Optimization of slit density distribution in shoe upper made of warp knitting to optimize the contact pressure

A method for finding the optimum distribution of the slit density in a shoe upper, such that the contact pressure with the foot approaches an ideal distribution is presented in this paper. In a shoe upper made of warp knitting, slits are made by discontinuously knitting the weft under a uniform arrangement of the warp fiber. In this study, the shoe upper is modeled as an orthotropic hyperelastic body in which the slit density is defined as the design variable for controlling the intermediate density for the maximum and minimum slit densities. Referring to a formulation of a topology optimization problem of the density variation type, an inverse problem for finding the optimum distribution of the slit density is formulated with the objective and constraint cost functions defined by the squared error norm between the actual and ideal contact pressures and the deviation of the rate of the material with the maximum slit density from a limit value, respectively. This problem is solved in the same way as a topology optimization problem. The validity of this method is confirmed by numerical examples using a simple finite element model, which resembles the domain around the big toe.