Its Power and Its Problem

A recent LinkedIn post by the creator of FEA Academy raised an important issue about the difference between an algorithm and an integrator. The image from the post is shown below.

In other corners of the FEA world, the algorithm and the integrator are a package deal–the Newton-Raphson method is commonly understood to mean load control.

So, the multiple choice options would throw any OpenSees aficionado for a loop. Newton-Raphson and Modified Newton are algorithms while arc-length is an integrator. Why would you choose an apple when the answer is an orange?

Also, notice that the arc-length image of the LinkedIn post shows a Newton-Raphson algorithm where the tangent stiffness is updated at points 1, 2, and 3 until reaching the solution. The image is conceptual, but the tangent stiffness at point 2 is shown to be close to singular.

To avoid a near singular tangent stiffness with Newton-Raphson, you can use arc-length with Modified Newton, as shown below.

However, you can get stuck with a bad tangent using Modified Newton as well–imagine the load step started very close to the limit point. The safest thing to do is to use Modified Newton with the initial stiffness, but this approach will require many iterations.

A handful of other continuation methods are implemented as static integrators in OpenSees. These integrators can be used with any equilibrium solution algorithm following an incremental-iterative framework.

The power of OpenSees is the number of options you have for the analysis algorithms, integrators, solvers, constraint handlers, etc. The myriad choices is also a problem for OpenSees. Not all options work well together, e.g., residual norm with penalty constraints, so it is important to know the differences between the numerical methods.

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The title of this post came from a comment Dr. Silvia Mazzoni made during an OpenSees Cafe where we discussed this LinkedIn quiz.