On the Interrelation between Stability and Computations

Abstract

This chapter focuses on the interrelation between stability and computations. A common method of computing results for nonlinear elasticity problems is to reduce them to an approximate set of nonlinear algebraic problems. The nonlinear algebraic problems are often solved by Newton's method. The resulting sequence of linearized problems can then be solved. The first iterations in Newton's method can be carried out by one approximate method and later iterations for small corrections can use another. On the other hand, problems with bifurcation points and limit points reduce to linear algebraic equations with singular matrices and Newton's method becomes slowly convergent unless a numerical device can be found to avoid zero determinants. For autonomous systems, truncated Fourier series solutions fail to give numerical results unless the frequency is known. The analysis shows the way to correct the approximate frequency in a simple fashion that probably could not be deduced from a purely numerical approach. The methods used for nonlinear buckling problems then carry over to nonlinear vibration problems much as they do for linear problems. Newton's method suggests an approximate solution that may check experiment.