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Converging to Something

Original Post - 08 Oct 2023 - Michael H. Scott

Visit Structural Analysis Is Simple on Substack.

Is it better to have converged and lost than never to have converged at all?

The displacement-based and force-based frame elements are both distributed plasticity formulations–just one is way better at simulating the spread of plasticity than the other.

Despite this fairly well known fact, I still see people use four, five, six, or more integration points per displacement-based element, believing the computed solution will improve with more integration points.

Consider a simple beam with a bilinear moment-curvature relationship, which, without loss of generality, is a simple proxy for whatever you plan to do with fiber-discretized sections.

The exact, closed-form moment-rotation (\(M-\theta\)) response is easily obtained using the principle of virtual forces–unit load, trapezoids, graphical integration, etc.

Then, a single displacement-based element with two, four, and six Gauss-Legendre integration points and a single force-based element with three, four, and six Gauss-Lobatto points give the computed moment-rotation response shown below.

With a single displacement-based element, the computed response is stronger and stiffer than the exact solution. On the other hand, a single force-based element gives the correct strength, but is generally more flexible than the exact solution.

While the single force-based element converges to the exact solution from below, the single displacement-based element converges to a wrong solution from above.

Yeah, converging to something is better than never converging at all.

But your results won’t really get any better using more than two integration points in displacement-based elements. The results will be more smooth and precise, just not accurate.

If you want to improve the computed solution with the displacement-based formulation, use more elements, i.e., refine the mesh, instead.