The ANM method was designed for solving PDE problems with one parameter in structure or in fluid mechanics [1],[2].
After discretization of these problems, usually with finite elements, one has to deal with a nonlinear algebraic system defined by F: IRn+1 -> IRn where 0=F(x)=F0 +Lx+Q(x,x) with x=(u,p), p the scalar parameter; the operator L is linear and Q is quadratic. In fact, any problem which can be brought in this form can be solved with the ANM.
The principle of the method is based on analytic perturbation. One looks for a solution of the form u(p)=u0 + u1p + u2p2 + ... + uNpN and, after comparing the powers of p in the equation F(u,p)=0, we get a sequence of N linear systems, all having the same matrix built from the Jacobian of F. Of course, for dealing with limit points (folds or bifurcations), one has to introduce another parameter in place of p, as for example the pseudo-arclength [4].
The interesting feature of the method is that the expansion of u is valid in a whole interval which does not need to be small [3], contrarly to the standard pseudo-arclength medthod [4]. Thus, proceeding by successive intervals, we get a (continuous) piecewise polynomial approximation of a solution of the problem . The usual criterion for determining the interval length requires that the first nonzero term of the residual is smaller than a given (small) number; one observes that, while using this criterion, very small steps could be needed, especially close to a bifurcation.
Some numerical experiments on classic examples [5] have shown the robustness of the ANM for computing isolated bifurcation branches. However, we observed switching from one branch to another one close to a bifurcation, and this even for a problem with n=2. The detection of bifurcations, as presented in some papers did not convince ourselves. We try to use the sign of the augmented Jacobian, with some success.
The ANM with the residuum criterion is not able to compute trivial branches (for which u(p)=u0, a constant) and to detect bifurcations on them. We set up a simple method, to solve this problem; based on the special "quadratic form" of the algebraic system, bifurcation on a trivial branch are given by solving a generalized eigenvalue problem with the inverse power method. For locating the bifurcations on nontrivial branches, we used either the sign of the augmented Jacobian of F or the "asymptotic" radius of convergence of the serie given in [3]. We were able to solve for the clamped 2D rod problem presented in [6]; a part of the bifurcation diagram is shown below, where we made a comparison to the solution obtained with the code AUTO [4].
In the light of our numerical experiments, we want to study the following points:
Establish a robust criterion for bifurcation detections
Estimate the error due to the truncation of the serie for u
Determine a steplength based on point 1. and 2.
Design an algorithm for an (almost) automatic computation of a priori unknown bifurcation diagrams
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May 29, 2004