Beam model

[WeiyiVirtonomy]

I haven't transplanted the code to SPHinXsys yet, but you could find the documentation and some test results in the markdown file.

I got stuck with this formulation. I wanted to reproduce the test case "4.3 out-of-plane bending of the cantilever to form a helix due to a concentrated moment and an out-of-plane force" in this reference: test. However, as observed in the pure bending case, the beam goes crazy after rotating for a certain degree. Unlike the 2d case where we can constrain the motions to in-plane, this is a 3d case.

It is also weird that the 2d pure bending case can only get a solution close to the reference for certain material parameters. The rotational angle is only related to $EI$ in this case, so as long as $EI$ is constant, the beam should be bent for a cycle.

With $GA=5000$ and $EI=100$ and $\rho=1$, the result agrees well with the analytical solution when the in-plane constraint is applied.
However, when we reduce $GA$ to 500, the simulation fails immediately. I tried to reduce the time step size by using a smaller density, but it didn't help.

With $GA=5e5$ and $EI=100$ and density increased to $1e6$, there is only a very small rotation:

By reducing the physical damping, it gets a larger deformation, but still much smaller than the analytical solution:

The default physical damping is calculated by the empirical formula $0.1 \sqrt{\rho * E} * r$.

[WeiyiVirtonomy]

@Xiangyu-Hu I have pushed the code based on the shell algorithm under tests/3d_examples/test_3d_SPHinXsys_beam

• Numerical damping
  When there is no physical damping, with a moment of 2.5 N.m, the numerical damping does help to stabilize. The left side is the one with numerical damping, while the right one is without damping. When the right one breaks, the left one is still stable.

• Instability in the pure bending
  See the video. The result can be reproduced by running pure_bending_test
  https://github.com/user-attachments/assets/e001f0e4-0b61-4371-83ab-fc8dd4798897

[Xiangyu-Hu]

@WeiyiVirtonomy have you tried the same problem but with (much) high resolution?

[WeiyiVirtonomy]

@WeiyiVirtonomy have you tried the same problem but with (much) high resolution?

I increased the resolution by 4 times, but it didn't help

[Xiangyu-Hu]

@WeiyiVirtonomy have you tried the same problem but with (much) high resolution?

I increased the resolution by 4 times, but it didn't help

Thanks for the information. This confirms that the issue is not due to resolution.

[Xiangyu-Hu]

@WeiyiVirtonomy seems issue is here https://doi.org/10.1016/S0045-7825(97)00059-5.

[Xiangyu-Hu]

@WeiyiVirtonomy https://doi.org/10.1016/S0045-7825(98)00249-7 a paper on dynamical simulation.

[Xiangyu-Hu]

@WeiyiVirtonomy what is the difference for v2 version of simo-reissner formulation?

[Xiangyu-Hu]

@WeiyiVirtonomy seems a bug in tangential operator.

[Xiangyu-Hu]

@WeiyiVirtonomy Also have you tries to slowly increase the load as in the references? https://doi.org/10.1016/j.ijsolstr.2024.112746

[WeiyiVirtonomy]

@WeiyiVirtonomy what is the difference for v2 version of simo-reissner formulation?

V2 is based on this paper: Explicit symplectic momentum-conserving time-stepping scheme for the dynamics of geometrically exact rods. They used a Verlet scheme based on position, and solved the rotation increment based on Newton-Raphson iteration, instead of obtaining the angular acceleration from the equation.
It didn't help to solve the instability.

[WeiyiVirtonomy]

@WeiyiVirtonomy Also have you tries to slowly increase the load as in the references? https://doi.org/10.1016/j.ijsolstr.2024.112746

I have tried to increase the load step by step, and only apply the next level of load when the beam has reached the steady state with the physical damping, but it didn't help

[WeiyiVirtonomy]

@WeiyiVirtonomy seems a bug in tangential operator.

Thanks, I'll try this!

[WeiyiVirtonomy]

@WeiyiVirtonomy seems a bug in tangential operator.

I ran the test again with the fixed T operator, but unfortunately, it still has the same problem.

[Xiangyu-Hu]

@WeiyiVirtonomy seems a bug in tangential operator.

I ran the test again with the fixed T operator, but unfortunately, it still has the same problem.

After I checked the code, I did not notice anything wrong either.
I think that we may use quaternion to handle rotation update.

[Xiangyu-Hu]

@WeiyiVirtonomy Also have you tries to slowly increase the load as in the references? https://doi.org/10.1016/j.ijsolstr.2024.112746

I have tried to increase the load step by step, and only apply the next level of load when the beam has reached the steady state with the physical damping, but it didn't help

Could it add full force load but increment torque?

[WeiyiVirtonomy]

@Xiangyu-Hu [Total and Updated Lagrangian Geometrically Exact Beam Elements](DOI:10.1007/1-4020-5370-3_658