Computation of Critical Boundaries on Equilibrium Manifolds

The study of various equilibrium phenomena leads to nonlinear equations (1) $F(y,u) = 0$, where y ∈ Rⁿ is a vector of behavior or state variables, u ∈ Rᵖ a vector of p 2 parameters or controls and F:D ⊂ Rⁿ × Rᵖ → Rⁿ a sufficiently differentiable map. The solution set of (1) in Rⁿ × Rᵖ is often called the equilibrium surface of the problem, and in many applications it is of interest to determine the critical boundary, that is, the set of solutions of (1) where the derivative Dy F(x) is singular. For example, in structural problems these points may represent buckling points. After characterizing the various properties of the problem, we present three different numerical methods which allow for a computational trace of paths in the critical boundary. These methods represent extensions of the earlier-developed locally parametrized continuation method for tracing regular paths on the equilibrium surface. They permit, for instance, a direct computational determination of the change of the buckling point of a structure under changes of the load regime. Numerical examples involving the roll stability of an airplane and the buckling of a circular arch are given which indicate the efficiency of the approach.