Issue 
Sci. Tech. Energ. Transition
Volume 78, 2023
Power Components For Electric Vehicles



Article Number  41  
Number of page(s)  10  
DOI  https://doi.org/10.2516/stet/2023037  
Published online  22 December 2023 
Regular Article
Multimaterial topology optimization of a flux switching machine
^{1}
Université ParisSaclay, ENS ParisSaclay, CNRS, SATIE, 4, Avenue des Sciences, 91190 GifsurYvette, France
^{2}
CY Cergy Paris Université, CNRS, SATIE, site de de Neuville, 5 mail Gay Lussac, 95031 Neuville sur Oise, France
^{3}
CNAM, LMSSC, Case 2D6R10, 2 rue Conté, 75003 Paris, France
^{4}
Université Paris Saclay, ENS Paris Saclay, CentraleSupélec, CNRS, LMPS, 91190 GifsurYvette, France
^{5}
ENS Rennes, CNRS, SATIE, Campus de Ker Lann, 11 Av. Robert Schuman, 35170 Bruz, France
^{*} Corresponding author: theodore.cherriere@ensparissaclay.fr
Received:
14
September
2023
Accepted:
8
November
2023
This paper investigates the topology optimization of the rotor of a 3phase fluxswitching machine with 12 permanent magnets located within the stator. The objective is to find the steel distribution within the rotor that maximizes the average torque for a given stator, permanent magnets, and electrical currents. The optimization algorithm relies on a density method based on gradient descent. The adjoint variable method is used to compute the sensitivities efficiently. Since the rotor topology depends on the current feedings, this approach is tested on several electrical periods and returns alternative topologies. Then, the method is extended to the multimaterial case and applied to optimize the nonmagnet part of the stator. When dealing with 3 phases, the algorithm returns the reference topology as well as a theoretical machine with no return conductor according to the set current angle. To illustrate the creativity of the method, the optimization is finally performed with a singlephase and returns a new topology.
Key words: Density method / Flux switching machine / Multimaterial topology optimization / Nonlinear magnetostatics
© The Author(s), published by EDP Sciences, 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
Topology optimization is a conception tool that makes no use of geometric parametrization. It was first developed in mechanical engineering by [1], then introduced in electrical engineering by [2], and has gained interest from engineers in the last decade with the development of additive manufacturing [3]. Among various techniques, such as the levelset method [4], or the phasefield approach [5] (see [6] for a recent overview), densitybased approaches [1, 7] are the most popular. The principle of this technique is explained hereafter.
The geometry to be optimized is represented by a socalled density field ρ, which can be interpreted as pixelation on N_{e} elements once discretized spatially, as shown in Figure 1. A density value of 0 represents air, and 1 represents iron in the corresponding mesh. The topology optimization problem then reads as(1)where f is the objective function to minimize (such as the cost, the mass, or in the present case the opposite of the average torque), and ρ is the vector of optimization variables, which contains the density of each mesh element. To avoid solving a combinatorial problem that may be intractable, density methods introduce intermediate materials (also called gray materials) associated with density values strictly between 0 and 1. Therefore, the material properties can be continuously interpolated between the actual candidate materials to use a fast gradientbased optimization algorithm.
Fig. 1 Principle of densitybased representation. a) Original domain, b) Densitybased representation. 
Although the presence of gray materials is not critical during optimization, they should not appear in the final geometry. Indeed, gray materials do not necessarily have a proper physical interpretation or may represent, in the best case, a microstructure that is complex to manufacture [8]. A common solution for removing gray materials proposed by the literature involves penalizing the material interpolation, such as the Solid Isotropic Materials with Penalization (SIMP) scheme [8, 9]. Yet topology optimization problems are generally illposed, and other numerical artifacts can occur, such as checkerboards that need regularization (e.g., numerical filtering [10]). Densitybased approaches have been applied to many magnetics problems, from the simple Ccore electromagnets [11, 12] to 3phase synchronous reluctant machines with several materials and physics [13, 14]. However, apart from [15], no such work has studied unconventional structures such as Flux Switching Machines (FSM), where the unbiased creativity of topology optimization may be the most useful.
This article aims to apply a densitybased topology optimization methodology to FSM. First proposed by [16], this type of machine contains a passive rotor with the field inductor located in the stator only, making its cooling easier [17] and suitable for highspeed applications. Several types of FSM exist in the literature; see [18]. The chosen test case is a 3phase Permanent Magnet Flux Switching (PMFS) machine with 12 permanent magnets evenly spaced within the stator, adapted from [19] and shown in Figure 2.
This paper is structured as follows. First, Section 2 recalls the magnetostatics equations. The optimization algorithm is then detailed in Section 3. Next, two numerical applications exploring unconventional topologies are presented and discussed. On the one hand, classical rotor optimizations are conducted under various electrical frequencies in Section 4. On the other hand, stator optimization, which requires a multimaterial framework, explores the topologies obtained with 1 and 3phase current feeding in Section 5. In conclusion, Section 6 summarizes the essential results and draws perspectives on this work.
2 Physical problem
Since densitybased topology optimization will be used, we first set up the interpolations of material properties in Section 2.1 in the Kennelly convention [20]. These quantities were then injected in the physical equation of magnetostatics recalled in Section 2.2, which is further discretized with the 2D Finite Element Method (FEM).
2.1 Material interpolation
There are at least two material properties of interest when dealing with a magnetostatics problem: the magnetic polarization of materials m^{1} and the current density. In the case of soft magnetic material, m is collinear to the flux density b and 0 when b = 0, so that it can be expressed in terms of a scalar reluctivity ν(b)(2)
Two possibilities are generally reported in the literature to interpolate the magnetic properties when dealing with soft magnetic materials. One can interpolate either the magnetic reluctivity ν as in [12, 21, 22], or the magnetic permeability μ = ν^{−1} as in [2, 23, 24], among many others. However, an additional magnetic quantity to interpolate (the remanence) should be introduced when dealing with polarized materials, as in [24, 25]. We choose an alternative, proposed in [26], which interpolates the magnetic polarization of a FeCo alloy (AFK1 from Aperam [27]) directly. The corresponding BH curve is plotted in Figure 3. Its interpolation reads in the general case of n_{m} materials(3)where P_{m} is a penalization function associated with the magnetic polarization, ω_{i} a shape function defined in [28] associated with the ivertex of a convex interpolation domain , and m_{i} the magnetic polarization of the imaterial. To address the topology optimization of the stator, the current density should also be interpolated with a similar scheme:(4)where P_{j} is a penalization function associated with the current density, and j_{i}, is the outofplane component of the current density of the material i. The chosen domain as well as the penalization functions are given in Sections 4 and 5, and an example of interpolation is drawn in Figure 4. More details on the interpolation scheme and associated algorithms can be found in [29].
Fig. 3 Anhysteretic BH curve of FeCo in logarithmic scale used for the computations. 
Fig. 4 Interpolation example of the current density amplitude between 7 materials. Note that in the real optimization problem in Section 5, the actual value of the current density at each electric angle is interpolated instead of its amplitude. 
2.2 Physical equations
From the static Maxwell’s equations, one can write the magnetostatics equation for a 2D problem [26]:(5)with and the x and y components of the interpolated magnetic polarization , the current density, and a_{z} the zcomponent of the magnetic vector potential a, related to the flux density by the formula b = ∇ × a. To be solved numerically, the strong equation (5) should be written in a weak form and discretized using, for instance, the FEM. Then, the physical problem reads as follows(6)where K is the finite element matrix, a the vector containing the discretized degrees of freedom a_{z}, and s righthand side related to source terms. Figure 5 draws the mesh used in this article. Homogeneous Dirichlet boundary conditions confined the flux inside the machine, and a lockedstep sliding band technique [30] is applied to emulate the rotation of the rotor without remeshing.
Fig. 5 Mesh of the simulation domain containing 30,666 firstorder elements and 16,329 nodes. 
The nonlinear FEM system (6) is solved with the NewtonRaphson scheme:(7)with r_{n} = Ka_{n} − s(ρ, a_{n}) is the residual of (6) at iteration n. After convergence, the torque can be computed from the flux density field within the airgap e by Arkkio’s method [31]:(8)where b_{r}, b_{θ} are the cylindrical components of the flux density, and R_{s} and R_{r} are the stator’s inner radius and the rotor’s outer radius, respectively. The axial length of the machine is normalized to L = 1 m.
3 Optimization algorithm
The purpose of the optimization is to solve the problem (1), i.e. to find an optimal density vector ρ that minimizes a given objective function f. In this article, the objective function is set to the opposite of the average torque (i.e., the average torque should be maximized), computed with (8) over 720 angular positions along one mechanical turn:(9)
As the density vector ρ contains many components (at least one per mesh element), the optimization algorithm should rely on the sensitivity d_{ρ}f. The optimization flowchart is given in Figure 6.
Fig. 6 Flowchart of the optimization algorithm. 
The efficient computation of d_{ρ}f components is nontrivial since f rarely depends on ρ explicitly. Actually, f rather depends on the system’s physical state, such as the magnetic field b in the present case. The latter depends implicitly on the density vector ρ through the physical equation (6), which involves the material interpolations (3) and (4).
3.1 Sensitivities computation
The Adjoint Variable Method (AVM) is the most efficient approach to compute firstorder derivatives from many variables. Once the FEM system (6) is solved, the sensitivities are computed using the adjoint state λ, which is the solution of the following adjoint system(10)then the sensitivities with respect to each design variable ρ_{i} can be calculated with(11)
Note that (10) is linear and should be solved only once per iteration. Consequently, the average singlethread computing time per optimization iteration of the sensitivities (41s) is almost negligible compared to the one of the nonlinear FEM (352s). This high computing time is caused by the high number of nodes and angular positions for the calculation.
3.2 Update
To accelerate the optimization to the real materials, each component of the descent direction associated with the mesh element is normalized as follows:(12)with the sensitivity vector of f to the part of the global density vector associated with the element . A simplified trustregion algorithm [32] adapts the step size heuristically according to a quality indicator, which compares the measured variation of the objective function Δf between two consecutive iterations with its variation predicted by the linearized model. It is defined as(13)where Δρ is the variation of the density vector between two consecutive iterations. If q is too low, the iteration is rejected, and the step size is reduced. If q is big enough, the iteration is accepted, and the step size can also be increased. The flowchart in Figure 7 gives more details on this algorithm that is applied in the following sections.
Fig. 7 Flowchart of the update algorithm. 
Then, ρ_{n+1} is projected orthogonally to remain within the bounds of . The algorithm stops if the step α becomes smaller than 5 × 10^{−4}, or after 100 iterations.
4 Rotor optimization
The presented topology optimization algorithm is first applied to FSM rotor. The optimization domain is shown in Figure 8. In order to explore new structures, different electrical frequencies are injected within the FSM. The optimization setup is presented in Section 4.1, and the results in Section 4.2.
Fig. 8 Optimization domain (gray) in case of a rotor optimization. 
4.1 Parametrization
There are only two candidate materials to distribute inside the rotor: soft magnetic steel (FeCo alloy AFK1 from Aperam [27]) and air. Therefore, the interpolation domain is a 1D line segment that joins both materials and supports the optimization variables contained in ρ; air corresponding to ρ = 0 and steel to ρ = 1. The density interpolates the magnetic polarization linearly:(14)and the resulting BH curves for intermediate materials are given in Figure 9. Note that the intermediate BH curves are equivalent to those of a 1D serial assembly of steel and air along the flux lines, ρ representing the linear fraction of steel^{2}. Initially, all densities are set to 0.5.
Fig. 9 Magnetic behavior used for the computation. 
Because there is no current neither in air nor steel, the current density inside the rotor is set to 0. However, the current density inside a stator conductor , following the enumeration in Table 1, reads:(15)where J = 10 A/mm^{2} is the current density amplitude, N is the number of electrical periods during one complete mechanical rotation, θ_{m} is the mechanical angle of the rotor, and ψ the electric angle set to 0° for the rotor optimization. The N values were tested between 1 and 20, corresponding to different electrical frequencies.
Materials names and indices.
4.2 Results
The final average torques of optimized machines are plotted in Figure 10. The highest torque is obtained for N = 10, corresponding to the current feeding of the reference machine shown in Figure 2. Therefore, a very similar rotor design shown in Figure 11a is obtained and develops about the same torque (〈T_{10}〉 = 2244 Nm/m) as the reference one (〈T_{ref}〉 = 2236 Nm/m) under the same current feeding.
Fig. 10 Average torque of optimized designs depending on N. 
Fig. 11 Resulting designs and flux maps with the highest torques for rotor optimization [15]. a) Final design N = 0, b) Flux density (N = 10), c) Final design N = 4, d) Flux density (N = 4), e) Final design N = 16, b) Flux density (N = 16). 
The results indicated in Figure 10 show that N = 4 and N = 16 can produce a lower torque than the reference but more significant than the rest of the N values. The associated designs are drawn in Figures 11c and 11e, and their convergence curves are compared to the one obtained with an unsuitable value N = 13 in Figures 12a and 12b. Although they reach a lower torque than the N = 10 structure, their topologies fundamentally differ from the reference rotor and are unusual for a human designer. Indeed, they contain disconnected parts, illustrating the ability of topology optimization to find innovative designs.
Fig. 12 Convergence curves during the optimization. a) Torque evolution, b) Step size evolution. 
The algorithm fails to find meaningful structures for the rest of the N values, which may indicate the impossibility of designing suitable rotors under these conditions. An example of such a “bad” optimized design for N = 13 is drawn in Figure 13, which leads to the conjecture that there exist suitable rotors only for N = 4 + 6n, n ∈ N.
Fig. 13 “Bad” structure obtained with N = 13. a) Final design, b) Flux density. 
5 Stator optimization
Densitybased methods can be extended to multimaterial problems, such as optimizing an FSM stator. Following the results of the previous section, we impose N = 10 with the reference rotor geometry shown in Figure 2. The magnets’ positions and orientations are also fixed, given the optimization zone drawn in Figure 14.
Fig. 14 Design domain (gray) in case of the stator optimization. 
5.1 Threephase machine
The optimization of a 3phase stator is first addressed. Since the permanent magnets are imposed, there are 8 candidate materials (steel, air, and conductors^{3}) listed in Table 1. A multimaterial interpolation formalism is necessary to address this problem. To handle this large amount of materials, the interpolations (3) and (4) are supported by a multidimensional space . We chose as a convex polytope that supports shape functions set {ω_{i}}_{i∈⟦0,7⟧} [28], represented in Figure 15. The diamondshaped domain allows the distribution of the conductors onto the same plane with respect to their electrical phase. The basis function values are computed with an optimized version of the code given in [33], available at [34].
Fig. 15 Interpolation domain that supports a set of shape functions in the 3phase stator case. a) Material placement, b) Values of a shape function associated with B^{+}. 
The chosen penalization functions P_{m} and P_{j} are adapted from [35] and read:(16) (17)
The optimization is then performed for several current angles ψ ∈ [0°, 60°]. The resulting designs are given in Figure 16 with their average torques. The current angle strongly influences the resulting topologies; when it is incorrectly set, it may return nonmanufacturable designs with no return conductors.
Fig. 16 Designs obtained after stator optimizations. a) ψ = 0°, 〈T〉 = 2444 Nm/m, b) ψ = 12°, 〈T〉 = 2522 Nm/m, c) ψ = 24°, 〈T〉 = 2747 Nm/m, d) ψ = 36°, 〈T〉 = 2798 Nm/m, e) ψ = 48°, 〈T〉 = 2643 Nm/m, f) ψ = 60°, 〈T〉 = 2444 Nm/m. 
Intermediate ψ values promote single conductors in each slot, as there is a factor between the total current amplitude in sliced conductors and plain conductor slots, as illustrated in Figure 17.
Fig. 17 Fresnel diagram of the current in sliced (black) and plain conductors (red). 
The structure obtained with ψ = 0° is very similar to the reference shown in Figure 2. However, the conductors are larger, carrying more current for the same current density, set to J = 10 A/mm^{2}. Consequently, the average torque of the optimized design is higher. Note that the order of conductors is inverted within the same slot. As topology optimization considers only electromagnetism performance, it places the conductors according to their electrical phase, resulting in distributed windings. However, Figure 2 proposes concentrated windings, probably for manufacturability reasons that are still ongoing research topics in topology optimization.
5.2 Singlephase machine
To illustrate the creativity of topology optimization, we now address an academic problem with no reference solution. More precisely, we want to optimize the stator of a 10/12 machine that has the same 10tooth rotor and 12magnet configuration, using only a single electrical phase, i.e., a positive and a negative conductor denoted as and . Singlephase FSMs are of interest for domestic and lowcost applications, and some classic topologies were compared in [36].
Since there are now only 4 candidate materials, the interpolation domain is a tetrahedron drawn in Figure 18a, and the basis functions are the barycentric coordinates as illustrated in Figure 18b. The penalization functions are the same as in Section 5.1.
Fig. 18 Interpolation domain supports a set of shape functions in the singlephase stator case. a) Materials placements, c) Values of a shape function associated with the FeCo steel. 
A parametric sweep on ψ was performed. The results are shown in Figure 19, and the corresponding torques are plotted in Figure 20. The case of ψ = 0° [60°] leads to manufacturable designs that reach 2200 Nm/m with a mixture of C and Ecore patterns [37, 38].
Fig. 19 Examples of singlephase optimized designs. a) ψ = 0°, 〈T〉 = 2200 Nm/m, b) ψ = 20°, 〈T〉 = 2200 Nm/m, c) ψ = 40°, 〈T〉 = 2198 Nm/m, d) ψ = 60°, 〈T〉 = 2198 Nm/m. 
Fig. 20 Average torque obtained for each ψ. 
5.3 Torque comparison
The dependence of the average torque with the current angle ψ is given in Figure 20 hereafter.
The rotor is almost insensitive to ψ because its optimized topology compensates for the electric phase shift by a mechanical rotation. This mechanical rotation is impossible for the stator optimization since the permanent magnet positions are imposed. For the threephase machine, the factor between the maximum and minimum average torque is because of the sliced conductors, as predicted in Section 5.1. Since there are no sliced conductors within the stator slots for the singlephase machine structures, its average torque remains almost constant.
Even if it was not considered in the optimization, the torque ripple is an important criterion when designing an electrical machine, and is given in Table 2. The corresponding instant torques of manufacturable machines (i.e., the structures obtained with ψ = 0°) are plotted in Figure 21.
Fig. 21 Instant torque obtained for the optimized designs for each ψ. 
Torques of the optimized designs (ψ = 0°).
The singlephase machine produces a pulsating torque with a high ripple but can still produce a significant average torque. The two other optimized designs outperform the reference, and the best performance is obtained by optimizing the 3phase stator.
In the end, the bestfound topology is similar to the reference, but our topology optimization approach has explored other unusual structures.
6 Conclusion
This work presents and details a densitybased topology optimization method applied to the torque maximization of a permanent magnet flux switching machine. Unconventional problems are addressed, such as the rotor optimization for several electrical frequencies and the stator optimization for different numbers of phases. Unusual structures were obtained, such as disconnected rotors or a hybrid singlephase stator made of C and Ecores. This variety of results from an empty optimization domain highlights the creativity and unbiasedness of a topology optimization approach. The bestobtained topology is similar to the reference design but with better torque performances. In other applications, the optimal topology might differ from the known references.
In conclusion, topology optimization is a valuable tool to help design electrical machines that can be improved further. Future research will add the permanent magnet to the set of candidate materials so that the stator can be optimized in its entirety. Such work will be the first step to the topology optimization of a complete flux switching machine with both rotor and stator simultaneously. Additional objective functions can be added to tackle industrial problems such as ripple and backelectromotive force. Finally, other physics such as thermics and mechanics may be considered in order to obtain more readytouse designs.
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All Tables
All Figures
Fig. 1 Principle of densitybased representation. a) Original domain, b) Densitybased representation. 

In the text 
Fig. 2 Reference PMSF machine adapted from [19]. 

In the text 
Fig. 3 Anhysteretic BH curve of FeCo in logarithmic scale used for the computations. 

In the text 
Fig. 4 Interpolation example of the current density amplitude between 7 materials. Note that in the real optimization problem in Section 5, the actual value of the current density at each electric angle is interpolated instead of its amplitude. 

In the text 
Fig. 5 Mesh of the simulation domain containing 30,666 firstorder elements and 16,329 nodes. 

In the text 
Fig. 6 Flowchart of the optimization algorithm. 

In the text 
Fig. 7 Flowchart of the update algorithm. 

In the text 
Fig. 8 Optimization domain (gray) in case of a rotor optimization. 

In the text 
Fig. 9 Magnetic behavior used for the computation. 

In the text 
Fig. 10 Average torque of optimized designs depending on N. 

In the text 
Fig. 11 Resulting designs and flux maps with the highest torques for rotor optimization [15]. a) Final design N = 0, b) Flux density (N = 10), c) Final design N = 4, d) Flux density (N = 4), e) Final design N = 16, b) Flux density (N = 16). 

In the text 
Fig. 12 Convergence curves during the optimization. a) Torque evolution, b) Step size evolution. 

In the text 
Fig. 13 “Bad” structure obtained with N = 13. a) Final design, b) Flux density. 

In the text 
Fig. 14 Design domain (gray) in case of the stator optimization. 

In the text 
Fig. 15 Interpolation domain that supports a set of shape functions in the 3phase stator case. a) Material placement, b) Values of a shape function associated with B^{+}. 

In the text 
Fig. 16 Designs obtained after stator optimizations. a) ψ = 0°, 〈T〉 = 2444 Nm/m, b) ψ = 12°, 〈T〉 = 2522 Nm/m, c) ψ = 24°, 〈T〉 = 2747 Nm/m, d) ψ = 36°, 〈T〉 = 2798 Nm/m, e) ψ = 48°, 〈T〉 = 2643 Nm/m, f) ψ = 60°, 〈T〉 = 2444 Nm/m. 

In the text 
Fig. 17 Fresnel diagram of the current in sliced (black) and plain conductors (red). 

In the text 
Fig. 18 Interpolation domain supports a set of shape functions in the singlephase stator case. a) Materials placements, c) Values of a shape function associated with the FeCo steel. 

In the text 
Fig. 19 Examples of singlephase optimized designs. a) ψ = 0°, 〈T〉 = 2200 Nm/m, b) ψ = 20°, 〈T〉 = 2200 Nm/m, c) ψ = 40°, 〈T〉 = 2198 Nm/m, d) ψ = 60°, 〈T〉 = 2198 Nm/m. 

In the text 
Fig. 20 Average torque obtained for each ψ. 

In the text 
Fig. 21 Instant torque obtained for the optimized designs for each ψ. 

In the text 
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