Robust shape optimization of electric devices based on deterministic optimization methods and finiteelement analysis with affine parametrization and design elements
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Abstract
In this paper, gradientbased optimization methods are combined with finiteelement modeling for improving electric devices. Geometric design parameters are considered by piecewise affine parametrizations of the geometry or by the design element approach, both of which avoid remeshing. Furthermore, it is shown how to robustify the optimization procedure, that is, how to deal with uncertainties on the design parameters. The overall procedure is illustrated by an academic example and by the example of a permanentmagnet synchronous machine. The examples show the advantages of deterministic optimization compared to standard and popular stochastic optimization procedures such as particle swarm optimization.
Keywords
Finiteelement analysis Genetic algorithms Gradient methods Electric machines Optimization methods Particle swarm optimization Permanentmagnet machines Quadratic programming1 Introduction
In almost all electric design procedures, numerical optimization is employed as one of the last design steps in order to optimize the device’s performance and efficiency, to minimize its weight and size and to save on material and manufacturing costs. Often, the quality of this optimization step indirectly determines the success of the product and, hence, the market position of the company. The reliability, accuracy and computational cost of the numerical optimization procedure becomes in itself a subject of competition. This paper illustrates that shape optimization can be improved substantially when finiteelement (FE) analysis procedures are equipped with piecewise affine parametrization or design elements, such that wellperforming deterministic optimization methods become applicable.
Impressive technical improvements have been achieved by numerical optimization on the basis of magnetic equivalent circuits or 2D and 3D FE models. All have led to highly optimized designs, for example, for permanentmagnet synchronous machines (PMSMs) in automotive applications. Since three decades, FEbased optimization has been addressed in several text books (see, e.g., [12]) and hundreds of journal articles (see, e.g., [14] and the references therein). Although originally, gradientbased methods were preferred (see, e.g., [48, 56, 58]), already for more than two decades, stochastic algorithms are more popular (see, e.g., [19, 33]). The majority of the proposed procedures opt for stochastic or populationbased optimization methods, such as genetic algorithms and particle swarm optimization (see, e.g., [34]), because they allow to use FE solvers as a black box, they can easily consider geometric parameters, their parallelization is straightforward and they are more likely to find the global optimum. Stochastic algorithms have been used for robust optimization, have been applied together with surrogate modeling and have been extended to multiobjective optimization problems [3, 23]. In particular for PMSMs, optimization with stochastic methods became the method of choice [2, 9, 51].
The trend toward stochastic optimization combined with FE analysis continues without restraint, as is illustrated by the number of according contributions at recent conferences. This paper partially counteracts this tendency by turning back to deterministic optimization algorithms. Deterministic optimization methods are known to converge faster than stochastic optimization methods, albeit possibly to a local optimum. Moreover, the analysis of gradientbased methods is more mature, allowing for a rigorous control of mesh discretization errors, for instance. The main drawback of many deterministic methods is, however, the necessity to provide derivatives, which is particularly cumbersome when optimization according to geometric parameters is pursued. This drawback is here addressed explicitly and is alleviated by piecewise affine parametrizations of the geometry or by the design element approach. The overall deterministic optimization routine is shown to outperform the most popular stochastic algorithms by factors. Moreover, the optimization method will be robustified to include uncertainties on the design parameters.
The paper is structured as follows: Sect. 2 recalls the basics of mathematical optimization. It clearly distinguishes between deterministic methods (Sect. 2.3) and particle swarm optimization as a relevant representative of stochastic methods (Sect. 2.4). Furthermore, an extension to robust optimization is discussed in Sect. 2.5. Section 3 deals with FE analysis of magnetodynamic fields. The core parts of the paper are Sect. 3.3.1 about affine parametrization and Sect. 3.3.2 about design elements, both facilitating and improving the calculation of derivatives with respect to geometric parameters. The superior performance of gradienttype deterministic optimization is illustrated for a benchmark example in Sect. 4 and for a PMSM in Sect. 5. Conclusions are formulated in Sect. 6.
2 Constrained optimization
2.1 Constrained optimization problem
The optimization is carried out with respect to I design parameters \(\mathbf{p}=(p_1,p_2,\ldots ,p_I)\) belonging to the admissible set \({\mathcal {P}}_{\mathrm {ad}} =\{\mathbf{p}\in {\mathbb {R}}^IG_m(\mathbf{p})\le 0,m=1,\ldots ,M\}\), where \(G_m(\mathbf{p})\) denote the constraints. The design parameters can be any continuous variables, for example material constants, excitation parameters and geometric sizes or positions. The constraints limit the admissible range of these parameters, for example to preserve the topology of the geometry or to set physical and operational constraints. Discrete design parameters are not considered in this work, although many methods apply, for example as part of a branchandbound technique, to mixedinteger optimization problems as well [21].
2.2 Optimization methods

Problem (1) considers a single optimization goal. For a multiobjective optimization problem, a Pareto front is calculated such that the relative importance of the optimization goals can be fixed later on [7, 12]. This paper does not further consider multiobjective optimization. Nonetheless, the developed techniques are applicable to multiobjective optimization as well.

A distinction is made between global optimization and local optimization, where the first strives to a global optimum, whereas the second may run into a local one. This paper is limited to methods for local optimization. In practice, if a global optimum is required, the methods may be repeated for different start values, or could be embedded as part of a global optimization scheme [24].

Especially when the evaluation of the objective function is computationally expensive, it is recommended to carry out the optimization method on the basis of a surrogate model (indirect optimization methods). Such a simplified model can be obtained by expert knowledge on the application [59], by design space reduction [17], by a response surface methodology [17] or by space mapping [29] or manifold mapping [15]. Here, a direct optimization procedure is used. All ideas presented here can, however, be used in combination with indirect optimization approaches as well [30].

The result from a nominal optimization is a set of optimized design parameters leading to an optimum of the objective function. The optimum may, however, become irrelevant when it is highly sensitive to uncertainties in the design parameters. One speaks about robust optimization when the optimization is carried out taking such uncertainties into account. In this paper, both nominal and robust optimization methods are considered. An approach for robustification is discussed in Sect. 2.5.

Two families of basic optimization methods exist: deterministic and stochastic methods. Among the stochastic methods, genetic algorithms [35], differential evolution [38] and particle swarm optimization (PSO) [27] are well known.
2.3 Gradientbased deterministic method
2.4 Particle swarm optimization
 1.
Maintain a part of the current velocity;
 2.
Head toward the particle’s best found point (\({\hat{\mathbf{p}}}_q\));
 3.
Head toward the swarm’s best found point (\({\hat{\mathbf{p}}}_\text {swarm}\)).
2.5 Robust optimization
In a nominal optimization procedure, one is looking for the minimum value of an objective function. However, during manufacturing, small deviations can occur on the parameters. As a consequence, the optimal solution may become suboptimal in reality. Robust optimization searches for an optimum that is not too much affected by the expected parameter deviations [41, 60].
3 Finiteelement model
The behavior of the devices under consideration is determined by magnetic field phenomena and is simulated using a FE model.
3.1 Magnetoquasistatic formulation
3.2 Finiteelement discretization
3.3 Geometry parametrization
3.3.1 Affine parametrization
This section discusses the idea of decomposing (a part of) the mesh such that geometrical changes can be represented by piecewise affine maps [30, 47]. The affine maps will be referred to by \(f^\ell _{\mathbf{p},\text {aff}}\).
3.3.2 Design element approach
For many geometry optimization tasks, the domain underlying geometry changes cannot be decomposed in triangles or tetrahedra with straight edges and faces, which excludes the use of affine parametrization. NURBS are a more general way to represent geometries and are widely used in CAD systems. Therefore, it seems natural to use the control points (and weights) of NURBS curves as design parameters [6, 49]. This approach has received considerable attention in recent years as new approaches, incorporating NURBS geometries into FE analysis, have emerged. Isogeometric analysis [25] and the NURBSenhanced FE method [50] are important examples. Here, NURBS are only used for the geometry parametrization. A triangular (tetrahedral) mesh is generated once and deformed using the concept of design elements [6, 26].
3.4 Sensitivities
4 Example 1: Die press mold
As a first example, a die press mold for radially magnetizing a segment of sintered magnetic powder (SMP) is considered [54]. This problem has been proposed as testing electromagnetic analysis methods (TEAM) benchmark problem 25 [53] and has been used in numerous papers for comparing optimization algorithms. The vast majority of these publications apply and compare stochastic optimization methods [32, 52], possibly combined with surrogate models [8], uncertainty quantification [39], multiobjective optimization or a combination of them [31]. Only a few papers (see, e.g., [1, 4]) choose deterministic methods, again possibly combined with surrogate models [20], uncertainty quantification [55] or multiobjective optimization. This paper addresses one of the main drawbacks of deterministic methods, which is the consideration of geometric parameters. For this example, the design element approach is used.
Results from the optimization of the die press mold with particle swarm optimization (PSO), trust region (TR) (with MATLAB\(^{\circledR }\)’s fmincon) and an own implementation of sequential quadratic programming (SQP) combined with the design element approach
Method  Minimizer \({\hat{\mathbf{p}}}_{\mathrm{min}}\)  Minimum  Iteration  Function calls  Time  

(in mm)  (in \(\text {T}^2\))  count  \(\mathrm {f}()\)  \(\nabla \mathrm {f}()\)  (in s)  
PSO  \(\begin{pmatrix} 5.1000\\ 16.0000 \\ 16.0000\\ 9.5000\end{pmatrix}\)  1.413498  7  280  N/A  56.63 
SQP (fmincon)  \(\begin{pmatrix} 5.1000\\ 16.0000 \\ 16.0000\\ 9.5000\end{pmatrix}\)  1.413498  4  7  7  31.61 
SQP (own implementation)  \(\begin{pmatrix} 5.1000\\ 16.0000 \\ 16.0000\\ 9.4999\end{pmatrix}\)  1.413498  2  3  2  12.84 
The performance of a standard algorithm for particle swarm optimization (PSO), of the sequential quadratic programming (SQP) method implemented in MATLAB\(^{\circledR }\)’s fmincon function [43] and of an own implementation of SQP is compared in Table 1. Both SQP implementations use the analytical gradients, the BFGS formula for updating the Hessian and a sufficient decrease condition in a merit function. For the PSO, a set of 40 particles is considered and the implementation is multithreaded, while the gradientbased methods are singlethread implementations. The termination criterion for the PSO algorithm is the number of stall iterations, which was set to 5. The PSO actually finds the optimum after 2 iterations. This is because the optimum is at a vertex of the boxshaped domain and all the particles leaving the admissible region are projected onto the boundary. All three methods converge to the same optimum. The deterministic algorithms are substantially faster than PSO, even though PSO exploits parallelization. On the same machine, an evaluation of the objective function \(J(\mathbf{p})\) is performed in 1.65 s, an analytical evaluation of the gradient \(\nabla J(\mathbf{p})\) in 4.69 s and a numerical evaluation of the gradient \(\nabla _{\text {num}} J(\mathbf{p})\) using a forward difference quotient in 7.48 s. All tests were done on a 64 GB RAM Intel\(^{\circledR }\) Xeon\(^{\circledR }\) E52630 v4 machine.
5 Example 2: Permanentmagnet synchronous machine (PMSM)
5.1 Design parameters
5.2 Objective function
5.3 Optimization problem
Numerical results obtained for a \(\varvec{\delta }=0.2~\hbox {mm}\) [5]
\(p_1\)  \(p_2\)  \(p_3\)  \(S_{\mathrm{pm}}\)  \(E_0\)  FE slv  Time  

(mm)  (mm)  (mm)  \(({\hbox {mm}^2})\)  (V)  calls  (s)  
Initial design  19.00  7.00  7.00  133  30.370  –  – 
Genetic algorithm  21.04  2.98  6.56  62.80  30.370  \(\approx 6760\)  520.5 
PSO with penalty term  20.60  3.09  5.91  63.71  30.370  \(\approx 3470\)  267.16 
PSO, own implementation  21.08  2.98  6.63  62.80  30.370  1765  217.52 
SQP, nominal optimization  21.07  2.98  6.61  62.80  30.370  34  2.0 
SQP, robust optimization  20.88  3.73  6.82  77.87  31.086  48  5.9 
5.4 Results
 1.
The first optimization run is carried out with the genetic algorithm implemented in MATLAB\(^{\circledR }\).
 2.The second optimization run is carried out with MATLAB\(^{\circledR }\)’s PSO implementation. To circumvent the restriction to boxshaped parameter domains, the admissible set is enforced by a penalty turn. The new objective function readswhere \(f(t)=e^{(4t^{0.1})}1\) was chosen heuristically such that \(J_\text {pen}\) grows exponentially if one of the constraints is violated. The function \(J_\text {pen}\) was called 4740 times, but was organized as to only evaluate the nonlinear constraint if all other constraints were satisfied. The number of particles was set to 30, the maximum number of stall iterations to \(N_\text {stall}=15\) and the function change tolerance to \(10^{6}\). The PSO characteristic constants are chosen to be \(\omega _0=0.5\) and \(\omega _1=\omega _2=1.49\). The algorithm took 157 iterations before termination.$$\begin{aligned} J_\text {pen}(\mathbf{p})&=J(\mathbf{p}) +2J(\mathbf{p}) \big ( f\left( \max (p_2+p_315,0)\right) \nonumber \\&\quad +f\left( \max (3p_12p_350,0)\right) \nonumber \\&\quad +f\left( \max (g(x),0)\right) \big ) \text {,}\end{aligned}$$(45)
 3.
The third optimization is carried out with an own PSO implementation, for the original objective function \(J(\mathbf{p})\) and applying the nonlinear constraints directly. Here, it is assumed that the admissible set is convex such that points inside the convex hull formed by all previous points do not need to be checked. Fifty particles were used. Termination was enforced after maximally \(N_\text {it,max}=100\) steps or when \(N_\text {stall,max}=15\) stall iterations were observed.
 4.
The fourth run was done with the deterministic method described in Sect. 2.3, relying upon FE simulations equipped with an affine parametrization of the geometry as described in Sect. 3.3.1.
 5.
The fifth run was done with the deterministic method for robust optimization expressed by (11) in Sect. 2.5, again with affine parametrization of the geometry.
The results of all optimization procedures are compared with the values of the initial design (Fig. 5). All routines achieve a substantial decrease in the PM size from \(133~\hbox {mm}^2\) up to about \(63~\hbox {mm}^2\). The price for robustness is a slightly larger size of about \(77~\hbox {mm}^2\). The deterministic methods outperform the stochastic ones by two orders of magnitude. This impressively illustrates the major message of this paper stating that deterministic optimization methods accompanied by FE analysis providing gradients with respect to geometric parameters should be favored over stochastic methods, at least for the here considered class of problems.
6 Conclusion
Affine parametrization and design element approaches are capable of parametrizing the geometry of finiteelement models such that accurate derivatives with respect to geometric parameters become available. This alleviates one of the major drawbacks of gradienttype deterministic optimization methods. For the example of a die mold press, standard sequential programming combined with the design element approach outperforms particle swarm optimization by more than a factor ten. The second example illustrates the applicability of gradienttype robust optimization combined with an affine parametrization of the geometry for a permanentmagnet synchronous machine. Supported by the substantial improvement in computational efficiency, this paper stands up for a revival of deterministic methods for numerical optimization in electrotechnical design procedures.
Notes
Acknowledgements
This work is supported by the German BMBF in the context of the SIMUROM project (grant nr. 05M2013) and the PASIROM project (grant nr. 05M2018), by the ’Excellence Initiative’ of the German Federal and State Governments and by the Centre and Graduate School Computational Engineering at TU Darmstadt.
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