Towards Customized Footwear with Improved Comfort

A methodology enabling the customization of shoes for comfort improvement is proposed and assessed. For this aim, 3D printed graded density inserts were placed in one of the critical plantar pressure zones of conventional insoles, the heel. A semi-automated routine was developed to design the 3D inserts ready for printing, which comprises three main stages: (i) the definition of the number of areas with different mesh density, (ii) the generation of 2D components with continuous graded mesh density, and (iii) the generation of a 3D component having the same 2D base mesh. The adequacy of the mesh densities used in the inserts was previously assessed through compression tests, using uniform mesh density samples. Slippers with different pairs of inserts embedded in their insoles were mechanically characterized, and their comfort was qualitatively assessed by a panel of users. All users found a particular pair, or a set, of prototype slippers more comfortable than the original ones, taken as reference, but their preferences were not consensual. This emphasizes the need for shoe customization, and the usefulness of the proposed methodology to achieve such a goal.

A high-resolution method based on off-step non-polynomial spline approximations for the solution of Burgers-Fisher and coupled nonlinear Burgers’ equations

Purpose This paper aims to develop a new high accuracy numerical method based on off-step non-polynomial spline in tension approximations for the solution of Burgers-Fisher and coupled nonlinear Burgers’ equations on a graded mesh. The spline method reported here is third order accurate in space and second order accurate in time. The proposed spline method involves only two off-step points and a central point on a graded mesh. The method is two-level implicit in nature and directly derived from the continuity condition of the first order space derivative of the non-polynomial tension spline function. The linear stability analysis of the proposed method has been examined and it is shown that the proposed two-level method is unconditionally stable for a linear model problem. The method is directly applicable to problems in polar systems. To demonstrate the strength and utility of the proposed method, the authors have solved the generalized Burgers-Huxley equation, generalized Burgers-Fisher equation, coupled Burgers-equations and parabolic equation in polar coordinates. The authors show that the proposed method enables us to obtain the high accurate solution for high Reynolds number. Design/methodology/approach In this method, the authors use only two-level in time-direction, and at each time-level, the authors use three grid points for the unknown function u(x,t) and two off-step points for the known variable x in spatial direction. The methodology followed in this paper is the construction of a non-polynomial spline function and using its continuity properties to obtain consistency condition, which is third order accurate on a graded mesh and fourth order accurate on a uniform mesh. From this consistency condition, the authors derive the proposed numerical method. The proposed method, when applied to a linear equation is shown to be unconditionally stable. To assess the validity and accuracy, the method is applied to solve several benchmark problems, and numerical results are provided to demonstrate the usefulness of the proposed method. Findings The paper provides a third order numerical scheme on a graded mesh and fourth order spline method on a uniform mesh obtained directly from the consistency condition. In earlier methods, consistency conditions were only second order accurate. This brings an edge over other past methods. Also, the method is directly applicable to physical problems involving singular coefficients. So no modification in the method is required at singular points. This saves CPU time and computational costs. Research limitations/implications There are no limitations. Obtaining a high accuracy spline method directly from the consistency condition is a new work. Also being an implicit method, this method is unconditionally stable. Practical implications Physical problems with singular and non-singular coefficients are directly solved by this method. Originality/value The paper develops a new method based on non-polynomial spline approximations of order two in time and three (four) in space, which is original and has lot of value because many benchmark problems of physical significance are solved in this method.