Software-Engineering Optimization with Mathematical Method Improvements for Numerical Reuleaux Method

Abstract

This paper presents an extension both in software optimization with simulations and detailed mathematical theory of Numerical Reuleaux Method based on previous publications. In the literature, there are a number of papers in Numerical Reuleaux Method and its applications (Aerospace, Helicopter Dynamics in Turbulence Conditions, Biomedical Engineering, Biomechanics, etc) since 2007. This contribution is a detailed presentation of the mathematical framework that constituted the basis for those articles along 2007-2020. The Classical Reuleaux Method (<strong>CRM</strong>) is frequently used in Physical Dynamics, Engineering Mechanics and Bioengineering to determine the Instantaneous Rotation Center (<strong>IRC</strong>) of a rigid body in arbitrary movement. The generic mathematical <strong>CRM </strong>only can be applied on rigid bodies, whose shape remains constant during the movement. If the solid in movement is a Pseudo-Rigid Body (<strong>PRB</strong>), the <strong>CRM </strong>has to be modified numerically to conform the shape changes and adapt on the density distribution variations of the <strong>PRB </strong>(we denominate it, in this case<sup>1</sup>, <strong>The Numerical Reuleaux Method</strong>, <strong>NRM</strong>). This Geometrical-Numerical Approximation Method is based on the division of the Pseudo-rigid body into small volume parts called voxels (roughly speaking parallelepipedic), namely, voxelization of the body subject to dynamics. The theoretical basis of the method is strictly shown in complementary details, with the necessary Theorems and Propositions of the model. Nonlinear Optimization Techniques that support the initial theory have been developed, and the Error boundaries with Error reduction techniques are determined. Computational Simulations have been carried out to prove the <strong>NRM </strong>Theoretical Model feasibility and numerical veracity of the Propositions, Theorems, and Error Boundaries. Appropriate software was made to carry out these simulations conveniently. The initial results agree to the theoretical calculations, and the IRC computation for 2 voxels shows to be simple and easy. Some initial guidelines for a theoretical development of this algorithm applied on large pseudo-rigid bodies, by using Monte-Carlo techniques, are sketched. Recent applications, Aerospace and Biomechanics, are also shown.