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Newton methods

ABAQUS - Does Newton's method affect the convergence rate of higher order incremental scheme such as RK4 implemented via UMAT?

Hello everyone,

I'm currently doing an analysis on the convergence rate of an incremental model that I have implemented in ABAQUS using a UMAT. Here's the BIG lines of the problem:

1) In UMAT : Solution of a differential equation using explicit Runge-Kutta 4 (RK4), which has a 4th order convergence rate, yields the stress state "S". (In my problem, stresses are applied)

Solving nonlinear equilibrium problem

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I am trying to solve (for equilibrium points) a geometrically nonlinear problem of the form

f(u) = F

where u is the displacement and F is the load. At first I tried the plain

Newton method, with, perhaps obviously, limited success. I then tried to use

linesearch and trustregion strategies to improve the situation. Eventually, however,

I realized that I needed to use some form of continuation method. Though the most

commonly used methods in mechanics seems to be based on gradually increasing the

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