Model order reduction approaches for the optimal design of permanent magnets in electro-magnetic machines

8th Vienna International Conference on Mathematical Modelling 8th Vienna18International Conference on Mathematical Modelling February - 20, 2015. Vienna University of Technology, Vienna, 8th Vienna Vienna International International Conference on Mathematical Mathematical Modelling 8th Conference on Modelling February 18 - 20, 2015. Vienna University of Technology, Vienna, Austria Available online at www.sciencedirect.com February 18 20, 2015. Vienna University of Technology, February 18 - 20, 2015. Vienna University of Technology, Vienna, Vienna, Austria Austria Austria

ScienceDirect

IFAC-PapersOnLine 48-1 (2015) 242–247

Model order reduction approaches for the Model order reduction approaches for the Model order reduction approaches for Model order reduction approaches for the the optimal design of permanent magnets in optimal design of permanent magnets in optimal design of permanent magnets optimalelectro-magnetic design of permanent magnets in in machines electro-magnetic machines electro-magnetic machines electro-magnetic machines ∗ ∗ ∗∗ ∗∗

A. Alla ∗ M. Hinze ∗ O. Lass ∗∗ S. Ulbrich ∗∗ A. Alla ∗∗ M. Hinze ∗∗ O. Lass ∗∗ S. Ulbrich ∗∗ A. A. Alla Alla M. M. Hinze Hinze O. O. Lass Lass ∗∗ S. S. Ulbrich Ulbrich ∗∗ ∗ at Hamburg, Bundesstr. 55, 20146, Hamburg, Germany ∗ Universit¨ Hamburg, 55, ∗ Universit¨ (e-mail:a michael.hinze}@uni-hamburg.de). ∗ Universit¨ attt{alessandro.alla, Hamburg, Bundesstr. Bundesstr. 55, 20146, 20146, Hamburg, Hamburg, Germany Germany Universit¨ a Hamburg, Bundesstr. 55, 20146, Hamburg, Germany (e-mail: {alessandro.alla, michael.hinze}@uni-hamburg.de). ∗∗ Universit¨ at Darmstadt, Dolivostr. 15, 64293 Darmstadt, (e-mail: {alessandro.alla, michael.hinze}@uni-hamburg.de). ∗∗ Technische (e-mail: {alessandro.alla, michael.hinze}@uni-hamburg.de). Universit¨ a t Darmstadt, Dolivostr. 15, 64293 Darmstadt, ∗∗ Technische Germany (e-mail: {lass, ulbrich}@mathematik.tu-darmstadt.de) ∗∗ Technische Universit¨ a Dolivostr. Technische Universit¨ att Darmstadt, Darmstadt, Dolivostr. 15, 15, 64293 64293 Darmstadt, Darmstadt, Germany (e-mail: {lass, ulbrich}@mathematik.tu-darmstadt.de) Germany Germany (e-mail: (e-mail: {lass, {lass, ulbrich}@mathematik.tu-darmstadt.de) ulbrich}@mathematik.tu-darmstadt.de) Abstract: In an electromagnetic machine with permanent magnets the excitation field is Abstract: In an electromagnetic machine with permanent magnets excitation is provided permanent magnet instead a coil. The center of thethe generator, thefield rotor, Abstract: In electromagnetic machine with permanent magnets the excitation field is Abstract:by Inaa an an electromagnetic machineof with permanent magnets the excitation field is provided by permanent magnet instead of aa coil. The center of the generator, the rotor, contains the magnet. Our optimization goal consists in finding the minimum volume of the provided by a permanent magnet instead of coil. The center of the generator, the rotor, provided by a permanent magnet instead of a coil. The center of the generator, the rotor, contains the magnet. Our optimization goal consists in finding the minimum volume of the magnet gives a Our desired electromotive Thisin in the an optimization problem for contains the optimization goal consists finding minimum of containswhich the magnet. magnet. Our optimization goalforce. consists inresults finding the minimum volume volume of the the which gives a desired electromotive force. This results in an optimization problem for amagnet parametrized partial differential equation (PDE). We propose a goal-oriented model order magnet which gives a desired electromotive force. This results in an optimization problem for magnet which gives a desired electromotive force. This results in an optimization model problem for a parametrized partial differential equation (PDE). We propose a goal-oriented order reduction approach to provide a reduced order surrogate model for the parametrized PDE a parametrized partial differential equation (PDE). We propose a goal-oriented model order a parametrized partial differential equationorder (PDE). We propose afor goal-oriented model order reduction approach to provide a reduced surrogate model the parametrized PDE which thenapproach is utilizedto the numerical optimization. Numerical tests provided in order reduction provide a order model for the parametrized PDE reduction approach toin provide a reduced reduced order surrogate surrogate model forwill thebe parametrized PDE which then is utilized in the numerical optimization. Numerical tests will be provided in order to show the effectiveness of the proposed method. which then is utilized in the numerical optimization. Numerical tests will be provided in order which then is utilized in the numerical optimization. Numerical tests will be provided in order to show the effectiveness of proposed method. to show the of the the method. to 2015, showIFAC the effectiveness effectiveness the proposed proposed method. © (International of Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Proper Orthogonal Decomposition, Optimization, Model Order Reduction, Static Keywords: Proper Orthogonal Decomposition, Optimization, Model Order Order Reduction, Static Static Maxwell’s equation Keywords: Proper Keywords: Proper Orthogonal Orthogonal Decomposition, Decomposition, Optimization, Optimization, Model Model Order Reduction, Reduction, Static Maxwell’s equation Maxwell’s Maxwell’s equation equation 1. INTRODUCTION to the model order reduction and its application to our 1. INTRODUCTION INTRODUCTION to the model problem. order reduction and its application to our optimization Numerical to demonstrate 1. to the order and its to 1. INTRODUCTION to the model model problem. order reduction reduction andresults its application application to our our optimization Numerical results to demonstrate effectiveness of the proposed methods are presented in optimization problem. Numerical results to demonstrate Permanent magnet synchronous machines play an increas- the optimization problem. Numerical results to demonstrate the proposed presented Permanent magnet synchronous synchronous machines playdynamic an increasincreas5. Finallyof conclusion drawn inare Section 6. in the effectiveness effectiveness ofathe the proposedismethods methods are presented in ing role in applications demanding a highly be- Section Permanent magnet machines play an the effectiveness of proposed methods are presented in Permanent magnet synchronous machines playdynamic an increasSection 5. Finally aathe conclusion is drawn in Section 6. ing role in applications demanding a highly beSection 5. Finally conclusion is drawn in Section 6. haviour of the drive and energy efficient speed and electroing role in applications demanding a highly dynamic beSection 5. Finally a conclusion is drawn in Section 6. ing role of in the applications demanding a highly dynamic behaviour drive and and energy efficient efficient speed and electroelectromotive force control. To guarantee optimal operation of the haviour of drive energy speed and haviour of the the drive and energy efficient speed and electromotive force control. To guarantee optimal operation of the 2. MODEL PROBLEM drive, simplified analytical modelsoptimal are used in the control motive force To operation of motive force control. control. To guarantee guarantee operation of the the 2. MODEL PROBLEM drive, simplified simplified analytical modelsoptimal are used in the the control 2. scheme of the machine. Designing a technical device is a drive, analytical models are used in control 2. MODEL MODEL PROBLEM PROBLEM drive, simplified analytical models are used in the control scheme of the machine. Designing a technical device is a complex process. For this purpose we model the machine scheme of the machine. Designing a technical device is a Our mathematical is built upon the Permanent scheme ofprocess. the machine. Designing a technical device is a Our mathematical model complex For this purpose we model the machine model is built (PMSM) upon the of Permanent by a magneto static Maxwell’s equation. A firstthe deep study Magnetic complex process. For this we machine Synchronous HennerOur mathematical model is upon Permanent complex process. For this purpose purpose we model model the machine Our mathematical modelMachines is built built (PMSM) upon the the of Permanent by aathis magneto static Maxwell’s equation. A first first deep study Magnetic Synchronous Machines Henneron model with an optimization problem is given in the by magneto static Maxwell’s equation. A deep study berger et al. (1997). This PMSM is used as base Magnetic Synchronous Synchronous Machines Machines (PMSM) (PMSM) of of the Hennerby a magneto static Maxwell’s equation. A first deep study Magnetic Henneron this this modelofwith with an optimization optimization problem is is given given in in the the berger et al. (1997). This PMSM is used as the base PhD Thesis Pahner (1998). on model an problem design, where some modifications have been added. The berger et al. (1997). This PMSM is used as the on this modelofwith an optimization problem is given in the berger et al. (1997). This PMSMhave is used as the base base PhD Thesis Pahner (1998). design, where some modifications been added. The PhD Thesis of Pahner (1998). stator contains six slots per pole. A double-layer winding design, where some modifications have been added. The PhD Thesis where some modifications have been added. The In this workofwePahner present(1998). an optimization problem related design, stator contains six slots per pole. A double-layer winding with two slots per pole per phase is used. Both, stator stator contains six slots per pole. A double-layer winding In this work we present an optimization problem related stator contains six slots per pole. A double-layer winding to the volume of the magnet. Due to the different configuIn this work we present an optimization problem related with two slots per pole per phase is used. Both, stator In thisvolume work we an optimization problemconfigurelated and are made of laminated whichBoth, is modeled with rotor two slots slots per pole pole per phase phasesteel is used. used. Both, stator to the the of present the magnet. magnet. Due to to theandifferent different two per per is stator rations of the magnet we will introduce affine decompoto of Due configuand rotor are made of laminated steel which is modeled to the volume volume of the the magnet. Due to the theandifferent configu- with = 500 with by a constant relative permeability of µ and rotor are made of laminated steel which is modeled rations of the magnet we will introduce affine decompor and rotor are made of laminated steel which is modeled sition which allows us to work on a fixed reference domain rations of the magnet we will introduce an affine decompo500 with by aa constant relative permeability of µ rations of theallows magnet we will introduce anreference affine decompor = vanishing conductivity. The length of the machine is 100 = 500 with by constant relative permeability of µ sition which us to work on a fixed domain r by a constant relative The permeability of µmachine as proposed in Rozza et al. (2008). sition which allows us to work on aa fixed reference domain r = 500iswith vanishing conductivity. length of the 100 sition which allows us to work on fixed reference domain mm. In this work we do not include saturation effects of vanishing conductivity. The length of the machine is as proposed in Rozza et al. (2008). vanishing conductivity. The length ofsaturation the machine is 100 100 as in et In this work we do not include effects as proposed proposed in Rozza Rozza et al. al. (2008). (2008). The rather high computational expense of the optimiza- mm. the material but rather concentrate on the development mm. In In this this work work we we do do not not include include saturation saturation effects effects of of mm. of The rather rather high computational expense of the the optimizaoptimizamaterial but rather concentrate on the tion problem leads us to reduce the dimension of the the of efficient optimization methods. Opposed todevelopment the original The high computational expense of the material but rather concentrate on the development The rather high computational expense of the optimizathe material but rather concentrate on the development tion problem problem leads ofus us to to reduce reduce the dimension dimension of the the of efficient optimization methods. Opposed the original problem by means Orthogonal Decomposition where five surface mounted magnetsto we tion leads the of of efficient optimization methods. Opposed to the original tion problem leads ofusProper to reduce the dimension of the design, of efficient optimization methods. Opposed toare theused, original problem by means Proper Orthogonal Decomposition design, where five surface mounted magnets are used, we (POD). The POD method allows to reduce the degrees of consider one buried permanent magnet in the rotor. This problem by means of Proper Orthogonal Decomposition design, where five surface mounted magnets are used, we problem The by means of Proper Orthogonal design, where five surface mounted magnets are used,This we (POD). POD method allows toGubisch reduceDecomposition the degrees of consider one buried permanent magnet in the rotor. freedom of the problem (see e.g., et al. (2013)) design provides greater mechanical strength for holding (POD). The POD method allows to reduce the degrees of consider one buried permanent magnet in the rotor. This (POD). The POD method allows to reduce the degrees of consider one buried permanent magnet in the rotor. This freedom of the problem (see e.g., Gubisch et al. (2013)) design provides greater mechanical strength for holding and allows to design which the magnet in place. Themechanical geometry isstrength shown in 1. freedom of problem (see e.g., et design provides greater for holding freedom of the the problemfast (seeoptimization e.g., Gubisch Gubischmethods et al. al. (2013)) (2013)) provides greater mechanical strength forFigure holding and allows allows to design design fast optimization methods which design the magnet in place. The geometry is shown in Figure 1. only need a few evaluations of the fully resolved objective The position of the rotor is fixed and not modeled as and to fast optimization methods which the magnet in place. The geometry is shown in Figure 1. and allows to design fast optimization methods which the magnet in of place. The geometry is shown in Figure as 1. only need a few evaluations of the fully resolved objective The position the rotor is fixed and not modeled function, and thus evaluations of the fully resolved PDE aThe transient process, i.e., a sequence of rotor positions. only need aa few evaluations of the fully resolved objective position of the rotor is fixed and not modeled as only need few evaluations of the fully resolved objective The positionprocess, of the i.e., rotora issequence fixed and not modeled as function, and thus evaluations of the fully resolved PDE a transient of rotor positions. model, seeand e.g.,thus APOD (Afanasiev et al. (2001)), TRPOD the process, loading method introducedof Rahmen et al. function, evaluations of fully resolved PDE transient i.e., rotor positions. function, thus evaluations of the the fully resolved PDE aInstead, aInstead, transient process, i.e., aa sequence sequence ofby rotor positions. model, et seeand e.g., APOD (Afanasiev et al. al. (2001)), TRPOD the loading method introduced by Rahmen et al. (Arian al. (2002),Zahr et al. (2014)). (1998) is utilized which exploits the frequency domain. model, see e.g., APOD (Afanasiev et (2001)), TRPOD Instead, the loading method introduced by Rahmen et model, seeal. e.g., APOD (Afanasiev et al. (2001)), TRPOD Instead, the loading method introduced by Rahmen et al. al. (Arian et (2002),Zahr et al. (2014)). (1998) is utilized which exploits the frequency domain. From Fourier analysis ofexploits the vector potential around the (Arian et et al. (2014)). (1998)a is utilized which the frequency domain. (Arian et al. al.of(2002),Zahr (2002),Zahr (2014)). is utilized which exploits the frequency domain. The outline this paper isetasal.follows. In Section 2 we give (1998) From aa Fourier of the vectorlike potential around the of analysis the stator the electromotive From analysis of the potential around outline of this this paper paper is model as follows. follows. In Section Section we give give inner From surface a Fourier Fourier analysis of quantities the vector vectorlike potential around the the aThe brief introduction of ouris problem. The optimizaThe outline of as In 222 we inner surface of the stator quantities the electromotive The outline of this paper is as follows. In Section we give force can be derived. inner surface of the stator quantities like the electromotive a brief brief introduction of our ourinmodel model problem. The optimizainner surface of the stator quantities like the electromotive problem is described Section 3. Section 4 is devoted ation introduction of problem. The optimizaforce can be derived. a brief introduction of ourinmodel problem. The be derived. tion problem is described described Section 3. Section Section 4 isoptimizadevoted force force can canare be described derived. by the magneto static approximation tion problem problem is is described in in Section Section 3. 3. Section 44 is is devoted devoted PMSMs PMSMs are described by the magneto static approxima This work is supported by the German BMBF in the context of tion of Maxwell’s equations. The model we consider in this PMSMs are described by magneto static approximaPMSMs are described by the the magneto static approxima This work is supported by the German BMBF in the context of tion of Maxwell’s equations. The model we consider in this  work is a parametrized elliptic partial differential equation the SIMUROM project (grant number 05M2013). tion of Maxwell’s equations. The model we consider in This work is supported by the German BMBF in the context of  This work is supported by the German BMBF in the context of tion ofisMaxwell’s equations. The modeldifferential we consider in this this work a parametrized elliptic partial equation the SIMUROM project (grant number 05M2013). work the work is is aa parametrized parametrized elliptic elliptic partial partial differential differential equation equation the SIMUROM SIMUROM project project (grant (grant number number 05M2013). 05M2013).

Copyright © 2015, 2015,IFAC IFAC (International Federation of Automatic Control) 242Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © Copyright © 2015, IFAC 242 Copyright © 2015, IFAC 242 Peer review under responsibility of International Federation of Automatic Copyright © 2015, IFAC 242Control. 10.1016/j.ifacol.2015.05.154

MATHMOD 2015 February 18 - 20, 2015. Vienna, Austria

D

-w

-v

-v

-v

A. Alla et al. / IFAC-PapersOnLine 48-1 (2015) 242–247

C

u -w

on the L domains inside the box obtained by the domain decomposition. The p dependency of K is now only in the weight functions θk which are easy to evaluate. Note, that the same decomposition is made for the right hand side. This affine decomposition of the linear system is essential to obtain fast reduced order models, see Section 4. Furthermore, we only simulate one pole due to the symmetry of the problem.

-v

-w

-u -u

-w

243

-u

3. OPTIMIZATION Given the geometry presented in Figure 1, the goal of the optimization is to minimize the required material for the permanent magnet while maintaining the electromotive force E0 . In the mathematical model this leads to a cost function of the form ¯ := p1 p2 + ρ max(0, E d − E0 (p, u(p))), (5) min J(p)

B

E

p∈Mad

Fig. 1. Geometry for the model problem and the region for the affine decomposition marked with dashed lines. (PDE) in terms of the magnetic vector potential (MVP) A, ∇ × (µ∇ × A(p)) = Jsrc (p) − ∇ × Hpm (p). (1) The boundary conditions are given by A|CD = A|EB = 0, A|BC = −A|DE ,

(2)

where Jsrc is the source current density and Hpm is the coercivity of the permanent magnet, and the meaning of B, C, D, E is explained in Figure 1. In the planar 2D case, the finite element (FE) discretization leads to following parametrized linear system, (3) K(p)u(p) = jsrc (p) + jpm (p). N For the MVP we use the ansatz A = i=1 ui ϕi , where ϕi are suitable ansatz functions. The FE system matrix K(p) emphasizes the dependency on the parameter p. The parameter p is related to the location and size of the permanent magnet. We consider a setting with a three dimensional parameter p = (p1 , p2 , p3 ). We use two parameters to describe the size of the permanent magnet, where p1 corresponds to its width and p2 to its height in mm. Additionally, since a PMSM with a buried permanent magnet is considered, we introduce the parameter p3 to describe the central perpendicular distance between the permanent magnet and the surface of the rotor in mm. To obtain a computationally fast model and to avoid remeshing when the parameter changes we require an affine decomposition. For this purpose we introduce a box around the permanent magnet (Figure 1, dashed line) in which we perform a domain decomposition, (see e.g. Rozza et al. (2008)) into L subdomains of triangular shape. As a consequence the system matrix can be written in the form K(p) = Kout +

L 

θk (p)Kk ,

(4)

k=1

where Kout is the system matrix for the domain outside the box and Kk , k = 1, . . . , L, are the system matrices 243

where E d is the desired electromotive force and Mad := {p ∈ R3 | p ≤ p ≤ p}

with p = (1, 1, 5) and p = (∞, ∞, 14) denotes the set of admissible parameters. The nonnegative parameter ρ is a penalization parameter and in our case set to 100. Note that for the computation of the electromotive force the solution to (3) is required, hence we have an optimization problem with a PDE constraint. Further, we only consider a 2D problem, hence we replace the volume by the area. In addition to the constraints defined in the admissible set we introduce the constraints 2 50 p2 + p3 ≤ 15 and p1 − p3 ≤ . 3 3 The first constraint is motivated by the affine decomposition. Since only a subdomain of the geometry is involved it is required that we stay within this region during the optimization process. The second constraint is a design constraint. It is required that the permanent magnet always has a certain distance to the rotor surface. Hence the depth of the magnet is linked to the width, which is expressed by this linear constraint. The cost function (5) is not smooth due to the max operator. For the optimization we seek a smooth formulation of the minimization problem in order to apply derivative based methods. Therefore, we rewrite the problem by introducing a slack variable ξ. Setting x = (p, ξ) we get the smooth optimization problem min J(x) := p1 p2 + ρξ

x∈R4

subject to

(6)



 p2 + p3 − 15 2 50   p 1 − p3 −   3 3  d   E − E0 (p, u(p)) − ξ     ≤ 0. 1 − p1 G(x) :=      1 − p2   5 − p3     −ξ p3 − 14

(7)

We now have an optimization problem that can be solved by standard methods presented in, e.g., Nocedal et al.

MATHMOD 2015 244 February 18 - 20, 2015. Vienna, Austria

A. Alla et al. / IFAC-PapersOnLine 48-1 (2015) 242–247

(2006), Hinze et al. (2009) or Tr¨ oltzsch (2010). Let us give a brief outline of the methods utilized in the present work, including the computation of the derivatives. We apply here the approach using the sensitivity equations. Although computationally more expensive than the adjoint approach this method will be beneficial in the model order reduction approach presented in Section 4. The sensitivities are required to compute the derivative of the electromotive force. Since E0 is a linear operator we get ∂E0 (p, u(p)) = E0 (p, si ) for i = 1, . . . , 3, ∂pi are the sensitivities. To obtain the where si = ∂u(p) ∂pi sensitivities, the linear sensitivity equations K(p)si (p) = fi ,

for i = 1, . . . , 3

(8)

have to be solved with fi = (jsrc (p) + jpm (p))pi − Kpi (p)u(p).

Here the subindex pi indicates the derivative with respect to the i-th parameter. Note, that these derivatives are easy to compute due to the previousely introduced affine decomposition (4). The derivative of the matrix K(p) is given by the derivatives of the functions θk (p). To perform the numerical optimization we use the Sequential Quadratic Programming (SQP) method. This method solves the optimization problem by a sequence of quadratic approximations and provides fast convergence. To avoid the computationally expensive evaluation of the Hessian we use an approximation based on the damped BFGS update, which belongs to the class of symmetric rank two updates. To guarantee the convergence an Armijo backtracking strategy using a 1 −penalty function is used (Nocedal et al. (2006)). 4. PROPER ORTHOGONAL DECOMPOSITION FOR PARAMETRIZED PROBLEMS In this section, we explain the POD method for the approximate solution of the parametrized equation (3). To begin with, let us suppose to know the finite element solution u(p) ∈ RN for given parameter p ∈ Mad . For this purpose let {pj }nj=1 be a grid in Mad and let u(pj ) denote the corresponding solutions to (3) for the grid points pj . We define the snapshots set V := span{u(p1 ), . . . , u(pn )} and determine a POD basis {ψ1 , . . . , ψ } of rank  by solving the following minimization problem:

min

ψ1 ,...,ψ

n  j=1

2        j j αj u(p ) − u(p ), ψi W ψi    i=1

s.t.ψj , ψi W = δij

j=1

Moreover  2 n  d        j j αj u(p ) − u(p ), ψi W ψi  = λi .   j=1

i=1

W

i=+1

4.1 POD approximation for state and sensitivities

We briefly recall how to generate the reduced order model by means of POD. Suppose we have computed the POD basis {ψi , . . . , ψ } of rank . The weight matrix W is given by W = K(¯ p) + M(¯ p), where p¯ is a fixed reference parameter and M denotes the mass matrix. Then, we define the POD ansatz for the  state u (p) := i=1 u ¯ i ψi . This ansatz in (3) leads to an −dimensional system for the unknown {¯ ui }i=1 , namely K (p)¯ u(p) = j src (p) + j pm (p).

(10)

Here the entries of the stiffness matrix K are given by ψi , K(p)ψj . The right hand side is composed of the projections jsrc (p), ψi  and jpm (p), ψi , respectively. Recall that due to the affine decomposition this projection has to be computed only once, and the system matrix can be written as L  θk (p)Kk, , K (p) = Kout, + k=1

where Kout, = ψi , Kout ψj  and Kk, = ψi , Kk ψj  with respect to the euclidean inner product. The same structure can be used for the right hand side. Note that this is very important in order to obtain an efficient reduced order model since the system can be set up for different values of p without the need of the original high dimensional matrices and right hand sides. In an analogous way we obtain the reduced sensitivity equation from (8). We need to make an ansatz for the sensitivities si and project the system onto the subspace spanned by the POD basis. Note that in the present work we use the same basis functions for the state u and the sensitivities s. A better approximation property is achieved by adding the solution of the sensitivity equation to the snapshots set. More details are given in the next section. 4.2 The POD method for optimization problem

W

for 1 ≤ i, j ≤ ,

operator R : Rn → Rn , i.e., Rψi = λi ψi with λi > 0, where R is defined as follows: n  αj u(pj ), ψW u(pj ) for ψ ∈ Rn . Rψ =

(9)

where αj are nonnegative weights, δij denotes the Kronecker symbol, W is a positive definite N × N matrix and ψi ∈ Rn . The weighted inner product used is defined as follows: u, vW = u Wv. It is well-known (see Gubisch et al. (2013)) that problem (9) admits a unique solition {ψ1 , . . . , ψ }, where ψi denotes the i−th eigenvector of the selfadjoint linear mapping 244

The selection of snapshots is crucial for the quality of the ROM. It is clear that the more information we provide about the system (i.e., the more snapshots details), the better and more accurate approximations we can achieve. Nevertheless, we want to avoid a too expensive offline stage. For this purpose we propose a goal-oriented approach for the computation of the POD basis functions. The goal is given by the difference between the electromotive force computed by the FE model and the reduced model, i.e., |E0 (p, u (p)) − E0 (p, u(p))|. The goal-oriented approximation approach aims at minimizing the error for a

MATHMOD 2015 February 18 - 20, 2015. Vienna, Austria

A. Alla et al. / IFAC-PapersOnLine 48-1 (2015) 242–247

particular output function instead of minimizing the error in the state. We start with a very coarse parameter space choosing only one parameter p0 and solve the full problem together with the sensitivity equations associated to this parameter. Then we compute the POD basis functions and perform the reduced optimization procedure. At the end of the process we find a new parameter p1 which is an approximation of the optimal desired design, we update the parameter set D = {p0 , p1 } ⊂ Mad , solve the full problem and the sensitivity equations related to the new parameter p1 . Then, we enlarge the snapshots set and compute new POD basis functions. We iterate this process until the stopping criteria |E(pk , u (pk ))−E(pk , u(pk ))| < tol is reached. The procedure is summarized in Algorithm 1. Algorithm 1 (Goal-Oriented POD optimization) Require: p0 , u(p0 ), V = [], k = 0, tol 1: while |E(pk , u (pk )) − E(pk , u(pk ))| > tol do 2: Compute sensitivity si (pk ), for i = 1, 2, 3. 3: Set Snapshots set V = [V, u(pk ), s1 (pk ), s2 (pk ), s3 (pk )] 4: Compute POD basis functions {ψi }i=1 with  = rank(V) 5: Find pk+1 solving the optimization problem utilizing the reduced order model 6: Compute u (pk+1 ) and E(pk+1 , u (pk+1 )). 7: Compute u(pk+1 ) and E(pk+1 , u(pk+1 )) 8: Set k=k+1 9: end while In our simulations this approach turns out to be the most efficient since it avoids long offline computations.

245

column, errf ) snapshots span. As expected, the error in the third column is less than the error in the second column when varying the number of POD basis functions. Here the relevance of the snapshots set for the approximation quality can clearly be seen. Table 1. Test 1: Comparison of maximum relative error of the reduced order model for different snapshots sets.  3 4 5 6 7

errc 0.1002 0.0657 0.0577 0.0592 0.0584

errf 0.0402 0.0522 0.0198 0.0238 0.0051

Figure 2 shows the behavior of the maximum relative error with respect to the number of POD basis functions. The Figure refers to the rich snapshots span (errf ). Moreover, we present the decay of the associated eigenvalues λi . From the decay of the eigenvalues it can be seen that we are able to reduce the dimension of the discrete state equation significantly. RELATIVE ERROR 2

10

MAXIMUM ERROR SINGULAR VALUES 1

10

0

10

−1

10

−2

10

−3

5. NUMERICAL EXPERIMENTS

10

This section presents numerical tests in order to show the performance of our proposed methods. All the numerical simulations reported in this paper are performed on an iMac with an Intel Core i5, 2.7 Ghz and 8GB RAM using R , R2013a. Although, the CPU is of multicore MATLAB architecture we only utilize a single thread to avoid effects from parallel computing. Hence, the CPU time corresponds to the wall-time. We will present three test cases. In the first experiment we show the performance of the reduced order model obtained by the POD method. The second and the third experiment focus on the optimization problem, where we first perform the optimization only using the FE method and compare the performance to the strategy proposed in Algorithm 1.

10

5.1 Test 1 The first test concerns the parametric model order reduction. We focus on two different snapshots sets: one is given by the PDE solutions corresponding to the corners of the parameter space, the second snapshots set is computed on an equidistant grid with grid size one. In the first case we have only eight snapshots, while in the second case we have 1170 snapshots. In Table 1 we show the maximum relative error in the POD approximation with the poor (second column, errc ) and with the rich (third 245

−4

0

5

10

15

20

25

30

35

40

45

50

NUMBER OF POD BASIS FUNCTIONS

Fig. 2. Decay of the relative error and the eigenvalues. Summarizing this experiment we can see that the POD method is able to generate a reduced order model of good quality. Furthermore, the choice of the snapshots are essential since the reduced order model is only valid in a neighbourhood of the parameter p chosen to build the reduced order model. This is also observed in the optimization procedure using Algorithm 1. 5.2 Test 2 In this experiment we present the results for the optimization process using the FE model. The initial parameter choice is p0 = (19, 7, 7), and the desired electromotive force is set to E d = 30.3702. Note that this particular choice corresponds to E d = E0 (p0 , u(p0 )). Further the volume (V = p1 p2 ) of the permanent magnet for the initial condition is 133. The iteration history of the optimization routine is shown in Table 2. The optimal design is obtained with popt = (21.071, 2.980, 6.607) and the corresponding volume is 62.798. This corresponds to a reduction of more than 50% compared to the initial configuration. Moreover,

MATHMOD 2015 February 18 - 20, 2015. Vienna, Austria 246

A. Alla et al. / IFAC-PapersOnLine 48-1 (2015) 242–247

Table 2 shows the norm of the gradient of the cost function in the third column. Once we are close to the optimal solution we can observe the fast quadratic convergence of the proposed method.

Solution uopt (E0 = 30.3702, V = 62.7965)

Table 2. Test 2: Iteration history of the optimization algorithm using the FE model. iter 0 1 2 3 4 5 6 7 8 9

J 133.000 105.248 85.601 75.466 59.608 62.196 62.741 62.796 62.796 62.796

∇J 102.0294 5.752917 1.705679 2.031486 3.716464 1.497623 0.197684 0.010143 0.000123 0.000000

(19.000 (19.136 (19.566 (20.106 (20.777 (21.163 (21.062 (21.071 (21.071 (21.071

p 7.000 5.500 4.375 3.753 2.869 2.939 2.979 2.980 2.980 2.980

7.000) 6.500) 6.446) 6.732) 6.166) 6.746) 6.593) 6.607) 6.607) 6.607)

In Figure 3 we show the initial configuration. Comparing this to the optimal design shown in Figure 4 the significant reduction in the volume can be clearly seen.

Fig. 4. Solution uopt to system (3) for the parameter popt obtaind by the optimization process.

Solution u0 (E0 = 30.3702, V = 133)

that the snapshots set was updated once. Note that in every update we have to solve the system (3) and the sensitivity equations (8) in the full space. This is the most expensive part of the proposed algorithm. We solve only eight PDEs in this approach. Every iteration is the result of an optimization routine computed in the reduced space which is cheap since the −dimensional problem is much smaller than the full problem. In fact each iteration in Table 3 considers the solution of four reduced equations. As we can see this algorithm needs to solve eight FE systems and 60 reduced systems. Compared to the computational costs of the optimization utilizing only the FE model we have a reduction in the computational cost of approximately a factor of five although the CPU time for the whole process is approximately 1s. Table 3. Test 3: Opimization history utilizing Algorithm 1 Fig. 3. Solution u0 to system (3) for the parameter p0 . Although the CPU time for our function evaluation is only 2.5s, this method is rather expensive since we needed 10 iterations to reach the optimal design. Each iteration needs the solution of 4 PDEs: one solution of (3) and three solves for the sensitivity equations (8). Summarizing, we have to solve 40 PDEs to achieve the optimal design. The dimension, in this example, is still rather moderate with 8128 discretization points. We note that the number of unknowns and the related computational expenses dramatically increase if we simulate the whole machine or consider the 3D case.

iter 0 1 2 3 4 5 6 7 8 0 1 2 3 4

J 133.000 105.248 85.706 76.332 59.643 62.691 63.053 63.072 63.072 63.072 62.906 62.795 62.797 62.797

∇J 102.0294 5.712287 1.767354 1.863950 3.352590 1.170096 0.133491 0.005441 0.000048 102.2290 0.134651 0.074026 0.002143 0.000047

p (19.000 7.000 7.000) (19.136 5.500 6.500) (19.590 4.375 6.482) (20.069 3.803 6.677) (20.751 2.874 6.126) (21.071 2.975 6.607) (21.007 3.001 6.511) (21.018 3.001 6.528) (21.018 3.001 6.527) 21.018 3.001 6.527) (21.088 2.983 6.632) (21.069 2.980 6.604) (21.070 2.980 6.606) (21.070 2.980 6.606)

5.3 Test 3 In the last experiment we combine the results from the previous two experiments and Algorithm 1. Table 3 shows 246

The results obtained by the reduced approach are almost the same as the one obtained by the more expensive FE problem. In Table 4 the values for the electromotive

MATHMOD 2015 February 18 - 20, 2015. Vienna, Austria

A. Alla et al. / IFAC-PapersOnLine 48-1 (2015) 242–247

force are compared when computed with the FE model and the reduced model. Here it can be seen that in the second iteration of Algorithm 1 the difference already is less than 10−4 , which is the chosen accuracy tolerance in this algorithm. Table 4. Test 3: Comparison of the electromotive force computed from the u and u during the optimization process utilizing Algorithm 1. iter 0 1 2

E0 (p, u(p)) 30.370231 30.393726 30.370245

E0 (p, u (p)) 30.370231 30.370231 30.370231

V 133.000000 63.072370 62.796700

6. CONCLUSION AND PROSPECTS We present an algorithm for the optimal design of a permanent magnet in electro-magnetic machines. The algorithm allows to reduce the computational cost of the problem and at the same to provide an accurate approximation of the design as confirmed by the numerical tests. Future work will include more advanced models that involve nonlinear permeability and hence take into account important properties such as saturation of iron. Additionally, the extension to a 3D model, where we expect to observe an improved speed-up of the optimization, need to be considered. REFERENCES K. Afanasiev and M. Hinze. Adaptive control of a wake flow using proper orthogonal decomposition. Lecture Notes in Pure and Applied Mathematics 216, 317-332. Shape Optimization & Optimal Design, Marcel Dekker, 2001. E. Arian, M. Fahl and E. Sachs. Trust-region proper orthogonal decomposition models by optimization methods. In Proceedings of the 41st IEEE Conference on Decision and Control, Las Vegas, Nevada, pages 33003305, 2002. M. Gubisch and S. Volkwein. Proper Orthogonal Decomposition for Linear-Quadratic Optimal Control. Submitted, 2013. M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich. Optimization with PDE Constraints. Mathematical Modelling: Theory and Applications, 23. Springer Verlag, 2009. S. Hennerberger, U. Pahner, K. Hameyer and R. Belmans. Computation of a highliy satured permanent magnet synchronous motor for a hybrid electric vehicle. IEEE Trans. Magn., vol. 33, no. 5, pp. 4086-4088, 1997. J. Nocedal and S.J. Wright. Numerical Optimization, second edition. Springer Series in Operation Research and Financial Engineering, 2006. U. Pahner. A general design tool for terical optimization of electromagnetic energy transducers. PhD Thesis, KU Leuven, 1998. M.A. Rahman and P. Zhou. Determination of saturated parameters of PM motors using loading magnetic fields. IEEE Trans. Magn., vol. 27, no. 5, pages 3947-3950, 1991. G. Rozza, D.B.P. Huynh and A.T. Patera. Reduced Basis Approximation and a Posteriori Error Estimation for 247

247

Affinely Parametrized Elliptic Coercive Partial Differential Equations. Arch. Comput. Methods. Eng., vol.15, pages 229-275, 2008. F. Tr¨oltzsch. Optimal Control of Partial Differential Equations: Theory, Methods and Application, American Mathematical Society, 2010. M. Zahr and C. Farhat. Progressive Construction of a Parametric Reduced-Order Model for PDE-Constrained Optimization. International Journal for Numerical Methods in Engineering, in press, 2014.