Shape optimization of running shoes with desired deformation properties
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INDUSTRIAL APPLICATION PAPER
Shape optimization of running shoes with desired deformation properties Mai Nonogawa1 · Kenzen Takeuchi2 · Hideyuki Azegami3 Received: 31 October 2019 / Revised: 26 February 2020 / Accepted: 27 February 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
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 H 1 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. Keywords Running shoes · Hyper-elastic material · Shape optimization · Boundary value problem · H 1 gradient method
1 Introduction Running is one of the most popular sports in the world. However, it has been reported that the risk of injuries to lower extremities is greater than other sports, because repeated loads are applied to lower extremities during running (Matheson et al. 1987). For this reason, running shoes should not only enhance the runner’s performance but also prevent injuries during running.
Responsible Editor: Seonho Cho Mai Nonogawa
[email protected] 1
Institute of Sports Science, Asics Corporation, 6-2-1, Takatsukadai, Nishiku, Kobe 651-2271, Japan
2
Faculty of Engineering and Design, Kagawa University, 1-1, Saiwai-cho, Takamatsu, 760-8521, Japan
3
Graduate School of Informatics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8601, Japan
Shoes have various requirement properties, such as cushioning property, shoe stability, grip property, and breathability (Cavanagh 1980; Nishiwaki 2008). Especially in the procedure designing sole of shoes, we often focus on the cushioning property and the shoe stability. The cushioning property means absorption of the impact from the ground at the contact phase. It is evaluated using time derivative of vertical ground reaction force (for instance Gard and Konz (200
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