# Geometrically nonlinear finite element problem and mesh distortion

In my research on topology optimization of fluid-structure interaction problems (2D), I am using a geometrically nonlinear model to represent the structure. Body/surface forces are extracted from the fluid solver and applied to the structure under plane strain conditions.

I am facing a problem with severe, yet localized, mesh distortions that is causing a particular element to be inverted and hence failure of convergence.

To illustrate the problem at hand, consider a simple structural problem where a 2m (width) x 1m (height) 2D rectangle is fixed at the bottom edge and loaded at the right edge with a traction force in the positive $$x$$ direction. For a traction force per unit area of 5.1e3 N/m^2, the element on the bottom right of the domain is severely distorted and on the verge of inversion (see below a screenshot of the deformed mesh zoomed in on the bottom right corner). The attached results are for quadratic Lagrange elements solved on COMSOL with a quadratic Lagrange geometric order and of course geometric nonlinearity enabled.

I noticed that the severe distortion is localized to this single element and everywhere else the mesh is relatively fine. In addition, I calculated the $$l^2$$ error estimates in COMSOL and they are fine everywhere except this particular element. Refining the mesh reduces the $$l^2$$ error but makes the problem even closer to non-convergence due to the softening effect of mesh refinement. Also, using bilinear elements have a similar result, given a fine enough mesh density.

My concerns are:

1. I am not sure how to characterize this phenomenon, does it fall under a certain category of numerical issues in finite element analysis? I am not sure what to search for to find more information/potential solutions for it.

2. Of the relatively easy solutions I could think of, is stiffening this particular finite element somehow. Given a decent mesh density, the effect on the overall result should not be significant. Is this approach a horrible desecration to the art of finite elements or is this acceptable given the localization of the mesh distortion?

• If that element distortion is due to a physical singularity, I think no amount of mesh refinement will help and infact might make the problem worse. Change the geometry so that there isn't a sharp corner.
– NNN
Commented May 18, 2023 at 11:35

1. In FEA, the basis functions are defined over a reference element, here a unit square. This is a "function basis factory" of sorts, as you then write a global basis restricted to a given element $$K$$ as that generic basis composed by $$F_K^{-1}$$, where $$F_K$$ is the element mapping of $$K$$.
In practical terms, you need to ensure that the Jacobian determinant of each element in the mesh remains positive, and preferably above a threshold (say $$0.1$$). This threshold may be of importance as perhaps your mesh is geometrically valid (determinant $$>0$$ everywhere), but barely so, in the sense that the linear solver they use is behaving poorly because of practically vanishing integrals in the mass matrix (if you're unlucky enough for a minimum of the determinant to be at a quadrature point) or such.