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Parallelintime optimization of induction motors
Journal of Mathematics in Industry volume 13, Article number: 6 (2023)
Abstract
Parallelintime (PinT) methods were developed to accelerate timedomain solution of evolutionary problems using modern parallel computer architectures. In this paper we incorporate one of the efficient PinT approaches, in particular, the asynchronous truncated multigridreductionintime algorithm, into a bound constrained optimization procedure applied to an induction machine. Calculation of an optimal motor geometry with respect to its efficiency in the steady state is thus parallelized at each iteration of the optimization algorithm. As a result, a more efficient motor model is obtained about 11 times faster compared to optimization using the standard sequential time stepping.
1 Introduction
Modern corporate design of electromagnetic devices such as electric motors is based on computeraided optimization of several key performance indicators such as, e.g., output power, losses, overall efficiency, etc. In this way, customer requirements can be incorporated already within the early design stages before a physical prototype is manufactured. In order to create an optimal digital prototype, one typically has to perform transient simulations of various multiphysical effects (e.g., magnetic and mechanical) in the time domain. Such calculations are often very time consuming due to the need to resolve the arising highfrequency field components, which despite their small amplitudes, can lead to big losses. Parallelintime (PinT) methods such as Parareal [1] or multigridreductionintime (MGRIT) [2] are powerful tools for an acceleration of these development stages, as it is shown for an induction motor in [3] and [4], respectively. The MGRIT algorithm is based on multilevel reduction [5] principles applied to the time dimension. In this process, time integration is applied in parallel to temporal subdomains at the finer levels and serially over the entire time interval at the coarsest level. One of the key advantages of the algorithm is its nonintrusive nature, which allows existing time integrators to be reused and embedded in a timeparallel framework. A variant of the MGRIT algorithm, the asynchronous truncated multigridreductionintime algorithm (ATMGRIT) [6], uses multiple independent overlapping local coarse grids at the coarsest level to increase the parallelism of the MGRIT algorithm.
In this paper, we optimize the geometry of a threephase squirrelcage induction motor with respect to its efficiency in the steady state. For this, a derivativefree algorithm BOBYQA for bound constrained optimization is used, which is based on quadratic interpolation [7, 8]. Each iteration of the optimization procedure includes the PinT timedomain computation with ATMGRIT until the steady state of the machine is obtained. The mechanical power and the Joule losses are calculated at the postprocessing step and used for the construction of the objective function, which is optimized in terms of the rotor bars size, i.e., their width and height.
This paper is organized as follows. In Sect. 2 we provide a mathematical partial differential equation (PDE) model of the electromagnetic phenomena taking place in an induction motor and discretize it using space and timedomain numerical methods. Section 3 formulates an optimization problem in terms of the motor’s efficiency in the steady state and describes the overall optimization procedure based on a timedomain solution. In Sect. 4 we describe the ATMGRIT method for accelerated PinT solution, which can be used at each iteration of the optimization algorithm. Application of the proposed methodology to a fourpole induction machine is illustrated in Sect. 5. Finally, the paper is completed with a conclusion in Sect. 6.
2 Simulation of induction machines
Electric motors are electric devices that transform electrical energy into mechanical energy. In this paper we consider a specific type of motors called threephase induction motors, which are among the most widespread electric motors within the power range under 500 kW.
Threephase induction motors are supplied with a threephase voltage source \(v_{k}\) given by
illustrated in Fig. 1, where f is the frequency and Û is the amplitude (or peak value) of voltage. The windings carrying threephase voltage are placed into the stator slots in the outer part of the motor called stator as depicted in Fig. 2 for a twodimensional (2D) induction motor model.
The inner part of the considered induction motor is a squirrelcage rotor, which consists of solid conductor bars, placed into the rotor slots and connected at both ends by the conducting end rings. The stator and rotor are separated by the air gap, which is traversed by the magnetic flux and allows for a current flow in the rotor bars. As a result, the Lorentz force acts on the squirrel cage and makes the rotor rotate with a mechanical speed \(\omega _{\mathrm{mech}}\). Finally, the produced electromagnetic torque \(T_{\mathrm{EM}}\) is transferred to the mechanical load through a shaft placed in the very inner part of the motor.
Solution of a dynamical system like an electric motor excited with a periodic signal (1) typically consists of a transient part, followed by a (periodic) steady state taking place once the transients are eventually damped out. The steadystate operating characteristics such as rotational speed and torque are important design criteria, especially during initial design stages [9]. Figure 3 illustrates the timedomain torque evolution of an induction motor rotating at the constant speed \(\omega _{\mathrm{mech}}=1420\) rpm. There, the steady state is reached at the tenth period of length \(T={0.02}\) s, i.e., on the interval \([(q1)T,qT]=[0.18,0.2]\) s, with \(q=10\). In this section we provide the theoretical basis and numerical approaches for the timedomain simulation of electromagnetic energy converters.
2.1 Mathematical model
The electromagnetic fields in induction motors are commonly modeled by a magnetoquasistatic (MQS) approximation of Maxwell’s equations [12], which is suitable for lowfrequency applications provided the wavelength is much larger than the problem size [13]. The MQS setting neglects the displacement currents as they are outweighed by the magnetic effects and the Joule losses. One can then derive for the magnetic vector potential (MVP) A the eddy current problem
with \(\Omega \subset \mathbb{R}^{3}\) denoting an open, bounded, simply connected domain with Lipschitz boundary and \(T_{\mathrm{end}}>0\). Here \(\sigma =\sigma (\mathbf{x} )\geq 0\) denotes the electric conductivity, \(\nu =\nu (\mathbf{x} ,\operatorname{curl}\mathbf{A} )>0\) is the magnetic reluctivity, and \(\mathbf{J}_{\mathrm{s}}\) is the source current density defined by
with each \(\boldsymbol{\chi}_{k}(\mathbf{x} )\in \mathbb{R}^{3}\) denoting a winding function [14], which spatially distributes the current \(i_{k}(t)\in \mathbb{R}\) flowing through the kth stranded conductor. The threephase input voltage (1) is coupled to the eddy current equation (2) via the relation [15, Sect. 6]
with \(R_{k}\) denoting the DC resistance of the stator stranded conductors. For a complete formulation we include the homogeneous Dirichlet boundary condition and an initial condition (IC)
where n denotes the outward normal vector to the boundary \(\Gamma =\partial \Omega \). Combining the equations (2) and (4), one obtains a coupled fieldcircuit system, which we will discretize in the following Sect. 2.2.
Remark 1
For the simulation of electric motors one often assumes that they are invariant under translation in the axial \(x_{3}\)direction. In this case, 2D models in the \(x_{1}x_{2}\)plane as the one from Fig. 2 are considered. This leads to the setting
which transforms the equation (2) for the vector quantity A into the equation
for the scalar quantity \(\mathbf{A}_{3}\), where \(\Omega _{\mathrm{2D}}\subset \mathbb{R}^{2}\) is an open, bounded, simply connected domain with Lipschitz boundary, [9].
2.2 Numerical solution
A standard approach to solve a spacetime dependent system is the method of lines, where one first discretizes the problem in space using, e.g., the finite element method (FEM), and then integrates the resulting timedependent system using a numerical time integrator such as, e.g., the implicit Euler method.
Discretization of the coupled system (2)–(4) in space using FEM with d degrees of freedom leads to a timedependent system on \((0,T_{\mathrm{end}}]\)
with respect to \(\mathbf{a}(t)\in \mathbb{R}^{d}\) and \(\mathbf{i}(t)=[i_{1}(t),i_{2}(t),i_{3}(t)]^{ \top} \in \mathbb{R}^{3} \). Here \(\mathbf{M}_{\sigma}\) and \(\mathbf{K}_{\nu}(\cdot )\) are the \((d\times d)\)dimensional mass and curlcurl matrices, respectively. Matrix \(\mathbf{X}\in \mathbb{R}^{d\times 3}\) is given by
with basis functions \(\mathbf{w}_{j}\) from the Hilbert space \(\mathbf{H}(\operatorname{curl};\Omega )\), see [16]. The matrix \(\mathbf{R}\in \mathbb{R}^{3\times 3}\) is a diagonal matrix of resistances \(R_{k}\) and \(\mathbf{v}(t)=[v_{1}(t),v_{2}(t),v_{3}(t)]^{ \top} \in \mathbb{R}^{3}\) is the threephase input voltage given by (1). Additionally, we have the prescribed ICs
which together with the system (9a)–(9b) form an initialvalue problem (IVP). The equation (9a) is in general a system of index1 differentialalgebraic equations (DAEs), since the matrix \(\mathbf{M}_{\sigma}\) is singular when the considered domain Ω includes nonconducting regions, i.e., regions where \(\sigma =0\). Therefore, implicit methods have to be applied for the time integration [17]. The implicit Euler discretization of the spacediscrete system (9a)–(9b) starting from the given values \(\mathbf{a}_{0}\) and \(\mathbf{i}_{0}\) with the step size \(\delta T=T_{\mathrm{end}}/N_{{t}}\) reads
where \(\mathbf{a}_{j}\) and \(\mathbf{i}_{j}\) denote approximate solutions at time step \(t_{j}=j\delta T\), with \(j=1,\dots ,N_{{t}}\). Since the equation (11a) is nonlinear, a linearization approach, e.g., the Newton method [18], has to be applied at each time step. In [3] it was shown that the implicit Euler method has an implicit projection property that the consistency of initial values known from DAE theory is not an issue.
2.3 Quantities of interest
The electromagnetic torque induced in the air gap and exerted on the rotor can be calculated using the formula [19], [20, Sect. 1.5]
where S is the surface enclosing the rotor, r is the position vector connecting the rotor origin to S, n is the unit normal vector to S, and σ is the Maxwell stress tensor [21, Sect. 6.3] given by
with the reluctivity in vacuum \(\nu _{0}\), the magnetic flux density B given by \(\mathbf{B} =\operatorname{curl}\mathbf{A} \), and the Kronecker delta \(\delta _{ij}\). The product of the produced torque and the rotational speed defines the mechanical power \(P_{\mathrm{mech}}\), i.e.,
Since a part of the input power is lost as heat, it is important to calculate also the Joule losses. For the 2Dsetting from Remark 1 and length \(\ell _{3}\) of the motor in the axial \(x_{3}\)direction, these losses are given by
where \(\mathbf{A} _{3}\) denotes the \(x_{3}\)component of the MVP A, see (7). The considered quantities of interest can be calculated in a postprocessing step of the simulation and will be used in an optimization procedure described in Sect. 3.
3 Optimization
In the optimization we use the height h and width w of the rotor bars as optimization variables and parametrize the domain \(\Omega = \Omega (p)\) with these two parameters \(p=(h,w)\). Our goal is to find the optimal width and height of the rotor bars, such that our objective function J is minimal under the constraint that the design variables lie in a set of admissible designs \(D_{\mathrm{ad}}\). Additionally, we require that the state equations (2) and (4) together with the boundary and initial conditions (5)–(6) are fulfilled. As an objective function we consider:
with
where \(q\in \mathbb{N}\) represents the period at which the steady state is reached and \(T>0\) is the length of the period (e.g., \(q=10\) and \(T={0.02}\) s in Fig. 3). The objective J can be seen as a negative measure of efficiency, as it is given by the quotient of the output and the input power on the righthand side in (16). Since \(P_{\mathrm{mech}}\) and \(P_{\mathrm{loss}}\) involve integrals over the parametrized domain (see (14), (12), and (15)), they depend on the design p. They both depend on the solution \(\mathbf{A} _{3}\) of the state equation, which depends on the design p itself. In the optimization, we know [22] that for every admissible design p, there is a unique solution to our state equation, which we call \(\mathbf{A} _{3}(p)\) and consider the reduced problem
which does not involve the state equation as a constraint anymore and where
with \(h_{\mathrm{l}},h_{\mathrm{u}},w_{\mathrm{l}},w_{\mathrm{u}}\in \mathbb{R}\).
To solve the optimization problem, we use the derivativefree optimization algorithm PyBOBYQA [8], which is a Python implementation of BOBYQA [7]. The idea of this algorithm is to use a model for the objective function and improve the model in every iteration to make it approximate the minimum of the objective function sufficiently. Specifically, a quadratic interpolation polynomial \(\mathcal{Q}^{(k)}(s)\approx \hat{J}(p^{(k)} + s)\) around the current iterate \(p^{(k)}\) is used, which coincides with the true objective function on an interpolation set \(Y=\{y_{0}=p^{(k)},\ldots ,y_{m1}\}\):
with \(c\in \mathbb{R},g\in \mathbb{R}^{n}\text{ and } H\in \mathbb{R}^{n \times n}\).
To improve the model, the current model \(\mathcal{Q}^{(k)}\) is minimized inside a trustregion \(\{s\in \mathbb{R}^{n} \colon \s\_{2}\le \Delta ^{(k)}\}\), where \(\Delta ^{(k)}>0\) is the trustregion radius:
The initial trustregion radius \(\Delta ^{(0)}\) is user supplied and has to fulfill \(p^{(0)}+\Delta ^{(0)}v\in D_{\mathrm{ad}}\) for all \(v\in \mathbb{R}^{2}\) with \(\v\_{2}=1\). It is called trustregion, because we hope, that when the radius \(\Delta ^{(k)}\) is small enough, we can trust the model \(\mathcal{Q}^{(k)}\) in a neighborhood of our current point \(p^{(k)}\).
The quality of the step s computed by solving (TRS) is assessed by calculating the ratio
If \(R^{(k)}\) exceeds a predefined threshold the step is accepted, one of the interpolation points \(y_{i}\in Y\) is replaced by \(p^{(k+1)}=p^{(k)}+s\) (the elements of Y then get reordered, such that \(y_{0}=p^{(k+1)}\)), the model \(\mathcal{Q}^{(k)}\) is updated and the trustregion radius is enlarged. If \(R^{(k)}\) is small or negative, which is the case when the true objective does not decline, or the model is a bad predictor, the step is rejected (\(p^{(k+1)}=p^{(k)}\)) and the trustregion radius \(\Delta ^{(k)}\) is reduced, since the model was inaccurate on the former trustregion. The algorithm terminates when the trustregion radius becomes smaller than a predefined tolerance \(\Delta ^{(\text{end})}\).
A flowchart of the optimization procedure, which iteratively updates the parameter p based on the timedomain calculation until the steady state, is illustrated in Fig. 4.
For a symmetric matrix H, the model \(\mathcal{Q}^{(k)}\) has \(m=\frac{1}{2}(n+1)(n+2)\) degrees of freedom and, thus, m interpolation points and objective evaluations (right hand side (20)) are needed to compute the model. For every evaluation of the objective function \(\hat{J}(p) = J(\mathbf{A} _{3}(p),p)\) we have to solve our state equation, which is expensive. We therefore want to use as few interpolation points as possible and the idea of BOBYQA is to use only \(m=2n+1\) and eliminate the rest of the degrees of freedom by requiring that the Hessian of the interpolation polynomial has minimal curvature. To compute such a minimal curvature model, the following optimization problem has to be solved:
When computing the initial model, we set \(H_{\mathrm{prev}}=0\). The initial interpolation set is chosen as \(Y=\{p^{(0)},p^{(0)}\pm \Delta ^{(0)} e_{1}, p^{(0)}\pm \Delta ^{(0)} e_{2} \}\), where \(e_{i}\) is a zero vector with a one in the ith entry. When the set Y is poised for interpolation, the solution to (22) is given by the solution of a linear system in dimension \(m+n+1\) (see [23] for details).
What we have omitted in the description of PyBOBYQA is, that the algorithm has a model improvement phase, in which the geometry of the interpolation set is improved when necessary, see [23].
By describing the geometry of the rotor bars with two design variables and using \(m=2n+1\) degrees of freedom for the quadratic model, we still have to solve the discrete timedomain problem (11a)–(11b) for five different designs to compute an initial interpolation model and one time in every optimization iteration. As this is the most expensive part in the optimization, we use a PinT algorithm to accelerate the solution of the discrete timedomain problem (11a)–(11b), which we describe in the following Sect. 4.
4 ATMGRIT
Consider an IVP of the form
which arises, for example, after the spatial discretization of a spacetime PDE. We discretize (23) on a uniform temporal grid \(t_{j} = j\delta T\), \(j=0,1,\dots ,N_{t}\), where \(N_{t}\) is the number of time steps, \(\delta T = T_{\mathrm{end}}/N_{t}\) and \(\mathbf{u}_{j}\approx \mathbf{u}(t_{j})\). Let \(\Phi _{j}\) be a time integrator which propagates the solution \(\mathbf{u}_{j1}\) from time point \(t_{j1}\) to time point \(t_{j}\), including all problemdependent forcing terms. Considering a onestep time integration method such as, e.g., implicit Euler, the timediscrete problem is given by
or, considering all time points at once, we obtain the spacetime system
We now describe the ATMGRIT algorithm for the timeparallel solution of the problem (24).
ATMGRIT is an iterative method for solving IVPs using multigrid reduction techniques [5]. It is therefore based on a hierarchy of temporal grids, restriction and prolongation operators for the transfer between temporal grids, and relaxation schemes. For a given (fine) time grid \(\mathcal{T}^{(0)} = \{j\delta T^{(0)} : j=0,1,\ldots,N_{t}^{(0)}\}\), with \(N_{t}^{(0)}=N_{t}\), and a given coarsening factor \(\widetilde{m}>1\), we define a splitting of all time points into F and Cpoints, such that every m̃th point is a Cpoint and all others are Fpoints. The Cpoints define a global coarse grid \(\mathcal{T}^{(1)} = \{j\delta T^{(1)} : j=0,1,\ldots,N_{t}^{(1)}\}\) with timestep size \(\delta T^{(1)}=\widetilde{m}\delta T^{(0)} \). For the coarsegrid operator we choose a rediscretization of the problem with step size \(\delta T^{(1)} \), although other approaches such as coarsening in space [24–26] can also be used. Applying this strategy recursively for \(L>1\) levels, we obtain a multilevel hierarchy of time grids \(\mathcal{T}^{(\ell )}\) with \(\ell = 0,1,\ldots,L1\). At the coarsest level, we define \(N_{t}^{(L1)}+1\) overlapping local coarse grids based on the global grid \(\mathcal{T}^{(L1)} \). For a given distance \(\widetilde{n} \in \mathbb{N}\), the ith local coarse grid is given by
An example of a threelevel timegrid hierarchy for \(N_{t}^{(0)}=21\), \(\widetilde{m}=2\) and \(\widetilde{n}=3\) is shown in Fig. 5. Note that all spacetime problems associated with the local grids are independent of each other and can be solved simultaneously.
We define two types of relaxation schemes, the socalled Frelaxation and the socalled Crelaxation. The Frelaxation propagates the solution from a Cpoint to all subsequent Fpoints up to the next Cpoint. Similarly, the Crelaxation propagates the solution to a Cpoint. Figure 6 illustrates the actions of F and Crelaxation. Both relaxation schemes are highly parallel and can be applied simultaneously to each interval of F or Cpoints, respectively. For the transfer between global temporal grids, we define restriction as an injection at Cpoints and the “ideal” prolongation, corresponding to the transpose of an injection at Cpoints followed by an Frelaxation. For the transfer from the global temporal grid to the local time grids at the coarsest level, we define both restriction and prolongation as injection.
Using the full approximation storage framework [27], the multilevel ATMGRIT algorithm can be written as in Algorithm 1. There, \(\mathcal{A}^{(\ell )}\mathbf{u}^{(\ell )}= \mathbf{g}^{(\ell )}\) and \(\mathcal{A}^{(\ell ,i)}\mathbf{u}^{(\ell ,i)}=\mathbf{g}^{(\ell ,i)}\) specifies the spacetime system of equations on levels \(\ell =0,1,\ldots ,L1\) and on the local coarse grids \(i=0,1,\ldots ,N_{t}^{(\ell )}\), respectively. The transfer between the global temporal grids is given by the operators \(R_{I}^{(\ell )}\) and \(P^{(\ell )}\) and the transfer from the global temporal grid to the local grids at the coarsest level by the operators \(R_{I}^{(\ell ,i)}\) and \(P_{I}^{(\ell ,i)} \). The relaxation scheme of the method can be controlled by the parameter ν, where \(\nu =1\), i.e., an Frelaxation followed by a Crelaxation and another Frelaxation is a typical choice for the ATMGRIT algorithm.
Note that all components of the algorithm are highly parallel and can be executed simultaneously. The algorithm solves for the exact discrete solution of the fine temporal grid after \(N_{t}^{(0)}/(2\widetilde{m})\) iterations for \(FCF\)relaxation. Furthermore, the algorithm is equivalent to the MGRIT method [2] if \(\widetilde{n}=N_{t}^{(L1)}+1\) and equivalent to Parareal [1] for \(L=2\), \(\nu =0\), and \(\widetilde{n}=N_{t}^{(1)}+1\), [6].
5 Numerical experiments
In this section we apply the optimization procedure described in Sect. 3 combined with the ATMGRIT algorithm from Sect. 4 to an induction machine. For this, we consider one pole of the 2D fourpole squirrelcage motor model depicted in Fig. 2, imposing periodic boundary conditions due to the symmetry [10]. A constant rotational speed \(\omega _{\mathrm{mech}}=1420\) rpm and a nonlinear material behavior are chosen for the problem setting. PinT simulation of this machine model was already performed, e.g., in [3, 4], which we now incorporate into a geometry optimization framework.
The motor is excited with a threephase sinusoidal voltage supply of frequency \(f={50}\) Hz and amplitude \(\hat{U}={311.1}\) V given by
where \(v_{k}\) is defined in (1) and
is an initial rampup used on the first two periods of length \(T=1/f={0.02}\) s to reduce the transient response [10]. One phase of the applied signal (25) is shown in Fig. 7.
The timedomain simulation is performed by solving (11a)–(11b) on a (fine) temporal grid with \(N_{t}=16{,}384\) points and end point \(T_{\mathrm{end}}={0.2}\) s. This corresponds to time stepping with a step size of \(\delta T\approx {1.2\cdot 10^{5}}\) s, starting from a homogeneous IC for the MVP, i.e., \(\mathbf{a}_{0}=\mathbf{0}\) in (10). Within this setting, the steady state is reached at the period \(q=10\) up to the tolerance of \(<5\cdot 10^{4}\) in terms of the relative error
denoting the average torque at the period q and \(N_{\mathrm{p}}=\lfloor N_{t}/q\rfloor \) being the number of time steps per period.
Based on the steadystate behavior of the induction motor, our goal is to optimize the rotor bars size, in particularly, their width and height using the objective function (18). For this, we set
as the admissible design bounds in (19), choose \(h^{(0)}=0.01425\) and \(w^{(0)}=0.002\) as an initial design depicted in Fig. 9 (left) and set the initial trustregion radius to \(\Delta ^{(0)}=10^{4}\). For each objective function evaluation, we generate a mesh representation of the current geometry using Gmsh [28, 29], where each mesh, depending on the geometry, consists of approximately \(4{,}500\) degrees of freedom. Afterwards the ATMGRIT algorithm of the Python package PyMGRIT [4, 30] is called based on a twolevel strategy with \(\widetilde{m}=64\), \(\widetilde{n}=100\), and Frelaxation. The time integration on the temporal grids of ATMGRIT is done by means of the external GetDP library [11], which implements the implicit Euler method for the time stepping and the Newton method with damping as a nonlinear solver. Furthermore, we choose an initial guess for the ATMGRIT algorithm based on a global coarse grid solve. This choice was shown in [6] to be a promising setting for the simulation of this model, since too large time steps on the coarse grid can otherwise lead to divergence of at least one nonlinear solve in GetDP. The ATMGRIT algorithm terminates when the maximums norm of the relative difference of the Joule losses of two successive iterations is less than 1%, see [4] for details.
The calculations were performed on an Intel Xeon Phi cluster consisting of four 1.4 GHz Intel Xeon Phi processors. The code can be found in the PyMGRIT repository [30]. The implementation uses a master/worker strategy for the optimization and the simulations, using one process for the optimization and 256 processes for each simulation, where all resources are used for temporal parallelization.
5.1 Optimization results
Figure 8 shows the negative values of the objective function evaluated during the optimization procedure on the left and an overview of the geometries considered on the right. In the 26 optimization iterations a total of 31 function evaluations were required: one evaluation in every iteration and an additional five in the first iteration for building the initial quadratic interpolation model. The optimal geometry p̄ with \(\bar{w}=0.00254\) and \(\bar{h}=0.01226\) was found in the 18th iteration. In the iterations after the minimum was found and added to the interpolation set, the solution of (TRS) was always a point worse than p̄, resulting in a negative ratio (21), a rejected step and a reduced trustregion radius. The algorithm then terminated when the trustregion radius \(\Delta ^{(k)}\) became smaller than our chosen tolerance \(\Delta ^{(\text{end})}=10^{6}\). Figure 9 shows the two geometries for the initial design (left) and the optimal design (right). Different structures of the geometry of the rotor bars (orange) can be seen: the bars in the optimal design are less high and significantly wider than those in the initial design. Overall, the optimal design increases the negative of the objective function and thus the efficiency of the electrical machine from 87.22% to 87.57%. Figure 10 gives a detailed comparison of the torque (left) and the Joule losses (right) between the initial and the optimal design in the steady state. Compared to the original design, both values were increased, with an average increase of 16.25% in torque and 20.03% in Joule losses for the optimal design.
In each step of the optimization algorithm we use the ATMGRIT algorithm to simulate the corresponding geometry. To determine the effect of the timeparallel method compared to sequential timestepping for solving the problem, we count the calls of the time integrator for both methods. In each iteration of the optimization algorithm, we achieve a theoretical speedup of up to 11.67 by using ATMGRIT compared to the standard sequential timestepping. Note that for this application, counting the serial solves of the ATMGRIT algorithm is a reasonable measure to determine the theoretical speedup, since the computational cost dominates the runtime cost of the algorithm and the communication cost is negligible compared to the computational cost.
6 Conclusion
In this paper we incorporated a parallelintime solution approach into a box constrained shape optimization of an induction motor. Each iteration of the optimization algorithm includes a timedomain solution until the steady state, which we parallelize using the asynchronous truncated multigridreductionintime method. The objective function is a measure of the motor’s efficiency and is constructed from the calculated steadystate characteristics such as the produced torque and the Joule losses. Thanks to the time parallelization an optimal geometry is obtained about 11 times faster compared to the standard sequential time stepping.
Availability of data and materials
The optimization workflow and the parallel implementation of the ATMGRIT algorithm are available in the PyMGRIT package at https://github.com/pymgrit/pymgrit.
Abbreviations
 PinT:

Parallelintime
 MGRIT:

multigridreductionintime
 ATMGRIT:

asynchronous truncated multigridreductionintime
 PDE:

partial differential equation
 2D:

twodimensional
 MQS:

magnetoquasistatic
 MVP:

magnetic vector potential
 IC:

initial condition
 FEM:

finite element method
 IVP:

initialvalue problem
 DAEs:

differentialalgebraic equations
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Acknowledgements
The authors thank Dr. Oliver Rain from Robert Bosch GmbH for sharing his expertise in electric motors and corporate engineering.
Funding
This work is supported by the European HighPerformance Computing Joint Undertaking (JU) under the grant agreement No 955701. The JU receives support from the European Union’s Horizon 2020 research and innovation programme and Belgium, France, Germany, Switzerland. Open Access funding enabled and organized by Projekt DEAL.
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The electric machine model was adapted by BP for geometry optimization. The optimization procedure was designed by IKR, BP, SF, SSc and JH, and implemented by JH. The manuscript was written by IKR, BP and JH, and edited by SF, SU, and SSc. All authors read and approved the final manuscript.
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Hahne, J., Polenz, B., KulchytskaRuchka, I. et al. Parallelintime optimization of induction motors. J.Math.Industry 13, 6 (2023). https://doi.org/10.1186/s13362023001345
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DOI: https://doi.org/10.1186/s13362023001345