements (TVFEM's) are utilized to formulate an ungauged vector-potential scheme in terms of the field method plus one very sparse "gradient matrix" with two nonzero integer or pointer entries per row. The suggested formalism is intended for use with iterative solvers. It combines the simplicity and modest memory requirements of the field formulation with the superior numerical convergence of the ungauged vector-potential scheme.

关键词： electromagnetic analysis finite-element methods iterative methods numerical analysis

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Time-Harmonic Induction-Machine Model Including Hysteresis and Eddy Currents in Steel Laminations

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IEEE TRANSACTIONS ON MAGNETICS 2009年第7期45卷 2981-2989页

作者： Pippuri, Jenni Arkkio, Antero Helsinki Univ Technol Dept Elect Engn FI-02015 Espoo Finland

We have studied electromagnetic losses of a frequency-converter-fed cage-induction motor by using a numerical machine model that includes eddy-current and hysteresis phenomena in electrical steel sheets. We used the model to solve the two-dimensional (2-D) time-harmonic field and winding equations of a cage-induction machine, utilizing a finite-element method and phasor variables. We used complex reluctivity to couple the hysteresis and eddy currents in the sheets with the 2-D analysis. The model modifies the absolute value of the reluctivity according to a one-dimensional (I-D) eddy-current solution developed in the lamination thickness. To define the argument of the reluctivity, we applied both the 1-D field solution and measured hysteresis data. We compared computations of additional electromagnetic losses in a 37-kW test machine due to the higher harmonics of a frequency-converter supply with experimental results. The agreement is found to be reasonable.

关键词： Eddy currents finite-element methods induction motors losses

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finite element Modeling of the Nano-scale Adhesive Contact and the Geometry-based Pull-off Force

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TRIBOLOGY LETTERS 2011年第1期41卷 65-72页

作者： Zhang, Xiangjun Zhang, Xiaohao Wen, Shizhu Tsinghua Univ State Key Lab Tribol Beijing 100084 Peoples R China

Nano-scale adhesive contact mediated by intermolecular van der Waals forces has become a typical fundamental problem in many areas. Interpretation and control of the strength and efficiency of the nano-scale adhesive contacts require a proper modeling considering the actual interfacial forces, the varying contact area, and clearance. In this article, the finite-element (FE) method is developed to model the nano-scale adhesive contact of elastic bodies with an adhesive pressure derived from the interatomic interaction Lennard-Jones potential, which permits numerical solutions for a variety of interface geometries. Compared with the analytical results from conventional Hertz, JKR, and DMT models, the validity of the FE model is verified. For nano-scale contact, the assumption of equivalent radius adopted in the Hertz model is initially investigated and proved to be improper for nano-scale adhesive contact due to the distribution variations of interfacial force caused by local contact geometry. Then adhesive contact behaviors of four typical nano-scale contact geometries inspired by tip shapes of bio-adhesive pads are investigated in detail, which are flat punch tip, sphere tip, mushroom tip, and empty cup tip. The simulation results indicate that the nano-scale tip geometry plays a dominant role on the pull-off strength. Within the investigated geometries, cup tip results in a highest adhesion efficiency followed by flat punch tip, sphere tip, and mushroom tip, respectively, which are highly geometry dependent and verified by former experimental results. The dominant effect is found coming from the contact area ratio of the adhesive area to the sticking area or the whole contact area. The FE modeling can serve a useful purpose in revealing the nano-scale geometry-based adhesion contact for surface topography design in MEMS to avoid stiction failure and for the artificial sticky feet in bionics to increase adhesion strength.

关键词： Adhesion contact mechanics finite-element methods Contact geometry Van der Waals forces

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Mass transport in electrochemical nanogap sensors

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ELECTROCHIMICA ACTA 2013年 112卷 943-949页

作者： Mathwig, Klaus Lemay, Serge G. Univ Twente MESA Inst Nanotechnol NL-7500 AE Enschede Netherlands

Nanofluidic thin-layer cells based on redox cycling allow for extremely sensitive electrochemical detection. Here we establish a physical mass-transfer model for analyte molecules in these transducers which takes into account advective and diffusive transport of both oxidized and reduced species as well as reversible dynamic adsorption at the sensor surfaces. We use finite-element modeling to determine the transient response of nanogap sensors;numerically we predict that the response time can be reduced substantially by pressure-driven advection while the faradaic limiting current remains unaffected by this flow for all experimentally accessible flow rates. (C) 2013 Elsevier Ltd. All rights reserved.

关键词： Electrochemical sensor Mass transport finite-element methods Redox cycling Nanofluidics

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A three-dimensional spherical mesh generator

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GEOPHYSICAL JOURNAL INTERNATIONAL 1997年第1期130卷 193-200页

作者： Everett, ME Department of Geology and Geophysics Texas A&M University College Station TX 77843 USA

A new spherical mesh generator is described. It represents an efficient, deterministic packing of tetrahedra into a solid sphere, a spherical shell, or both. The mesh can be used for finite-element solutions to a wide variety of global numerical modelling problems in the geosciences. The nodes within the mesh are distributed uniformly, and long, thin tetrahedra are avoided. The method proposed here offers several advantages over 3-D Delaunay algorithms for finite-element mesh generation. For the related problem of trivariate scattered data interpolation, which is not considered here, the 3-D Delaunay algorithms are the method of choice.

关键词： finite-element methods

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Understanding the treatment of waves in atmospheric models. Part 1: The shortest resolved waves of the 1D linearized shallow-water equations

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QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY 2014年第682期140卷 1426-1440页

作者： Ullrich, P. A. Univ Calif Davis Dept Land Air & Water Resources Davis CA 95616 USA

This article provides an intercomparison of the dispersive and diffusive properties of several standard numerical methods applied to the 1D linearized shallow-water equations without the Coriolis term, including upwind and central finite-volume, spectral finite-volume, discontinuous Galerkin, spectral element, and staggered finite-volume. All methods are studied up to tenth-order accuracy, where possible. A consistent framework is developed which allows for direct intercomparison of the ability of these methods to capture the behaviour of linear gravity waves. The Courant-Friedrichs-Lewy (CFL) condition is also computed, which is important for gauging the stability of these methods, and leads to a measure of approximate equal error cost. The goal of this work is threefold: first, to determine the shortest wavelength which can be considered `resolved' for a particular method;second, to determine the effect of increasing the order of accuracy on the ability of a method to capture wave-like motion;and third, to determine which numerical methods offer the best treatment of wave-like motion.

关键词： high-order dispersion analysis finite-volume methods finite-element methods linear gravity waves dynamical core

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