Linearized elasticity (static problem)- application on a cantilever beam (extnesion on step-8)

[Mohammed Abdelmonem Hassan Mohammed]

Hello everyone,

I am quite new to dealii and FEM field in general, so apologies if my questions seems trivial.

In an effort to understand linearized elasticity, I was comparing FEM solution of step-8 (after changing the mesh to cantilever beam of the below specs) to euler-bernouli static beam deflection ode Euler–Bernoulli beam theory.

Mesh : 

- slender rod of length 0.15 m (length is direction of x )

- square cross-section (0.0025x0.0025) (yz direction in which the origin is the center of the square),

- fixed at x=0, free at x = 0.15.

- mesh resolution in x, y and z is 1000,1,1 respectively.

 I noticed that when I use linear shape functions with an increasing mesh x resolution, the solution seems to approach euler-bernoulli ode solution. however, when I use higher order shape functions, fem solution do not converge as the linear shape function case.

I suspect that maybe it is due to the missing boundary values on the free end (e.g 2nd derivative = 0). how would that apply in the context of linearized elasticity in 3d and dealii framework?

I would also be grateful if I am pointed at the correct direction if this line of thinking is incorrect.

[Wolfgang Bangerth]

>  I noticed that when I use linear shape functions with an increasing mesh x
> resolution, the solution seems to approach euler-bernoulli ode solution.
> however, when I use higher order shape functions, fem solution do not converge
> as the linear shape function case.

This would be concerning. The finite element approximation should converge to
the same solution whether you use linear or quadratic elements. It should just
converge faster with quadratic elements, but the limit point of the
convergence needs to be the same.

If that's not the case, it's worth exploring why that would be so, perhaps
starting with articulating how exactly the two limits differ.

Best
W.

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

[Mohammed Abdelmonem Hassan Mohammed]

Hello Professor Bangerth,

Thanks for the reply. After doing multiple review on the code, I found I had some errors in the assembly. some brackets were misaligned and off so the global contributions were added at each quadrature point!

Now the linear, quadratic and cubic shape functions all approach the same solution as Euler-Bernoulli ode.

Many thanks,

Mohammed

[Wolfgang Bangerth]

On 3/30/24 23:21, Mohammed Abdelmonem Hassan Mohammed wrote:
>
> Now the linear, quadratic and cubic shape functions all approach the same
> solution as Euler-Bernoulli ode.

Great, that's good news!