In this paper we consider (hierarchical, La-grange)reduced basis approximation anda posteriori error estimation for linear functional outputs of affinely parametrized elliptic coercive partial differential equa-tions. The essential ingredients are (primal-dual)Galer-kin projection onto a low-dimensional space associated with a smooth “parametric manifold” - dimension re-duction; efficient and effective greedy sampling meth-ods for identification of optimal and numerically stable approximations - rapid convergence;a posteriori er-ror estimation procedures - rigorous and sharp bounds for the linear-functional outputs of interest; and Offine-Online computational decomposition strategies - min-imummarginal cost for high performance in the real-time/embedded (e.g., parameter-estimation, control)and many-query (e.g., design optimization, multi-model/ scale)contexts. We present illustrative results for heat conduction and convection-diffusion,inviscid flow, and linear elasticity; outputs include transport rates, added mass,and stress intensity factors.

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Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations

The text is focused on formulation, analysis, and computational procedures for reduced basis approximation and a posteriori error estimation for parametrized PDEs in the real-time and many-query contexts. The book also serves as background and (certain specific chapters) documentation for the companion rbMIT © MIT Software package. The book is intended to serve both "Developers" — numerical analysts and computational tool-builders — who wish to further develop the methodology, and "Users" — computational engineers and educators — who wish to apply the methodology to new applications. ("End Users" interested in the Worked Problems will not need the material in the book.) The book consists of General Preliminaries followed by nine Parts: each successive Part addresses a new class of problems in each case introduced by an appropriate abstraction and particular examples. We begin with parametrically coercive compliant problems and then proceed to general coercive elliptic problems (such as conduction, forced convection, and elasticity) and non-coercive elliptic problems (such as frequency-domain acoustics and elastodynamics); we next discuss parabolic problems (such as transient convection-diffusion); we then consider non-affine (in parameter) problems, and finally certain nonlinear problems (such as the incompressible Navier-Stokes equations).

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Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations

The paper presents an accurate analytical subdomain model for computation of the open-circuit magnetic field in surface-mounted permanent-magnet machines with any pole and slot combinations, including fractional slot machines, accounting for stator slotting effect. It is derived by solving the field governing equations in each simple and regular subdomain, i.e., magnet, air-gap and stator slots, and applying the boundary conditions to the interfaces between these subdomains. The model accurately accounts for the influence of interaction between slots, radial/parallel magnetization, internal/external rotor topologies, relative recoil permeability of magnets, and odd/even periodic boundary conditions. The back-electromotive force, electromagnetic torque, cogging torque, and unbalanced magnetic force are obtained based on the field model. The relationship between this accurate subdomain model and the conventional subdomain model, which is based on the simplified one slot per pole machine model, is also discussed. The investigation shows that the proposed accurate subdomain model has better accuracy than the subdomain model based on one slot/pole machine model. The finite element and experimental results validate the analytical prediction.

An Accurate Subdomain Model for Magnetic Field Computation in Slotted Surface-Mounted Permanent-Magnet Machines

A modelforalarge network of"neurons" withagraded response (orsigmoid input-output relation) is studied. Thisdeterministic system hascollective properties in veryclose correspondence withtheearlier stochastic model based onMcCulloch-Pitts neurons. Thecontent-addressable memoryandother emergent collective properties oftheorigi- nalmodelalso arepresent inthegraded response model. The idea that suchcollective properties areusedinbiological sys- temsisgiven addedcredence bythecontinued presence ofsuch properties formorenearly biological "neurons." Collective analog electrical circuits ofthekinddescribed will certainly function. Thecollective states ofthetwomodels haveasimple correspondence. Theoriginal modelwill continue tobeuseful forsimulations, because its connection tograded response sys- temsisestablished. Equations that include theeffect ofaction potentials inthegraded response system arealsodeveloped. Recent papers (1-3) haveexplored theability ofasystem of highly interconnected "neurons" tohaveuseful collective computational properties. Theseproperties emergesponta- neously inasystem having alarge numberofelementary "neurons." Content-addressable memory(CAM)isoneof thesimplest collective properties ofsucha system. The mathematical modeling hasbeenbasedon"neurons" that aredifferent bothfromreal biological neurons andfromthe realistic functioning ofsimple electronic circuits. Someof these differences aremajor enough thatneurobiologists and circuit engineers alike havequestioned whether real neural orelectrical circuits wouldactually exhibit thekindofbe- haviors foundinthemodelsystem evenifthe"neurons" wereconnected inthefashion envisioned. Twomajor divergences between themodelandbiological orphysical systems stand out.Realneurons (andreal physi- caldevices suchasoperational amplifiers that might mimic them) havecontinuous input-output relations. (Action po- tentials areomitted until Discussion.) Theoriginal modeling usedtwo-state McCulloch-Pitts (4)threshold devices having outputs of0or1only. Realneurons andreal physical circuits haveintegrative timedelays duetocapacitance, andthetime evolution ofthestate ofsuchsystems should berepresented byadifferential equation (perhaps withaddednoise). The original modeling usedastochastic algorithm involving sud- den0-1or1-0changes ofstates ofneurons atrandom times. Thispaper showsthat theimportant properties oftheorigi- nalmodelremain intact whenthese twosimplifications of themodeling areeliminated. Although itisuncertain wheth- ertheproperties ofthese newcontinuous "neurons" areyet close enough totheessential properties ofrealneurons (and/or their dendritic arborization) tobedirectly applicable toneurobiology, amajor conceptual obstacle hasbeenelimi- nated. Itiscertain that aCAM constructed onthebasic ideas

Neurons withgraded response havecollective computational properties likethoseoftwo-state neurons

This paper systematically covers the significant developments of the last decade, including surrogate modeling of electrical machines and direct and stochastic search algorithms for both single- and multi-objective design optimization problems. The specific challenges and the dedicated algorithms for electric machine design are discussed, followed by benchmark studies comparing response surface (RS) and differential evolution (DE) algorithms on a permanent-magnet-synchronous-motor design with five independent variables and a strong nonlinear multiobjective Pareto front and on a function with eleven independent variables. The results show that RS and DE are comparable when the optimization employs only a small number of candidate designs and DE performs better when more candidates are considered.

A Review of Recent Developments in Electrical Machine Design Optimization Methods With a Permanent-Magnet Synchronous Motor Benchmark Study

Finite element method (FEM) reveals to be one of the most valuable tools in electric machines design as it accurately takes into account non-linear materials and complex geometry. However, this is by nature an analysis tool, which makes the design procedure very iterative, hence time consuming. Furthermore, the final result is quite uncertain. This main drawback can be eliminated by combining FEM with response surface method (RSM). RSM is a statistical tool to derive experimentally an analytical relationship between a response and some input variables. It is particularly suitable when the underlying phenomenon linking them is not well known or too complex to be modeled mathematically. The relationship is derived by regression from a small number of observations of the response that can be provided by FEM. Genetic algorithm (GA) can then worked directly on this model to optimize the device with a decrease in computational cost. This paper presents the combination of FEM, RSM and GA to optimize the rotor geometry of an interior permanent magnet synchronous motor for fast and accurate design procedure.

Design optimization of permanent magnet motors using response surface methodology and genetic algorithms

A method of modeling and numerical simulation of a brushless permanent-magnet dc motor using time-stepping finite-element technique is presented. In the proposed model, the electromagnetic field equations, the stator circuit equation, and the motion equation are solved simultaneously at each time step; thus, the eddy-current effect, the saturation effect, the rotor movement, and the nonsinusoidal quantities can all be taken into account directly in the system of equations. Dynamic conditions of the motor at starting, step voltage variation, and load torque changes are investigated using the proposed dynamic model.

Modeling and numerical simulation of a brushless permanent-magnet DC motor in dynamic conditions by time-stepping technique

This paper discusses the application of response surface methodology (RSM) to design optimization for brushless dc permanent magnet motors for a hard disk driver to reduce cogging torque. The feasibility of using RSM with the finite-element method in practical engineering problems is investigated with computational examples and comparison between the fitted response and the results obtained from an analytical solution.

Design optimization for cogging torque minimization using response surface methodology

A field-circuit coupled time-stepping finite-element method for the study of electric machines is presented. In reality, various circuits such as stator windings, rotor cage bars, and the control electronics are directly coupled with the electromagnetic field of the electric machines, hence, the physical quantities relating to these circuits as well as the field should be taken into account simultaneously. The method is used to simulate the electromagnetic performance of a permanent magnet synchronous motor (PMSM). The dynamic performance for PMSM, both computational and experimental results, is presented. A good match between the two sets has been achieved.

Time stepping finite element analysis for the dynamic performance of a permanent magnet synchronous motor

This paper proposes an optimization procedure for interior permanent magnet synchronous machines to achieve wide speed operation. This method applies on motors with finite maximum speed. The method takes into account the effect of saturation. Non-linear models of the d- and q-axis flux linkages are obtained from finite element analysis and are used to derive the power capability of the machine. These models are combined with response surface method and genetic algorithms to optimize its constant power speed range

M.A. Jabbar

Papers

Design optimization of permanent magnet motors using response surface methodology and genetic algorithms

Modeling and numerical simulation of a brushless permanent-magnet DC motor in dynamic conditions by time-stepping technique

Design optimization for cogging torque minimization using response surface methodology

Time stepping finite element analysis for the dynamic performance of a permanent magnet synchronous motor

Optimization of the constant power speed range of a saturated permanent magnet synchronous motor