Geometry dependent torque optimization for small spindle motors based on reduced basis finite element formulation

GEOMETRY-DEPENDENT TORQUE OPTIMIZATION FOR SMALL SPINDLE MOTORS BASED ON REDUCED BASIS FINITE ELEMENT FORMULATION AZMI BIN AZEMAN NATIONAL UNIVERSITY OF SINGAPORE 2003 GEOMETRY-DEPENDENT TORQUE OPTIMIZATION FOR SMALL SPINDLE MOTORS BASED ON REDUCED BASIS FINITE ELEMENT FORMULATION AZMI BIN AZEMAN (B.A. (Hons.), M. Eng., Cantab) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2003 Acknowledgement This work would not have been possible without the strong support and motivation of my research supervisor Assoc. Prof M. A. Jabbar. His gift in unravelling the complexity of electromagnetics has made this subject an enjoyable and less painful learning experience than it would otherwise have been. His talent in teaching is only exceeded by his devotion to his students. I would also like to record my appreciation to Mr. Woo and Mr. Chandra of the Machines and Drives Lab, NUS, for their amazing ability to make things happen and their dedication to work which has never failed to amaze. I would also like to acknowledge the help and advice given by Prof. A. Patera from MIT, Ms. Sidrati Ali and Mr. Ang Wei Sin from the Singapore-MIT Alliance that has made it possible for me to come to grasp with the complexity of the Reduced-Basis Method. Their patience is truly divine. Last but not the least, I would like to extend my deepest appreciation to Mr. Jeffrey Lau for patiently ploughing through endless pages of this thesis, correcting it and giving helpful hints to make it better. One can ask for no better proof-reader. i Table of Contents ACKNOWLEDGEMENT ..................................................................................... I SUMMARY .........................................................................................................V LIST OF SYMBOLS.........................................................................................VII LIST OF FIGURES............................................................................................IX LIST OF TABLES ...........................................................................................XIII 1 2 3 INTRODUCTION......................................................................................... 1 1.1 CLASSIFICATION OF OPTIMISATION METHODS .......................................................................... 2 1.2 OVERVIEW OF FINITE ELEMENT METHOD ................................................................................. 5 1.3 INTRODUCTION TO REDUCED BASIS TECHNIQUE ...................................................................... 9 1.4 THE COGGING TORQUE PROBLEM ........................................................................................... 10 1.5 ORGANIZATION OF THESIS ....................................................................................................... 13 INCORPORATING GEOMETRY-DEPENDENCIES................................ 15 2.1 MAGNETOSTATIC PROBLEM DEFINITION................................................................................. 15 2.2 MAGNETIZATION MODEL OF PERMANENT MAGNET ............................................................... 16 2.3 THE STANDARD FINITE ELEMENT PROBLEM ........................................................................... 20 2.4 APPLICATION OF AFFINE TRANSFORMATION ON STATIC MESH .............................................. 24 2.5 TRANSFORMATION OF WEIGHT FUNCTIONS ............................................................................ 36 COMPUTATIONAL ASPECTS ................................................................ 39 3.1 FINITE ELEMENT DISCRETIZATION .......................................................................................... 39 3.2 TRIANGULAR ISOPARAMETRIC ELEMENT ................................................................................ 45 3.3 TWO-DIMENSIONAL NUMERICAL INTEGRATION ..................................................................... 51 3.4 LINEAR SYSTEM SOLUTION BY THE NEWTON-RAPHSON METHOD ......................................... 57 3.5 TORQUE COMPUTATION ........................................................................................................... 62 ii 4 5 3.6 REDUCED-BASIS FORMULATION ............................................................................................. 65 3.7 COMPUTATIONAL ADVANTAGE OF SEGMENTATION ............................................................... 69 SOFTWARE TECHNIQUE AND ALGORITHM ....................................... 72 4.1 A BRIEF INTRODUCTION TO THE BRUSHLESS DC MOTOR ...................................................... 72 4.2 SOFTWARE PLATFORM (MATLAB) ........................................................................................ 75 4.3 SUPPORTING APPLICATION (FLUX2D) ................................................................................... 77 4.4 SOFTWARE STRUCTURE AND FLOW-CHART ............................................................................ 86 4.4.1 Offline Algorithm................................................................................................................ 88 4.4.2 Online Algorithm................................................................................................................ 95 RESULTS AND ANALYSIS ..................................................................... 97 5.1 OFFLINE TORQUE COMPUTATION .......................................................................................... 100 5.1.1 Comparison data for 8-pole/6-slot Spindle Motor .......................................................... 100 5.1.2 Comparison data for 8-pole/9-slot Spindle Motor (Two Meshes) .................................. 109 5.2 ONLINE TORQUE COMPUTATION ............................................................................................ 118 5.3 TECHNICAL CONSIDERATIONS .............................................................................................. 121 5.3.1 Mesh Distribution............................................................................................................. 121 5.3.2 Singularity and Ill-Conditioning...................................................................................... 123 5.3.3 Span Size Selection........................................................................................................... 125 5.3.4 Speed of Computational Analysis .................................................................................... 126 REFERENCES............................................................................................... 129 LIST OF PUBLICATIONS.............................................................................. 133 APPENDIX A: MATHEMATICAL PROOFS AND DERIVATION ................. 134 A.1 NODAL BASIS ................................................................................................................................ 134 A.2 LINEAR AND BILINEAR TERMS ..................................................................................................... 136 A.3 APPLICATIONS OF LINEARITY AND BI-LINEARITY ........................................................................ 137 A.4 ONE DIMENSIONAL GAUSS QUADRATURE SCHEME ..................................................................... 138 A.5 ORTHOGONAL POLYNOMIALS ....................................................................................................... 140 iii APPENDIX B: SPINDLE MOTOR STRUCTURE AND MATERIAL DATA.. 144 B.1 MACHINE DIMENSIONS ................................................................................................................. 145 B.2 MATERIAL DATA ........................................................................................................................... 147 B.3 FINITE ELEMENT QUALITY FACTOR .............................................................................................. 150 iv Summary This thesis looks at the novel technique of Reduced-Basis Method applied to the optimisation of electromagnetic problems. Iterative optimisation in a wide search space can be time consuming, regardless of whether statistical or deterministic methods are employed in the search. A fast method of computing the cost function, in this case the cogging torque, is developed which takes advantage of the accuracy of finite element method but with faster computation time. The method is applied to the problem of predicting the cogging torque in a brushless DC permanent magnet machine. Cogging torque is known to be highly geometry dependent and the variation of cogging torque with permanent magnet dimensions (radial thickness and arc angle) is investigated. Results are compared against that obtained using FLUX2D, a commercially available Electromagnetic Finite Element Package. A comparison is made between FLUX2D predictions against the Offline reduced-basis torque computation using one static mesh. This is a crucial verification step as the Online fast approximation to the cogging torque problem uses the Offline set of vector potentials as the basis on which it computes the torque. Random set of geometry variables are then tested on the Online torque computation module to compute torque and compared against the actual value predicted by FLUX2D. Analysis of the results is performed to determine the accuracy of the method and more importantly the range of accuracy for the case of a single static mesh. v Results from the reduced-basis finite element solution are close to the results obtained from FLUX2D for a similar machine within a certain accuracy window. The accuracy of the method can be extended over a wider range by using multiple static meshes. The increase in the number of static meshes does not inhibit computation speed as this computation effort is done offline and independent of the optimisation and stored in memory for future retrieval. vi List of Symbols r F Lorentz force r B Magnetic flux density vector Bn Normal magnetic flux density Bt Tangential magnetic flux density µo Permeability of free space r J Current density vector ro Locus of points invariant to the affine transformation lmo The original magnet thickness of the static mesh dm Depth of magnet lmn The new required magnet thickness r The distance of mesh node in polar coordinate (r ,θ ) form given in the static mesh r% The transformed polar coordinate of the same node given the change in magnet length vii θo Locus of points invariant to the affine transformation θ mo The original mechanical pitch angle of the static mesh θ mn The new required magnet pitch angle θ The angle between the line joining the pole (origin) to the node position and the positive x-axis θ% The transformed polar coordinate of the same node given the change in magnet pitch angle viii List of Figures Fig. 1.1: General Optimization Iteration Step..................................................... 4 Fig. 1.2: The basic steps of the finite element method ...................................... 7 Fig. 2.1: Actual characteristic of a permanent magnet .................................... 16 Fig. 2.2: General triangular element................................................................. 20 Fig. 2.3: Radial Magnet of arbitrary thickness lmn before transformation ........ 24 Fig. 2.4: Transformed radial length with fixed thickness lmo ............................ 25 Fig. 2.5: Magnet of arc θ mn before transformation ............................................ 26 Fig. 2.6: Transformed magnet arc angle with fixed arc θ mo ............................. 26 Fig. 3.1: Linear approximation of permanent magnet B-H curve..................... 44 Fig. 3.2: Triangular Isoparametric Element...................................................... 45 Fig. 3.3: Sample problem for isoparametric transformation............................. 49 Fig. 3.4: Location of evaluation points for 16-point 2D Gauss Quadrature ..... 54 Fig. 3.5: Convergence rate for different number of Gauss Points ................... 56 Fig. 3.6: Computing torque in air gap elements ............................................... 64 Fig. 4.1: 8-pole/6-slot spindle motor with invariant lines in bold ...................... 73 Fig. 4.2: 8-pole/9-slot spindle motor with invariant lines in bold ..................... 74 Fig. 4.3: Area distortion for different magnet thickness and arc angle .......... 76 Fig. 4.4: Triangular mesh plot (8-pole/6-slot) imported from FLUX2D............. 81 ix Fig. 4.5: Triangular mesh plot (8-pole/9-slot) imported from FLUX2D............. 82 Fig. 4.6: Flux plot for 8-pole/6-slot spindle motor produced by Flux 2D ......... 83 Fig. 4.7: Flux plot for 8-pole/9-slot spindle motor produced by Flux 2D .......... 83 Fig. 4.8: Cogging torque profile for 8-pole/6-slot motor from FLUX2D ............ 85 Fig. 4.9: Cogging torque profile for 8-pole/9-slot motor from FLUX2D ............ 85 Fig. 4.10: Inter-linkages between FLUX2D and MATLAB Modules................. 87 Fig. 4.11: Prediction of un for a desired ld and arc angle θ d .......................... 89 Fig. 4.12: Off-line procedure to compute n x N Matrix Z n× N ............................ 91 Fig. 4.13: Stiffness matrix assembly for geometry-dependent regions............ 94 Fig. 4.14: On-line Procedure for computing torque......................................... 96 Fig. 5.1: Variation of cogging torque with arc angle (FLUX2D) ....................... 98 Fig. 5.2: Variation of cogging torque with radial thickness (FLUX2D) ............. 98 Fig. 5.3: Variation of cogging torque with arc angle from FLUX2D ................. 99 Fig. 5.4: Variation of cogging torque with radial thickness by FLUX2D........... 99 Fig. 5.5: Current magnet arc angle 30o .......................................................... 101 Fig. 5.6: Current magnet arc angle 30.5o ....................................................... 101 Fig. 5.7: Current magnet arc angle 31.0o ....................................................... 102 Fig. 5.8: Current magnet arc angle 31.5o ....................................................... 102 Fig. 5.9: Current magnet arc angle 32.0o ....................................................... 103 Fig. 5.10: Current magnet arc angle 32.5o ..................................................... 103 x Fig. 5.11: Current magnet arc angle 33.0o ..................................................... 104 Fig. 5.12: Current magnet radial length 1.000 mm ........................................ 104 Fig. 5.13: Current magnet radial length 1.025 mm ........................................ 105 Fig. 5.14: Current magnet radial length 1.050 mm ........................................ 105 Fig. 5.15: Current magnet radial length 1.075 mm ........................................ 106 Fig. 5.16: Current magnet radial length 1.100 mm ........................................ 106 Fig. 5.17: Current magnet radial length 1.125 mm ........................................ 107 Fig. 5.18: Current magnet radial length 1.150 mm ........................................ 107 Fig. 5.19: Current magnet radial length 1.175 mm ........................................ 108 Fig. 5.20: Current magnet radial length 1.200 mm ........................................ 108 Fig. 5.21: Static mesh (0.75 mm, 33.7o ), current angle 28.6o ....................... 109 Fig. 5.22: Static mesh (0.75 mm, 33.7o ), current angle 30.3o ........................ 110 Fig. 5.23: Static mesh (0.75 mm, 33.7o ), current angle 32.0o ........................ 110 Fig. 5.24: Static mesh (0.75 mm, 33.7o ), current angle 33.7o ........................ 111 Fig. 5.25: Static mesh (0.75 mm, 33.7o ), current angle 35.4o ........................ 111 Fig. 5.26: Static mesh (0.75 mm, 33.7o ), current angle 37.1o ........................ 112 Fig. 5.27: Static mesh (0.75 mm, 33.7o ), current angle 38.8o ........................ 112 Fig. 5.28: Static mesh (0.60 mm, 33.7o ), current angle 28.6o ....................... 113 Fig. 5.29: Static mesh (0.60 mm, 33.7o ), current angle 30.3o ........................ 113 Fig. 5.30: Static mesh (0.60 mm, 33.7o ), current angle 32.0o ....................... 114 xi Fig. 5.31: Static mesh (0.60 mm, 33.7o ), current angle 33.7o ........................ 114 Fig. 5.32: Static mesh (0.60 mm, 33.7o ), current angle 35.4o ........................ 115 Fig. 5.33: Static mesh (0.60 mm, 33.7o ), current angle 37.1o ........................ 115 Fig. 5.34: Static mesh (0.60 mm, 33.7o ), current angle 38.8o ........................ 116 Fig. 5.35: Regions of accuracy for different static meshes ............................ 118 Fig. 5.36: Static mesh composition for geometry-dependent regions ........... 122 Fig. 5.37: Static mesh composition for geometry-independent regions......... 122 Fig. 5.38: Curve-fitted torque computation for different N ............................ 125 Fig. 5.39: Number of operation count for various N values.......................... 127 xii List of Tables Table 2.1: Values of weight function coefficients ............................................. 22 Table 3.1: Weights for Gauss Quadrature ....................................................... 53 Table 3.2: Evaluation Points for Gauss Quadrature ........................................ 54 Table 5.1: Torque comparison for different rotor angle ................................. 119 Table 5.2: Torque comparison for different rotor angle (multiple meshes).... 120 xiii 1 Introduction Design of a new electromechanical device is a complex process requiring a careful balance between performance, manufacturing effort and cost of materials. Performance of the device can be determined from the solution of the coupled continuum physics equations governing the electrical, mechanical, electromagnetic and thermal behaviour. To arrive at the optimal design within the wide range of physical and economic constraints requires a lifetime of experience and an ability to recognize a good solution. Designers with such abilities are rare, thus giving rise to many varieties of computational tools and expert systems designed to aid in the design process. Finite element analysis has become one of the most popular methods for solving the electromagnetic field equations in electromechanical devices. The flexibility of the method makes it comparatively simple to model the complex geometry and non-linear material properties, including external circuits and provide accurate results with an acceptable use of computing power. Today’s designers have access to a menu driven graphical interface, a wide range of two or three dimensional analysis tools solving field problems, userprogrammable pre- and post processing facilities and parameterized geometric modelling [1]. 1 The need for optimisation tools is found in almost every branch of mathematical modelling. Mathematically, this can be expressed as Minimize F ( x) where F is termed the objective function and x is the constrained parameter space vector of F . An example of an objective function, F , may be the cogging torque per unit volume of machine. The parameter space, x , is then the array of variables that define the behaviour of F . The variables may be discrete, such as the number of pole/slot combinations, or continuous – the width of the magnet in the motor or its arc angle in the case of an arc magnet, or a mixture of both. There may also be constraints placed on the variables – the maximum outer diameter of the motor could be made no larger than a certain fixed dimension, there must be minimum clearance in the air gap due to manufacturing tolerance or the maximum mechanical pitch angle of an 8-pole machine can be no larger than 45o . In electromechanical devices, variables are not just confined to geometry. The properties of materials, current density and choice of magnetic materials could also be included as variables. 1.1 Classification of Optimisation Methods 2 Optimisation methods are divided into two classes – deterministic and stochastic. The difference between the two is that, for a defined set of initial values of x , deterministic methods always follow the same path to the (local) minimum value of F , while stochastic methods include the element of randomness that should arrive at a similar solution each time via a different route. The randomness of stochastic processes has its intrinsic charm in that it allows the algorithm to have a wider search of the problem space, thus guaranteeing that the global minimum is found. There are advantages to both types of optimisation. Deterministic methods are relatively inexpensive and find the local minimum comparatively easily, while stochastic methods may find the region of global minimum more slowly but is effective if the ultimate objective is to obtain the global minimum. In general, deterministic methods such as the conjugate gradient method, modified Newton-Raphson etc. rely on gradients to determine the next value of x , which can be a problem if F does not happen to be an analytic function that is differentiable [3],[4]. Stochastic methods such as simulated annealing [2], [16] and genetic algorithm [3],[4],[5] overcome this shortcoming by drawing analogy to natural processes occurring in nature – based on the idea of minimizing the total energy level in the case of simulated annealing and on the idea of natural selection and competition in the case of genetic algorithm – to determine the viable choices and control the explosion of evaluations necessary to sample the entire parameter space. 3 Nevertheless, regardless of whether deterministic or stochastic methods are used to evaluate the next value of x , the objective function F has to be evaluated at each point of the iteration in order to determine whether to continue to search or to stop and declare a success or failure. The general optimisation iteration is shown in Fig. 1.1. New variable set Deterministic/ Stochastic Algorithm Solution of F Is F minimized? x No Yes END Fig. 1.1: General Optimization Iteration Step In the example of a cogging torque minimization problem, the solution of F can entail the solution of the finite element problem to find the cogging torque 4 in the machine structure. The finite element method has found favour in optimisation techniques because of three main developments in finite element software package. Firstly, the availability of variational modelling controlled by a set of user-defined parameters allows the optimisation or experimental design algorithm to generate a new version of the geometry [1]. Secondly, reliable automatic meshing and adaptive solvers produce a mesh that can be analysed and return a solution with reasonable accuracy [1], [12]-[17]. Finally, complex post-processing to determine the value of the objective function can be pre-programmed by the user, employing the design parameters to compute values in correct relation to the most recent update to the geometry [1]. Essentially the three developments above take away the need for human intervention, allowing the optimisation algorithm to interface with the finite element sub-module uninterrupted until a global minimum of F is found. 1.2 Overview of Finite Element Method Finite element solution method can be summarised into three basic layers of operations as shown in Fig. 1.2. In the pre-processing step, the geometry is modified to take into account the new parameters and the geometry is then meshed. In the solution stage, the mesh and material data are then used to solve the problem. In the post-processing stage, the finite element solution is then used to find the value of the objective function. While the three developments in finite element computing mentioned above have managed to make the finite element solution process automatic, combined computational 5 effort will be relatively time consuming and will form a key bottleneck in the optimisation process shown in Fig. 1.1. 6 NEW PARAMETER SET X (PRE-PROCESSING) GENERATE NEW MESH (SOLUTION PHASE) SOLVE FINITE ELEMENT PROBLEM (POST-PROCESSING) OBTAIN SOLUTION FOR F Fig. 1.2: The basic steps of the finite element method The key idea in this project is to firstly maintain the advantage of accuracy that comes with the use of finite-element analysis without sacrificing the speed necessary to make the optimisation process practical and manageable. Finite element method involves the solution of large linear system of equations, which is in itself time consuming if iterative methods such as the NewtonRaphson or the conjugate gradient method (CGM) are used to invert the 7 matrix. Consequently, the use of finite element method as shown in Fig. 1.1 involves iteration within an iteration, which requires a large computational effort. Computational effort can be reduced if the iterative finite element solution process can be avoided in the optimisation algorithm altogether. Secondly, the computing effort can be more manageable if the three layers of the finite-element process shown in Fig. 1.2 can be compressed into a single functional layer. This would be possible if the finite element algorithm can be made into a function of the variables x such that a change in x would lead automatically to a change in F without the need for geometry modification, remeshing and solving again for F . 8 1.3 Introduction to Reduced Basis Technique The reduced basis method is essentially a scheme for approximating segments of a solution curve or surface defined by a system containing a set of free variables. For each curve segment an approximate manifold is constructed that is “close” to the actual curve or surface. The computational effectiveness of this method is derived from the fact that it is often possible to obtain accurate approximations when the dimension of the approximate manifold is many orders smaller than that of the original system. The basic idea of the reduced basis method was introduced in 1977 [6] for the analysis of trusses. The idea was then revived three years later in a series of papers [7], [8] to deal with other structural applications. The method has since then been applied to the solution of heat transfer problem in a thermal fin [9]. The development of the reduced-basis method to incorporate variations in geometric parameters was motivated by the need to combine the accuracy of finite element solution with the computational effectiveness of reduced-basis approximation [10] for the purpose of optimisation. This led to the idea of “offline” and “online computation. In the offline computation, the problem space not affected by geometric transformation can be pre-computed. The contribution of the geometry-dependent regions, on the other hand, has to be computed every time the reduced-basis method is applied to a new point in the parameter space [11]. But provided that the parameter dependent region constitutes only a small fraction of the total problem domain, it can be 9 expected that the whole procedure of matrix construction and solution to be relatively inexpensive. 1.4 The Cogging Torque Problem The application of the reduced-basis method to electromagnetic problems is new. Most workers in this field [12]-[18] rely on standard finite element packages because of its accuracy and cost-effectiveness as new designs can be economically tested without the need for costly prototyping in the initial design stages. Much work to incorporate finite element solution to electrical machine optimisation work has brought improvement to finite element software modules as described earlier, mainly done to remove the human element in the iterative process, yet have basically left the requirement to undergo the basic processes of pre-processing to post-processing relatively intact. With the reduced-basis approach, a new paradigm is possible. Much of the tedium of finite element computing can be done “off-line” and stored for future recall. Geometry transformation over a limited region removes the requirement for re-meshing in order to approximate a solution, and the desired evaluation of the objective function is obtained from the approximate reduced-basis space which is close to the actual solution space, through a process which is computationally less costly. 10 In this research project, the example of cogging torque evaluation in a brushless DC machine has been selected as the cost function to minimize. Cogging torque is produced in a permanent magnet machine by the magnetic attraction between the rotor mounted permanent magnets and the stator. It is a pulsating torque which does not contribute to the net effective torque. In fact it is considered an undesired effect that contributes to the torque ripple, vibration and noise and it is therefore a major design goal to eliminate or reduce this cogging effect. The motivation for selecting cogging torque as a case study of the reducedbasis method is the fact that it is highly dependent on the machine geometry. The variation of cogging torque with geometry has been a subject of extensive research [12]-[18]. Dr. Jabbar et. al concluded in his papers in 1992 and 1993 that smaller cogging torque results if the pole-slot combination is not “simple” i.e., for even-odd pole-slot combinations such as 8-pole/9-slot or 8-pole/15slot. On the other hand, higher cogging is expected for “simple” combinations such as 6-pole/6-slot, 8-pole/12-slot and 8-pole/6-slot [12]. He also concluded that apart from slot-pole combination, another effective method of reducing cogging torque and ripple torque is by shaping the poles, resulting in less fluctuation of the torque wave [13]. Since then, other workers in this field such as C.C. Hwang et. al. [17] have reported the variation of the cogging effect with different combinations of the least common multiple of pole and slot and the ratio of armature teeth to magnet pole arc, both variables affecting the machine geometry. The cogging 11 torque results were computed using standard finite element method, which were computationally tedious given the many parameter combinations required. Chang Seop Koh et. al. [16] on the other hand, studied the effect of shaping the pole shape to minimize the cogging torque. He employed a sophisticated evolutionary simulated annealing algorithm interfaced with a standard finite element package. In his work, he defined the stator tooth shape dimensions as variables which were varied by the optimisation algorithm to search for the best combination. He concluded in his report that one of the most important factors influencing cogging torque was the pole shape of the armature core. In fact, there is a general rule to estimate the cogging torque magnitude periodicity based on the combination of slots and magnet poles [12]. The larger the smallest common multiple between the slot number and the pole number, the smaller is the amplitude of the cogging torque. The smallest common factor between the magnet pitch angle and the slot angle gives the polar angle periodicity of the cogging torque effect. It is a novel approach to study the variation of cogging torque with changes in certain geometric parameters by the reduced-basis method. In this project, the cogging torque variation is studied, taking the permanent magnet radial length and its pole arc angle as the variable parameters. Two variations of spindle motor are under study. The first is an 8-pole/6-slot brushless DC machine and 12 the second is an 8-pole/9-slot brushless DC machine as shown in Fig. 4.1 and Fig. 4.2 in Appendix B, with dimensions chosen to correspond with the machine dimension reported in [17] for comparison purposes. The cogging torque for this particular machine is also computed using commercial software [1] to check against the result produced by the “off-line” computation of torque. 1.5 Organization of Thesis This thesis is organized in the following way. The following two chapters discuss the theoretical aspects of the reduced basis method. In chapter one, the basic framework of the finite element method is explained. Following the use of affine geometrical transformation, mathematical modification to the standard finite element codes is derived based on standard linear algebra and vector calculus for problems in two dimensions. The objective of the transformations applied is to map meshes with the required geometric parameters into a “template” static mesh. In the second chapter, the finite element stiffness matrix and forcing function for the transformed finite element formulae are developed. Isoparametric transformation and Gauss Quadrature technique are elaborated; these are critical steps for the numerical evaluation of the forcing function and stiffness matrix. The computation is then performed in Matlab [20], an ideal platform for handling matrix problems. In this chapter, the Newton-Raphson method is introduced as a means of solving problems with non-linear materials, with the 13 necessary modifications developed to account for the geometric transformations. Lastly, for the purpose of torque computation, the Maxwell Stress Method is chosen due to convenience of computing in the circular air gap, though there are other possible methods of computing torque [21], [23], [24]. In chapter three, the formulae derived in the preceding chapters are encoded into Matlab programs [26]. An algorithm for importing mesh data from FLUX2D, computing the offline vector potential values at the nodes and for predicting the online vector potential for a specific parameter set is shown in flow-chart forms for clarity. Actual cogging torque values are also computed using FLUX2D for the purpose of verification of the Reduced-Basis Offline technique. 14 2 Incorporating Geometry-Dependencies 2.1 Magnetostatic Problem Definition Starting from Maxwell equation ∇ × H = J , the magnetostatic problem can be modelled by Poisson’s Equation [21], [27]. In 2-D Cartesian coordinate system, the Poisson equation is given by ∂ 2u ∂ 2 u + = f ( x, y ) in Ω ∂x 2 ∂y 2 (2.1) In equation (2.1), u is the exact vector potential at the nodes and f ( x, y ) is the forcing function which will be derived in the next section. f ( x, y ) consists of magnet equivalent current in the case of cogging torque minimization. Ω is the problem space which is subject to Dirichlet and Neumann boundary conditions such that u = 0 on Γ e ∂u = 0 on Γ n ∂n 15 where n is the outward normal unit vector at the boundary and Γ e and Γ n are the Dirichlet and Neumann boundaries. For a well-posed problem, the total boundary is given by Γ = Γ e ∩ Γ n over the domain Ω . 2.2 Magnetization Model of Permanent Magnet The permanent magnet can be modelled as an equivalent current source in the element [21], [22],[28]. The demagnetization curve of a permanent magnet is shown in Fig. 2.1. Mo B HC H Fig. 2.1: Actual characteristic of a permanent magnet Computationally it is not necessary to assume a linear permanent magnet as the non-linear behaviour of the permanent magnet and iron can be taken into account either by using a look-up table or by employing the Newton-Raphson method in the iteration process. However, to simplify the analysis, the B-H property of a permanent magnet in the second quadrant can be reasonably 16 approximated as a straight line. In this case only two parameters Br and Hc are required in order to fully define the magnetic characteristics. Therefore B = µ o {(1 + xm ) H + M } where xm is the magnetic susceptibility, M = Br / µ the magnetization vector (amperes/meter) and H is the externally applied field. Defining the reluctivity as v = 1 µ o (1 + xm ) , the equation reduces to vB = H + vµ o M and taking the curl of both sides of the equation and noting that ∇ × H = J and ∇ × u = B , the equation transforms to ∇ × v(∇ × u ) = J + ∇ × (vµ o M ) Defining a (2.2) F = ∇ ×ν ( ∇ × u ) − J − ∇ × (νµ o M ) , functional the optimized computational solution for vector potential, u% , can be obtained by minimizing the error of the product of the functional F (u% ) and weight function W over the problem region Ω such that ∫∫ W ⋅ ( ∇ × v ( ∇ × u% ) )∂x∂y − ∫∫ W ⋅ J ∂x∂y − ∫∫ W ⋅ (∇ × ( vµ M ) )∂x∂y = 0 o Ω Ω (2.3) Ω Equation (2.3) can be re-arranged as ∫∫ ∇ × (ν∇ × u% −νµ M ) ⋅W ∂xdy − ∫∫ J ⋅W ∂x∂y = 0 o Ω (2.4) Ω 17 Integrating by parts we can write the first term of equation (2.4) as ∫∫ ( ∇ × (ν∇ × u% ) −νµ M ) ⋅W ∂x∂y = o Ω ∫∫ (ν∇ × u% −νµ M ) ⋅ ( ∇ × W ) ∂x∂y + ∫∫ ∇ ⋅ ( (ν∇ × u% −νµ M ) × W ) ∂x∂y o o Ω (2.5) Ω By applying the Divergence Theorem on the last term of equation (2.5), the area integral is transformed into a line integral over the boundary C enclosing the area ˆ ∫∫ ∇ ⋅ (ν∇ × u% −νµ M ) ×W ∂x∂y = ∫ ⎡⎣(ν∇ × u% −νµ M ) × W ⎤⎦ ⋅ ndC o o Ω (2.6) C ( F × G ) ⋅ T = F ⋅ (G × T ) the line By applying identities F × G = −G × F and integral on the right-hand side of equation (2.6) reduces to ) ) ∫ {(ν∇ × u% −νµ M ) × W } ⋅ ndC = ∫ W ⋅{(ν∇ × u% −νµ M ) × n}dC o o C (2.7) C By imposing a homogeneous boundary condition, the integral in equation (2.7) in turn reduces to zero and finally equation (2.4) reduces to ∫∫ v ( ∇ × u% ) ⋅ ( ∇ × W ) = ∫∫ vµ M ⋅ ( ∇ × W ) ∂x∂y + ∫∫ W ⋅ J ∂x∂y o Ω Ω (2.8) Ω 18 after substituting J = ∇ × {v ( ∇ × A ) − vµ o M )} into equation (2.4). For two dimensional Cartesian case, ⎛ ∂u% ∂W ∂u% ∂W + ∂x ∂y ∂y ∫∫ν ⎜⎝ ∂x Ω ⎛ ⎞ ⎛ ∂W ∂W −My ⎟dxdy = ∫∫ ⎜νµ o ⎜ M x ∂y ∂x ⎠ ⎝ Ω ⎝ ⎞ ⎞ ⎟ + J ⋅ W ⎟dxdy ⎠ ⎠ (2.9) The forcing function term representing the permanent magnet in equation (2.9) is ∫∫ vµ Ω o ⎛ ∂W ∂W ⎞ −My ⎜Mx ⎟dxdy ∂y ∂x ⎠ ⎝ (2.10) The forcing function term representing current injection is ∫∫ J ⋅Wdxdy (2.11) Ω 19 2.3 The Standard Finite Element Problem y u3( x3 , y3 ) ( x2 , y2 ) u2 u1 ( x1 , y1 ) x Fig. 2.2: General triangular element Discretization of the domain Ω for finite element analysis [27],[20] is performed using first-order triangular element which has three nodes at the vertices of the triangle and the linear interpolation of the vector potential within the element domain Ωe is given in Cartesian coordinate system as u% = α + β x + γ y = [1 x ⎡α ⎤ y ] ⎢⎢ β ⎥⎥ ⎢⎣ γ ⎥⎦ (2.12) 20 where α , β and γ are the constants to be determined. The interpolation function should represent the values of the potential at the nodes and consequently ⎡ u%1 ⎤ ⎡1 x1 ⎢u% ⎥ = ⎢1 x 2 ⎢ 2⎥ ⎢ ⎢⎣ u%3 ⎥⎦ ⎢⎣1 x3 y1 ⎤ ⎡α ⎤ y2 ⎥⎥ ⎢⎢ β ⎥⎥ y3 ⎥⎦ ⎢⎣ γ ⎥⎦ Inverting the matrix the values of the coefficients can be determined ⎡α ⎤ ⎡ x2 y3 − x3 y2 ⎢β ⎥ = 1 ⎢ y − y 3 ⎢ ⎥ 2A ⎢ 2 ⎢⎣ γ ⎥⎦ ⎢⎣ x3 − x2 x3 y1 − x1 y3 y3 − y1 x1 − x3 x1 y2 − x2 y1 ⎤ ⎡ u%1 ⎤ y1 − y2 ⎥⎥ ⎢⎢u%2 ⎥⎥ x2 − x1 ⎥⎦ ⎢⎣ u%3 ⎥⎦ (2.13) where ⎡1 x1 1 A = det ⎢⎢1 x2 2 ⎢⎣1 x3 y1 ⎤ y2 ⎥⎥ y3 ⎥⎦ The magnitude of A is equal to the area of the linear triangular element. However its value will be positive of the element numbering is in the anticlockwise direction and negative otherwise. 21 Substituting the coefficients from equation (2.13) into equation (2.12) and rearranging the equation, the vector potential value in the triangular element is then given by u% = W1 ( x, y )u%1 + W2 ( x, y )u%2 + W3 ( x, y )u%3 (2.14) Wi ( x, y ) is called the shape function for linear triangular element and is given by W1 ( x, y ) = 1 [a1 x + b1 y + c1 ] 2A W2 ( x, y ) = 1 [a2 x + b2 y + c2 ] 2A W3 ( x, y ) = 1 [a3 x + b3 y + c3 ] 2A Table 2.1: Values of weight function coefficients i =1 i=2 i=3 ai y2 − y3 y3 − y1 y1 − y2 bi x3 − x2 x1 − x3 x2 − x1 ci x2 y3 − x3 y2 x3 y1 − x1 y3 x1 y2 − x2 y1 22 ⎡W1 ⎤ Writing W = ⎢⎢W2 ⎥⎥ ⎣⎢W3 ⎦⎥ with W1 = W1 ( x, y ) , W2 = W2 ( x, y ) and W3 = W3 ( x, y ) It is then possible to express vector potential u in equation (2.14) in the form of u% = [W1 W2 ⎡ u%1 ⎤ W3 ] ⎢⎢u%2 ⎥⎥ ⎢⎣ u%3 ⎥⎦ The partial differentials of W and u% with respect to x and y are also given by ⎡ ∂∂Wx1 ⎤ ⎡ a1 ⎤ 1 ⎢ ⎥ ∂W ⎢ ∂W2 ⎥ =⎢ ⎥= a2 ∂x ⎢ ∂∂Wx ⎥ 2∆ e ⎢ ⎥ 3 ⎢⎣ a3 ⎥⎦ ⎢⎣ ∂x ⎥⎦ ⎡ ∂∂Wy1 ⎤ ⎡ b1 ⎤ 1 ⎢ ⎥ ∂W ⎢ ∂W2 ⎥ =⎢ ⎥= b2 ∂y ⎢ ∂y ⎥ 2∆ e ⎢ ⎥ ∂W ⎢⎣ b3 ⎥⎦ ⎢⎣ ∂y3 ⎥⎦ 23 ∂u% ∂W1 = [ ∂x ∂x ∂u% ∂W1 = [ ∂y ∂y ∂W2 ∂x ⎡ u%1 ⎤ ∂W3 ⎢ ] u% ⎥ = [a1 ∂x ⎢ 2 ⎥ ⎢⎣ u%3 ⎥⎦ a2 ⎡ u%1 ⎤ a3 ] ⎢⎢u%2 ⎥⎥ ⎢⎣ u%3 ⎥⎦ ∂W2 ∂y ⎡ u%1 ⎤ ∂W3 ⎢ % ⎥ b2 ∂y ] ⎢u2 ⎥ = [b1 ⎢⎣ u%3 ⎥⎦ ⎡ u%1 ⎤ b3 ] ⎢⎢u%2 ⎥⎥ ⎢⎣ u%3 ⎥⎦ 2.4 Application of Affine Transformation on Static Mesh r-ro ro r O lmn Fig. 2.3: Radial Magnet of arbitrary thickness lmn before transformation 24 ro r% O lmo Fig. 2.4: Transformed radial length with fixed thickness lmo Fig. 2.3 shows a radial magnet region of radial thickness lmn and distance ro from the origin O . The bold arc line represents the locus of points which are invariant i.e. mesh points which are unchanged under any transformation. Consider the radial affine transformation. r% = ro + lmo (r − ro ) lmn Under this non-linear transformations, the coordinates of points in the magnet region are transformed into the points within a magnet of fixed radial thickness lmo . The effect of this transformation is shown in Fig. 2.4. In a similar fashion, the general mechanical arc angle of the magnet pole can θ be transformed by the affine transformation θ% = θ o + mo (θ − ϑo ) . θ mn 25 Under this transformation, shown in Fig. 2.5 (before) and in Fig. 2.6 (after), the effect of transformation is to increase the arc angle of the magnet. The magnet enlarges about the invariant line with polar coordinate angle θ o . θ mn θ θo Fig. 2.5: Magnet of arc θ mn before transformation θ mo θ% θo Fig. 2.6: Transformed magnet arc angle with fixed arc θ mo 26 The affine transformation may be re-arranged as follows r% = ro + lmo l l (r − ro ) = (1 − mo ) ro + mo r . lmn lmn lmn Letting β = (2.15) lmo l and γ = 1 − mo = 1 − β , the radial transformation in equation lmn lmn (2.15) could be written simply as r% = γ ro + β r and the inverse transformation from r% → r could be written as r = r% − γ ro β . Similarly θ% = θ o + θ mo θ θ (θ − θ o ) = (1 − mo )θ o + mo θ θ mn θ mn θ mn Letting α = . (2.16) θ θ mo and η = 1 − mo = 1 − α , the angular affine transformation in θ mn θ mn equation (2.16) could be written as θ% = ηθ o + αθ and its corresponding inverse transformation written as θ = θ% − ηθ o . α 27 The affine transformations are taken into account using standard partial differential results from calculus [19]. The partial differentials for r = f (r%,θ% ) and r% = f% (r ,θ ) are given as follow ∂r% =β ∂r ∂r 1 = ∂r% β ∂r% =0 ∂θ (since r% is not a function of θ ) ∂r =0 ∂θ% Similarly, the partial differentials for θ = f (r%,θ% ) and θ% = f% (r ,θ ) could be written as follows ∂θ% =0 ∂r ∂θ% =α ∂θ ∂θ =0 ∂r% ∂θ 1 = ∂θ% α The Jacobian of this transformation is given by ∂r ∂r% J= ∂r ∂θ% 1 ∂θ β ∂r% = ∂θ 0 ∂θ% 0 1 = 1 αβ α 28 Applying transformation from Cartesian coordinate system to a polar coordinate system, the RHS of equation (2.1) is transformed into ∫∫ Ω ∂W ∂u% ∂W ∂u% ∂W ∂u% 1 ∂W ∂u% + ∂x∂y = ∫∫ r + ∂r∂θ ∂x ∂x ∂y ∂y ∂r ∂r r ∂θ ∂θ Ω (2.17) Applying the above affine transformations from θ → θ% and from r → r% , equation (2.17) then transforms to ∫∫ r Ω ∂W ∂u% 1 ∂W ∂u% + ∂r ∂θ ∂r ∂r r ∂θ ∂θ ⎛ r% − γ ro ⎞ ⎛ ∂W ∂r% ∂W ∂θ% ⎞ ⎛ ∂u% ∂r% ∂u% ∂θ% ⎞ 1 = ∫∫ ⎜ + + ⎜ ⎟⎜ ⎟ β ⎠⎟ ⎝ ∂r% ∂r ∂θ% ∂θ ⎠ ⎝ ∂r% ∂r ∂θ% ∂θ ⎠ αβ Ω ⎝ + β ⎛ ∂W ∂r% ∂W ∂θ% ⎞⎛ ∂u% ∂r% ∂u% ∂θ% ⎞ 1 + + ∂r%∂θ% ⎜ ⎟⎜ ⎟ r% − γ ro ⎝ ∂r% ∂θ ∂θ% ∂θ ⎠⎝ ∂r% ∂θ ∂θ% ∂θ ⎠ αβ = ∫∫ r% − γ ro ⎛ 2 ∂W ∂u% ⎞ 1 ⎛ 2 ∂W ∂u% ⎞ % + β ⎟ ⎜α ⎟ ∂r%∂θ 2 ⎜ ∂r% ∂r% ⎠ α ( r% − γ ro ) ⎝ αβ ⎝ ∂θ% ∂θ% ⎠ = ∫∫ r% − γ ro ⎛ ∂W ∂u% ⎞ α ⎛ ∂W ∂u% ⎞ ∂r%∂θ% ⎜ % % ⎟+ % α ⎝ ∂r ∂r ⎠ ( r − γ ro ) ⎜⎝ ∂θ% ∂θ% ⎟⎠ Ω Ω (2.18) Equation (2.18) is the transformed Poisson equation after taking into account the change in geometry due to the change in the magnet radial thickness and pitch angle. To facilitate the solution of this problem using standard finite 29 element technique, equation (2.18) is transformed back into Cartesian coordinate system by applying the following transformations y% x% = r% cos θ% and y% = r% sin θ% with r% 2 = x% 2 + y% 2 and tan θ% = . x% The following partial differentials are then obtained ∂x% x% = cos θ% = ∂r% r% ∂r% x% = ∂x% r% ∂x% = − r% sin θ% = − y% ∂θ% ∂θ% y% =− 2 ∂x% r% y% ∂y% = sin θ% = r% ∂r% ∂r% y% = ∂y% r% ∂y% = r% cos θ% = x% ∂θ% ∂θ% x% = ∂y% r% 2 The Jacobian of this transformation is given by J= ∂r% ∂x% ∂r% ∂y% ∂θ% ∂x% ∂θ% ∂y% = x% r% - y% r% 2 y% r% x% r% 2 x% 2 + y% 2 1 = = r% 3 r% Therefore the integral equation reduces to 30 ∫∫ Ω r% − γ ro ⎛ ∂W ∂u% ⎞ α ⎛ ∂W ∂u% ⎞ ∂r%∂θ% ⎜ % % ⎟+ % α ⎝ ∂r ∂r ⎠ ( r − γ ro ) ⎜⎝ ∂θ% ∂θ% ⎟⎠ = ∫∫ r% − γ ro ⎛ ∂W ∂x% ∂W ∂y% ⎞ ⎛ ∂u% ∂x% ∂u% ∂y% ⎞ 1 + + ∂x%∂y% α ⎜⎝ ∂x% ∂r% ∂y% ∂r% ⎟⎠ ⎜⎝ ∂x% ∂r% ∂y% ∂r% ⎟⎠ r% + ∫∫ ⎛ ∂W ∂x% ∂W ∂y% ⎞⎛ ∂u% ∂x% ∂u% ∂y% ⎞ 1 α + + ⎜ ⎟ ∂x%∂y% ( r% − γ ro ) ⎝ ∂x% ∂θ% ∂y% ∂θ% ⎟⎜ ⎠⎝ ∂x% ∂θ% ∂y% ∂θ% ⎠ r% = ∫∫ r% − γ ro ⎛ x% ∂W y% ∂W ⎞⎛ x% ∂u% y% ∂u% ⎞ + + ⎜ ⎟⎜ ⎟ ∂x%∂y% r%α ⎝ r% ∂x% r% ∂y% ⎠⎝ r% ∂x% r% ∂y% ⎠ Ω Ω Ω α + ∫∫ ⎛ ∂W ∂W ⎞⎛ ∂u% ∂u% ⎞ − y% + x% − y% + x% ⎟ ∂x%∂y% ⎜ ⎟⎜ ∂x% ∂y% ⎠⎝ ∂x% ∂y% ⎠ r% ( r% − γ ro ) ⎝ = ∫∫ %% ∂W ∂u% xy %% ∂W ∂u% y% 2 ∂W ∂u% ⎞ r% − γ ro ⎛ x% 2 ∂W ∂u% xy + + + ⎜ ⎟ ∂x%∂y% α r% ⎝ r% 2 ∂x% ∂x% r% 2 ∂x% ∂y% r% 2 ∂y% ∂x% r% 2 ∂y ∂y ⎠ Ω Ω + ∫∫ Ω α ⎛ 2 ∂W ∂u% ∂W ∂u% ∂W ∂u% ∂W ∂u% ⎞ %% %% − xy − xy + x% 2 y% ⎜ ⎟ ∂x%∂y% ∂x% ∂x% ∂x% ∂y% ∂y% ∂x% ∂y ∂y ⎠ r% ( r% − γ ro ) ⎝ ⎛ x% 2 ( r% − γ ro ) y% 2α ⎞ ∂W ∂u% = ∫∫ ⎜⎜ + ∂x%∂y% ⎟⎟ 3 % % % % % − ∂ ∂ r r r r x x α γ ( ) o ⎠ Ω ⎝ ⎛ xy %% ( r% − γ ro ) %%α ⎞ ⎛ ∂W ∂u% ∂W ∂u% ⎞ xy + ∫∫ ⎜⎜ − + ⎟⎜ ⎟ ∂x%∂y% 3 r% α r% ( r% − γ ro ) ⎟⎠ ⎝ ∂x% ∂y% ∂y% ∂x% ⎠ Ω ⎝ ⎛ y% 2 ( r% − γ ro ) ⎞ ∂W ∂u% x% 2α + ∫∫ ⎜⎜ + ∂x%∂y% ⎟⎟ 3 r% α r% ( r% − γ ro ) ⎠ ∂y% ∂y% Ω ⎝ ⎛ ∂W ∂u% ∂W ∂u% ⎞ ∂W ∂u% ∂x%∂y% + ∫∫ ϕ 2 ( x%, y% ) ⎜ + ⎟ ∂x%∂y% + ∂x% ∂x% ⎝ ∂x% ∂y% ∂y% ∂x% ⎠ Ω Ω ∂W ∂u% ∫∫Ω ϕ3 ( x%, y% ) ∂y% ∂y% ∂x%∂y% = ∫∫ ϕ1 ( x%, y% ) (2.19) 31 After making the substitutions ϕ1 ( x% , y% ) = x% 2 ( r% − γ ro ) y% 2α + r% 3α r% ( r% − γ ro ) ϕ2 ( x% , y% ) = %% ( r% − γ ro ) xy %%α xy − 3 r% α r% ( r% − γ ro ) ϕ3 ( x% , y% ) = y% 2 ( r% − γ ro ) x% 2α + r% 3α r% ( r% − γ ro ) On the right-hand side of equation (2.9), the term representing the permanent magnet is given by equation (2.10) and is reproduced here ∫∫ vµ Ω o ⎛ ∂W ∂W ⎞ −My ⎜Mx ⎟dxdy ∂y ∂x ⎠ ⎝ Appling affine transformation to the magnet region, the first partial differential in the above equation is transformed ∫∫ Ω ∂W ∂W ⎛ ∂W ⎞ dxdy = ∫∫ ⎜ r sin θ + cos θ ⎟ drdθ ∂y ∂r ∂θ ⎠ Ω ⎝ ⎛ θ% − ηθ ⎞⎛ ∂W ∂θ% ∂W ∂r% ⎞ ⎛ r% − γ ro ⎞ ⎛ θ% − ηθ ⎞⎛ ∂W ∂r% ∂W ∂θ% ⎞ 1 % θ% sin cos + + ⎜ ⎟⎜ ⎟ ⎜ α ⎟⎜ ∂θ% ∂θ + ∂r% ∂θ ⎟ drd 2 ⎟ ⎝ αβ ⎠ ⎝ α ⎠⎝ ∂r% ∂r ∂θ% ∂r ⎠ αβ ⎝ ⎠⎝ ⎠ = ∫∫ ⎜ Ω ⎛ r% − γ ro ⎞ ⎛ θ% − ηθ sin ⎜ 2 ⎟ ⎝ αβ ⎠ ⎝ α = ∫∫ ⎜ Ω ⎞ ⎛ ∂W ⎞ 1 ⎛ θ% − ηθ β cos + ⎟ ⎜ ∂r% ⎟ αβ ⎜ α ⎠ ⎠⎝ ⎝ ⎞ ⎛ ∂W ⎞ % θ% ⎟ ⎜ ∂θ% α ⎟ drd ⎝ ⎠ ⎠ 32 = ⎛ θ% − ηθ ⎞ ∂W ⎛ θ% − ηθ ⎞ ∂W % θ% + α cos ⎜ ⎟ ⎟ ∂θ% drd α ⎠ ∂r% α ⎝ ⎠ 1 ( r% − γ ro ) sin ⎜ αβ ∫∫ ⎝ Ω Transforming back into Cartesian coordinate system, 1 αβ = = ⎛ θ% − ηθ ⎞ ∂W ⎛ θ% − ηθ ⎞ ∂W % θ% + α cos ⎟ ∂r% ⎜ α ⎟ ∂θ% drd α ⎝ ⎠ ⎝ ⎠ ∫∫ ( r% − γ ro ) sin ⎜ Ω 1 αβ ∫∫ ( r% − γ ro ) sin ⎛ θ% − ηθ o ⎞ ⎛ ∂W 1 αβ ∫∫ + α ( r% − γ ro ) sin ⎛ θ% − ηθo ⎞ ⎛ ∂W r% Ω = ⎜ ⎝ r% Ω ∂x% ∂W ∂y% ⎞ α ⎛ θ% − ηθ o ⎞ ⎛ ∂W ∂x% ∂W ∂y% ⎞ % % + + cos + ⎟ ⎟ dxdy ⎟⎜ ⎜ ⎟⎜ ⎠ ⎝ ∂x% ∂r% ∂y% ∂r% ⎠ r% ⎝ α ⎠ ⎝ ∂x% ∂θ% ∂y% ∂θ% ⎠ ⎜ ⎝ α x% ∂W y% ⎞ α ⎛ θ% − ηθ o + cos + ⎟ ⎜ ∂x% r% ∂y% r% ⎟ r% ⎜ α ⎠⎝ ⎝ ⎠ ∂W ⎞ ⎛ ∂W ⎟ ⎜ − y% ∂x% + x% ∂y% ⎠⎝ 1 ⎛ ⎛ r% − γ ro ⎞⎛ x% ⎞ ⎛ θ% − ηθ o ⎞ ⎛ α y% ⎞ ⎛ θ% − ηθ o ⎞ ⎞ ∂W % % sin ⎜ cos ⎜ dxdy ⎟−⎜ ⎟ ⎟⎟ ⎜ ⎟ ⎟ ⎟ r% ⎠ ⎝ r% ⎠ ⎝ α ⎠ ⎝ r% ⎠ ⎝ α ⎠ ⎠ ∂x% Ω ⎝ 1 ⎛ ⎛ r% − γ ro ⎞⎛ y% ⎞ ⎛ θ% − ηθ o ⎞ ⎛ α x% ⎞ ⎛ θ% − ηθ o ⎞ ⎞ ∂W % % sin ⎜ cos ⎜ dxdy ⎟+⎜ ⎟ ⎟⎟ ⎜ ⎟ ⎟ ⎟ % % % % r r r y α α ∂ ⎝ ⎠ ⎝ ⎠ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎞ % % ⎟ dxdy ⎠ ⎜ αβ ∫∫ ⎜ ⎜⎝ ⎜ αβ ∫∫ ⎜ ⎜⎝ Ω The second term in the equation can also be transformed ∫∫ Ω ∂W ∂W ⎛ ∂W ⎞ dxdy = ∫∫ ⎜ r cos θ − sin θ ⎟ drdθ ∂x ∂r ∂θ ⎠ Ω ⎝ ⎛ θ% − ηθ ⎞⎛ ∂W ∂r% ∂W ∂θ% ⎞ 1 ⎛ θ% − ηθ ⎞⎛ ∂W ∂θ% ∂W ∂r% ⎞ ⎛ r% − γ ro ⎞ % θ% − + + cos ⎜ sin ⎜ ⎟⎜ ⎟ ⎟⎜ ⎟ drd 2 ⎟ ⎝ αβ ⎠ ⎝ α ⎠⎝ ∂r% ∂r ∂θ% ∂r ⎠ αβ ⎝ α ⎠⎝ ∂θ% ∂θ ∂r% ∂θ ⎠ = ∫∫ ⎜ Ω 33 ⎛ θ% − ηθ ⎛ r% − γ ro ⎞ cos ⎜ 2 ⎟ ⎝ αβ ⎠ ⎝ α = ∫∫ ⎜ Ω = ⎞ ⎛ ∂W ⎞ 1 ⎛ θ% − ηθ ⎟ ⎜ ∂r% β ⎟ − αβ sin ⎜ α ⎠ ⎠⎝ ⎝ ⎛ θ% − ηθ ⎞ ∂W ⎛ θ% − ηθ − α sin ⎜ ⎟ α ⎠ ∂r% ⎝ α 1 (r% − γ r ) cos ⎜ αβ ∫∫ ⎝ o Ω ⎞ ⎛ ∂W ⎞ % θ% ⎟ ⎜ ∂θ% α ⎟ drd ⎠ ⎠⎝ ⎞ ∂W % θ% ⎟ % drd ⎠ ∂θ Transforming back into Cartesian coordinate system, ⎛ θ% − ηθ ⎞ ∂W ⎛ θ% − ηθ ⎞ ∂W % θ% − α sin ⎜ drd ⎟ ⎟ α ⎠ ∂r% ⎝ α ⎠ ∂θ% 1 ( r% − γ ro ) cos ⎜ αβ ∫∫ ⎝ Ω = = 1 ∫∫ αβ ( r% − γ ro ) cos ⎛ θ% − ηθ o ⎞ ⎛ ∂W 1 αβ ∫∫ ⎜ ⎝ r% Ω α ∂x% ∂W ∂y% ⎞ α ⎛ θ% − ηθ o ⎞ ⎛ ∂W ∂x% ∂W ∂y% ⎞ % % sin + − ⎜ ⎟ ⎟ ∂x% ∂r% ∂y% ∂r% ⎜ α ⎟ ⎜ ∂x% ∂θ% + ∂y% ∂θ% ⎟ dxdy ⎠⎝ ⎝ ⎠⎝ ⎠ r% ⎠ ( r% − γ ro ) cos ⎛ θ% − ηθ o ⎞ ⎛ ∂W Ω r% ⎜ ⎝ α x% ∂W y% ⎞ α ∂W ⎛ θ% − ηθ o ⎞ ⎛ ∂W ⎟ ⎜ ∂x% r% + ∂y% r% ⎟ − r% sin ⎜ α ⎟ ⎜ − y% ∂x% + x% ∂y% ⎠⎝ ⎝ ⎠⎝ ⎠ = ⎛ ⎛ r% − γ ro ⎞⎛ x% ⎞ ⎛ θ% − ηθ o ⎞ ⎛ α y% ⎞ ⎛ θ% − ηθ o ⎞ ⎞ ∂W % % cos dxdy ⎜ ⎜ ⎟+⎜ ⎟ ⎟⎟ ⎟ sin ⎜ ⎜ ⎜ r% ⎟⎠ ⎜⎝ r% ⎟⎠ αβ ∫∫ ⎝ α ⎠ ⎝ r% ⎠ ⎝ α ⎠ ⎠ ∂x% Ω ⎝⎝ + ⎛ ⎛ r% − γ ro ⎞ ⎛ x% ⎞ ⎛ θ% − ηθ o ⎞ ⎛ α y% ⎞ ⎛ θ% − ηθ o ⎞ ⎞ ∂W % % cos dxdy ⎜ ⎜ ⎟−⎜ ⎟ ⎟⎟ ⎟ sin ⎜ ⎜ ⎜ r% ⎟⎠ ⎜⎝ r% ⎟⎠ αβ ∫∫ ⎝ α ⎠ ⎝ r% ⎠ ⎝ α ⎠ ⎠ ∂y% Ω ⎝⎝ ⎞ % % ⎟ dxdy ⎠ 1 1 Combining the terms and substituting, ∫∫ vµ Ω o ⎛ ∂W ∂W ⎞ −My ⎜Mx ⎟dxdy ∂y ∂x ⎠ ⎝ = ∫∫ vµo M x Ω ∂W ∂W dxdy − ∫∫νµo M y dxdy ∂y ∂x Ω 34 = ⎛ ⎛ r% − γ ro ⎞⎛ x% ⎞ ⎛ θ% − ηθ o ⎞ ⎛ α y% ⎞ ⎛ θ% − ηθ o ⎞ ⎞ ∂W % % sin cos v µ M dxdy − ⎜ ⎜ ⎟ ⎜ ⎟ ⎟⎟ o x ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎝ r% ⎠ ⎝ r% ⎠ % % αβ ∫∫ α r α x ∂ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎠ Ω ⎝ − ⎛ ⎛ r% − γ ro ⎞⎛ x% ⎞ ⎛ θ% − ηθ o ⎞ ⎛ α y% ⎞ ⎛ θ% − ηθ o ⎞ ⎞ ∂W % % cos v µ M dxdy ⎜ ⎜ ⎟+⎜ ⎟ ⎟⎟ o y ⎜ ⎟ ⎟ sin ⎜ ⎜ ⎟ ⎜ ⎝ r% ⎠ ⎝ r% ⎠ % % αβ ∫∫ α r α x ∂ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎠ Ω ⎝ + ⎛ ⎛ r% − γ ro ⎞⎛ y% ⎞ ⎛ θ% − ηθ o ⎞ ⎛ α x% ⎞ ⎛ θ% − ηθ o ⎞ ⎞ ∂W % % sin cos v µ M dxdy + ⎜ ⎜ ⎟ ⎜ ⎟ ⎟⎟ o x ⎜ ⎟ ⎜ ⎟ ⎜ ⎜⎝ r% ⎟⎠ ⎝ r% ⎠ αβ ∫∫ ⎝ α ⎠ ⎝ r% ⎠ ⎝ α ⎠ ⎠ ∂y% Ω ⎝ − ⎛ ⎛ r% − γ ro ⎞⎛ x% ⎞ ⎛ θ% − ηθ o ⎞ ⎛ α y% ⎞ ⎛ θ% − ηθ o ⎞ ⎞ ∂W % % cos v µ M dxdy ⎜ ⎜ ⎟−⎜ ⎟ ⎟⎟ o y ⎜⎜ ⎟ sin ⎜ ⎟⎜ ⎟ αβ ∫∫ ⎝ α ⎠ ⎝ r% ⎠ ⎝ α ⎠ ⎠ ∂y% Ω ⎝ ⎝ r% ⎠ ⎝ r% ⎠ 1 1 1 1 Letting ⎛ ⎛ r% − γ ro ⎞⎛ x% ⎞ ⎛ θ% − ηθ o ⎞ ⎛ α y% ⎞ ⎛ θ% − ηθ o ⎞ ⎞ sin ⎜ cos ⎜ ⎟−⎜ ⎟ ⎟⎟ ⎜ ⎟ ⎟ ⎟ ⎝ α ⎠⎠ ⎝ ⎝ r% ⎠ ⎝ r% ⎠ ⎝ α ⎠ ⎝ r% ⎠ ϑ1 ( x% , y% ) = M x ⎜⎜ ⎜ ⎛ ⎛ r% − γ ro ⎞ ⎛ x% ⎞ ⎛ θ% − ηθ o ⎞ ⎛ α y% ⎞ ⎛ θ% − ηθ o ⎞ ⎞ −M y ⎜ ⎜ cos ⎜ ⎟+⎜ ⎟ ⎟⎟ ⎜ ⎟ ⎟ sin ⎜ ⎟ ⎜ ⎝ α ⎠ ⎝ r% ⎠ ⎝ α ⎠ ⎠ ⎝ ⎝ r% ⎠ ⎝ r% ⎠ and writing ⎛ ⎛ r% − γ ro ⎞⎛ y% ⎞ ⎛ θ% − ηθ o ⎞ ⎛ α x% ⎞ ⎛ θ% − ηθ o ⎞ ⎞ sin ⎜ cos ⎜ ⎟+⎜ ⎟ ⎟⎟ ⎜ ⎟ ⎟ ⎟ ⎝ α ⎠⎠ ⎝ ⎝ r% ⎠ ⎝ r% ⎠ ⎝ α ⎠ ⎝ r% ⎠ ϑ2 ( x% , y% ) = M x ⎜⎜ ⎜ ⎛ ⎛ r% − γ ro ⎞ ⎛ x% ⎞ ⎛ θ% − ηθ o ⎞ ⎛ α y% ⎞ ⎛ θ% − ηθ o ⎞ ⎞ −M y ⎜ ⎜ cos ⎜ ⎟−⎜ ⎟ ⎟⎟ ⎜ ⎟ ⎟ sin ⎜ ⎟ ⎜ ⎝ α ⎠ ⎝ r% ⎠ ⎝ α ⎠ ⎠ ⎝ ⎝ r% ⎠ ⎝ r% ⎠ 35 We can therefore write ∫∫ vµ Ω o ⎛ νµ o vµ ∂W ∂W ⎞ ∂W ∂W % % + o ϑ2 ( x% , y% ) % % −My ϑ1 ( x% , y% ) dxdy dxdy ⎜Mx ⎟dxdy = ∫∫ ∂y ∂x ⎠ ∂x% ∂y% αβ αβ ⎝ Ω (2.20) 2.5 Transformation of Weight Functions W1 ( x, y ) = 1 [a1 x + b1 y + c1 ] 2A ∂W1 1 ⎡ ⎛ ∂x ∂r ∂x ∂θ ⎞ ⎛ ∂y ∂r ∂y ∂θ ⎞ ⎤ a1 ⎜ = + + b1 ⎜ + ⎟ ⎟⎥ ⎢ ∂x% 2 A ⎣ ⎝ ∂r ∂x% ∂θ ∂x% ⎠ ⎝ ∂r ∂x% ∂θ ∂x% ⎠ ⎦ = 1 ⎡ ∂x ⎛ ∂r ∂r% ∂r ∂θ% ⎞ ∂x a1 ⎢ ⎜ + ⎟+ 2 A ⎢⎣ ∂r ⎝ ∂r% ∂x% ∂θ% ∂x% ⎠ ∂θ ⎛ ∂θ ∂r% ∂θ ∂θ% ⎞ ⎤ + ⎜ ⎟⎥ % ⎝ ∂r% ∂x% ∂θ ∂x% ⎠ ⎥⎦ + 1 ⎡ ∂y ⎛ ∂r ∂r% ∂r ∂θ% ⎞ ∂y ⎛ ∂θ ∂r% ∂θ ∂θ% ⎞ ⎤ b1 ⎢ ⎜ + + ⎟+ ⎜ ⎟⎥ 2 A ⎢⎣ ∂r ⎝ ∂r% ∂x% ∂θ% ∂x% ⎠ ∂θ ⎝ ∂r% ∂x% ∂θ% ∂x% ⎠ ⎥⎦ since x = f (r ,θ ) and y = f (r ,θ ) when the equations are transformed to polar coordinates. Affine transformation provides the second set of relations i.e. r = g (r%,θ% ) and θ = g (r%,θ% ) while the third transformation converts the system of equations back to Cartesian coordinate system through the relation r% = h( x% , y% ) and θ% = h( x% , y% ) ⎛ 1 x% ⎞ ⎛ 1 x% ⎞ 1 ⎡ ∂W1 ⎛ 1 y% ⎞ ⎤ 1 ⎡ ⎛ 1 y% ⎞ ⎤ a1 ⎢ cos θ ⎜ b r θ θ sin cos = + r sin θ ⎜ + + ⎥ ⎢ ⎟ ⎜ ⎟ 1 ⎟ ⎜ % 2 ⎟⎥ 2 ∂x% 2A ⎣ ⎝ α r% ⎠ ⎦ 2 A ⎣ ⎝ α r ⎠⎦ ⎝ β r% ⎠ ⎝ β r% ⎠ 36 ⎡ ⎛1 1 ∂W2 a2 ⎢ cos θ ⎜ = 2A ⎣ ∂x% ⎝β x% ⎞ ⎛1 + r sin θ ⎜ %r ⎟⎠ ⎝α ⎡ ⎛1 y% ⎞ ⎤ 1 b2 ⎢sin θ ⎜ + 2 ⎟⎥ r% ⎠ ⎦ 2 A ⎣ ⎝β x% ⎞ ⎛1 + r cos θ ⎜ %r ⎟⎠ ⎝α y% ⎞ ⎤ ⎟⎥ r% 2 ⎠ ⎦ ⎡ ⎛ 1 x% ⎞ ⎛ 1 x% ⎞ ∂W3 1 ⎛ 1 y% ⎞ ⎤ 1 ⎡ ⎛ 1 y% ⎞ ⎤ a3 ⎢cos θ ⎜ b3 ⎢sin θ ⎜ = + r sin θ ⎜ + + r cosθ ⎜ ⎟ ⎟ 2 ⎟⎥ 2 ⎟⎥ 2A ⎣ ∂x% ⎝ α r% ⎠ ⎦ 2 A ⎣ ⎝ α r% ⎠ ⎦ ⎝ β r% ⎠ ⎝ β r% ⎠ Similarly, the differentiation of W with respect to y% is given by ⎛ ∂y ∂r ∂y ∂θ ⎞ ⎤ ∂W1 1 ⎡ ⎛ ∂x ∂r ∂x ∂θ ⎞ = + + b1 ⎜ + ⎢ a1 ⎜ ⎟ ⎟⎥ ∂y% 2 A ⎣ ⎝ ∂r ∂y% ∂θ ∂y% ⎠ ⎝ ∂r ∂y% ∂θ ∂y% ⎠ ⎦ = 1 ⎡ ∂x ⎛ ∂r ∂r% ∂r ∂θ% ⎞ ∂x ⎛ ∂θ ∂r% ∂θ ∂θ% ⎞ ⎤ a1 ⎢ ⎜ + + ⎟+ ⎜ ⎟⎥ 2 A ⎢⎣ ∂r ⎝ ∂r% ∂y% ∂θ% ∂y% ⎠ ∂θ ⎝ ∂r% ∂y% ∂θ% ∂y% ⎠ ⎥⎦ + 1 ⎡ ∂y ⎛ ∂r ∂r% ∂r ∂θ% ⎞ ∂y ⎛ ∂θ ∂r% ∂θ ∂θ% ⎞ ⎤ b1 ⎢ ⎜ + + ⎟+ ⎜ ⎟⎥ 2 A ⎢⎣ ∂r ⎝ ∂r% ∂y% ∂θ% ∂y% ⎠ ∂θ ⎝ ∂r% ∂y% ∂θ% ∂y% ⎠ ⎥⎦ = ⎛ 1 y% ⎞ ⎛ 1 y% ⎞ 1 ⎡ ⎛ 1 x% ⎞ ⎤ 1 ⎡ ⎛ 1 x% ⎞ ⎤ a1 ⎢cosθ ⎜ b1 ⎢sin θ ⎜ − r sin θ ⎜ + + r cosθ ⎜ ⎟ ⎟ 2 ⎟⎥ 2 ⎟⎥ 2A ⎣ ⎝ α r% ⎠ ⎦ 2 A ⎣ ⎝ α r% ⎠ ⎦ ⎝ β r% ⎠ ⎝ β r% ⎠ Similarly, ⎡ ⎡ ⎛ 1 y% ⎞ ⎛ 1 y% ⎞ ∂W2 1 ⎛ 1 x% ⎞ ⎤ 1 ⎛ 1 x% ⎞ ⎤ a2 ⎢ cos θ ⎜ b2 ⎢sin θ ⎜ = − r sin θ ⎜ + + r cos θ ⎜ ⎟ ⎟ 2 ⎟⎥ 2 ⎟⎥ 2A ⎣ ∂y% ⎝ α r% ⎠ ⎦ 2 A ⎣ ⎝ α r% ⎠ ⎦ ⎝ β r% ⎠ ⎝ β r% ⎠ ⎡ ⎛ 1 y% ⎞ ⎛ 1 y% ⎞ ∂W3 1 ⎛ 1 x% ⎞ ⎤ 1 ⎡ ⎛ 1 x% ⎞ ⎤ a3 ⎢cos θ ⎜ b3 ⎢sin θ ⎜ = − r sin θ ⎜ + + r cos θ ⎜ ⎟ ⎟ 2 ⎟⎥ 2 ⎟⎥ 2A ⎣ ∂y% ⎝ α r% ⎠ ⎦ 2 A ⎣ ⎝ α r% ⎠ ⎦ ⎝ β r% ⎠ ⎝ β r% ⎠ 37 By making the following substitutions ⎛ 1 x% ⎞ ⎛ 1 y% ⎞ + r sin θ ⎜ ⎟ 2 ⎟ ⎝ α r% ⎠ ⎝ β r% ⎠ ψ 1 ( x%, y% ) = cosθ ⎜ ⎛ 1 x% ⎞ ⎛ 1 y% ⎞ + r cosθ ⎜ ⎟ 2 ⎟ ⎝ α r% ⎠ ⎝ β r% ⎠ ψ 2 ( x% , y% ) = sin θ ⎜ ⎛ 1 y% ⎞ ⎛ 1 y% ⎞ − r sin θ ⎜ ⎟ 2 ⎟ ⎝ α r% ⎠ ⎝ β r% ⎠ ψ 3 ( x%, y% ) = cosθ ⎜ ⎛ 1 y% ⎞ ⎛ 1 x% ⎞ + r cosθ ⎜ ⎟ 2 ⎟ ⎝ α r% ⎠ ⎝ β r% ⎠ ψ 4 ( x% , y% ) = sin θ ⎜ We can write the equations as ∂W1 1 1 = a1ψ 1 ( x%, y% ) + b1ψ 2 ( x% , y% ) 2A 2A ∂x% ∂W2 1 1 = a2ψ 1 ( x% , y% ) + b2ψ 2 ( x%, y% ) 2A 2A ∂x% ∂W3 1 1 a3ψ 1 ( x%, y% ) + b3ψ 2 ( x% , y% ) = 2A 2A ∂x% ∂W1 1 1 = a1ψ 3 ( x% , y% ) + b1ψ 4 ( x%, y% ) 2A 2A ∂y% ∂W2 1 1 = a2ψ 3 ( x%, y% ) + b2ψ 4 ( x% , y% ) 2A 2A ∂y% ∂W3 1 1 a3ψ 3 ( x% , y% ) + b3ψ 4 ( x% , y% ) = % 2A 2A ∂y 38 3 Computational Aspects 3.1 Finite Element Discretization In finite element analysis, the problem space is broken down into smaller domains called finite elements, in this case into discrete triangular elements . The Poisson equation discussed in the preceding section is solved over each triangular element space Ωe . Writing the weight function in matrix form, we have ⎡W1 ⎤ W = ⎢⎢W2 ⎥⎥ ⎢⎣W3 ⎥⎦ The first order differentials with respect to transformed coordinate system x% and y% are then given by ⎡ a1ψ 3 + b1ψ 4 ⎤ ⎡ a1ψ 1 + b1ψ 2 ⎤ 1 ⎢ 1 ⎢ ∂W ∂W ⎥ a2ψ 1 + b2ψ 2 ⎥ and a2ψ 3 + b2ψ 4 ⎥⎥ = = ⎢ ⎢ ∂y% 2 A ∂x% 2 A ⎢⎣ a3ψ 3 + b3ψ 4 ⎦⎥ ⎣⎢ a3ψ 1 + b3ψ 2 ⎦⎥ Approximating the vector potential solution in an element u% = W1u%1 + W2u%2 + W3u%3 or u% = [W1 W2 ⎡ u%1 ⎤ W3 ] ⎢⎢u%2 ⎥⎥ ⎢⎣ u%3 ⎥⎦ 39 in the x and y coordinate system, the differentials of the vector potential in the x% and y% coordinate system are then given by ∂u% ⎡ ∂W1 = ∂x% ⎣⎢ ∂x% ∂W2 ∂x% 1 [a1ψ 1 + b1ψ 2 = 2A ∂u% ⎡ ∂W1 = ∂y% ⎢⎣ ∂y% ∂W2 ∂y% 1 = [a1ψ 3 + b1ψ 4 2A ⎡ u%1 ⎤ ∂W3 ⎤ ⎢ ⎥ u%2 ∂x% ⎥⎦ ⎢ ⎥ ⎢⎣ u%3 ⎥⎦ a2ψ 1 + b2ψ 2 ⎡ u%1 ⎤ a3ψ 1 + b3ψ 2 ] ⎢⎢u%2 ⎥⎥ ⎣⎢ u%3 ⎦⎥ ⎡ u%1 ⎤ ∂W3 ⎤ ⎢ ⎥ u%2 ∂y% ⎥⎦ ⎢ ⎥ ⎢⎣ u%3 ⎥⎦ a2ψ 3 + b2ψ 4 ⎡ u%1 ⎤ a3ψ 3 + b3ψ 4 ] ⎢⎢u%2 ⎥⎥ ⎢⎣u%3 ⎥⎦ The left-hand side of the Poisson Integral equation (2.19) affected by affine transformation is given by ν ∫∫ ϕ1 ( x% , y% ) Ω ⎛ ∂W ∂u% ∂W ∂u% ⎞ ∂W ∂u% ∂W ∂u% ∂x%∂y% + ν ∫∫ ϕ 2 ( x% , y% ) ⎜ + ∂x%∂y% + ν ∫∫ ϕ 3 ( x%, y% ) ∂x%∂y% ⎟ ∂x% ∂x% ∂y% ∂y% ⎝ ∂x% ∂y% ∂y% ∂x% ⎠ Ω Ω The reluctance ν is approximately constant (in a small element region) and can be taken out of the double integral equation. Considering each term of the integral equation separately, we have 40 ν ∫∫ ϕ1 ( x% , y% ) Ωe ∂W ∂u% % % dxdy ∂x% ∂x% ⎡ a1ψ 1 + b1ψ 2 ⎤ == ϕ ( x% , y% ) ⎢⎢ a2ψ 1 + b2ψ 2 ⎥⎥ [ a1ψ 1 + b1ψ 2 2 ∫∫ 1 4 A Ωe ⎣⎢ a3ψ 1 + b3ψ 2 ⎦⎥ ν a2ψ 1 + b2ψ 2 ⎡ u%1 ⎤ % % a3ψ 1 + b3ψ 2 ] ⎢⎢u%2 ⎥⎥ dxdy ⎣⎢ u%3 ⎦⎥ The second term of the integral is written as ⎛ ∂W ∂u% ∂W ∂u% ⎞ % % + ⎟dxdy ⎝ ∂x% ∂y% ∂y% ∂x% ⎠ ν ∫∫ ϕ 2 ( x%, y% ) ⎜ Ωe = ν ∫∫ ϕ 2 ( x% , y% ) Ωe ∂W ∂u% ∂W ∂u% % % + ν ∫∫ ϕ 2 ( x% , y% ) % % dxdy dxdy ∂x% ∂y% ∂y% ∂x% Ωe ⎡ a1ψ 1 + b1ψ 2 ⎤ = ϕ ( x%, y% ) ⎢⎢ a2ψ 1 + b2ψ 2 ⎥⎥ [ a1ψ 3 + b1ψ 4 2 ∫∫ 2 4 A Ωe ⎢⎣ a3ψ 1 + b3ψ 2 ⎥⎦ ν ⎡ a2ψ 3 + b1ψ 4 ⎤ + 2 ∫∫ ϕ 2 ( x% , y% ) ⎢⎢ a2ψ 3 + b2ψ 4 ⎥⎥ [ a1ψ 1 + b1ψ 2 4 A Ωe ⎢⎣ a3ψ 3 + b3ψ 4 ⎥⎦ ν a2ψ 3 + b2ψ 4 ⎡ u%1 ⎤ % % a3ψ 3 + b3ψ 4 ] ⎢⎢u%2 ⎥⎥ dxdy ⎢⎣ u%3 ⎥⎦ a2ψ 1 + b2ψ 2 ⎡ u%1 ⎤ % % a3ψ 1 + b3ψ 2 ] ⎢⎢u%2 ⎥⎥ dxdy ⎢⎣ u%3 ⎥⎦ The last term on the left-hand side can be similarly written in matrix form as ν ∫∫ ϕ3 ( x% , y% ) Ωe ∂W ∂u% % % dxdy ∂y% ∂y% 41 ⎡ a1ψ 3 + b1ψ 4 ⎤ ϕ ( x%, y% ) ⎢⎢ a2ψ 3 + b2ψ 4 ⎥⎥ [ a1ψ 3 + b1ψ 4 = 2 ∫∫ 3 4 A Ωe ⎣⎢ a3ψ 3 + b3ψ 4 ⎦⎥ ν a2ψ 3 + b2ψ 4 ⎡ u%1 ⎤ % % a3ψ 3 + b3ψ 4 ] ⎢⎢u%2 ⎥⎥ dxdy ⎣⎢ u%3 ⎦⎥ Numerical integration is required to evaluate all three terms on the left-hand side of the Poisson equation, applying first isoparametric element transformation followed by Gauss Quadrature. These methods will be discussed further in the next two sections. After numerical integration, the sum of the three matrices above results in the stiffness matrix of the element and can be simply written as ⎡ sii sij s ji 4 A2 ⎢ ⎢ ski ⎣ s jj skj ν ⎢ sik ⎤ ⎡ u%i ⎤ ⎥ s jk ⎥ ⎢⎢u% j ⎥⎥ skk ⎥⎦ ⎢⎣u%k ⎥⎦ (3.1) On the right-hand side, νµo αβ ∫∫ ϑ ( x%, y% ) 1 Ωe νµo = 2 Aαβ νµ ∂W % %+ o dxdy ∂x% αβ ∫∫ ϑ ( x%, y% ) 2 Ωe ∂W % % dxdy ∂y% ⎡ a1ψ 1 + b1ψ 2 ⎤ ⎢ ⎥ % % + νµ o ∫∫Ω ϑ1 ( x%, y% ) ⎢a2ψ 1 + b2ψ 2 ⎥ dxdy 2 Aαβ e ⎢⎣ a3ψ 1 + b3ψ 2 ⎥⎦ ⎡ a1ψ 3 + b1ψ 4 ⎤ ⎢ ⎥ % % ∫∫Ω ϑ2 ( x%, y% ) ⎢a2ψ 3 + b2ψ 4 ⎥ dxdy e ⎢⎣ a3ψ 3 + b3ψ 4 ⎥⎦ 42 1 = 2 µ r Aαβ ⎡ a1ψ 1 + b1ψ 2 ⎤ 1 % %+ ϑ1 ( x%, y% ) ⎢⎢ a2ψ 1 + b2ψ 2 ⎥⎥ dxdy ∫∫ 2µ r Aαβ Ωe ⎣⎢ a3ψ 1 + b3ψ 2 ⎦⎥ ⎡ a1ψ 3 + b1ψ 4 ⎤ % % ϑ2 ( x%, y% ) ⎢⎢ a2ψ 3 + b2ψ 4 ⎥⎥ dxdy ∫∫ Ωe ⎣⎢ a3ψ 3 + b3ψ 4 ⎦⎥ Note that the right-hand side term exists only for elements in the magnet regions, so called the magnet equivalent current component attributed to the permanent magnets. Computing the double-integral in the element, the righthand side term can be combined and simply written as ⎡ J eq ,1 ⎤ 1 ⎢ ⎥ J eq ,2 ⎥ ⎢ 2 µ r Aαβ ⎢ J eq ,3 ⎥ ⎣ ⎦ (3.2) Here the J eq components are the results of computation of the double integrals. In the case of magnet regions, where the right-hand side of the Poisson Equation is non-zero due to the existence of magnet equivalent current above, we will assume that the magnet is operating in the second quadrant of its B-H curve and is approximately linear as shown in Fig. 3.1. Consequently, the relative permeability of permanent magnet µ r is taken as a constant and given by the relative permeability of the magnet material used. 43 0.8 0.6 Flux Density, Tesla 0.4 0.2 -600 -500 -400 -300 -200 -100 Field Intensity kA/m Fig. 3.1: Linear approximation of permanent magnet B-H curve The assumption that reluctance ν is constant is valid only when computing the integrals in a small finite element. Over the entire problem space, which consists of magnet and non-linear iron regions, the reluctance ν is not constant. There will be variation in the reluctance as it is a function of its magnetic flux density. An example of a non-linear material is iron with B-H curve shown in Fig. B.3. To solve problems involving non-linear materials, the Newton-Raphson method is employed. Its application in this project will be discussed later. 44 3.2 Triangular Isoparametric Element Given the fact that the integrands cannot be analytically solved, the integration of the stiffness matrix and forcing function has to be done numerically over the region of the finite element triangle of arbitrary shape. Given irregular shapes of elements, numerical integration can be slow as it takes into account each integration region. On the other hand, if the arbitrary triangular element can be first transformed into a standard triangle, the integration will have fixed limits and a standard algorithm can be used to compute for all elements in the problem space. An isoparametric triangular element [20] is ideal for this purpose. Fig. 3.2 shows the template triangular element that all arbitrary elements will be transformed into prior to numerical integration. η 3 1 η=1 ε =1 2 ε Fig. 3.2: Triangular Isoparametric Element 45 Shape functions of the linear triangular isoparametric element shown in Fig. 3.2 are given by H1 = 1 − ε − η H2 = ε H3 = η The values of x and y in the isoparametric coordinate system are then given by x = H1 x1 + H 2 x2 + H 3 x3 y = H1 y1 + H 2 y2 + H 3 y3 where ( x1 , y1 ) , ( x2 , y2 ) and ( x3 , y3 ) are the coordinates in the Cartesian coordinate system. It follows that by differentiating the above transformations with respect to ε and η , the following partial differentials are obtained: ∂x = − x1 + x2 ∂ε ∂x = − x2 + x3 ∂η ∂y = y2 − y1 ∂ε ∂y = y3 − y2 ∂η 46 The Jacobian of the transformation J and its inverse R are then given by ⎡ ∂x ⎢ ∂ε [ J ] = ⎢ ∂x ⎢ ⎢⎣ ∂η ∂y ⎤ ∂ε ⎥ ⎥ ∂y ⎥ ∂η ⎥⎦ and [ R] = 1 det J ⎡ y3 − y2 ⎢x − x ⎣ 2 3 y1 − y2 ⎤ x2 − x1 ⎥⎦ where det J = ( x2 − x1 )( y3 − y2 ) − ( x3 − x2 )( y2 − y1 ) It also follows from the definition of H1 , H 2 and H 3 ∂H1 = −1 ∂ε ∂H 2 =1 ∂ε ∂H 3 =0 ∂ε ∂H1 = −1 ∂η ∂H 2 =0 ∂η ∂H 3 =1 ∂η ⎡ ∂∂Hxi ⎤ ⎡ ∂∂Hεi ⎤ Given that ⎢ ∂H i ⎥ = [ R ] ⎢ ∂H i ⎥ ⎢⎣ ∂y ⎥⎦ ⎢⎣ ∂η ⎥⎦ 47 The following transformations are obtained y2 − y3 y2 − y1 ⎡ ∂∂Hx1 ⎤ ⎡ −1⎤ ⎡ ( x2 − x1 )( y3 − y2 )−( x3 − x2 )( y2 − y1 ) + ( x2 − x1 )( y3 − y2 )−( x3 − x2 )( y2 − y1 ) ⎤ ⎥ ⎢ ∂H1 ⎥ = [ R ] ⎢ ⎥ = ⎢ x3 − x2 x1 − x2 1 − ⎣ ⎦ ⎢⎣ ( x2 − x1 )( y3 − y2 )−( x3 − x2 )( y2 − y1 ) + ( x2 − x1 )( y3 − y2 )−( x3 − x2 )( y2 − y1 ) ⎥⎦ ⎢⎣ ∂y ⎥⎦ y3 − y2 ⎡ ∂∂Hx2 ⎤ ⎡1 ⎤ ⎡ ( x2 − x1 )( y3 − y2 ) −( x3 − x2 )( y2 − y1 ) ⎤ ⎥ ⎢ ∂H 2 ⎥ = [ R ] ⎢ ⎥ = ⎢ x2 − x3 0 ⎣ ⎦ ⎢⎣ ( x2 − x1 )( y3 − y2 ) −( x3 − x2 )( y2 − y1 ) ⎥⎦ ⎢⎣ ∂y ⎥⎦ y1 − y2 ⎡ ∂∂Hx3 ⎤ ⎡0 ⎤ ⎡ ( x2 − x1 )( y3 − y2 )− ( x3 − x2 )( y2 − y1 ) ⎤ ⎥ ⎢ ∂H 3 ⎥ = [ R ] ⎢ ⎥ = ⎢ x2 − x1 1 ⎢ ⎥ ⎢⎣ ∂y ⎥⎦ ⎣ ⎦ ⎣ ( x2 − x1 )( y3 − y2 )− ( x3 − x2 )( y2 − y1 ) ⎦ The vector potential within an isoparametric element is given by A = H1 A1 + H 2 A2 + H 3 A3 The partial differentials with respect to x and y are given by ∂H 3 ∂A ∂H ∂H 2 = A1 1 + A2 + A3 ∂x ∂x ∂x ∂x and ∂H 3 ∂H ∂H 2 ∂A = A1 1 + A2 + A3 ∂y ∂y ∂y ∂y 48 As a way of showing how useful this transformation really is, a simple example of a double-integration is given below over a single triangular element. y (2,2) (4,0) x Fig. 3.3: Sample problem for isoparametric transformation Consider the problem of finding the solution to the double integral ∫∫ ( x + 2)dxdy ∆ over the triangular region ∆ bounded by the lines as shown in the Fig. 3.3 above. Working out the solution analytically, 49 2 x 4 − x+4 0 0 2 ∫∫ ( x + 2)dxdy = ∫ ∫ ( x + 2)dydx + ∫ ∆ 2 4 0 2 ∫ ( x + 2) dydx 0 = ∫ x( x + 2)dx + ∫ ( x + 2)(4 − x)dx 2 4 = ∫ ( x 2 + 2 x)dx + ∫ (− x 2 + 2 x + 8)dx 0 2 ⎛ x3 ⎞ + ⎜ − + x 2 + 8 x ⎟ 42 ⎝ 3 ⎠ 8 64 8 = + 4 − + 16 + 32 + − 4 − 16 3 3 3 48 = 3 = 16 = 3 x + x2 3 2 0 Now applying the isoparametric transformation, ∫∫ ( x + 2)dxdy ∆ 1 − ε +1 = 8∫ ∫ 0 [(1 − ε − η ) x1 + ε x2 + η x3 + 2]dη d ε 0 1 − ε +1 = 8∫ ∫ 0 (4ε + 2η + 2)dηd ε 0 1 = 8∫ [4ε (1 − ε ) + (1 − ε ) 2 + 2(1 − ε )]d ε 0 1 = 8∫ [4ε − 4ε 2 + 1 − 2ε + ε 2 + 2 − 2ε ]d ε 0 1 = 24 ∫ [1 − ε 2 ] 0 = 24[ε − ε3 3 1 = 24[1 − ] 3 = 16 ] after substituting x = H1 x1 + H 2 x2 + H 3 x3 , with ( x1 , y1 ) , ( x2 , y2 ) and ( x3 , y3 ) equal to (0, 0) , (4, 0) and (2, 2) respectively and Jacobian 50 J = ( x2 − x1 )( y3 − y2 ) − ( x3 − x2 )( y2 − y1 ) = 8 . Therefore it can be seen that the method of isoparametric transformation yields the same answer as the direct solution of the problem. 3.3 Two-Dimensional Numerical Integration The general form of the double integration of equations (2.19) and (2.20) after isoparametric transformation is given by 1 − ε +1 ∫∫ 0 f (ε ,η )dη d ε 0 The above integration is best solved using two-dimensional numerical integration using Gauss-Quadrature scheme [29] for its high convergence and accuracy as compared to the trapezium rule. In a Quadrature scheme, any double integral function can be approximated by 1 − ε +1 ∫∫ 0 0 n ⎡ n ⎤ f (ε ,η )dη d ε ≈ ∑ wi ⎢ ∑ w j f (ε i ,η j ) ⎥ i =1 ⎣ j =1 ⎦ (3.3) where ( wi , ε i ) is the set of weights and evaluation points obtained for 1dimensional Gauss Quadrature integrating from ε = 0 to ε = 1 and ( w j ,η j ) is a 51 series of weights and evaluation points for integrating from η = 0 to η = −ε + 1 . To illustrate this technique, consider a 4-point two-dimensional Gaussian Quadrature Scheme. The weights and evaluation points for integration from ε = 0 to ε = 1 are found to be Wtouter ⎡0.9306 ⎤ ⎡0.0694 ⎤ ⎡ 0.1739 ⎤ ⎢0.6700 ⎥ ⎢0.3300 ⎥ ⎢ 0.3261⎥ ⎢ ⎥ ⎢ ⎥ ⎥ = and ε outer = and ηouter = −ε outer + 1 = ⎢ ⎢ 0.3261⎥ ⎢0.3300 ⎥ ⎢0.6700 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ 0.1739 ⎦ ⎣0.0694 ⎦ ⎣0.9306 ⎦ For each ε outer , we need to evaluate 1 ⎡ 1 ⎢ c (x ) c (x ) 1 2 ⎢ 1 1 ⎢ M M ⎢ L ⎣⎢ cn −1 ( x1 ) L L O L 1 ⎡ ⎤ ⎢ ηouter ⎥ 1 ⎤ ⎡ w1 ⎤ ⎢ c1 ( x)dx ⎥ ⎥ c1 ( xn ) ⎥⎥ ⎢⎢ w2 ⎥⎥ ⎢ ∫0 ⎥ =⎢ M ⎥⎢ M ⎥ ⎢ M ⎥ ⎥ ⎢ ⎥ ⎢ηouter ⎥ cn −1 ( xn ) ⎦⎥ ⎣⎢ wn ⎦⎥ ⎢ c ( x)dx ⎥ ⎢⎣ ∫0 n −1 ⎥⎦ to obtain the weights and solving for the roots of cn ( x) such that ηouter ∫ cn ( x)ck ( x)dx = 0 for all k < n 0 In order to make use of the Legendre Polynomials ( cn ( x) = Pn ( x) ) given in the appendix, there is a need to make a simple transformation in order to change 52 the default limit from x = −1 to x = 1 to limit x = 0 t o x = a . Let y = Therefore x = 2y dx 2 − 1 and = . When x = −1 , y = 0 and when x = 1 , y = a . a dy a ⎛ 2y ⎞ ⎛ 2y ⎞ 2 ∫−1 Pn ( x) Pk ( x)dx = 0 ⇒ ∫0 Pn ⎜⎝ a − 1⎟⎠ Pk ⎜⎝ a − 1⎟⎠ a dy = 0 1 a a x+ . 2 2 a ηouter weights can be solved by computing ∫ 0 for all k < n and the ⎛ 2x ⎞ 2 Pk ⎜ dx on the right-hand − 1⎟ ⎝ ηouter ⎠ ηouter side of the linear system of equations and substituting for the roots of 2x 2 cn ( x) = ηouter Pn ( ηouter − 1) on the left-hand side (see appendix for further discussion on the Gaussian Quadrature technique). Table 3.1: Weights for Gauss Quadrature ε outer 0.0694 0.3300 0.6700 0.9306 0.1619 0.1165 0.0574 0.0121 0.3034 0.2185 0.1076 0.2260 0.3034 0.2185 0.1076 0.2260 0.1619 0.1165 0.0574 0.0121 53 Table 3.2: Evaluation Points for Gauss Quadrature ε outer 0.0694 0.3300 0.6700 0.9306 0.0646 0.0465 0.0229 0.0048 0.3071 0.2211 0.1089 0.0229 0.6235 0.4489 0.2211 0.0465 0.8660 0.6235 0.3071 0.0646 Fig. 3.4: Location of evaluation points for 16-point 2D Gauss Quadrature Consider a simple example to demonstrate the application of the above 16point Gauss Quadrature over the triangular region shown in Fig. 3.4. Working the solution out analytically, 54 1 1−ε ∫ ∫ εη dη d ε 0 0 1 1 = ∫ ε (1 − ε ) 2 d ε 2 0 1 1 = ∫ ( ε − 2ε 2 + ε 3 )d ε 20 1 1 ⎡ ε 2 2ε 3 ε 4 ⎤ = ⎢ − + ⎥ 2⎣ 2 3 4 ⎦0 1 ⎡1 2 1⎤ − + 2 ⎢⎣ 2 3 4 ⎥⎦ 1 = 24 ≈ 0.0417 = Computing the same problem with a 4-point Gauss Quadrature, y = 0.1739 × (0.9306 × (0.0121 × 0.0048 + 0.2260 × 0.0229 + 0.2260 × 0.0465 + 0.0121 × 0.0646)) + 0.3261 × (0.6700 × (0.0574 × 0.0229 + 0.1076 × 0.1089 + 0.1076 × 0.2211 + 0.0574 × 0.3071)) + 0.3261 × (0.3300 × (0.1165 × 0.0465 + 0.2185 × 0.2211 + 0.2185 × 0.4489 + 0.1165 × 0.6235)) + 0.1739 × (0.0694 × (0.1619 × 0.0646 + 0.3034 × 0.3071 + 0.3034 × 0.6235 + 0.1619 × 0.8660)) = 0.0417 The computation yields a numerical error of much less than 0.1% from the actual value. As can be seen from the following figure, the scheme has already converged to the actual solution with a 2-point Quadrature. Further refinements using higher Quadrature schemes (in this example using 4 or 8 55 points respectively) do not yield any appreciable dividends in terms of accuracy. Fig. 3.5: Convergence rate for different number of Gauss Points As this project requires the use of numerical Quadrature to evaluate the element stiffness matrix developed in the previous sections, it is important to decide on the Quadrature order early in order to ensure acceptable numerical error (less than 0.1%) without impeding computing speed. Fig. 3.5 shows the convergence rate for various Gauss Points for the example problem. It can be seen that the algorithm converges even when a 2-point Gauss Quadrature scheme was used. Therefore, in order to maintain accuracy and minimize 56 computation cost, a 4-Point Quadrature scheme is selected to compute all the double integrals in this project. 3.4 Linear System Solution by the Newton-Raphson Method The general Newton-Raphson method to obtain x, y and z such that F ( x, y, z ) = 0 can be simply written as ∇ F ⋅ ∆r = − F ( x , y , z ) In equation (3.4), (3.4) ∇F = ∂F ∂F ∂F i+ j+ k and ∆r = ∆xi + ∆yj + ∆zk . ∇F , the ∂x % ∂y % ∂z % % % % gradient function of F, is a vector that points to the steepest gradient of the function F at any particular point. The finite element equation for each element is given by equations (3.1) and (3.2) can be written as ⎡ sii ν ⎢ s ji 4 A2 ⎢ ⎢ ski ⎣ sij s jj skj sik ⎤ ⎡ ui ⎤ 1 ⎥ s jk ⎥ ⎢⎢u j ⎥⎥ = 2 µ r Aαβ skk ⎥⎦ ⎢⎣uk ⎥⎦ ⎡ J eq ,i ⎤ ⎢ ⎥ ⎢ J eq , j ⎥ ⎢ J eq ,k ⎥ ⎣ ⎦ (3.5) 57 Let ⎡ sii F= ν ⎢ s ji 4 A2 ⎢ ⎢ ski ⎣ sij s jj skj sik ⎤ ⎡ ui ⎤ 1 ⎥ s jk ⎥ ⎢⎢u j ⎥⎥ − 2 µ r Aαβ skk ⎥⎦ ⎣⎢uk ⎦⎥ ⎡ J eq ,i ⎤ ⎢ ⎥ ⎢ J eq , j ⎥ ⎢ J eq , k ⎥ ⎣ ⎦ ∇F = and ∆r = ∆ui i + ∆u j j + ∆uk k and defining % % % ⎡ sii ⎡ sii ⎤ ν ⎢ ⎥ 1 ∂ν ⎢ ∂F = s ji + s ji ∂ui 4 A2 ⎢ ⎥ 4 A2 ∂ui ⎢ ⎢ ski ⎢⎣ ski ⎥⎦ ⎣ ⎡ sii ⎤ ⎡ sii ⎢⎣ ski ⎥⎦ ⎢ ski ⎣ ν ⎢ ⎥ 1 ∂ν ∂B 2 ⎢ s ji + s ji = 4 A2 ⎢ ⎥ 4 A2 ∂B 2 ∂ui ⎢ sij s jj skj (3.6) ∂F ∂F ∂F i+ j+ k ∂ui % ∂u j % ∂uk % sik ⎤ ⎡ ui ⎤ ⎥ s jk ⎥ ⎢⎢u j ⎥⎥ skk ⎥⎦ ⎢⎣uk ⎥⎦ sij s jj skj ⎡ k ⎤ ⎢ ∑ sinun ⎥ ⎢ n =i ⎥ ⎡ sii ⎤ 2 ⎥ ν ⎢ ⎥ 1 ∂ν ∂B ⎢ k = s + 2 s jnun ⎥ ∑ ⎢ 2 ⎢ ji ⎥ 2 4A 4 A ∂B ∂ui ⎢ n =i ⎥ ⎣⎢ ski ⎦⎥ ⎢ k ⎥ ⎢ ∑ sknun ⎥ ⎣ n =i ⎦ sik ⎤ ⎡ ui ⎤ ⎥ s jk ⎥ ⎢⎢u j ⎥⎥ skk ⎥⎦ ⎢⎣uk ⎥⎦ (3.7) Similarly, 58 ⎡ k ⎤ ⎢ ∑ sinun ⎥ ⎢ n =i ⎥ ⎡ sij ⎤ 2 ⎥ ∂F ν ⎢ ⎥ 1 ∂ν ∂B ⎢ k = s + 2 s jnun ⎥ ∑ ⎢ 2 ⎢ jj ⎥ 2 ∂u j 4 A 4 A ∂B ∂u j ⎢ n =i ⎥ ⎢ skj ⎥ ⎣ ⎦ ⎢ k ⎥ ⎢ ∑ sknun ⎥ ⎣ n =i ⎦ (3.8) and ⎡ k ⎤ ⎢ ∑ sinun ⎥ ⎢ n =i ⎥ ⎡ sik ⎤ 2 ⎥ ∂F ν ⎢ ⎥ 1 ∂ν ∂B ⎢ k = s + 2 s jnun ⎥ ∑ ⎢ 2 ⎢ jk ⎥ 2 ∂uk 4 A 4 A ∂B ∂uk ⎢ n =i ⎥ ⎢⎣ skk ⎥⎦ ⎢ k ⎥ ⎢ ∑ sknun ⎥ ⎣ n =i ⎦ Substituting into equation (3.4), ∂F ∂F ∂F ∆ui + ∆u j + ∆uk = − F (ui , u j , uk ) ∂ui ∂u j ∂uk Writing this in matrix form, we have ⎡ sii ν ⎢ s ji 4 A2 ⎢ ⎢ ski ⎣ sij s jj skj ⎡ k ⎢ ∑ sinun ⎢ n =i sik ⎤ ⎡ ∆ui ⎤ 1 ∂ν ⎢ k ⎥ s jk ⎥ ⎢⎢ ∆u j ⎥⎥ + 2 ∑ s jnun 4 A ∂B 2 ⎢⎢ n =i ⎥ skk ⎦ ⎢⎣ ∆uk ⎥⎦ ⎢ k ⎢ ∑ skn un ⎣ n =i k ∑s n =i k ∑s u jn n n =i k ∑s n =i ⎤ u ⎥ n =i ⎥ k ⎥ s jnun ⎥ × ∑ n =i ⎥ k ⎥ sknun ⎥ ∑ n =i ⎦ k u in n u kn n ∑s in n 59 ⎡ ∂∂2uB ⎢ i ⎢0 ⎢ ⎢⎣ 0 0 ∂ B ∂u j 2 0 0 ⎤ ⎡ ∆ui ⎤ ⎥ −ν 0 ⎥ ⎢⎢ ∆u j ⎥⎥ = 4 A2 ⎥ ∂ 2 B ⎢ ∆u ⎥ ∂uk ⎥ ⎦⎣ k⎦ ⎡ sii ⎢ ⎢ s ji ⎢ ski ⎣ sij s jj skj sik ⎤ ⎡ ui ⎤ 1 ⎥ s jk ⎥ ⎢⎢u j ⎥⎥ + 2 µ r Aαβ skk ⎥⎦ ⎢⎣uk ⎥⎦ ⎡ J eq ,i ⎤ ⎢ ⎥ ⎢ J eq , j ⎥ ⎢ J eq , k ⎥ ⎣ ⎦ The vector potentials here are obtained from the previous iteration. (3.9) ∂ν is ∂B 2 found from the saturation curve representation of Fig. B.3 using cubic spline representation.. To obtain ⎛ ∂u ⎞ ⎛ ∂u ⎞ B = ⎜ ⎟ +⎜ 2 ⎟ ⎝ ∂x% ⎠ ⎝ ∂y% ⎠ 2 ∂B 2 , we proceed as follow ∂ui 2 2 For first order elements, u = Wi u1 + W2u2 + W3u3 From the previous section on transforming weight functions, we have derived ∂W1 1 1 = a1ψ 1 ( x%, y% ) + b1ψ 2 ( x% , y% ) ∂x% 2A 2A ∂W2 1 1 = a2ψ 1 ( x% , y% ) + b2ψ 2 ( x%, y% ) ∂x% 2A 2A ∂W3 1 1 = a3ψ 1 ( x%, y% ) + b3ψ 2 ( x% , y% ) ∂x% 2A 2A ∂W1 1 1 = a1ψ 3 ( x% , y% ) + b1ψ 4 ( x%, y% ) ∂y% 2A 2A ∂W2 1 1 = a2ψ 3 ( x%, y% ) + b2ψ 4 ( x% , y% ) ∂y% 2A 2A 60 ∂W3 1 1 = a3ψ 3 ( x% , y% ) + b3ψ 4 ( x% , y% ) ∂y% 2A 2A So ∂W2 ∂W ∂u ∂W1 ∂W2 ∂W ∂u ∂W1 = u1 + u2 + 1 u3 = u1 + u2 + 1 u3 and ∂y% ∂y% ∂y% ∂y% ∂x% ∂x% ∂x% ∂x% and therefore we have ∂W2 ∂W ⎞ ⎛ ∂W ∂W2 ∂W ⎞ ⎛ ∂W B 2 = ⎜ 1 u1 + u2 + 1 u3 ⎟ + ⎜ 1 u1 + u2 + 1 u3 ⎟ % % % % % ∂x ∂x ⎠ ⎝ ∂y ∂y ∂y% ⎝ ∂x ⎠ 2 2 giving the relations ∂W ⎛ ∂W ∂W2 ∂W ⎞ ∂W ⎛ ∂W ∂W2 ∂W ⎞ ∂B 2 u2 + 1 u3 ⎟ + 2 1 ⎜ 1 u1 + u2 + 1 u3 ⎟ = 2 1 ⎜ 1 u1 + % % % % % % % ∂ui ∂x ⎝ ∂x ∂x ∂x ⎠ ∂y ⎝ ∂y ∂y ∂y% ⎠ (3.10) ∂B 2 ∂W ⎛ ∂W ∂W2 ∂W ⎞ ∂W ⎛ ∂W ∂W2 ∂W ⎞ u2 + 1 u3 ⎟ + 2 2 ⎜ 1 u1 + u2 + 1 u3 ⎟ = 2 2 ⎜ 1 u1 + ∂u j ∂x% ⎝ ∂x% ∂x% ∂x% ⎠ ∂y% ⎝ ∂y% ∂y% ∂y% ⎠ (3.11) ∂W ⎛ ∂W ∂W ⎛ ∂W ∂W2 ∂W ⎞ ∂W2 ∂W ⎞ ∂B 2 = 2 3 ⎜ 1 u1 + u2 + 1 u3 ⎟ + 2 3 ⎜ 1 u1 + u2 + 1 u3 ⎟ ∂uk ∂x% ⎝ ∂x% ∂x% ∂x% ⎠ ∂y% ⎝ ∂y% ∂y% ∂y% ⎠ (3.12) Given that the equations (3.10) to (3.12) are functions of x% and y% , we can obtain the mean values in each element by integrating the differentials at the centroid of the elements. 61 3.5 Torque Computation There are many methods to approach the problem of computing torque from a finite element solution of vector potential at the nodes. From the literature on this subject [21],[23] the methods can be classified from the starting point of their derivation. The Maxwell Tensors Method, for example, is derived on the basis of Ampere’s Force Law r r r dF = J × B while the derivation of the Energy methods, such as the Virtual Work Method [21], the Energy Method [23] and the Simplified Energy Method [25], comes from the relation T =− ∂W ∂θ i.e. the torque is obtained from the derivative of the magnetostatic energy Wm with respect to the rotation angle θ . The differences mainly lie in the chosen path of computation, for example, in the case of the Maxwell Tensors Method, computation is normally done in the circular air gap region enclosing the interior stator. In the case of the simplified energy method, on the other hand, computation is preferably done along the boundary of the slots [25]. 62 In this project, the Maxwell Tensors’ method is used as the mesh data from the commercial software FLUX2D used in computing the reduced basis torque provides a middle air gap layer as shown in Fig. 3.6. The force density formulas for this method is well known and given by pt = Bn Bt pt = Bn2 − Bt2 2µo µo (3.13) (3.14) The force contribution of an element in the air gap can thus be computed as follows Ft = pt × d m × lairgap (3.15) Fn = pn × (( x1 − x2 ) 2 + ( y1 − y2 ) 2 ) × d m (3.16) 63 (an) (x2,y2) (at) (x1,y1) Origin Fig. 3.6: Computing torque in air gap elements for the case of the element shown in Fig. 3.6. For a rotating machine, torque is obtained by taking the product of the force and the perpendicular distance from the axis of rotation (origin). The perpendicular distance rp is measured from the origin to the middle of the air gap layer as shown in Fig. 3.6. Fn does not contribute to the torque in this case as it is a radial force and always parallel to rp . Therefore the torque contribution of this element is then given by 64 Te = rp × Ft and the net torque TR is obtained by summing up all the torque contributions in the middle air gap layer as TR = ∑ Te . 3.6 Reduced-Basis Formulation Equation (2.9) is reproduced here ⎛ ⎞ ⎛ ∂u% ∂W ∂u% ∂W ⎞ ⎛ ∂W ∂W ⎞ + −My ⎟dxdy = ∫∫ ⎜νµ o ⎜ M x ⎟ + J ⋅ W ⎟dxdy ∂x ∂y ∂y ⎠ ∂y ∂x ⎠ ⎝ Ω ⎝ ⎠ ∫∫ν ⎜⎝ ∂x Ω The above equation is in the form of a(u%, W ) = l (W ) as it can be shown that the left-hand side (LHS) of the equation is a bilinear equation while the right-hand side (RHS) is a linear equation (see appendix for definition). ⎛ ∂u% ∂W ∂u% ∂W ⎞ a (u% , W ) = ∫∫ν ⎜ + ⎟dxdy ⎝ ∂x ∂x ∂y ∂y ⎠ Ω and ⎛ ⎞ ⎛ ∂W ∂W ⎞ l (W ) = ∫∫ ⎜νµ o ⎜ M x −My ⎟ + J ⋅ W ⎟dxdy ∂y ∂x ⎠ ⎝ Ω ⎝ ⎠ Letting the parameter space ηi = (lmn ,θ mn ) we can write the above equation as 65 a (u (ηi ), W ;η ) = l (W ) In a finite element problem, both u (ηi ) and W belongs to the same solution space and the solution may be written as a linear sum of basis ς i for i = 1....n for an n-dimensional problem. Truncating to a reduced-basis N [...]... Fig 5.35: Regions of accuracy for different static meshes 118 Fig 5.36: Static mesh composition for geometry- dependent regions 122 Fig 5.37: Static mesh composition for geometry- independent regions 122 Fig 5.38: Curve-fitted torque computation for different N 125 Fig 5.39: Number of operation count for various N values 127 xii List of Tables Table 2.1: Values of weight function coefficients... solution of large linear system of equations, which is in itself time consuming if iterative methods such as the NewtonRaphson or the conjugate gradient method (CGM) are used to invert the 7 matrix Consequently, the use of finite element method as shown in Fig 1.1 involves iteration within an iteration, which requires a large computational effort Computational effort can be reduced if the iterative finite. .. the reduced- basis method to incorporate variations in geometric parameters was motivated by the need to combine the accuracy of finite element solution with the computational effectiveness of reduced- basis approximation [10] for the purpose of optimisation This led to the idea of “offline” and “online computation In the offline computation, the problem space not affected by geometric transformation... pre-computed The contribution of the geometry- dependent regions, on the other hand, has to be computed every time the reduced- basis method is applied to a new point in the parameter space [11] But provided that the parameter dependent region constitutes only a small fraction of the total problem domain, it can be 9 expected that the whole procedure of matrix construction and solution to be relatively... standard linear algebra and vector calculus for problems in two dimensions The objective of the transformations applied is to map meshes with the required geometric parameters into a “template” static mesh In the second chapter, the finite element stiffness matrix and forcing function for the transformed finite element formulae are developed Isoparametric transformation and Gauss Quadrature technique are... critical steps for the numerical evaluation of the forcing function and stiffness matrix The computation is then performed in Matlab [20], an ideal platform for handling matrix problems In this chapter, the Newton-Raphson method is introduced as a means of solving problems with non-linear materials, with the 13 necessary modifications developed to account for the geometric transformations Lastly, for the... Fig 4.13: Stiffness matrix assembly for geometry- dependent regions 94 Fig 4.14: On- line Procedure for computing torque 96 Fig 5.1: Variation of cogging torque with arc angle (FLUX2D) 98 Fig 5.2: Variation of cogging torque with radial thickness (FLUX2D) 98 Fig 5.3: Variation of cogging torque with arc angle from FLUX2D 99 Fig 5.4: Variation of cogging torque with radial thickness by FLUX2D... therefore a major design goal to eliminate or reduce this cogging effect The motivation for selecting cogging torque as a case study of the reducedbasis method is the fact that it is highly dependent on the machine geometry The variation of cogging torque with geometry has been a subject of extensive research [12]-[18] Dr Jabbar et al concluded in his papers in 1992 and 1993 that smaller cogging torque. .. computation of torque 1.5 Organization of Thesis This thesis is organized in the following way The following two chapters discuss the theoretical aspects of the reduced basis method In chapter one, the basic framework of the finite element method is explained Following the use of affine geometrical transformation, mathematical modification to the standard finite element codes is derived based on standard... need for geometry modification, remeshing and solving again for F 8 1.3 Introduction to Reduced Basis Technique The reduced basis method is essentially a scheme for approximating segments of a solution curve or surface defined by a system containing a set of free variables For each curve segment an approximate manifold is constructed that is “close” to the actual curve or surface The computational .. .GEOMETRY-DEPENDENT TORQUE OPTIMIZATION FOR SMALL SPINDLE MOTORS BASED ON REDUCED BASIS FINITE ELEMENT FORMULATION AZMI BIN AZEMAN (B.A (Hons.), M Eng., Cantab) A THESIS SUBMITTED FOR THE... STANDARD FINITE ELEMENT PROBLEM 20 2.4 APPLICATION OF AFFINE TRANSFORMATION ON STATIC MESH 24 2.5 TRANSFORMATION OF WEIGHT FUNCTIONS 36 COMPUTATIONAL ASPECTS 39 3.1 FINITE. .. Electromagnetic Finite Element Package A comparison is made between FLUX2D predictions against the Offline reduced- basis torque computation using one static mesh This is a crucial verification step as the Online

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