Integrating with curved manifolds

[Alex Cumberworth]

I am working with a mesh generated from subdivided_cylinder, and am wondering whether the area and volume should be assumed to match that of a true cylinder, or if it only converges to that when many refinements are made. I would like to calculate the total energy of the system by integrating over the whole volume, as well as calculate shear forces by integrating over faces on the surface.

I am trying to compare the results of the FEM code to that of Euler-Bernoulli beam theory, similarly to how was done in this question. Basically, I use Dirichlet conditions and hold one end in place, while the other end is displaced by small amount in the positive y direction. In this case, I use a radius of 1, a total length of 20, and a displacement of 0.1. When I ran this calculation with a regular flat manifold (i.e., a standard rectangular cuboid beam), I found that I only needed to refine along the beam axis (x) in order to converge to the analytical results from beam theory. However, here I am finding that no matter how much I refine along the beam axis, the result does not converge, and in fact barely changes at all. When I refine in all directions, then the energy and shear force increase, although they then overshoot the analytical result.

To be clear, for calculating the beam theory result, I used the same expression as for the rectangular cuboid beam, but with a different second moment of area, and this was in agreements with the FEM calculations with the rectangular cuboid beam. I am using quadratic elements.

Any thoughts would be greatly appreciated,

Alex

[Wolfgang Bangerth]

On 6/24/21 8:50 AM, Alex Cumberworth wrote:
> I am working with a mesh generated from subdivided_cylinder, and am wondering
> whether the area and volume should be assumed to match that of a true
> cylinder, or if it only converges to that when many refinements are made. I
> would like to calculate the total energy of the system by integrating over the
> whole volume, as well as calculate shear forces by integrating over faces on
> the surface.

That depends on whether you use a mapping argument in the integration or not.
If not, then you are implicitly using a Q1 mapping for all cells, which is not
going to provide a particularly accuracte approximation of volume and area. On
the other hand, if you use a MappingQ(4) or even better MappingManifold, and
if you use a sufficiently high order quadrature formula, then you should be
very close to the exact volume/area already on a fairly coarse mesh.

In no case will things be exact: Even with MappingManifold, you are still
computing integrals that you approximate via quadrature.

I have no opinion on what might be wrong with your beam example :-)

Cheers
W.

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Wolfgang Bangerth email: bang...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/