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COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Emerald Article: A robust objective function for topology optimization Min Li, David A. Lowther Article information: To cite this document: Min Li, David A. Lowther, (2011),"A robust objective function for topology optimization", COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, Vol. 30 Iss: 6 pp. 1829 - 1841 Permanent link to this document: http://dx.doi.org/10.1108/03321641111168138 Downloaded on: 15-05-2012 References: This document contains references to 15 other documents To copy this document: permissions@emeraldinsight.com This document has been downloaded 197 times. Access to this document was granted through an Emerald subscription provided by UNIVERSITY OF NEWCASTLE For Authors: If you would like to write for this, or any other Emerald publication, then please use our Emerald for Authors service. 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The current issue and full text archive of this journal is available at www.emeraldinsight.com/0332-1649.htm Topology A robust objective function optimization for topology optimization Min Li and David A. Lowther Department of Electrical and Computer Engineering, McGill University, 1829 Montreal, Canada Abstract Purpose – Robust design is very important for manufacturers to ensure the quality of the ﬁnished product. Therefore, a robustness measure is needed for the topological design of electromagnetic problems which may be sensitive to parameter variations. The purpose of this paper is to propose a robust objective function for topological design problems. Design/methodology/approach – In this paper, a robust objective function for topology optimization is deﬁned on an uncertainty set using the worst case analysis. The robustness of a topological design is deﬁned as the worst response due to the variations of the location of the topology change. The approach is based on the deﬁnition of a topological gradient. Findings – The robust topology optimization (RTO) was applied to eddy current crack reconstruction problems. The numerical applications showed that this method can provide more reliable results for the reconstruction in the presence of signiﬁcant noise in the measured signal. Research limitations/implications – The RTO may be applied to some more complicated design problems; however large computational costs may result. Originality/value – This paper has deﬁned a robustness metric for topology design and a robust design model is proposed for topology optimization problems. Keywords Sensitivity analysis, Topology optimization, Robust design, Electromagnetism, Eddy currents Paper type Research paper I. Introduction In recent years, computer-aided design has become the ﬁrst choice of design engineers when new electromagnetic devices are required. The physical problem is often mapped into a parametric search space, and the ﬁnal design is achieved through an automated design process which determines the optimal values of the parameters using an efﬁcient search algorithm. The physical prototype can then be implemented based on the speciﬁcation. If the newly designed device is to be manufactured, the tolerance problem must be taken into account. For example, in reality, small variations to the structure which can occur during the manufacturing process are not negligible, and these variations can sometimes cause serious deterioration in the performance of a device. Therefore, possible variations of the design variables should be handled and integrated into the mathematical formulation of the optimal design. Such a procedure is usually referred to as “robust design” since the performance of the resulting product is more stable. The robust design of electromagnetic devices has been studied for more than ten COMPEL: The International Journal years. Several practical approaches have been proposed in the past. One statistically for Computation and Mathematics in based formulation of robust design is to minimize the nominal value of the Electrical and Electronic Engineering Vol. 30 No. 6, 2011 performance function and its variation at the same time. Unfortunately, the statistical pp. 1829-1841 information is sometimes not available and it can be computationally expensive to q Emerald Group Publishing Limited 0332-1649 calculate the standard deviation. First-order sensitivity was used to approximate the DOI 10.1108/03321641111168138 COMPEL variance and the multi-objective cost optimization was solved using a weighting factor 30,6 in Yoon et al. (1999). Later, an alternative formulation based on the uncertainty set and worst performance analysis was proposed (Alotto et al., 2003; Steiner et al., 2004). A robust objective function was formulated as the worst performance of the function values in the small neighborhood representing the possible variations of the design variables. Examples of solving worst case tolerance design problems using 1830 evolutionary algorithms can be found in Spagnuolo (2003). On the other hand, statistical methods (e.g. Monte Carlo analysis) have also been employed for the robust design of electromagnetic devices (Cavaliere et al., 2003; Ciofﬁ et al., 2004). Most recently, a novel approach which treated robust optimization as one of conﬂict design was presented (Guimaraes et al., 2006). This method combines the ideas of multi-objective robust design formulation and the worst performance analysis. In this formulation, the worst case performance and the nominal value of the objective function can be minimized simultaneously. Therefore, more robust design solutions were provided to the designer and decisions can be made as a tradeoff between performance and reliability. Topology optimization is an important tool for device synthesis. Unlike shape optimization methods which must be based on an existing device and where the parameters are only allowed to change within a relatively small range, the optimal material distribution (OMD) method is able to represent all the potential layouts of the devices in the design space (Dyck and Lowther, 1996). Thus, topology optimization methods can provide innovative designs starting from an empty space without any assumptions. In practice, OMD is often used as a pre-processor for a boundary shape optimization which generates ﬁner details of the body. The grey-scale picture of the material density can be converted to a solid structure using image processing techniques (Choi et al., 1998). However, if the original topology is sensitive to small parameter variations, it may result in large changes in the ﬁnal shape obtained as well as performance deviations. Unfortunately, topics on managing uncertainties during topological design have rarely been discussed. The idea of reliability-based topology optimization (RBTO) was proposed and applied to the electromagnetic problems, e.g. the design of a c-core (Kang et al., 2004). Unfortunately, RBTO requires some probabilistic information and only the uncertainties of the environment are handled in the formulation. Rather than minimize the variation of the objective, RBTO focuses on the probability of the system response based on the distributions of the random parameters. Although there exist a variety of robust design formulations, they cannot be combined directly with traditional topological optimization methods which are based on homogenization or material distribution. Topological shape optimization, also known as the “bubble method”, emerged 30 years ago and has recently received intensive attention. The topological sensitivity analysis has been effectively applied to many different electromagnetic problems (Kim et al., 2008; Li and Lowther, 2010a, b). The topology of the design object, as well as the shape, is allowed to change during the design process. In addition, this formulation provides the possibility of incorporating a robustness measure into the design process. In this paper, we propose a robust design model for topology optimization. A robust objective function is deﬁned for the design problem based on the uncertainty set and the worst performance analysis. The robustness measure for the topology optimization is deﬁned for the ﬁrst time. The robust target function remains the same for both the Topology topological design and the shape optimization stages, despite the change of the design optimization parameters. The proposed robust topology optimization (RTO) formulation is also compared to the RBTO methods. II. Robust design formulation A. Non-robust design formulation 1831 The design of an electrical machine is typically stated as an optimization problem where an objective function is minimized under certain requirements of the performance and physical constraints. A general formulation of the design optimization problem can be expressed mathematically as: min fðxÞ s:t: gi ðxÞ # 0 i ¼ 1; 2; . . .m: ð1Þ where f is an objective function which indicates the performance of the design, x is a vector of the design variables; and gi are the inequality constraints on x. Usually, the performance of the design must meet a desired speciﬁcation. From the robust design point of view, f and x are just the nominal values of a design. In the presence of uncertainties, i.e. if there is a small variation to the nominal value of x, the value of f may no longer indicate a performance which meets the speciﬁcations, or the non-robust solution may become infeasible in the case of constrained problem. B. Worst performance due to perturbation Unlike the conventional formulation of design optimization, a robust design formulation must take account of the variation of the design variables. An uncertainty set U(x) is deﬁned as a compact sub-domain of the design space containing all the perturbations to the nominal values of the design variables x: Uðx0 Þ ¼ {j [ Rn : x0 2 D # j # x0 þ D}: ð2Þ where the vector D ¼ {D1 D2 . . . Dn}represents the largest variation to the nominal value x0 of the design variables. Sometimes the variations to the design variables may be deﬁned as proportional to the nominal value of every dimension, especially when the vector of design variables contains different physical quantities. Thus, the uncertainty set U(x) can be deﬁned as: Uðx0 Þ ¼ {j [ Rn : ð1 2 Di Þx0;i # ji # ð1 þ Di Þx0;1 }: ð3Þ where Di represents the largest variation to the ith component of vector x0 in per unit terms. Figure 1 shows a rectangular shape of the uncertainty set in 2D. ∆2* X0.2 X0 (X0.1, X0.2) ∆2 * X0.2 Figure 1. Rectangular uncertainty set ∆1* X0.1 ∆1* X0.1 COMPEL The advantage of the rectangular uncertainty set is the simplicity of its implementation and it does not require any statistical information about the design variables. 30,6 The worst performance of the objective function f due to the perturbation of the design variable x is given as: fw ¼ max fðjÞ ð4Þ j[UðxÞ 1832 Nevertheless, if the solution x * is located close to the boundary of the feasible region, the shape of the uncertainty set may be altered by the constraints. To ensure feasibility robustness, a robust constraint function is deﬁned as: max gi ðjÞ # 0 ð5Þ j[UðxÞ Overall, a robust design can be formulated as minimizing the worst performance of the objective function value within the uncertainty set: min max fðjÞ s:t: max gi ðjÞ # 0: x ð6Þ j[UðxÞ j[UðxÞ Unlike the non-robust function, the value of the robust objective function is determined by the worst performance, i.e. it is not sensitive to the perturbation of the nominal values of design variables. C. Multi-objective approach In the early attempts to tackle the robust design problem, Taguchi has developed a quality loss function (QLF) using the mean squared deviation to estimate the monetary loss as the performance deviation from the nominal value (Taguchi et al., 1989). A more general robust design formulation derived from Taguchi’s QLF is deﬁned as a multi-objective problem: " # E½fðxÞ min ð7Þ V½fðxÞ where E[f(x)] and V[f(x)] are the expectation (mean) and the variance of the objective function based on statistical models, respectively. In order to solve the problem without computing the statistical information, we may use the nominal value and the deviation of the worst performance instead of the expectation and the variance in equation (7) (Guimaraes et al., 2006): " # fðxÞ min : ð8Þ fw ðxÞ 2 fðxÞ The nominal value of the objective function is used to represent the mean, and the deviation of the performance is considered as the second objective. The problem in equation (8) can be solved using a weighting factor a: C ¼ ð1 2 aÞðfðxÞ þ aðfw ðxÞ 2 fðxÞÞ ¼ ð1 2 2aÞfðxÞ þ afw ðxÞ: ð9Þ If we choose a ¼ 0.5, C ¼ 0.5fw(x), and the worst case scenario is just one of the solutions to the multi-objective optimization. Using the multi-objective formulation, the robust design problem is treated as a Topology classical conﬂict design and, therefore, a pareto front of the potential robust solutions optimization can be generated. The most suitable solution is then chosen by the designer. III. Robust topology optimization A. Topological shape optimization In the conventional formulation of topology optimization based on the theory of 1833 homogenization or material distribution, the design space is usually divided into cells. And the design variables are the dimensions of the discretized elements. The possible values of these design variables are numbers between 0 and 1 which represent the intermediate states of the material properties. If only solid material is allowed in the design, there cannot be any variation to the nominal value of the design variables, since they must be either 0 or 1 at the end of the design. Therefore, it is not possible to apply the robust design formulation to conventional topology optimization methods. Topological shape optimization has been successfully applied to electromagnetic problems (Kim et al., 2008). This method has two stages: ﬁrst, generate topology changes to the domain of the design by creating a hole in the material based on the values of the topological gradient; second, optimize the shape of boundary of the hole using a standard shape optimizer (Figure 2). The objective function for the topological design is denoted as C in domain V. Consider an inﬁnitely small hole B(x, r) in V, where x is the center and r is the radius of the hole. The topological gradient TG(x) is deﬁned in (Li and Lowther 2010a) as: CðVÞ 2 CðV\Bðx; rÞÞ TGðxÞ ¼ lim ð10Þ r!0 dðrÞ where C is an objective function deﬁned on V, V\B(x, r) is the domain excluding the small hole B, and d(r) is the volume of B with a negative sign. In the topology optimization stage, the topology gradient is computed over the problem domain V. A small hole B(x, r) is drilled in V where the value of TG at x is greater than zero (Li and Lowther, 2010a). The performance function after this topology change is denoted as C(V\B(x, r)). The design variable of the topology design problem is the 3D coordinate of the center of the hole. Any variations of x may result in a potential topology change. The radius r is usually set as a ﬁxed length determined by the size of the cells in the grids or the size of the elements in the mesh. And the size and shape of the hole will be determined in the shape optimization stage for Γ x r Γr Ω Figure 2. A topology change to the problem domain COMPEL which the robust formulation will be discussed later. We deﬁne a scalar function 30,6 J(x) ¼ C(V\B(x, r)), and the worst performance function is given as: fw ¼ max JðjÞ: ð11Þ j[UðxÞ B. Robust formulation of the topological gradient 1834 The worst performance of the objective function can be predicted with the assistance of the topological gradient. Equation (10) can be re-arranged using an expansion on the objective function J near the point x0: JðxÞ ¼ CðV\ Bðx; rÞÞ ¼ CðVÞ þ dðrÞ · TGðxÞ þ oðrÞ: ð12Þ Since C(V) and d(r) are constants with respect to x (note that d(r) has a negative sign), J(j) has the largest value where TG(j) is the smallest. Therefore, the worst performance of J due to the perturbation of the design variables is determined by the point j in the uncertainty set, where TG(j) has the smallest value. Hence, we can use the value of TG(j) to replace the nominal value of TG(x) and a robust formulation for the TG can be obtained as: TGR ðxÞ ¼ min TGðjÞ: ð13Þ j[UðxÞ As a result, the robust formulation of the topological gradient TGR can be computed and used as a decision criterion on where the topology changes will take place. C. Worst vertex prediction for shape optimization After the topological analysis stage has determined where to drill a small hole in the problem domain, topological shape optimization is used to perform the second stage. The boundary of the hole is discretized and the shape of the boundary is optimized. As shown in Figure 3, several control points are created and the boundary is described using a B-spline curve. At the shape design stage, the design variables P ¼ [P1, P2, P3, P4, P5] are the coordinates of the control points on the boundary of the hole. The uncertainty set can be deﬁned as the perturbation to the nominal value of P, and the worst performance function can be applied in a similar fashion as in equation (4). As mentioned earlier, a hyper-rectangular uncertainty set is chosen. We have the assumption that the worst value of the performance function can be only found at one P1 Control points P5 P2 Figure 3. Boundary B-Spline parameterization P4 P3 of the vertices of the uncertainty set (Steiner et al., 2004), and the worst vertex can be Topology predicted as: optimization 0 1 signðfðx1þ Þ 2 fðx12 ÞÞ · D1 B . . .C B . . . C: xpred ¼ x0 þ B . . .C ð14Þ @ A 1835 signðfðxxþ Þ 2 fðxx2 ÞÞ · Dx Hence the worst performance is approximated by the function value at the worst vertex xpred: max fðjÞ < fðxpred Þ: ð15Þ j[UðxÞ This approximation requires only k ¼ 2n þ 1 times function evaluations, where n is the number of the design variables. It has seriously reduced the computational cost of the robust design optimization compared to that required by statistical analysis. D. Robust topology optimization A simple algorithm for RTO based on topological shape optimization is described as follows: . Set the iteration number k ¼ 0. . Calculate the robust topological gradient TGR at the center of each element. . Deﬁne the new domain Vk where the topology changes take place by removing the material in the elements where TGR is greater than zero. . Apply standard shape optimization method with a robust objective function to determine the shape of the boundary. . Check convergence and exit if the optimality condition is satisﬁed. . Set k ¼ k þ 1. Uncertainties in both the topology and shape are being handled throughout the entire design process. E. Optimization of the robust objective function Figure 4 shows a sample robust objective function determined by the worst case of the objective value. The robust objective function is continuous if the original cost function is continuous. However, the derivative of the robust objective function may not always exist. As can be shown in Figure 4, the graph of the objective function forms a sharp turn and its derivative changes sign. However, this may not be the case for a realistic problem since the objective functions are usually highly non-linear and are never that simple. In most of the cases, it is possible to use the derivatives at the predicted worst vertex to approximate the gradient of the robust objective function and to employ some special treatments at certain points. This should work for practical problems although a rigorous mathematical proof would be complicated. COMPEL 30,6 ∆ ∆ Robust objective function Continuous Non differentiable 1836 Non-robust objective function Figure 4. Continuous A robust objective Differentiable function IV. RBTO vs RTO The design formulations that take account of the uncertainties can be divided into two major categories: the reliability-based approaches and the robust design methods. The former focuses on the product safety with respect to probabilistic events while the latter tends to maintain the performance level against the environmental variations. A RBTO for an electromagnetic problem is formulated as: min fðxÞ s:t: Pf ðUÞ ¼ P½Gðxi ; Uj Þ , 0 # Pt 0 # xi # 1 ð16Þ i ¼ 1. . .n and j ¼ 1; 2; 3 U ¼ ½U1 ; U2 ; U3 ¼ ½m; Js ; Hc : Where the design variables x ¼ [x1, x2, . . . , xn] are the density functions in each ﬁnite element as opposed to x as dimensions in the previous section; G ¼ C-Cmin, C is the magnetic energy; Pf is the probability of failure and Pt is a preset limit; and U is the vector of the stochastic parameters. If G is greater than zero, the system fails, therefore the probabilistic constraint must be satisﬁed at all times. Although, RBTO is an effective tool for managing uncertainties in the topology design, it is essentially different to the RTO based on robust design. In RBTO, the variance of the performance is not being minimized during the design. Also the design variables are the density functions in the cells, therefore only variations of the external stochastic parameters, such as the permeability of the material, the source current density and the coercive force, are reﬂected in the formulations. On the other hand, RTO improves the quality of the product by minimizing the effect caused by the perturbation of the design itself. From a statistical point of view, the RTO concentrates more on the probabilistic distribution near the mean value rather than the behavior of the function at the extreme events. V. Numerical example The robust design formulation is tested with a typical topology optimization problem: eddy current imaging for ﬂaws buried under the surface of a conductor. A simple 2D model is used instead of the full 3D simulation for fast ﬁeld calculation. Figure 5 shows Topology an ECT imaging system consisting of ﬁve coils and an aluminum block containing optimization buried ﬂaws. The thickness of the block is 10 mm. An excitation frequency of 60 Hz is used to guarantee a deep enough penetration of the induced eddy currents. When one of the coils is excited with a current, the induced voltages of all ﬁve coils are measured. The ECT physical problem is simulated using (MagNet User’s Manual 2011) and the ﬁelds are solved using ﬁnite-element method. The target signals are 1837 obtained from a simulation model with a preset crack. And the goal of the RTO is to reconstruct the ﬂaw by matching the target signal with reconstructed signal. Thus, the objective function F is formulated as minimizing the squared errors between the two signals: 1XX 5 5 Z Target F¼ Vij 2 Vij ; ð17Þ 2 i¼1 j¼1 where Vij denotes the voltage of coil i induced by coil j. The topological gradient for the eddy current problem is given as (Li and Lowther, 2010b): 3s1 ðs1 2 s2 Þ È É TG ¼ Re E · EA : ð18Þ 2s1 þ s2 where s1 is the electrical conductivity of the material and s2 is the electrical conductivity of air, EA is the electrical ﬁeld computed from an adjoint problem. The robust topological gradient given in equation (13) is computed in order to estimate the location of the potential crack in the problem domain and to provide an initial crack for the robust shape optimization which in turn optimizes the shape of the crack. The crack reconstruction is an inverse problem; therefore, a unique solution is sought for. In the numerical tests, the value of the weighting factor as well as the size of the uncertainty set are determined using the knowledge of cracks in order to ﬁnd the most robust solution in the presence of noises. The results from both non-robust and robust formulations are compared. A. Single-crack reconstruction in the presence of noise The reconstruction results of a single crack are shown in Figure 6. The target crack is indicated with solid lines and the reconstructed cracks are revealed with dashed lines. A uniformly distributed random noise about 1 percent of the magnitude of the signal was added to the target voltages. Coil 1 Coil 2 Coil 3 Coil 4 Coil 5 Coil 1 Coil 2 Coil 3 Coil 4 Coil 5 Aluminum block Buried flaw Figure 5. An eddy current imaging system in 2D COMPEL 30,6 1838 (a) Crack reconstruction by non-robust function Figure 6. Reconstruction of a single crack (b) Crack reconstruction by robust function The single crack was successfully reconstructed by both the non-robust and robust formulations. Since there is only one ﬂaw, the initial topology estimated by the topological gradient is not sensitive to variations and the noise in the signal. B. Reconstruction of two cracks close together The second test uses a test specimen containing two cracks close together, shown in Figure 7. Because the two cracks are scanned simultaneously within the detection range of the coils, the response voltage signals from the different cracks can interact with each other. As a matter of fact, the signals generated by a second crack can be treated as noise in the signals from the ﬁrst crack. Figure 8 shows the contour plot of both non-robust and robust topological gradient values. As can be shown in Figure 8(b), there are two peaks in the values of the robust TG which correspond to the location of the two cracks in the test plate. While in Figure 8(a), the non-robust TG plot only showed one signiﬁcant peak, which implies only one crack may exist in the test plate. The reconstruction results of the two cracks in Figure 7, are shown in Figure 9 with the target crack shape displayed by solid lines and the reconstructed crack shapes indicated by dashed lines. The reconstruction result from Figure 9(a) shows that the non-robust formulation is unable to differentiate the two cracks. The reconstructed shape of the shallow and wide crack produces the closest matching signals for the target, based on the initial topology generated by the non-robust topological gradient. On the other hand, the RTO method Figure 7. Two cracks at a close distance 0 Topology –2 optimization –4 –6 –7 –20 –15 –10 –5 0 5 10 15 20 (a) Non-robust TG contour plot 1839 0 –2 –4 –6 Figure 8. –7 Topological gradient –20 –15 –10 –5 0 5 10 15 20 contour plots (b) Robust TG contour plot (a) Cracks reconstruction by non-robust formulation Figure 9. Reconstruction of two cracks (b) Cracks reconstruction by robust formulation demonstrates the capability to distinguish the two cracks in the present of signiﬁcant noise. The estimated initial topology is robust and it leads to a successful reconstruction of the shapes of the two cracks. VI. Conclusion This paper discussed the issues of managing uncertainties in topological design optimization. A robustness measure of a topology was deﬁned using the worst response due to the variations of the design variables. The robustness of the design can be evaluated during the entire design process of a topological shape optimization where the topology and shape of the object are being optimized simultaneously. The robust objective function was formulated based on the worst case analysis over an uncertainty set. The worst performance caused by the variations of the design parameters can be predicted using the topological gradient without the evaluation of the cost function. Therefore, this method has a huge computational advantage. COMPEL The proposed RTO was compared to the RBTO. RTO does not rely on the 30,6 probabilistic information and it has the capability of handling the uncertainties caused by the design variations as well as external stochastic parameters. Numerical applications of RTO to eddy current crack reconstruction problems showed that this method can provide reliable results in the presence of signiﬁcant noises. 1840 References Alotto, P., Magale, C., Renhart, W., Steiner, G. and Weber, A. (2003), “Robust target functions in electromagnetic design”, COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, Vol. 22 No. 3, pp. 549-60. Cavaliere, V., Ciofﬁ, M., Formisano, A. and Martone, R. (2003), “Robust design of high ﬁeld magnets through Monte Carlo analysis”, COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, Vol. 22 No. 3, pp. 589-602. Choi, S.H.E., Lowther, D.A. and Dyck, D.N. (1998), “Determining boundary shapes from the optimized material distribution system”, Magnetics, IEEE Transactions on Magnetics, Vol. 34 No. 5, pp. 2833-6. Ciofﬁ, M., Formisano, A. and Martone, R. (2004), “Stochastic handling of tolerances in robust magnets design”, IEEE Transactions on Magnetics, Vol. 40 No. 2, pp. 1252-5. Dyck, D.N. and Lowther, D.A. (1996), “Automated design of magnetic devices by optimizing material distribution”, IEEE Transactions on Magnetics, Vol. 32 No. 3, pp. 1188-93. Guimaraes, F.G., Lowther, D.A. and Ramirez, J.A. (2006), “Multiobjective approaches for robust electromagnetic design”, IEEE Transactions on Magnetics, Vol. 42 No. 4, pp. 1207-10. Kang, J., Kim, C. and Wang, S. (2004), “Reliability-based topology optimization for electromagnetic systems”, COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, Vol. 23 No. 3, pp. 715-23. Kim, D.H., Lee, S.B., Kwank, B.M., Kim, H.G. and Lowther, D.A. (2008), “Smooth boundary topology optimization for electrostatic problems through the combination of shape and topological design sensitivities”, IEEE Transactions on Magnetics, Vol. 44 No. 6, pp. 1002-5. Li, M. and Lowther, D.A. (2010a), “The application of topological gradients to defect identiﬁcation in magnetic ﬂux leakage-type NDT”, IEEE Transactions on Magnetics, Vol. 46 No. 8, pp. 3221-4. Li, M. and Lowther, D.A. (2010b), “Topological sensitivity analysis for steady state eddy current problems with an application to nondestructive testing”, Proceedings of 14th Biennial IEEE Conference Electromagnetic Field Computation (CEFC), Chicago, IL, USA. MagNet User’s Manual (2011), MagNet User’s Manual, available at: www.infolytica.ca Spagnuolo, G. (2003), “Worst case tolerance design of magnetic devices by evolutionary algorithms”, IEEE Transactions on Magnetics, Vol. 39 No. 5, pp. 2170-8. Steiner, G., Weber, A. and Magele, C. (2004), “Managing uncertainties in electromagnetic design problems with robust optimization”, IEEE Transactions on Magnetics, Vol. 40 No. 2, pp. 1094-9. Taguchi, G., Elsayed, E. and Hsiang, T. (1989), Quality Engineering in Production Systems, McGraw-Hill, New York, NY. Yoon, S.B., Jung, I.S., Hyun, D.S., Hong, J.P. and Kim, Y.J. (1999), “Robust shape optimization of electromechanical devices”, IEEE Transactions on Magnetics, Vol. 35 No. 3, pp. 1710-3. About the authors Topology Min Li received a BEng in Computer Science from Shanghai Jiaotong University, Shanghai, China, in 1999 and an MEng in Electrical Engineering from McGill University, Montreal, Canada, optimization in 2007. He is currently a PhD candidate in the Department of Electrical and Computer Engineering at McGill University. David A. Lowther received his BSc (Eng.) Hons from King’s College, London in 1970 and his PhD from Brighton Polytechnic in 1973. He currently holds a James McGill Chair in Electrical Engineering at McGill University in Canada. His research interests include computational 1841 electromagnetics, intelligent design systems, optimization and the design of electrical machines. David A. Lowther is the corresponding author and can be contacted at: david.lowther@mcgill.ca To purchase reprints of this article please e-mail: reprints@emeraldinsight.com Or visit our web site for further details: www.emeraldinsight.com/reprints

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