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Level set topology optimization of synchronous reluctance machines using a body-fitted mesh representation

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Abstract

This paper presents a framework for the topology optimization of electro-mechanical design problems. While the design is parametrized by means of a level set function defined on a fixed mesh of the design domain, mesh adaptation is used to generate a second mesh that conforms to the domain delineated by the iso-zero of the level set function. This body-fitted mesh is used in the finite element simulation of the physical problem in order to accurately represent the electromagnetic interface phenomena. An appropriate combination of the two geometry representations is obtained through the velocity field to ensure a consistent design space as the topology optimization process unfolds. The method is applied to the joint electro-mechanical optimization of a synchronous reluctance machine (SynRM).

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Notes

  1. If the performance function is a pointwise value, the expression of \(F(\tau , \varvec{A} ^{\star }, \varvec{u}^{\star })\) will then involve a Dirac function.

References

  • Abe K, Kazama S, Koro K (2007) A boundary element approach for topology optimization problem using the level set method. Commun Numer Methods Eng 23(5):405–416

    Article  MathSciNet  MATH  Google Scholar 

  • Alauzet F (2010) Size gradation control of anisotropic meshes. Finite Elem Anal Design 46(1–2):181–202

    Article  MathSciNet  Google Scholar 

  • Allaire G, Dapogny C, Frey P (2013) A mesh evolution algorithm based on the level set method for geometry and topology optimization. Struct Multidisc Optimi 48(4):711–715

    Article  MathSciNet  Google Scholar 

  • Allaire G, Dapogny C, Frey P (2014) Shape optimization with a level set based mesh evolution method. Comput Methods Appl Mech Eng 282:22–53

    Article  MathSciNet  MATH  Google Scholar 

  • Allaire G, Jouve F, Toader AM (2002) A level-set method for shape optimization. Comptes Rendus Mathematique 334(12):1125–1130

    Article  MathSciNet  MATH  Google Scholar 

  • Barcaro M, Meneghetti G, Bianchi N (2013) Structural analysis of the interior pm rotor considering both static and fatigue loading. IEEE Trans Ind Appl 50(1):253–260

    Article  Google Scholar 

  • Bendsoe MP, Sigmund O (2003) Topology optimization: theory, methods, and applications. Springer Science & Business Media

    MATH  Google Scholar 

  • Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods App Mech Eng 71(2):197–224

    Article  MathSciNet  MATH  Google Scholar 

  • Biedinger J, Lemoine D (1997) Shape sensitivity analysis of magnetic forces. Magnet IEEE Transact 33(3):2309–2316

    Article  Google Scholar 

  • Binder A, Schneider T, Klohr M (2006) Fixation of buried and surface-mounted magnets in high-speed permanent-magnet synchronous machines. IEEE Trans Ind Appl 42(4):1031–1037

    Article  Google Scholar 

  • Boglietti A, Cavagnino A, Pastorelli M, Staton D, Vagati A (2006) Thermal analysis of induction and synchronous reluctance motors. IEEE Trans Ind Appl 42(3):675–680

    Article  Google Scholar 

  • Bossavit A (1998) Computational electromagnetism: variational formulations, complementarity, edge elements. Academic Press

    MATH  Google Scholar 

  • Braibant V, Fleury C (1984) Shape optimal design using b-splines. Comput Methods Appl Mech Eng 44(3):247–267

    Article  MATH  Google Scholar 

  • Chai F, Li Y, Liang P, Pei Y (2016) Calculation of the maximum mechanical stress on the rotor of interior permanent-magnet synchronous motors. IEEE Trans Ind Electron 63(6):3420–3432

    Article  Google Scholar 

  • Choi KK, Chang KH (1994) A study of design velocity field computation for shape optimal design. Finite Elem Anal Des 15(4):317–341

    Article  MATH  Google Scholar 

  • Credo A, Fabri G, Villani M, Popescu M (2020) Adopting the topology optimization in the design of high-speed synchronous reluctance motors for electric vehicles. IEEE Trans Ind Appl 56(5):5429–5438

    Article  Google Scholar 

  • Dapogny C, Dobrzynski C, Frey P (2014) Three-dimensional adaptive domain remeshing, implicit domain meshing, and applications to free and moving boundary problems. J Comput Phys 262:358–378

    Article  MathSciNet  MATH  Google Scholar 

  • Deaton JD, Grandhi RV (2014) A survey of structural and multidisciplinary continuum topology optimization: post 2000. Struct Multidisc Optimi 49(1):1–38

    Article  MathSciNet  Google Scholar 

  • Di Nardo M, Galea M, Gerada C, Palmieri M, Cupertino F (2015) Multi-physics optimization strategies for high speed synchronous reluctance machines. In: 2015 IEEE Energy Conversion Congress and Exposition (ECCE), IEEE, (pp 2813–2820)

  • Di Nardo M, Galea M, Gerada C, Palmieri M, Cupertino F, Mebarki S (2015) Comparison of multi-physics optimization methods for high speed synchrnous reluctance machines. In: IECON 2015-41st Annual Conference of the IEEE Industrial Electronics Society, IEEE, (pp 002771–002776)

  • Di Nardo M, Calzo GL, Galea M, Gerada C (2017) Design optimization of a high-speed synchronous reluctance machine. IEEE Trans Ind Appl 54(1):233–243

    Article  Google Scholar 

  • van Dijk NP, Maute K, Langelaar M, Van Keulen F (2013) Level-set methods for structural topology optimization: a review. Struct Multidisc Optimi 48(3):437–472

    Article  MathSciNet  Google Scholar 

  • Duysinx P, Sigmund O (1998) New developments in handling stress constraints in optimal material distribution. Proceedings of the 7th AIAA/USAF/NASAISSMO Symp Multidiscip Anal Optimiz 1:1501–1509

    Google Scholar 

  • Emmendoerfer H Jr, Fancello EA (2016) Topology optimization with local stress constraint based on level set evolution via reaction-diffusion. Comput Methods Appl Mech Eng 305:62–88

    Article  MathSciNet  MATH  Google Scholar 

  • Feppon F, Allaire G, Bordeu F, Cortial J, Dapogny C (2019) Shape optimization of a coupled thermal fluid-structure problem in a level set mesh evolution framework. SeMA J 76(3):413–458

    Article  MathSciNet  MATH  Google Scholar 

  • Florez S, Shakoor M, Toulorge T, Bernacki M (2020) A new finite element strategy to simulate microstructural evolutions. Comput Mater Sci 172

  • Fratta A, Toglia G, Vagati A, Villata F (1995) Ripple evaluation of high-performance synchronous reluctance machines. IEEE Ind Appl Magaz 1(4):14–22

    Article  Google Scholar 

  • Gangl P, Langer U, Laurain A, Meftahi H, Sturm K (2015) Shape optimization of an electric motor subject to nonlinear magnetostatics. SIAM J Sci Comput 37(6):B1002–B1025

    Article  MathSciNet  MATH  Google Scholar 

  • Geiss MJ, Boddeti N, Weeger O, Maute K, Dunn ML (2019) Combined level-set-xfem-density topology optimization of four-dimensional printed structures undergoing large deformation. J Mech Des 141(5)

  • Geuzaine C, Remacle JF (2009) Gmsh: A 3-D finite element mesh generator with built-in pre-and post-processing facilities. Int J Numer Methods Eng 79(11):1309–1331

    Article  MathSciNet  MATH  Google Scholar 

  • Gilmanov A, Sotiropoulos F (2005) A hybrid cartesian/immersed boundary method for simulating flows with 3d, geometrically complex, moving bodies. J Comput Phys 207(2):457–492

    Article  MATH  Google Scholar 

  • Ha SH, Cho S (2008) Level set based topological shape optimization of geometrically nonlinear structures using unstructured mesh. Comput Str 86(13–14):1447–1455

    Article  Google Scholar 

  • Halbach A (2017) Sparselizard-the user friendly finite element c++ library: http://www.cenaero.be

  • Hassani B, Tavakkoli SM, Ghasemnejad H (2013) Simultaneous shape and topology optimization of shell structures. Struct Multidisc Optimi 48(1):221–233

    Article  MathSciNet  MATH  Google Scholar 

  • Henrotte F (2004) Handbook for the computation of electromagnetic forces in a continuous medium. Int. Compumag Soc Newsletter 24(2):3–9

    Google Scholar 

  • Hermann R et al (1964) Harley flanders, differential forms with applications to the physical sciences. Bull Am Mathem Soc 70(4):483–487

    Article  Google Scholar 

  • Hintermüller M, Laurain A (2008) Electrical impedance tomography: from topology to shape. Control Cybernet 37(4)

  • Hiptmair R, Li J (2013) Shape derivatives in differential forms i: An intrinsic perspective. Annali di Matematica Pura ed Applicata 192(6):1077–1098

    Article  MathSciNet  MATH  Google Scholar 

  • Hiptmair R, Li J (2017) Shape derivatives in differential forms ii: Shape derivatives for scattering problems. SAM Seminar for Applied Mathematics, ETH, Zürich, Switzerland, Research Report

  • Jansen M (2019) Explicit level set and density methods for topology optimization with equivalent minimum length scale constraints. Struct Multidisc Optim 59(5):1775–1788

    Article  MathSciNet  Google Scholar 

  • Kostko J (1923) Polyphase reaction synchronous motors. J Am Inst Elect Eng 42(11):1162–1168

    Article  Google Scholar 

  • Kuci E, Henrotte F, Duysinx P, Geuzaine C (2017) Design sensitivity analysis for shape optimization based on the Lie derivative. Comput Methods Appl Mech Eng 317:702–722

    Article  MathSciNet  MATH  Google Scholar 

  • Kuci E, Henrotte F, Geuzaine C, Dehez B, Gréef CD, Versèle C, Friebel C (2020) Design Optimization of Synchronous Reluctance Machines for Railway Traction Application Including Assembly Process Constraints. In: 2020 International Conference on Electrical Machines (ICEM), vol. 1, pp. 117–123. https://doi.org/10.1109/ICEM49940.2020.9270859. ISSN: 2381-4802

  • Kwack J, Min S, Hong JP (2010) Optimal stator design of interior permanent magnet motor to reduce torque ripple using the level set method. IEEE Trans Magnet 46(6):2108–2111

    Article  Google Scholar 

  • Lazarov BS, Sigmund O (2011) Filters in topology optimization based on helmholtz-type differential equations. Int J Numer Methods Eng 86(6):765–781

    Article  MathSciNet  MATH  Google Scholar 

  • Le C, Bruns T, Tortorelli D (2011) A gradient-based, parameter-free approach to shape optimization. Comput Methods Appl Mech Eng 200(9–12):985–996

    Article  MathSciNet  MATH  Google Scholar 

  • Lindh P, Tehrani MG, Lindh T, Montonen JH, Pyrhönen J, Sopanen JT, Niemelä M, Alexandrova Y, Immonen P, Aarniovuori L et al (2016) Multidisciplinary design of a permanent-magnet traction motor for a hybrid bus taking the load cycle into account. IEEE Trans Ind Electron 63(6):3397–3408

    Article  Google Scholar 

  • Madlib (2009) Mesh adaptation library, Cenaero, Belgium. https://sites.uclouvain.be/madlib/

  • Van Miegroet L, Duysinx P (2007) Stress concentration minimization of 2d filets using x-fem and level set description. Struct Multidisc Optim 33(4):425–438

    Article  Google Scholar 

  • Morfeo version 3.1.0 (2019) a Manufacturing ORiented Finite Element sOftware, Cenaero, Belgium. http://www.cenaero.be

  • Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46(1):131–150

    Article  MathSciNet  MATH  Google Scholar 

  • Novotny AA, Feijóo RA, Taroco E, Padra C (2003) Topological sensitivity analysis. Comput Methods Appl Mech Eng 192(7):803–829

    Article  MathSciNet  MATH  Google Scholar 

  • Noël L, Miegroet LV, Duysinx P (2016) Analytical sensitivity analysis using the extended finite element method in shape optimization of bimaterial structures. Int J Numer Methods Eng 107(8):669–695

    Article  MathSciNet  MATH  Google Scholar 

  • Olhoff N, Bendsøe MP, Rasmussen J (1991) On cad-integrated structural topology and design optimization. Comput Methods Appl Mech Eng 89(1–3):259–279

    Article  MATH  Google Scholar 

  • Osher S, Sethian JA (1988) Fronts propagating with curvature-dependent speed: algorithms based on hamilton-jacobi formulations. J Comput Phys 79(1):12–49

    Article  MathSciNet  MATH  Google Scholar 

  • Palmieri M, Perta M, Cupertino F (2016) Design of a 50.000-r/min synchronous reluctance machine for an aeronautic diesel engine compressor. IEEE Trans Ind Appl 52(5):3831–3838

    Article  Google Scholar 

  • Park IH, Coulomb JL, Hahn SY (1993) Implementation of continuum sensitivity analysis with existing finite element code. Magnet IEEE Trans 29(2):1787–1790

    Article  Google Scholar 

  • Sigmund O, Maute K (2013) Topology optimization approaches. Struct Multidisc Optim 48(6):1031–1055

    Article  MathSciNet  Google Scholar 

  • Sokolowski J, Zochowski A (2003) Optimality conditions for simultaneous topology and shape optimization. SIAM J Control Optim 42(4):1198–1221

    Article  MathSciNet  MATH  Google Scholar 

  • Sokolowski J, Zolesio JP (1992) Introduction to shape optimization. In: Introduction to shape optimization, Springer, (pp 5–12)

  • Svanberg K (1987) The method of moving asymptotes- a new method for structural optimization. Int J Numer Methods Eng 24(2):359–373

    Article  MathSciNet  MATH  Google Scholar 

  • Svanberg K (2002) A class of globally convergent optimization methods based on conservative convex separable approximations. SIAM J Optim pp 555–573

  • Tanaka I, Nitomi H, Imanishi K, Okamura K, Yashiki H (2012) Application of high-strength nonoriented electrical steel to interior permanent magnet synchronous motor. IEEE Trans Magnet 49(6):2997–3001

    Article  Google Scholar 

  • Tseng YH, Ferziger JH (2003) A ghost-cell immersed boundary method for flow in complex geometry. J Comput Phys 192(2):593–623

    Article  MathSciNet  MATH  Google Scholar 

  • Villanueva CH, Maute K (2017) Cutfem topology optimization of 3d laminar incompressible flow problems. Comput Methods Appl Mech Eng 320:444–473

    Article  MathSciNet  MATH  Google Scholar 

  • Wang MY, Wang X, Guo D (2003) A level set method for structural topology optimization. Comput Methods Appl Mech Eng 192(1):227–246

    Article  MathSciNet  MATH  Google Scholar 

  • Yaji K, Otomori M, Yamada T, Izui K, Nishiwaki S, Pironneau O (2016) Shape and topology optimization based on the convected level set method. Struct Multidisc Optim 54(3):659–672

    Article  MathSciNet  Google Scholar 

  • Yamasaki S, Kawamoto A, Nomura T (2012) Compliant mechanism design based on the level set and arbitrary lagrangian eulerian methods. Struct Multidisc Optim 46(3):343–354

    Article  MathSciNet  MATH  Google Scholar 

  • Yamasaki S, Kawamoto A, Nomura T, Fujita K (2015) A consistent grayscale-free topology optimization method using the level-set method and zero-level boundary tracking mesh. Int J Numer Methods Eng 101(10):744–773

    Article  MathSciNet  MATH  Google Scholar 

  • Yamasaki S, Nomura T, Kawamoto A, Sato K, Nishiwaki S (2011) A level set-based topology optimization method targeting metallic waveguide design problems. Int J Numer Methods Eng 87(9):844–868

    Article  MathSciNet  MATH  Google Scholar 

  • Yamasaki S, Yamanaka S, Fujita K (2017) Three-dimensional grayscale-free topology optimization using a level-set based r-refinement method. Int J Numer Methods Eng 112(10):1402–1438

    Article  MathSciNet  Google Scholar 

  • Zhang J, Zhang W, Zhu J, Xia L (2012) Integrated layout design of multi-component systems using xfem and analytical sensitivity analysis. Comput Methods Appl Mech Eng 245:75–89

    Article  MathSciNet  MATH  Google Scholar 

  • Zienkiewicz C, Taylor RL (1990) The finite element method Vol. 1: Basic formulation and linear problems. No. 3 in finite element method series. Wiley

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Acknowledgements

The first author warmly thanks Dr. Alexandre Halbach for his support with sparselizard.

Funding

This work was supported in part by the Walloon Region of Belgium under Grant PIT 7706 Traction2020. The present research benefited from computational resources made available on the Tier-1 supercomputer of the Fédération Wallonie-Bruxelles, infrastructure funded by the Walloon Region under the Grant Agreement No 1117545.

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  1. Centre de recherche en aéronautique (Cenaero), Rue des Frères Wright 29, Gosselies, 6041, Belgium

    Erin Kuci, Miche Jansen & Olivier Coulaud

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  1. Erin Kuci
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  2. Miche Jansen
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  3. Olivier Coulaud
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Correspondence to Erin Kuci.

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On behalf of all authors, the corresponding author states that there is no conflict of interest.

Replication of results

Our work relies on several programming languages (Python, C++) and also several other codes. Specifically, we use the described topology optimization with body-fitted mesh representation based on the in-house software Morfeo (2019). The mesh adaptations are performed with MadLIB (2009, https://sites.uclouvain.be/madlib/) and the nonlinear magnetostatics is solved with Sparselizard (Halbach 2017). Rather than providing a source code package which would only work under very strict platform requirements, we instead opt to aid the reader in reproducing our results by satisfying the reasonable and responsible demands for the open source codes underpinning the present article.

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Kuci, E., Jansen, M. & Coulaud, O. Level set topology optimization of synchronous reluctance machines using a body-fitted mesh representation. Struct Multidisc Optim 64, 3729–3745 (2021). https://doi.org/10.1007/s00158-021-03049-0

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