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Shape optimization of running shoes with desired deformation properties

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Abstract

The present paper describes a shape optimization procedure for designing running shoes, focusing on two mechanical properties, namely, the shock absorption and the stability keeping the right posture. These properties are evaluated from two deformations of a sole at characteristic timings during running motion. We define approximate planes for the deformations of sole’s upper boundary by least squares method. Using the planes, we choose the tilt angle in the shoe width direction at the mid stance phase of running motion as an objective function representing the stability, and the sunk amount at the contact phase of running motion as a constraint function representing the shock absorption. We assume that the sole is a bonded structure of soft and hard hyper-elastic materials, and the bonding and side boundaries are variable. In this study, we apply the formulation of nonparametric shape optimization to the sole considering finite deformation and contact condition of the bottom of the sole with the ground. Shape derivatives of the cost (objective and constraint) functions are obtained using the adjoint method. The H1 gradient method using these shape derivatives is applied as an iterative algorithm. To solve this optimization problem, we developed a computer program combined with some commercial softwares. The validity of the optimization method is confirmed by numerical examples.

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References

  • Areblad M, Nigg BM, Ekstrand J, Olsson KO (1990) Three-dimensional measurement of rearfoot motion during running. J Biomech 23:933–940

    Article  Google Scholar 

  • Azegami H (2016) Shape optimization problems (in Japanese). Morikita Publishing, Tokyo

    MATH  Google Scholar 

  • Azegami H (2017) Solution of shape optimization problem and its application to product design. In: Itou H, Kimura M, Chalupecký V, Ohtsuka K, Tagami D, Takada A (eds) Proceedings of the international conference CoMFoS15 Mathematical analysis of continuum mechanics and industrial applications, Springer Singapore, Mathematics for industry, 26: 83–98. https://doi.org/10.1007/978-981-10-2633-1

  • Azegami H, Takeuchi K (2006) A smoothing method for shape optimization: Traction method using the Robin condition. Int J Comput Methods 3(1):21–33

    Article  MathSciNet  Google Scholar 

  • Azegami H, Wu ZQ (1996) Domain optimization analysis in linear elastic problems: Approach using traction method. JSME Int J Ser A 39(2):272–278

    Google Scholar 

  • Cardoso JB, Arora JS (1988) Variational method for design sensitivity analysis in nonlinear structural mechanics. AIAA J 26:595– 603

    Article  MathSciNet  Google Scholar 

  • Cavanagh P (1980) The running shoe book. Anderson World, California

    Google Scholar 

  • Choi KK, Chang KH (1994) A study on the velocity computation for shape design optimization. J Finite Elem Anal Des 15:317–347

    Article  Google Scholar 

  • Clarke TE, Frederick EC, Cooper LB (1983) Effects of running shoes on ground reaction forces. Int J Sports Med 4:247–251

    Article  Google Scholar 

  • Dassault Systèmes (2018) Abaqus Analysis User’s Guide 2018. Vélizy-Villacoublay

  • Delfour MC, Zolésio JP (2011) Shapes and geometries: Metrics, analysis, differential calculus and optimization, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia

    Book  Google Scholar 

  • Gard SA, Konz RJ (2004) The effect of a shock-absorbing pylon on the gait of persons with unilateral transtibial amputation. J Rehabil Res Dev 40:109–124

    Article  Google Scholar 

  • Gibson LJ, Ashby MF (1980) Cellular solids structure and properties, Cambridge University Press, Cambridge, England

  • Hisada T (1995) Recent progress in nonlinear fem-based sensitivity analysis. JSME Int J 38:301–310

    Google Scholar 

  • Ihara H, Azegami H, Shimoda M (1999) Solution to shape optimization problems considering material nonlinearity. Comput Aided Opt Des Struct 40:87–95

    Google Scholar 

  • Iwai T, Sugimoto A, Aoyama T, Azegami H (2010) Shape optimization problem of elastic bodies for controlling contact pressure. JSIAM Lett 2:1–4

    Article  MathSciNet  Google Scholar 

  • Kaneko I, Majer G (1981) Optimal design of plastic structures under displacement constraints. Comput Methods Appl Mech Engrg 27:369–391

    Article  Google Scholar 

  • Kim NH, Choi KK, Chen JS (2000) Shape design sensitivity analysis and optimization of elasto-plasticity with frictional contact. AIAA J 398:1742–1753

    Article  Google Scholar 

  • Matheson GO, Clement DB, Makenzie DC, Taunton J, Lioyd-Smith D, Maclntyre JG (1987) Stress fractures in athletes: A study of 320 cases. Am J Sports Med 15:46–68

    Article  Google Scholar 

  • Mills NJ (2007) Polymer foams handbook. Butterworth-Heinemann, Oxford

    Google Scholar 

  • Nakabe N, Nishiwaki T (2002) Development of simplified numerical foot model for sole stability design. Eng Sport 4:824–830

    Google Scholar 

  • Nigg BM (1980) Biomechanics of running shoes. Human Kinetics, Champaign, Illinois

  • Nigg B M, Bahlsen HA, Luethi SM, Stokes S (1987) The influence of running velocity and midsole hardness on external impact forces in heel-toe running. J Biomech 20:951–959

    Article  Google Scholar 

  • Nigg BM, Herzog W, Read LJ (1988) Effect of viscoelastic shoe insoles on vertical impact forces in heel-toe running. Am J Sports Med 16:70–76

    Article  Google Scholar 

  • Nishiwaki T (2008) Running shoe sole stiffness evaluation method based on eigen vibration analysis. Sports Technol 1:76–82

    Article  MathSciNet  Google Scholar 

  • Oriwol D, Sterzing T, Milani TL (2011) The position of medial dual density midsole elements in running shoes does not influence biomechanical variables. Footwear Sci 3:107–116

    Article  Google Scholar 

  • Ryu YS, Haririan M, Wu CC, Arora JS (1985) Structural design sensitivity analysis of nonlinear response. Comput Struct 21:245–255

    Article  Google Scholar 

  • Stacoff A, Reinschmidt C, Stüssi E (1992) The movement of the heel within a running shoe. Med Sci Sports Exerc 24:695–701

    Article  Google Scholar 

  • Tanaka M, Noguchi H (2004) Structural shape optimization of hyperelastic material by discrete force method. Theor Appl Mech Japan 53:83–91

    Google Scholar 

  • Tsay JJ, Arora JS (1990) Nonlinear structural design sensitivity analysis for path dependent problems, part1: General theory. Comput Methods Appl Mech Engrg 81:183–208

    Article  MathSciNet  Google Scholar 

  • Vidal CA, Haber RB (1993) Design sensitivity analysis for rate-independent elastoplasticity. Comput Methods Appl Mech Engrg 107:393–431

    Article  MathSciNet  Google Scholar 

  • Vidal CA, Lee HS, Haber RB (1991) The consistent tangent operator for design sensitivity analysis of history-dependent response. Comput Syst Engrg 2:509–523

    Article  Google Scholar 

  • Woensel WV, Cavanagh PR (1992) A perturbation study of lower extremity motion during running. Int J Sport Biomech 8:30–47

    Article  Google Scholar 

  • Yamazaki K, Shibuya K (1998) Sensitivity analysis of nonlinear material and its application to shape optimization. In: Collect Tech Pap AIAA/ASME/ASCE/AHS/ASC Struct Struct Dyn Mater Conf, pp 1706–1713

  • Yuge K, Kikuchi N (1995) Optimization of a frame structure subjected to a plastic deformation. Struct Optim 10:197–208

    Article  Google Scholar 

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Authors and Affiliations

  1. Institute of Sports Science, Asics Corporation, 6-2-1, Takatsukadai, Nishiku, Kobe, 651-2271, Japan

    Mai Nonogawa

  2. Faculty of Engineering and Design, Kagawa University, 1-1, Saiwai-cho, Takamatsu, 760-8521, Japan

    Kenzen Takeuchi

  3. Graduate School of Informatics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8601, Japan

    Hideyuki Azegami

Authors
  1. Mai Nonogawa
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  2. Kenzen Takeuchi
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  3. Hideyuki Azegami
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Correspondence to Mai Nonogawa.

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Responsible Editor: Seonho Cho

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Replication of results

The results described in this paper can be replicated by implementing the formulas and algorithms described in this paper. Regarding the foot pressure distributions and material parameters for the sole model used in the numerical examples, the authors want to withhold its replication for commercial purposes.

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Nonogawa, M., Takeuchi, K. & Azegami, H. Shape optimization of running shoes with desired deformation properties. Struct Multidisc Optim 62, 1535–1546 (2020). https://doi.org/10.1007/s00158-020-02560-0

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  • DOI: https://doi.org/10.1007/s00158-020-02560-0

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