Measure of error in smoothness of approximation of sphere

big-listgeometric-measure-theorygeometrymeshingparameter estimation

I'm meshing a sphere and am solving a physics problem on this. What I want to show is that the error in the model scales like$$
\varepsilon = \epsilon^p,
$$
where $\epsilon$ is the "error" in the approximation of the smoothness of the sphere, but I was wondering how to define this $\epsilon$. The mesh itself is made out of piecewise linear elements, so for example a coarse mesh would be a cube, while a fine one would be more like a disco-ball.

At first I thought of relating this error to the curvature of the mesh, but the coarse mesh has the same curvature as the disco-ball (infinite in the elements, undefined on the mesh interfaces), so this takes me nowhere. Still, the disco-ball is smoother and $\epsilon$ should be smaller.

Another thing I could think of is measuring the difference in volume or surface area, but this also does not help since the mesh nodes are not perfectly distributed on the surface of the sphere, i.e. for a cube with side length $L=\sqrt{\frac{2}{3}\pi}\ R$ would have $\epsilon=0$, while the cube itself does not resemble the sphere.

A third idea (which is quite nice), is a norm of the form $$
\epsilon^2 = \int\left|p(\textbf{x})-p^h(\textbf x)\right|^2\mathrm dA
$$
where $p$ is a parametrization of the surface of the sphere, and $p^h$ the parametrization of the mesh, but I'm not sure if this norm is a good norm. Say the approximation is a perfect sphere, but the center is slightly shifted, the error would be large, while for my model it would be $0$. The norm is a good measure, but not a fool-proof one.

$\Large{TL;DR}:$

My question is, do you have an idea of an error estimator which penalizes not the exact approximation of a mesh, but the sharpness of the angles at the element interfaces? I'm not looking for $\textit{the}$ answer, but looking for a collection of possible error measures.

Best Answer

Here are a few possible error estimators:

  1. (surface area to volume ratio of polyhedron) minus (surface area to volume ratio of sphere)

  2. (ratio of the diameter of the largest sphere that can be inscribed in the meshed polyhedron to the diameter of the smallest sphere that contains the meshed polyhedron) minus one

  3. $2\pi$ minus (the solid angle in steradians of the vertex of the polyhedron whose solid angle is smallest)

(1) has the downside that adding a long, sharp spike to the polyhedron which is too narrow to significantly affect its surface area or its volume may make it look less sphere-like without much affecting its surface area to volume ratio.

(2) is the most difficult to calculate

(3) has the downside that it would rank a cube and an elongated cuboid as equally sphere-like

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