Future e-mobility calls for efficient electrical machines. For different
areas of operation, these machines have to satisfy certain desired properties
that often depend on their design. Here we investigate the use of multipatch
Isogeometric Analysis (IgA) for the simulation and shape optimization of the
electrical machines. In order to get fast simulation and optimization results,
we use non-overlapping domain decomposition (DD) methods to solve the large
systems of algebraic equations arising from the IgA discretization of
underlying partial differential equations. The DD is naturally related to the
multipatch representation of the computational domain, and provides the
framework for the parallelization of the DD solvers

Gangl, Peter

Langer, Ulrich

Mantzaflaris, Angelos

Schneckenleitner, Rainer

English

arXiv

International audienceFuture e-mobility calls for efficient electrical machines. For different areas of operation, these machines have to satisfy certain desired properties that often depend on their design. Here we investigate the use of multipatch Isogeometric Analysis (IgA) for the simulation and shape optimization of the electrical machines. In order to get fast simulation and optimization results, we use non-overlapping domain decomposition (DD) methods to solve the large systems of algebraic equations arising from the IgA discretization of underlying partial differential equations. The DD is naturally related to the multipatch representation of the computational domain, and provides the framework for the parallelization of the DD solvers

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Isogeometric Simulation and Shape Optimization with Applications to Electrical Machines

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HAL Id: hal-02275658https://hal.inria.fr/hal-02275658Submitted on 1 Sep 2019HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.Isogeometric Simulation and Shape Optimization withApplications to Electrical MachinesPeter Gangl, Ulrich Langer, Angelos Mantzaflaris, Rainer SchneckenleitnerTo cite this version:Peter Gangl, Ulrich Langer, Angelos Mantzaflaris, Rainer Schneckenleitner. Isogeometric Simulationand Shape Optimization with Applications to Electrical Machines. 12th International Conference onScientific Computing in Electrical Engineering, Sep 2018, Taormina, Italy. ￿10.1007/978-3-030-44101-2_4￿. ￿hal-02275658￿Isogeometric Simulation and ShapeOptimization with Applications to ElectricalMachinesPeter Gangl1, Ulrich Langer2,4, Angelos Mantzaflaris3, and RainerSchneckenleitner4Abstract Future e-mobility calls for efficient electrical machines. For different ar-eas of operation, these machines have to satisfy certain desired properties that of-ten depend on their design. Here we investigate the use of multipatch IsogeometricAnalysis (IgA) for the simulation and shape optimization of the electrical machines.In order to get fast simulation and optimization results, we use non-overlappingdomain decomposition (DD) methods to solve the large systems of algebraic equa-tions arising from the IgA discretization of underlying partial differential equations.The DD is naturally related to the multipatch representation of the computationaldomain, and provides the framework for the parallelization of the DD solvers.1 IntroductionIsogeometric Analysis (IgA) is a relatively new approach for discretizing partialdifferential equations (PDEs). IgA was introduced in [2]. It can be seen as an al-ternative to the more classical Finite Element Method (FEM). The idea in IgA isto use the same basis functions for both representing the geometry of the computa-Peter GanglInstitute of Applied Mathematics, TU Graz, Steyrergasse 30, 8010 Graz, Austria e-mail:gangl@math.tugraz.atUlrich LangerInstitute of Computational Mathematics, JKU Linz, Altenberger Straße 69, 4040 Linz, Austria e-mail: ulanger@numa.uni-linz.ac.atAngelos MantzaflarisInstitute of Applied Geometry, JKU Linz, Altenberger Straße 69, 4040 Linz, Austria e-mail: ange-los.mantzaflaris@jku.atRainer SchneckenleitnerRICAM, Austrian Academy of Sciences, Altenberger Straße 69, 4040 Linz, Austria e-mail:rainer.schneckenleitner@ricam.oeaw.ac.at12 Peter Gangl, Ulrich Langer, Angelos Mantzaflaris, and Rainer Schneckenleitnertional domain and solving the PDEs. This aspect makes IgA especially interestingfor design optimization procedures. In practice, it is often the case that one per-forms design optimization and geometric modeling simultaneously. State-of-the-artcomputer aided design (CAD) software uses B-splines or Non-Uniform Rational B-splines (NURBS) for the modeling process whereas the design optimization requiresan analysis suitable representation of the model. So far the design optimization ismainly done using FEM as discretization method. Hence, the B-spline or NURBSrepresentation of the geometric model has to be converted into a suitable mesh forthe Finite Element Analysis. This conversion is in general very computationally de-manding. The new IgA paradigm circumvents these problems. Therefore, IgA isvery beneficial for the simulation and optimization when the representation of thecomputational domain comes from CAD software; see [1, 11] for applications toelectrical machines.Since practical optimization problems tend to be very large, the numerical solu-tion of the underlying PDEs becomes computationally very expensive. Moreover,in PDE-constrained shape optimization processes, there are more than one PDE tosolve. In particular, line search requires to solve the magnetostatic PDE constraintseveral times. In order to get fast optimization results, we use Dual-Primal Iso-gEometric Tearing and Interconnecting (IETI-DP) methods for the solution of thelinear algebraic systems arising from the IgA discretization. The IETI-DP solversare non-overlapping domain decomposition methods; see [6, 7]. IETI-DP methodsare closely related to the FEM-based FETI-DP methods; see, e.g., [10] and thereferences therein. We show that IETI-DP methods are superior to sparse directsolvers with respect to computational time and memory requirement. Moreover,IETI-DP provides a natural framework for parallelization. Indeed, our numericalexperiments on a distributed memory computer show an excellent scaling behaviorof this method.The remainder of the paper is organized as follows. In Section 2, we describeour model problem and the shape optimization method that is based on the shapederivative. Section 3 is devoted to the IETI-DP solver and its performance on parallelcomputers. Finally, in Section 4, we use IETI-DP within the interior point optimizerIpopt [12] yielding an efficient shape optimization procedure.2 Shape optimization via gradient descent2.1 Problem descriptionWe investigate the simulation and shape optimization of an interior permanent mag-net (IPM) electric motor by means of IgA. The IgA approach seems to be very at-tractive for such practical problems. The most beneficial aspect of IgA in the contextof optimization is the fact that the same basis functions which are used to representthe geometry of the IPM electric motor are also exploited to solve the underlyingIsogeometric Simulation and Shape Optimization 3PDEs. In the optimization procedure, we want to optimize the shape of the mo-tor in order to maximize the runout performance, i.e. to maximize the smoothnessof the rotation of the motor. An example of an IPM electric motor is given in Fig-ure 1 (left). One possible way to optimize the runout performance of an IPM electricmotor is to minimize the squared L2-distance between the radial component of themagnetic flux B in the air gap and a desired smooth reference function Bd. Theresulting optimization problem is subject to the 2d magnetostatic PDE as constraint.Mathematically, the arising optimization problem can be expressed as follows:minDJ(u) :=∫Γ|B(u) ·nΓ−Bd|2ds =∫Γ|∇u · τΓ−Bd|2ds (1)s.t. u ∈ H10 (Ω) :∫ΩνD(x)∇u ·∇η dx = 〈F,η〉 ∀η ∈ H10 (Ω), (2)where J denotes the objective function, Γ is the midline of the air gap, Ω denotesthe whole computational domain, and D is the domain of interest also called de-sign domain. The variational problem (2) is nothing but the 2d linear magnetostaticproblem with the piecewise constant magnetic reluctivity νD(x) = χΩ f (D)(x)ν1 +χΩmag(x)νmag + χΩair(D)(x)ν0. Here, Ω f , Ωmag and Ωair denote the ferromagnetic,permanent magnet and air subdomains, respectively, and ν1, νmag and ν0 denotethe corresponding reluctivity values. Note that the shape D enters the optimizationproblem via the function νD and influences the objective function via the solution u.The right hand side F ∈ H−1(Ω) in (2) is defined by the linear functional〈F,η〉 :=∫Ω(J3η +νmagM⊥ ·∇η) dx (3)for all η ∈ H10 (Ω). Here, M⊥ denotes the perpendicular of the magnetization M,which is indicated in Fig. 1 and vanishes outside the permanent magnets, andJ3 is the third component of the impressed current density in the coils. Notethat the solution u is the third component of the magnetic vector potential, i.e.B(u) = curl((0,0,u)T ). Moreover, nΓ = (n1,n2,0)T and τΓ = (τ1,τ2)T denote theoutward unit normal and unit tangential vectors along the air gap, respectively.We are interested in the radial component of the magnetic flux density along theair gap due to the permanent magnetization. For that reason, we set J3 = 0 and con-sider the coil regions as air. Figure 1 (right) shows a quarter of a cross section ofa simplified IPM electric motor that is provided by CAD software. Hence, this ge-ometry representation is suitable for IgA simulation. The red-brown areas representferromagnetic material (Ω f ), the blue areas consist of air (Ωair), the yellow areas arethe permanent magnets (Ωmag). The air gap of the motor is highlighted in light blue.In this initial model for the optimization, the design domain D is the ferromagneticarea right above the permanent magnets. In order to get a smoother rotation we arelooking for a better shape of this part D.4 Peter Gangl, Ulrich Langer, Angelos Mantzaflaris, and Rainer SchneckenleitnerFig. 1 Real world IPM electric motor (see Acknowledgement) on the left, and a quarter of thecross section of a similar electric motor with 8 magnetic poles that is used in our numerical testson the right.2.2 The shape derivativeFor the optimization of the IPM electric motor, we use gradient based optimizationtechniques. Hence, we need the derivative of the objective J with respect to a changeof the current shape. The shape derivative in tensor form [4,8,11] of our optimizationproblem is given bydJ(D)(φ) =∫ΩS (D,u, p) : ∂φdx, ∀φ ∈ H10 (Ω , IR2) (4)with S (D,u, p) = (νD(x)∇u ·∇p−νmag∇p ·M⊥)I +νmag∇p⊗M⊥−νD(x)∇p⊗∇u−νD(x)∇u⊗∇p, where I denotes the identity, the state u solves the constraint(2), and p solves the adjoint problem∫ΩνD(x)∇p ·∇η dx =−2∫Γ(B(u) ·nΓ−Bd)(B(η) ·nΓ) ds ∀η ∈ H10 (Ω). (5)In (4), S (D,u, p) : ∂φ means Frobenius’ scalar product of the 2× 2 matricesS (D,u, p) and ∂φ = ( ∂φi∂x j)2i, j=1, defined by A : B := ∑ni=1 ∑nj=1 ai jbi j for generaln× n matrices A = (ai j)ni, j=1 and B = (bi j)ni, j=1, whereas a⊗ b := (aib j)ni, j=1 forvectors a = (ai)ni=1 and b = (b j)nj=1 from IRn.Isogeometric Simulation and Shape Optimization 52.3 Numerical shape optimizationWe used a continuous Galerkin (cG) IgA discretization for both the simulation andoptimization problems. The implementation is done in G+Smo1. Figure 2 (left)shows a possible computational domain suitable for cG. The shown multipatch do-main consists of 93 patches. For each of these patches, we used a B-spline mappingfrom a reference patch with splines of degree 3. For the optimization, we need theshape gradient ∇J ∈V := H10 (Ω , IR2) which can be computed by solving the auxil-iary problem: find ∇J ∈V such thatb(∇J,ψ) =−dJ(D)(ψ) ∀ψ ∈V. (6)The expression on the right hand side of (6) is the negative shape derivative whereasthe expression b(·, ·) on the left hand side is some V -elliptic, V -bounded bilinearform which must be chosen appropriately. For our studies, we usedb(φ ,ψ) =∫Ωφ ·ψ dx+∫Ωα(∂φ : ∂ψ) dx (7)with a patchwise constant function α ∈ L∞(Ω).Fig. 2 Initial and final design of an IPM motor.In the right picture of Fig. 2, we can see the optimized shape with respect tothe runout performance compared to the initial domain on the left. We were able toreduce the objective from 4.236 ·10−4 down to 2.781 ·10−4.1 Mantzaflaris, A. et al.: G+Smo (geometry plus simulation modules) v0.8.1.,http://gs.jku.at/gismo, 2017 Jun 19 20186 Peter Gangl, Ulrich Langer, Angelos Mantzaflaris, and Rainer Schneckenleitner3 Fast numerical solutions by IETI-DPUp to now, we have solved the arising PDEs by means of a sparse direct solver. Onedrawback of a direct solution method is that it is rather slow for large-scale sys-tems. In particular, in shape optimization, we have to solve the state equation (2),the adjoint equation (5), and the auxiliary problem (6) for the shape gradient, whichdecouples into two scalar problems, in every iteration of the optimization algorithm.Moreover, during a line search procedure, it might be the case that the state equa-tion has to be solved several times. To overcome the issue of a slow performance,we were looking for a fast and suitable solver for our simulation and optimizationprocesses. We chose the IETI-DP technique for solving the PDEs [6]. IETI-DP is anon-overlapping domain decomposition technique which introduces local subspaceswhich are then again coupled using additional constraints. A comparison betweenthe sparse direct solver SuperLU [3] and IETI-DP for solving the state equation(2) on a full cross section of an IPM electric motor clearly shows that the recentlydeveloped IETI-DP method [6] performs much better as can be seen in Table 1.The numerical results displayed in Table 1 and Table 2 were obtained on RADON1(https://www.ricam.oeaw.ac.at/hpc/overview/) a high performance computing clus-ter with 1168 computing cores and 10.7 TB of memory. Table 1 also shows that,with an increasing number of degrees of freedom, the proposed IETI-DP techniquesolves the problem much faster than the sparse direct solver. Moreover, it can beseen that, with too many degrees of freedom, the sparse direct solver ran out ofmemory whereas IETI-DP could provide the solution to the problem. The solutionto the state equation is shown in Fig. 3 (right).Table 1 SuperLU vs. IETI-DP on a single core.# dofs SuperLU IETI-DP speedup72 572 36.0 sec 17.0 sec 2.12250 844 193.0 sec 69.8 sec 2.77928 796 1943.0 sec 463.0 sec 4.203 570 332 – 1179.0 sec –Moreover, IETI-DP provides a natural framework for parallelization. Becauseof the multipatch structure of the computational domains in IgA, each patch canbe seen as a subdomain in the IETI-DP approach. Then one can create suitablesubdomains consisting of a certain number of patches for each processor, e.g., onepossible choice is to group the patches to subdomains according to their numberof degrees of freedom which means that the degrees of freedom are almost evenlydistributed over the number of processors. Table 2 shows the strong scaling behaviorof the IETI-DP solver. In this experiment, we solved the constraint equation (2) onthe full cross section of an IPM electric motor with 3 570 332 degrees of freedom.Isogeometric Simulation and Shape Optimization 7Fig. 3 Whole initial cross section as well as the solution.From Table 2, we can see the expected performance, i.e., if we double the numberof processors the computation time reduces nearly by a factor of two.Table 2 Strong scaling with IETI-DP and 3 570 332 dofs.# cores 1 2 4 8 16 32 64 128time [sec] 1179 577 325 164 89 43 22 14rate – 2.04 1.78 1.98 1.84 2.07 1.95 1.574 Shape optimization based on Ipopt and IETI-DPIn this section, we point out the usage of the interior point optimizer Ipopt [12],for the shape optimization using IETI-DP as underlying PDE solver. If we performshape optimization without any additional considerations, then we might run intotroubles. More precisely, it can happen that we get self-intersections in the finalshape even if the objective decreases.To prevent such self-intersections, we consider the Jacobian determinant of thegeometry transformation in the design domain and its neighboring air regions. TheJacobian determinant of these patches must have the same sign in each iteration.If the sign changes from one iteration to the next, then we reduce the step sizeuntil the Jacobian determinant of the new design has the same sign as in the initialconfiguration. In this way, we are able to ensure that the shape is technically feasible.In the first naive approach, all control coefficients of the multipatch domain areconsidered as design variables, and the vector field computed by (6) is applied glob-8 Peter Gangl, Ulrich Langer, Angelos Mantzaflaris, and Rainer Schneckenleitnerally. The computational effort for the optimization can be reduced by applying thecomputed vector field only on the important interfaces between the design domainand the neighboring air regions. This reduces the number of design variables fromapproximately 28000 to 128 in the coarsest setting. The inner control coefficientsof the design area and the bordering air regions are rearranged via a spring patchmodel [9].In a first test setting, Ipopt stops at an optimal solution after 95 iterations usinga BFGS method. We set the NLP error tolerance to 10−6, the relative error in theobjective change to the same value, and we decided to exit the optimization loopafter three iterations within these error bounds. The objective value dropped from4.266 ·10−4 down to 2.587 ·10−4Fig. 4 Optimal shape after 130 iterations with relaxed bounds (left), zoom into one of the designregions (right)Furthermore, we tried an additional experiment were we relaxed the bounds onthe constraints a bit. In particular, we set the bound relax factor in Ipopt to 1.The result of this experiment can be seen in Fig. 4. We may observe from Fig. 4 thatwe get a very smooth final shape with even a smaller objective value of 2.436 ·10−4.We point out that, if we adjust the different optimization parameters, we may getdifferent optimal shapes and different objective values in the end.Acknowledgements This work was supported by the Austrian Science Fund (FWF) via the grantsNFN S117-03 and the DK W1214-04. We also acknowledge the permission to use the Photo inFig. 1 (left) taken by the Linz Center of Mechatronics (LCM). The motor was produced by HanningElektro-Werke GmbH & Co KG.Isogeometric Simulation and Shape Optimization 9References1. Bontinck, Z., Corno, J., Schöps, S., De Gersem, H.: Isogeometric analysis and harmonicstator–rotor coupling for simulating electric machines. Comput. Methods Appl. Mech. En-grg., 334, pp. 40-55, 20182. Cottrell, J.A., Hughes, T.J.R., Bazilevs, Y.: Isogeometric analysis: CAD, finite elements,NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Engrg., 194,pp. 4135-4195, 20053. Demmel, J.W., Eisenstat, S. C., Gilbert, J.R., Li, X. S., Liu, J. W. H.: A supernodal approach tosparse partial pivoting. SIAM J. Matrix Analysis and Applications, 20(3), pp. 720-755, 19994. Gangl, P.: Sensitivity-Based Topology and Shape Optimization with Application to ElectricalMachines. Ph.D. thesis, Johannes Kepler University Linz, 2016.5. Gangl, P., Langer, U., Laurain, A., Meftahi, H., Sturm, K.: Shape Optimization of an ElectricMotor Subject to Nonlinear Magnetostatics. SIAM J. Sci. Comput., 37(6), pp. B1002–B1025,20156. Hofer, C., Langer, U.: Dual-Primal Isogeometric Tearing and Interconnecting solvers for mul-tipatch dG-IgA equations. Comput. Methods Appl. Mech. Engrg., 316, pp. 2–21, 20177. Kleiss, S., Pechstein, C., Jüttler B., Tomar S.L: IETI–isogeometric tearing and interconnect-ing. Comput. Methods Appl. Mech. Engrg., 247, pp. 201–215, 20128. Laurain, A., Sturm, K.: Distributed shape derivative via averaged adjoint method and applica-tions. ESAIM:M2AN, 50 (4), pp. 1241–1267, 20169. Nguyen, D. M., Gravesen, J., Evgrafov, A.: Isogeometric analysis and shape optimization inelectromagnetism. Ph.D. thesis, Technical University of Denmark, 2012.10. Pechstein, C.: Finite and boundary element tearing and interconnecting solvers for multiscaleproblems. Springer, Berlin 201311. Schneckenleitner, R.: Isogeometrical Analysis based Shape Optimization. Master thesis, Jo-hannes Kepler University Linz, 2017.12. Wächter, A., Biegler, L.T.: On the implementation of an interior-point filter line search algo-rithm for large scale nonlinear programming. Math. Program., 106(1):25–57, 2006

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