Note that the computation of F* usually takes no more than three iterations if the relative error, ,, is set at . The algorithm converges rapidly during the loading phase simulation, and the initial value of Fi at any given time step should be the F* from the previous | and the final resolution of each variable is final fS where d number of design variables in space or d-dimensional hypercube . Newton-Raphson Search Algorithm The solution of F F which satisfies a function f F 0 can be obtained by a method using the Newton-Raphson algorithm described in Eq. and shown in Fig. . The slope of the curve at point Fi is f Fi . Fi is the value of the independent variable at iteration i. The iteration process continues until Fi and Fi 1 converge to the solution of the function in which f Fi 1 approaches zero. The error function e shown in Eq. can also be used to check for the numerical convergence. _ F - F 1 7 Fi -Fi l . V let f Fj i 0. then F . 1 F . - . 2 f F Fig. Newton-Raphson Iterative Method e I F 1 - F . F . 3 Note that the computation of F usually takes no more than three iterations if the relative error e is set at . The algorithm converges rapidly during the loading phase simulation and the initial value of Fi at any given time step should be the F from the previous time step. 2002 by CRC Press LLC LOADING AND UNLOADING SIMULATION Fig. shows the loading and unloading curves for the struck 1 and striking 2 vehicles. The loading phase is defined by a power curve F kDn and the unloading by a constant slope. In the dynamic simulation during the loading phase an efficient numerical technique using the Newton-Raphson method is used to compute the individual deflections of the two non-linear springs EA and during the unloading phase the relative deformations of the two EA s and the corresponding unloading force are explicitly computed. Fig. Loading and Unloading Phases of Two Vehicles Loading Phase Simulation In general there are two non-linear springs involved in the force-deflection computation. An interactive scheme using the Newton-Raphson algorithm is used. This is described as follows. Given a total deflection of the two springs obtained from the numerical .
