Advancing Structural Optimization of an Electric Motor Rotor through Mesh Morphing Techniques

The electric motor has been one of the most important inventions of the last two centuries and has become increasingly important in the last 10 years thanks to its efficiency and its ability to reduce pollution. Given this increasing importance of electrical motors, an interest in design and optimization approaches for motor components is arising. Given the different physics involved in the transformation from electrical to mechanical energy, a valuable and reliable approach has to be applied in the optimization process. In past years, Mesh Morphing based on Radial Basis Functions (RBF) has largely proved its validity in generating shape modification for Finite Element Method (FEM) models. In this work, authors will demonstrate the advantages in applying two opposite approaches for shape optimization using RBF Mesh Morphing applied to electrical motors rotors. These components need to be accurately designed in order to guarantee an adequate life duration by reducing stresses in the rotating components. The two approaches that will be described are the parameter based one, in which a set of design parameters will be varied to generate shape modification used to feed a meta-model that will be used to identify an optimal configuration, and a parameter-less approach, in which the shape modification will be driven by FEM analysis results, with the aim to reduce stress hot-spots.


Introduction
The Electric Motor (EM) has been one of the most important inventions of the last two centuries.It was invented almost two hundred years ago and since then it has undergone many evolution.In the last years, the electric motor has become increasingly important in motor vehicles such as electric cars thanks to its efficiency and its ability to reduce pollution.This is not only due to its ability to operate without producing harmful emissions for the environment, but also to the fact that the electric motor is much more efficient than internal combustion engines because it converts electrical energy into mechanical energy with very high efficiency.
In the last decade, the automotive industry put a lot of effort in transitioning from internal combustion engine vehicle to electrical motor vehicles.The increased interest is obviously pushed by increasing environmental concerns, but the adoption of this new engine for everyday mobility will introduce also other benefits.EMs can produce higher torque and power with respect to a thermal engine of equal size or weight, emit lower noises, have lower maintenance and operating costs.
There are several typology of EM, but recently Interior Permanent Magnet (IPM) motors have become a predominant choice for driving modern electric vehicles (EVs).Capitalizing on IOP Publishing doi:10.1088/1757-899X/1306/1/012033 2 their high efficiency and torque capabilities, IPMs serve as an ideal response to the automotive sector's transition from internal combustion engines to electric drivetrains.Contemporary IPM rotor designs are the result of extensive research spanning several decades, striking a careful equilibrium between performance, efficiency, affordability, and reliability.At high speeds up to 20000 RPM, rotors experience considerable centrifugal forces.Thus, it's crucial for the rotor's laminates to firmly anchor the magnets, ensuring they remain intact and undamaged.Proper rotor engineering is key to mitigating such potential challenges.
Numerical tools developed to support and push further design of mechanical components can be employed in the EM rotors design too.In particular Finite Element Method (FEM) can provide a reliable way to investigate both thermal and structural behaviour of such components.As a matter of facts, each FEM model can represent the behaviour of a single rotor configuration, and in order to identify the optimal configuration, a set of models have to be prepared to be analysed.Each analysis can be very time consuming, and, to obtain results faster, each configuration can be generated using Radial Basis Functions (RBF) mesh morphing [1,2].This mesh-less technique allows to generate modified configuration using shape modifications without the need to generate the underlying modified geometry.RBF mesh morphing has been proved to be reliable and time-saving in several advanced workflows, such as Fluid-Structure Interaction (FSI) [3,4,5,6,7,8], or crack front evolution [9,10], but also in stress recovery in composite plates [11,12] in fracture mechanics applications [13], FEM results upscaling [14] and shape optimization [15].
The shape modification imposed by RBF mesh morphing to the numerical model can be defined according to user input or can be driven by numerical model itself, exploiting, for example evaluated surface stress.This approach, which can be defined parameter-less, is at the basis of the Biological Growth Method (BGM) [16], which mimics the behaviour of natural tissues when stress and forces are applied on surfaces.The application of this parameter-less approach has already proven its effectiveness even compared with parameter based optimizations methods and other parameter-less ones ( [17] and [18]).
In the present work, both the parameter based and the parameter-less procedures are used to optimize stresses on a EM rotor.The numerical tools used are the ones included in the ANSYS Workbench Finite Element Analysis (FEA) framework [19], including RBF Morph Structures [20], that has been used to apply mesh morphing in FEM models.
It is worth to remark that, given the complexity of an electric motor design, the numerical simulations should include all physics involved.In this work, the attention has been focused on the optimization of the structural model, in order to give a prompt and clear view of the benefits achievable with the proposed approaches.Due to the meshless nature of the RBF based mesh morphing, the introduction of multi-physics analysis in the optimization workflow will not invalidate it, since the same shape modification can be applied to each different numerical grid.
In the next sections a brief introduction on RBF based mesh morphing and BGM will be given (section 2), then the numerical model will be described (section 3), finally the optimizations results and the conclusion will be reported (sections 4 and 5).

Mesh Morphing and Biological Growth Method
In this section an overview of the RBF mesh morphing and of the BGM will be given, together with an insight of how these two techniques can be used and combined to perform both Parameter-less and Parameter Based optimizations.

Background on RBF Mesh Morphing
RBF were used as an interpolation tool to fit data with no prescription about the number of dimensions on which this data is defined [21]: they are a set of scalar functions capable IOP Publishing doi:10.1088/1757-899X/1306/1/0120333 to interpolate data defined in the so called source points, in which known values are defined.Generally speaking, the interpolation function can be written as: In equation (1) x k i are the source points defined in the space R n and x are the points at which the function is evaluated, called also target points.φ is the radial function.A detailed description of the RBF theory and its application was given in [16].

Background BGM
BGM was inspired by how biological tissues growth is influenced by external actions and forces: layers are added when an activation stress is reached.The studies from which this method has been developed are those by Heywood [22], Mattheck and Burkhardt [23] and Waldman and Heller [24].
In the present work another formulation for BGM is used, which has been presented in [16] and in [25].The displacement applied to each target node (S node ) is aligned with the surface normal direction and can be both inward and outward.To evaluate it, equation ( 2) is used, in which σ node is the stress value for each node, σ th is a threshold value for stress chosen by user, σ max and σ min are the maximum and minimum value for stress evaluated in the current set of source nodes.d is the maximum offset between the nodes on which the maximum and the minimum stress are evaluated.This parameter is defined by the user to control the nodes displacement whilst limiting the possible distortion of the mesh: Equation ( 2) allows to impose a displacement for nodes on the surface to be optimized that can be either inward, in case the stress for the current target node is lower than the stress chosen by user as threshold, or outward, in case stress on target node is higher than threshold one.

Parameter-less Based Optimisation
Combining RBF based mesh morphing (section 2.1) and BGM (section 2.2) can result in an automatic surface sculpting procedure driven by FEM analysis results and needing no shape modification parameters.The numerical procedure is described in [16] and summarized in the following steps: (i) stress from baseline configuration are evaluated by means of FEM simulation; (ii) BGM routines retrieve surface stresses and evaluate each target node displacement using equation ( 2); (iii) RBF problem is set up using BGM evaluated displacement and, if needed, additional RBF source points displacements defined by user; (iv) baseline model is then morphed and FEM solution evaluated again; (v) updated stresses are analyzed to decide if further optimization is needed; if so, the workflow can be iterated form point (ii), otherwise, the optimized configuration is reached.
In the described automatic surface sculpting procedure, the analyst has only to define two parameters: σ th and d of equation (2).

Parameter based Shape Optimisation
The parameter based optimization can be performed with RBF mesh morphing by prescribing actions to specific groups of nodes (scaling, translation, surface offset, ...) so that the shape of surfaces and of the volume mesh is updated accordingly.The entity of such actions (scaling factors, component of translation, amount of offset ...) are then combined and controlled so that a certain number of new shapes is generated by Design of Experiment (DoE) and optimal performances are then computed on the response surface [17].

FEM Model
In this section a description of the numerical model used to perform optimization is given.Numerical analyses were performed in the framework of Ansys Workbench, since it has all the numerical tools needed for these tasks: a FEM modeler and solver (Ansys Mechanical) an RBF based mesh morphing tool (RBF Morph Structural) which also has implemented the BGM routines, and a surface response based optimization tool (otpiSLang).

Numerical Model Description
Due to the peculiar geometry of the electric motor rotor, the modelling has taken advance of two simplifications based on the symmetry conditions applicable to the model.Only a sector of the rotor has been modelled, corresponding to a 45 • span of the rotor (see Figure 1), rotor external radius is 70mm whilst internal radius is 23.5mm.The other symmetry condition applied is the planar strain of the analysed component, which is acceptable considering the electric motor rotor configuration and the boundary conditions applied.Due to this, the thickness of the model is 5 • 10 −4 m, which is required to generate 1 element through the thickness.In particular all FEM analyses were performed considering a constant rotational velocity equal to 1885rad/s and all symmetry faces are constrained using frictionless support condition.Contact between rotor and the four magnets has been modeled using bonded contact condition.

Figure 1. Rotor sector geometrical model
After the meshing process (i.e. the discretization of the solids) the FEM model was composed by 342875 nodes and 47756 elements.Mesh refinements rules were applied in order to have more resolution in the areas in which high surface stress levels are located (hot-spots), as can be seen in Figure 2.
The identified hot-spots were three, located in the fillet area of the three magnet housing, as depicted in Figures 3, 4 and 5, respectively.Maximum equivalent stress values are reported in Table 1.

Parameter Based Setup
In order to perform the parameter based optimization a particular RBF set-up has been defined.The RBF set-up is referred to the source points definition, i.e. the points on which displacement (null or not) is imposed and on which the displacement field is evaluated.It has been decided to define the shape of the magnet housing fillet using three parameters.On the circular arc portion of the fillet, three points (at start, middle and end of the arc) have been selected and moved radially using a cylindrical local reference system (Figure 6).Exploiting the Hierarchical morph features of RBF Morph Structures, the RBF field obtained with these three point displacement is then used to drive the shape modification of the entire circular arc fillet portion, as depicted in Figure 7.Then, maintaining fixed the points of the fillet near the magnet contact area, the magnet housing fillet is morphed accordingly, as previewed in Figure 8.In Figure 9 the mesh in the baseline configuration is reported and compared with the same mesh portion in the morphed configuration (Figure 10) after the modification of the three shape parameters (i.e. points displacement depicted in Figure 6).To finalize the parameter based optimization setup, three Response Surface Optimization tasks were executed action on a single hot-spot separately (Figure 11).The Design of Experiment (DoE) tables were set up using the Latin Hypercube Sampling with CCD Samples algorithm, which generated 15 Design Points (DP) for each hot-spot.Each DP is generated by prescribing a different combination of the radial displacement of the three control points defined at each fillet.The Response Surface built on the outcome of the DoE was the Genetic Aggregation one and the optimization method used was the Multi-Objective Genetic Algorithm (MOGA).Each candidate point identified was finally re-evaluated using FEM, obtaining the final optimized shape for the three hot-spots after 48 FEM simulations.To complete the parameter-less set-up the sequential morphing procedure was prepared, using Ansys Workbench parameters to instruct RBF Morph Structure to use previous FEM results to morph the current model.Letting the code to sculpt the hot-spot surfaces using surface equivalent stress levels, the procedure required 17 steps to reach the minimum value for hotspot 1 and 2 and 7 steps to reach the minimum value for hot-spot 3 (Figure 13).In particular, maximum equivalent stress in hot-spot 1 is 143M P a (Figure 14), in hot-spot 2 is 181M P a (Figure 16) and in hot-spot 3 is 294M P a (Figure 18), obtaining maximum equivalent stress reduction of 31.25%,18.78% and 7.84% respectively.In particular, maximum equivalent stress in hot-spot 1 is 121M P a (Figure 20), in hot-spot 2 is 156M P a (Figure 22) and in hot-spot 3 is 278M P a (Figure 24), obtaining maximum equivalent stress reduction of 41.82%, 27.44% and 12.85% respectively.
In Table 2 all optimizations results are presented and compared.

Conclusions
In the present work two optimization approaches for EM rotors are presented, both based on RBF mesh morphing.The two optimization methods were tested on a generic rotor, which presented three regions in which high equivalent stress levels were identified.These regions (hot-spots) were located in the fillets of magnet housing.The first approach presented is the Parameter based one: the numerical model mesh in the hot-spots areas are made parametric by means of RBF mesh morphing by properly chose the fixed and moving source points of the RBF problem.The parametric model is then used generate all the FEM models in morphed configurations corresponding to the DP of a DoE on which the Response Surface Optimization was executed.
The second approach presented, the Parameter-Less one, combined the RBF based mesh morphing with the BGM, creating an automatic surface sculpting tool driven by surface stress levels Both approaches proven to be effective in reducing stresses in the identified hot-spots.Stress reduction obtained with the Parameter Based approach on the three hot spots were respectively 31, 25%, 18, 78% and 7, 84%.Stress reduction obtained with the Parameter-Less approach on the three hot spots were respectively 41, 82%, 27, 44% and 12, 85%.
The Parameter Based approach optimization reached lower stress reduction rate at a higher computational cost: 48 FEM simulation were run, besides the response surface calculation cost.However optimized shapes results smoother if compared with ones obtained with the other approach.
The Parameter-Less approach required a lower computational cost (17 FEM simulation were run) to obtain optimized configuration.
The present work focused attention on the structural analysis only, but it is worth to remark that both approaches can be used in a complex multi-physics workflow EM design, since mesh morphing field evaluated to obtain the optimized shape can be used to morph other physic numerical models, so that all numerical model shape can be synchronized and, eventually, forces and interaction can be updated and optimization workflow iterated to increase accuracy of results.

Figure 12 .
Figure 12.Hot-spot surfaces selected for BGM sculpting

Figure 13 .
Figure 13.Sequential morph set-up and maximum equivalent stress evolution for the three hot-spots .

Table 1 .
Maximum equivalent stress in identified hot-spots

Table 2 .
Optimizations results comparison