Sensitivity-Based Topology and Shape Optimization with Application to Electric Motors

  • Peter GanglEmail author
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 163)


In many industrial applications, one is interested in finding an optimal layout of an object, which often leads to PDE-constrained shape optimization problems. Such problems can be approached by shape optimization methods, where a domain is altered by smooth deformation of its boundary, or by means of topology optimization methods, which in addition can alter the connectivity of the initial design. We give an overview over established topology optimization methods and focus on an approach based on the sensitivity of the cost function with respect to a topological perturbation of the domain, called the topological derivative. We illustrate a way to derive this sensitivity and discuss the additional difficulties arising in the case of a nonlinear PDE constraint. We show numerical results for the optimization of an electric motor which are obtained by a combination of two methods: a level set algorithm which is based on the topological derivative, and a shape optimization method together with a special treatment of the evolving material interface which assures accurate approximate solutions to the underlying PDE constraint as well as a smooth final design.


  1. 1.
    G. Allaire. Shape optimization by the homogenization method. Applied mathematical sciences. Springer, New York, 2002.zbMATHCrossRefGoogle Scholar
  2. 2.
    H. Ammari and H. Kang. Polarization and Moment Tensors. Springer-Verlag New York, 2007.Google Scholar
  3. 3.
    S. Amstutz. Sensitivity analysis with respect to a local perturbation of the material property. Asymptotic analysis, 49(1), 2006.Google Scholar
  4. 4.
    S. Amstutz. Analysis of a level set method for topology optimization. Optimization Methods and Software - Advances in Shape an Topology Optimization: Theory, Numerics and New Application Areas, 26(4–5):555–573, 2011.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    S. Amstutz and H. Andrä. A new algorithm for topology optimization using a level-set method. Journal of Computational Physics, 216(2):573–588, 2006.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    S. Amstutz and A. Bonnafé. Topological derivatives for a class of quasilinear elliptic equations. Journal de mathématiques pures et appliquées.Google Scholar
  7. 7.
    T. Belytschko, R. Gracie, and G. Ventura. A review of extended/generalized finite element methods for material modeling. Model. Simul. Mater. Sci. Eng., 17(4), 2009.CrossRefGoogle Scholar
  8. 8.
    M. P. Bendsøe. Optimal shape design as a material distribution problem. Structural Optimization, 1(4):193–202, 1989.CrossRefGoogle Scholar
  9. 9.
    M. P. Bendsoe and N. Kikuchi. Generating optimal topologies in structural design using a homogenization method. Comput. Methods Appl. Mech. Eng., 71(2):197–224, Nov. 1988.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    M. P. Bendsøe and O. Sigmund. Topology Optimization: Theory, Methods and Applications. Springer, Berlin, 2003.zbMATHGoogle Scholar
  11. 11.
    A. Binder. Elektrische Maschinen und Antriebe: Grundlagen, Betriebsverhalten. Springer-Lehrbuch. Springer, 2012.CrossRefGoogle Scholar
  12. 12.
    M. Burger and R. Stainko. Phase-field relaxation of topology optimization with local stress constraints. SIAM J. Control Optim., 45(4):1447–1466, 2006.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    E. Burman, S. Claus, P. Hansbo, M. G. Larson, and A. Massing. CutFEM: Discretizing geometry and partial differential equations. International Journal for Numerical Methods in Engineering, 104(7):472–501, 2015.MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    F. Campelo, J. Ramırez, and H. Igarashi. A survey of topology optimization in electromagnetics: considerations and current trends. 2010.Google Scholar
  15. 15.
    A. N. Christiansen, M. Nobel-Jørgensen, N. Aage, O. Sigmund, and J. A. Bærentzen. Topology optimization using an explicit interface representation. Structural and Multidisciplinary Optimization, 49(3):387–399, 2014.MathSciNetCrossRefGoogle Scholar
  16. 16.
    M. C. Delfour and J.-P. Zolésio. Shapes and geometries, volume 22 of Advances in Design and Control. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, second edition, 2011. Metrics, analysis, differential calculus, and optimization.Google Scholar
  17. 17.
    H. A. Eschenauer, V. V. Kobelev, and A. Schumacher. Bubble method for topology and shape optimization of structures. Structural optimization, 8(1):42–51, 1994.CrossRefGoogle Scholar
  18. 18.
    S. Frei and T. Richter. A locally modified parametric finite element method for interface problems. SIAM J. Numer. Anal., 52(5):2315–2334, 2014.MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    T.-P. Fries and T. Belytschko. The extended/generalized finite element method: An overview of the method and its applications. Int. J. Numer. Meth. Eng., 84(3):253–304, 2010.MathSciNetzbMATHGoogle Scholar
  20. 20.
    P. Gangl. Sensitivity-based topology and shape optimization with application to electrical machines. PhD thesis, Johannes Kepler University Linz, 2016.Google Scholar
  21. 21.
    P. Gangl, U. Langer, A. Laurain, H. Meftahi, and K. Sturm. Shape optimization of an electric motor subject to nonlinear magnetostatics. SIAM Journal on Scientific Computing, 37(6):B1002–B1025, 2015.MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    H. Garcke, C. Hecht, M. Hinze, and C. Kahle. Numerical approximation of phase field based shape and topology optimization for fluids. SIAM Journal on Scientific Computing, 37(4):A1846–A1871, 2015.MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    A. Hansbo and P. Hansbo. An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems. Computer Methods in Applied Mechanics and Engineering, 191(47–48):5537 – 5552, 2002.MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    R. Hiptmair, A. Paganini, and S. Sargheini. Comparison of approximate shape gradients. BIT, 55(2):459–485, 2015.MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    A. Laurain and K. Sturm. Distributed shape derivative via averaged adjoint method and applications. ESAIM: M2AN, 50(4):1241–1267, 2016.MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Z. Li. The immersed interface method using a finite element formulation. Appl. Num. Math., 27:253–267, 1998.MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    S. Osher and J. A. Sethian. Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations. Journal of Computational Physics, 79(1):12–49, 1988.MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    C. Pechstein and B. Jüttler. Monotonicity-preserving interproximation of B-H-curves. J. Comp. App. Math., 196:45–57, 2006.MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    O. Sigmund and K. Maute. Topology optimization approaches: A comparative review. Structural and Multidisciplinary Optimization, 48(6):1031–1055, 2013.MathSciNetCrossRefGoogle Scholar
  30. 30.
    O. Sigmund and J. Petersson. Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Structural Optimization, 16(1):68–75, 1998.CrossRefGoogle Scholar
  31. 31.
    J. Sokołowski and A. Zochowski. On the topological derivative in shape optimization. SIAM Journal on Control and Optimization, 37(4):1251–1272, 1999.MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    J. Sokołowski and J.-P. Zolésio. Introduction to shape optimization, volume 16 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 1992. Shape sensitivity analysis.zbMATHCrossRefGoogle Scholar
  33. 33.
    K. Sturm. Minimax Lagrangian approach to the differentiability of nonlinear PDE constrained shape functions without saddle point assumption. SIAM Journal on Control and Optimization, 53(4):2017–2039, 2015.MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    N. P. van Dijk, K. Maute, M. Langelaar, and F. van Keulen. Level-set methods for structural topology optimization: a review. Structural and Multidisciplinary Optimization, 48(3): 437–472, 2013.MathSciNetCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Graz University of TechnologyInstitute of Applied MathematicsGrazAustria

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