Introduction to Nonlinear Aeroelasticity

7Numerical Continuation

7.1 Introduction

Numerical continuation is a collection of methodologies for solving nonlinear algebraic problems. It first became formalised in the 1980s (Allgower and Georg 1990) but has really come of age since the 2000s when software packages such as AUTO (Doedel et al. 2000) and Matcont (Govaerts 2000a; Govaerts et al. 2005) became popular. This chapter presents an introduction to numerical continuation and discusses the intricacies of its application to aeroelasticity, but does not aim to cover all the possible methodologies and algorithms; the reader can consult the specialised literature for a complete overview.

The problem statement is the following:

• Most nonlinear algebraic equations do not have general solutions.
• Nonlinear problems have many solutions that have to be determined numerically.
• Most numerical solution techniques are iterative; they start at an initial guess and finish near the true solution.
• Initial guesses that are close enough to the true solutions are required.
• Many problems in science and engineering depend on system parameters.
• Even if all the solutions have been determined at one value of the system parameters, they are unknown for the other values.

Numerical continuation addresses some of these issues by searching solutions that are neighbours in parameter space. In Section 3.9.1 we showed how to use the Newton–Raphson technique to solve algebraic equations of the form . Here we will address a parameter‐dependent problem. ...