Numerical Computation for Nonlinear Problems

Abstract

The overwhelming feeling you get from the preceding chapter is that the only thing you can really hope to do with a complex nonlinear system is to compute its critical loads and the corresponding modes with the linearized buckling theory. The examples we have seen have clearly demonstrated that nonlinear systems do not have to be very complicated before we find ourselves unable to find a closed-form solution to the problem of finding the equilibrium paths. Even for some rather modest one-dimensional problems, the possibility of finding a closed-form solution is a dismal prospect. Often, even if we do find a closed-form solution, it is so complicated that the only way to appreciate it is to evaluate the expression at a number of discrete points and plot the bifurcation diagram by connecting those points. There is little motivation for executing monumental feats of algebra if there is an alternative means of generating the discrete points along the path. An incremental nimierical solution method provides such a tool.