Applied Composite Materials

, Volume 26, Issue 2, pp 575–590 | Cite as

Shepard Interpolation Based on Geodesic Distance for Optimization of Fiber Reinforced Composite Structures with Non-Convex Shape

  • Ye Tian
  • Junxin Mou
  • Tielin Shi
  • Qi XiaEmail author


In the present paper, we propose to define the Shepard interpolation by using the geodesic distance. The geodesic distance between a pair of points is the length of the shortest geodesic line, and geodesic line is the generalization of straight line in the Euclidean geometry to general spaces, for instance 3D surfaces. When the shape of structure is non-convex, the geodesic distance is more reasonable than the Euclidean distance to define influence domain for interpolation points. In view of the fact that the length of a geodesic line can be considered as the time it takes to go with a certain velocity from one point to another, the fast marching method is used to compute the geodesic distance. In the design optimization of fiber reinforced composite structures, the proposed Shepard interpolation based on geodesic distance is used to construct a continuous global function that represents the fiber angle arrangement. The design variables to be optimized are the angles at scattered design points. Several examples with in-plane load are investigated. In the simple representative numerical examples, the proposed method shows good performance.


Shepard interpolation Geodesic distance Non-convex shape Design optimization Curved fiber Composite structures 



This research work is supported by the National Natural Science Foundation of China (Grant No. 51575203) and the Natural Science Foundation for Distinguished Young Scholars of Hubei province (Grant No. 2017CFA044).


  1. 1.
    Ghiasi, H., Pasini, D., Lessard, L.: Optimum stacking sequence design of composite materials part I: Constant stiffness design. Compos. Struct. 90, 1–11 (2009)CrossRefGoogle Scholar
  2. 2.
    Ghiasi, H., Fayazbakhsh, K., Pasini, D., Lessard, L.: Optimum stacking sequence design of composite materials part II: Variable stiffness design. Compos. Struct. 93, 1–13 (2010)CrossRefGoogle Scholar
  3. 3.
    Lozano, G.G., Tiwari, A., Turner, C., Astwood, S.: A review on design for manufacture of variable stiffness composite laminates. Proc. IME. B. J. Eng. Manufact. 230(6), 981–992 (2016)CrossRefGoogle Scholar
  4. 4.
    Sabido, A., Bahamonde, L., Harik, R., van Tooren, M.J.L.: Maturity assessment of the laminate variable stiffness design process. Compos. Struct. 160, 804–812 (2017)CrossRefGoogle Scholar
  5. 5.
    Sigmund, O., Maute, K.: Topology optimization approaches. Struct. Multidiscip. Optim. 48, 1031–1055 (2013)CrossRefGoogle Scholar
  6. 6.
    Deaton, J.D., Grandhi, R.V.: A survey of structural and multidisciplinary continuum topology optimization: post 2000. Struct. Multidiscip. Optim. 49, 1–38 (2014)CrossRefGoogle Scholar
  7. 7.
    Xia, L., Xia, Q., Huang, X.D., Xie, Y.M.: Bi-directional evolutionary structural optimization on advanced structures and materials: a comprehensive review. Arch. Computat. Methods Eng. 25, 437–478 (2018)CrossRefGoogle Scholar
  8. 8.
    Xia, Q., Shi, T.: Topology optimization of compliant mechanism and its support through a level set method. Comput. Methods Appl. Mech. Eng. 305, 359–375 (2016a)CrossRefGoogle Scholar
  9. 9.
    Xia, Q., Shi, T.: Optimization of structures with thin-layer functional device on its surface through a level set based multiple-type boundary method. Comput. Methods Appl. Mech. Eng. 311, 56–70 (2016b)CrossRefGoogle Scholar
  10. 10.
    Xia, Q., Xia, L., Shi, T.L.: Topology optimization of thermal actuator and its support using the level set based multiple–type boundary method and sensitivity analysis based on constrained variational principle. Struct. Multidiscip. Optim. 57, 1317–1327 (2018)CrossRefGoogle Scholar
  11. 11.
    Pedersen, P.: On thickness and orientational design with orthotropic materials. Struct. Optim. 3, 69–78 (1991)CrossRefGoogle Scholar
  12. 12.
    Luo, J.H., Gea, H.C.: Optimal bead orientation of 3d shell/plate structures. Finite Elem. Anal. Des. 31, 55–71 (1998a)CrossRefGoogle Scholar
  13. 13.
    Luo, J.H., Gea, H.C.: Optimal orientation of orthotropic materials using an energy based method. Struct. Optim. 15, 230–236 (1998b)CrossRefGoogle Scholar
  14. 14.
    Stegmann, J., Lund, E.: Discrete material optimization of general composite shell structures. Int. J. Numer. Meth. Eng. 62(14), 2009–2027 (2005)CrossRefGoogle Scholar
  15. 15.
    Bruyneel, M.: SFP–a new parameterization based on shape functions for optimal material selection: application to conventional composite plies. Struct. Multidiscip. Optim. 43, 17–27 (2011)CrossRefGoogle Scholar
  16. 16.
    Gao, T., Zhang, W.H., Duysinx, P.: A bi-value coding parameterization scheme for the discrete optimal orientation design of the composite laminate. Int. J. Numer. Meth. Eng. 91, 98–114 (2012)CrossRefGoogle Scholar
  17. 17.
    Gao, T., Zhang, W.H., Duysinx, P.: Simultaneous design of structural layout and discrete fiber orientation using bi-value coding parameterization and volume constraint. Struct. Multidiscip. Optim. 48, 1075–1088 (2013)CrossRefGoogle Scholar
  18. 18.
    Niu, B., Olhoff, N., Lund, E., Cheng, G.: Discrete material optimization of vibrating laminated composite plates for minimum sound radiation. Int. J. of Solids Struct. 47, 2097–2114 (2010)CrossRefGoogle Scholar
  19. 19.
    Hvejsel, C.F., Lund, E.: Material interpolation schemes for unified topology and multimaterial optimization. Struct. Multidiscip. Optim. 43, 811–825 (2011)CrossRefGoogle Scholar
  20. 20.
    Olmedo, R., Gürdal, Z.: Buckling Response of Laminates with Spatially Varying Fiber Orientations. In: 34Th Structures, Structural Dynamics and Materials Conference, La Jolla, CA, U.S.A (1993)Google Scholar
  21. 21.
    Blom, A., Setoodeh, S., Hol, J.M.: Design of variable-stiffness conical shells for maximum fundamental eigenfrequency. Comput. Struct. 86, 870–878 (2008)CrossRefGoogle Scholar
  22. 22.
    Blom, A., Tatting, B.F., Hol, J.M.: Fiber path definitions for elastically tailored conical shells. Compos. Part B-Eng. 40, 77–84 (2009)CrossRefGoogle Scholar
  23. 23.
    Parnas, L., Oral, S., Ceyhan, U.: Optimum design of composite structures with curved fiber courses. Compos. Sci. Technol. 63, 1071–1082 (2003)CrossRefGoogle Scholar
  24. 24.
    Liu, B., Haftka, R.T., Trompette, P.: Maximization of buckling loads of composite panels using flexural lamination parameters. Struct. Multidiscip. Optim. 26, 28–36 (2004)CrossRefGoogle Scholar
  25. 25.
    Ijsselmuiden, S.T., Abdalla, M., Gürdal, Z.: Implementation of strength-based failure criteria in the lamination parameter design space. AIAA J. 46, 1826–1834 (2008)CrossRefGoogle Scholar
  26. 26.
    Bohrerolar, R.Z.G., Almeida, S.F.M.D., Donadon, M.V.: Optimization of composite plates subjected to buckling and small mass impact using lamination parameters. Compos. Struct. 120, 141–152 (2015)CrossRefGoogle Scholar
  27. 27.
    Bruyneel, M., Zein, S.: A modified fast marching method for defining fiber placement trajectories over meshes. Comput. Struct. 125, 45–52 (2013)CrossRefGoogle Scholar
  28. 28.
    Brampton, C.J., Wu, K.C., Kim, H.A.: New optimization method for steered fiber composites using the level set method. Struct. Multidiscip. Optim. 52, 493–505 (2015)CrossRefGoogle Scholar
  29. 29.
    Lemaire, E., Zein, S., Bruyneel, M.: Optimization of composite structures with curved fiber trajectories. Compos. Struct. 131, 895–904 (2015)CrossRefGoogle Scholar
  30. 30.
    Kiyono, C.Y., Silva, E.C.N., Reddy, J.N.: A novel fiber optimization method based on normal distribution function with continuously varying fiber path. Compos. Struct. 160, 503–515 (2016)CrossRefGoogle Scholar
  31. 31.
    Nomura, T., Dede, E.M., Lee, J., Yamasaki, S., Matsumori, T., Kawamoto, A., Kikuchi, N.: General topology optimization method with continuous and discrete orientation design using isoparametric projection. Int. J. Numer. Meth. Eng. 101, 571–605 (2015)CrossRefGoogle Scholar
  32. 32.
    Setoodeh, S., Abdalla, M.M., Gürdal, Z.: Combined topology and fiber path design of composite layers using ellular automata. Struct. Multidiscip. Optim. 30, 412–421 (2005)CrossRefGoogle Scholar
  33. 33.
    Xia, Q., Shi, T.L.: Optimization of composite structures with continuous spatial variation of fiber angle through shepard interpolation. Compos. Struct. 182, 273–282 (2017)CrossRefGoogle Scholar
  34. 34.
    Shepard, D.: A two-dimensional interpolation function for irregularly-spaced data. In: Proceedings of the 23rd National Conference, pp 517–523. ACM, New York (1968)Google Scholar
  35. 35.
    Brodlie, K., Asim, M., Unsworth, K.: Constrained visualization using the Shepard interpolation family. Computer Graphics Forum 24, 809–820 (2005)CrossRefGoogle Scholar
  36. 36.
    Lodha, S.K., Franke, R.: Scattered Data Techniques for Surfaces. In: Scientific Visualization Conference, pp 189–230. Dagstuhl, Germany (1997)Google Scholar
  37. 37.
    Belytschko, T., Lu, Y.Y., Gu, L.: Element–free Galerkin methods. Int. J. Numer. Methods Eng. 37, 229–256 (1994)CrossRefGoogle Scholar
  38. 38.
    Belytschko, T., Krongauz, Y., Fleming, M., Organ, D., Liu, W.K.: Smoothing and accelerated computations in the element free Galerkin method. J. Comput. Appl. Math 74, 111–126 (1996)CrossRefGoogle Scholar
  39. 39.
    Organ, D., Fleming, M., Terry, T., Belytschko, T.: Continuous meshless approximations for nonconvex bodies by diffraction and transparency. Comput. Mech. 18, 225–235 (1996)CrossRefGoogle Scholar
  40. 40.
    Bronstein, A.M., Bronstein, M.M., Kimmel, R.: Numerical geometry of non-rigid shapes. Springer (2008)Google Scholar
  41. 41.
    Kimmel, R., Amir, A., Bruckstein, A.M.: Finding shortest paths on surfaces using level sets propagation. IEEE Trans. Pattern Anal. Mach. Intell. 17(6), 635–640 (1995)CrossRefGoogle Scholar
  42. 42.
    Sethian, J.A.: A fast marching level set method for monotonically advancing fronts. Proc. Natl. Acad. Sci. USA 93, 1591–1595 (1996)CrossRefGoogle Scholar
  43. 43.
    Sethian, J.A. Cambridge Monographs on Applied and Computational Mathematics: Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision and Materials Science, 2nd. Cambridge University Press, Cambridge (1999)Google Scholar
  44. 44.
    Kang, Z., Wang, Y.Q.: Structural topology optimization based on non-local Shepard interpolation of density field. Comput. Methods Appl. Mech. Eng. 200, 3515–3525 (2011)CrossRefGoogle Scholar
  45. 45.
    Kang, Z., Wang, Y.Q.: A nodal variable method of structural topology optimization based on Shepard interpolant. Int. J. Numer. Meth. Eng. 90, 329–342 (2012)CrossRefGoogle Scholar
  46. 46.
    Luo, Z., Zhang, N., Wang, Y., Gao, W.: Topology optimization of structures using meshless density variable approximants. Int. J. Numer. Meth. Eng. 93, 443–464 (2013)CrossRefGoogle Scholar
  47. 47.
    Wang, Y.Q., Kang, Z., He, Q.Z.: Adaptive topology optimization with independent error control for separated displacement and density fields. Comput. Struct. 135, 50–61 (2014)CrossRefGoogle Scholar
  48. 48.
    Wang, Y.Q., Kang, Z., He, Q.Z.: An adaptive refinement approach for topology optimization based on separated density field description. Comput. Struct. 117, 10–22 (2013)CrossRefGoogle Scholar
  49. 49.
    Farwig, R.: Rate of convergence of Shepard’s global interpolation formula. Math Comp 46, 577–590 (1986)Google Scholar
  50. 50.
    Xia, Q., Shi, T.L.: A cascadic multilevel optimization algorithm for the design of composite structures with curvilinear fiber based on shepard interpolation. Compos. Struct. 188, 209–219 (2018)CrossRefGoogle Scholar
  51. 51.
    Xia, Q., Shi, T.L.: Constraints of distance from boundary to skeleton: For the control of length scale in level set based structural topology optimization. Comput. Methods Appl. Mech. Eng. 295, 525–542 (2015)CrossRefGoogle Scholar
  52. 52.
    Sethian, J.A.: Evolution, implementation, and application of level set and fast marching methods for advancing fronts. J. Comput. Phys. 169, 503–555 (2001)CrossRefGoogle Scholar
  53. 53.
    Wang, X.M., Wang, M.Y., Guo, D.M.: Structural shape and topology optimization in a level-set framework of region representation. Struct. Multidiscip. Optim. 27, 1–19 (2004)CrossRefGoogle Scholar
  54. 54.
    Mei, Y., Wang, X.: A level set method for structural topology optimization and its applications. Adv. Eng. Softw. 35, 415–441 (2004)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.The State Key Laboratory of Digital Manufacturing Equipment and TechnologyHuazhong University of Science and TechnologyWuhanChina

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