Test cantilever beam with shell

[FabienPean-Virtonomy]

Description

The cantilever beam can be found here. In the Euler-Bernoulli formalism, a rigid cross-section orthogonal to the centerline is assumed, so for isotropic linear elastic material, it is closest to the analytical value when Poisson's ratio is 0. In this test case, the maximum expected deflection $\delta=\tfrac{qL^4}{8EI}=7.246$ mm.

Material properties:

• Linear elastic
  • E=21GPa (10x smaller than steel because too slow otherwise)
  • nu=0.49, because it starts to fail somewhere between 0.48 and 0.49
• Density: 7800 (like steel, tested with 780 but does not changing anything)

Results

	Analytical	Abaqus (steel)	SPHinxsys	SPHinxsys		SPHinxsys	SPHinxsys	SPHinxsys	SPHinxsys	SPHinxsys	SPHinxsys	SPHinxsys	SPHinxsys	SPHinxsys	SPHinxsys	SPHinxsys	SPHinxsys
Deflection (mm)	7.25		9.51	8.61	8.18	8.65	8.18	7.96	4.36	7.03	7.86	3.63	6.68	7.49	4.42	7.47	8.04
Error (%)	0	1.17	31.2	18.8	12.9	19.4	12.9	9.8	39.9	3.0	8.5	49.9	7.8	3.4	39.0	3.1	11.0
Runtime (s)			175	822	4369	194	840	4537	175	786	4321	195	835	4520	194	837	4509
Resolution			0.004	0.002	0.001	0.004	0.002	0.001	0.004	0.002	0.001	0.004	0.002	0.001	0.004	0.002	0.001
Cell or Node location			Cell	Cell	Cell	Node	Node	Node	Cell	Cell	Cell	Node	Node	Node	Node	Node	Node
Extending BC			FALSE	FALSE	FALSE	FALSE	FALSE	FALSE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
Exact volume			N.A.	N.A.	N.A.	FALSE	FALSE	FALSE	N.A.	N.A.	N.A.	FALSE	FALSE	FALSE	TRUE	TRUE	TRUE

Discussion

It seems to converge close to analytical value (~3% error at dp=2mm), but it must be confirmed with simulation running at smaller resolution. Running at smaller resolution (dp = thickness/2) does not converge towards analytical.
A main source of error is the boundary condition that must be carefully adjusted in order to have physical location x=0 to be fixed.
While writing this issue, it also occurred to me that there is some shear of the cross-section as the shell formulation follows the Mindlin–Reissner theory, if I understood it correctly. So unless I constrain the pseudo-normal, the closest beam formulation to follow would actually be the Timoshenko beam theory. All things considered it might get close enough.

The pain point is when I run at lower Poisson ratio. Simulation fails already somewhere in-between 0.48 and 0.49. Your help here @DongWuTUM would be appreciated. (Just look at the last commit in the PR, it looks like so much changes because it is based on another PR by Prof. Hu yet to be merged))

[DongWuTUM]

1. When the Poisson ratio is between 0.48-0.49, the simulation crashes. You can try to decrease the time step size slightly. (I have tried to reduce the dt, and the simulation is Okay). This issue is relevant to numerical stability, and we are looking for a method to fix it.
2. Whether the particle is on the boundary or not, the Volume should be dp * dp, which is not the problem of the exact volume and is relevant to the SPH collocation. The derivative operator, for example, is otherwise incorrect. Thus, we suggest the boundary particles are 0.5*dp away from the physical boundary.
3. Seems like the results of tests 1 and 2 are converging as increasing the resolution (decreasing the dp). We also do like the way in these two tests.
4. If you want to extend the fixed boundary, you can try to only constrain the displacement as you said (free the pseudo normal). Otherwise, there may be the hourglass modes.

[DongWuTUM]

Youngs_modulus should be 207e9 / scaling. (N = kg*m/s^2)

[DongWuTUM]

linear_load should be 100.