The concept of the sensitivity of a design solution to small perturbations of the design variables can be exploited in manifold ways: e.g. it could be evaluated just at the start of the optimization procedure, using computer-based experiments, in order to identify a reduced set of design variables and so discard the less sensitive ones . Alternatively, sensitivity can be evaluated at the end of the design process, in order to assess the robustness of the optimized solution; moreover, when sensitivity is incorporated in the objective or constraint functions, there is an extra cost at each iteration for simulating the local perturbation, and the cost increase might be substantial in the case of a multi-dimensional objective space. A full overview is presented in  and . Here, a cost-effective formulation inspired by perturbation hypervolume in the objective space is proposed.
2 Experimental procedure
The proposed procedure relies on the following tools: Finite-Element (FE) analysis is used for simulating the magnetic field in the motor, the sensitivity is considered as an additional design criterion, the problem of improving the design criteria while reducing the sensitivity is formulated as a multi-objective optimization problem.
2.1 Sensitivity-based approach
In order to compute sensitivity, a small perturbation (e.g. 1 mm for the case study) is considered for each design variable. The perturbation can either increase the value of the design variable (PP) or decrease it (MP). Considering the Plackett-Burman design  with two levels and three factors, for each solution the cases in Table 1 are considered:
The cases in Table 1 lead to four runs for each considered solution, instead of six runs in the case of a full perturbation scheme. Correspondingly, in the objective function space the five dots (four from the Plackett-Burman perturbation scheme, and one being the non-perturbed solution) can originate different configurations, as shown in Figure 1, which refers to the optimal shape-design problem presented in the subsequent Section 3.
Suppose to minimize a pair of objectives and simultaneously evaluate their sensitivity against small perturbations, δx, around solution x. Two cases are possible: either the unperturbed solution (i.e. the nominal solution) dominates the perturbed ones (case A), or the unperturbed solution is dominated (case B, in which two solutions dominate the non-perturbed one).
In particular, referring to case B, it is interesting to note in Figure 1 that a perturbed solution might, by chance, dominate the nominal solution, and so be preferable to it. If this happened, the nominal solution, x, should be replaced by the dominant perturbed solution in the relevant optimization iteration.
In both cases the area, A, of the convex polygon associated with the nominal solution is considered as a metric criterion of sensitivity: the smaller the area, the less sensitive the solution.
In order to control the sensitivity, reducing the polygon area is not completely satisfactory, in fact, there could be “small” polygons exhibiting a substantial deformation in shape. In such a case, an anisotropic effect on sensitivity would arise, meaning that for the same amount of perturbation of the design variables, the corresponding variation in the objective space could be biased along a paramount direction. In contrast, an isotropic sensitivity would be preferable.
To this end, the deployment criterion, D, which rules the symmetry of the polygon in two-dimensional objective space, is introduced:
where the square matrix G is computed as
and, in turn, H is a (N,2) matrix calculated as follows
where (f1g, f2g) are the coordinates of the center of gravity of the polygon, R1i and R2i are the scalarizing distances of the i-th vertex of the polygon from the center of gravity, and N is the number of vertices of the polygon.
The proposed approach gives rise to a cost-effective formulation [fk(x), A(x,δx), D(x,δx)], k=1, nf, incorporating only nf+2 objectives. In fact, according to a standard formulation of design sensitivity, the k-th objective should be attributed the relevant variation Δ with respect to all variables [fk(x), Δfk(x)], which gives rise to 2nf objectives in total, against nf+2 according to our approach.
A cost-effective formulation is particularly suitable for handling complex optimization problems e.g. multi-physics problems [6,7,8,9] or, in general, optimizations based on a forward problem the solution of which requires a FE simulation [10,11,12].
2.2 Cases study: switched reluctance motor (SRM)
Phase configuration consists of four series-connected coils per phase, driven by a Direct Current (DC) equal to 1 A. Both rotor and stator are made of M27 material, which is a standard material used for lamination of electrical machines  and . The finite-element model  of the SRM has been implemented by means of a commercial code1.
2.3 Optimization problem
Geometric parameters ROR, RIR and BIT (Figure 2) are selected as the design variables, subject to a set of constraints: overall diameter of the motor not exceeding 140 mm; air-gap length (AG) equal to 0.25 mm. Constraints are handled by means of the exterior penalty method. In terms of the design vector x = [ROR, RIR, BIT], the design criteria are defined by the following two functions:
i.e. the static torque is to be maximized, and
i.e. the iron losses are to be minimized . In (4) and (5) P is the specific power loss in the ferromagnetic material subject to induction B; S is the iron volume; r0 denotes the vector joining the rotation axis to a mesh node in the rotor region; T̿ indicates Maxwell’s stress tensor, and Γ represents a closed surface surrounding the rotor. Both objective functions f1 and f2 are calculated per meter length.
In order to assess the sensitivity of the solutions in the objective space, two criteria are considered
the area criterion, A, i.e. the area of the convex hull (more generally, the hypervolume in nf>2 dimensions) built on the set of five points (the nominal solution and the four perturbed solutions as shown in Table 1);
In general, there are two ways of taking into account criteria A and D: either as additional objectives, or as constraints . In this paper the latter has been chosen.
3 Results and discussion
The bi-objective problem, considering only functions (4) and (5) is solved by means of Non-dominated Sorting Genetic Algorithm-II (NSGA-II), without taking the sensitivity into account. The following set of parameters tuning NSGA-II was considered: 20 individuals, 20 iterations, mu = 0.2 and mum = 0.2. The results are shown in Figure 3.
Two remarks can be put forward: the average distance between the starting points and the Pareto front is relatively small, because the extent of the feasible region is severely limited by the problem constraints, which is typical of most industrial problems. Moreover, the sensitivity of optimal solutions is not controlled, because NSGA-II is applied without considering the sensitivity: to bridge this gap, the two previously defined criteria, A and D, are then applied. In particular, the bi-objective optimization problem (4)-(5) is solved implementing A and D as inequality constraints: the constraint threshold value is set as the average of A and D values of the starting solutions. The results of this optimization are shown in Figure 4 and Table 2. As an example, three solutions and three starting points are highlighted. The relevant polygon, area, A, and deployment, D, are also specified for each solution.
The solutions fulfill the constraints for both area, A, and deployment, D, while the starting points could not fulfill these constraints. It can be noted that the solutions are characterized, in general, by smaller areas and smaller D coefficients. These leads to more robust solutions.
In Figure 6 the Pareto solutions of the optimization with only D constrained are shown. As an example, three solutions and three starting points are highlighted. The relevant polygon, the area, A, and the deployment, D, are also specified for each solution.
Because A is not constrained, the solutions show, in general, larger areas than the starting points. On the other hand, the D value of the solutions is smaller than that characterizing the starting points.
In Figure 7 three starting points are shown with their relevant polygons; it can be noted that perturbing one of the starting points can lead to a better solution in the Pareto sense . In Table 3 the design variables and the objective functions of the three starting points and the corresponding non-dominated solutions, belonging to the same polygon, are reported.
In this paper, the sensitivity of optimal solutions is controlled in an efficient way: thanks to the use of area, A, and deployment, D, criteria, the sensitivity is controlled both in magnitude and isotropy. Moreover, the use of the Plackett-Burman design allows reduction of the computational costs. Independently of the number of objective functions characterizing the design problem, and thanks to the concept of sensitivity polygon, only two additional criteria A and D have to be computed. The proposed method has been validated by means of a shape design for a switched reluctance motor.
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About the article
Published Online: 2017-12-29
Citation Information: Open Physics, Volume 15, Issue 1, Pages 924–928, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2017-0112.
© 2017 Paolo Di Barba et al.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0