[Study] Multi time resolution

[WeiyiVirtonomy]

A 2d test case of multi-time resolution contact problem

Description

The left brick represents a catheter with higher stiffness, while the right brick represents the softer arterial wall. The catheter and the aorta have contact interaction.

By using single time step size, the configuration is:

By using a dual loop, the aorta particles are strongly oscillating:

[WeiyiVirtonomy]

@Xiangyu-Hu I have pushed a 2D test case for multi-time-resolution problems. I would appreciate it if we could schedule some time to discuss how to deal with this problem.

[WeiyiVirtonomy]

@DongWuTUM Could you explain the left Cauchy damping to me again? Is the damping stress elastic_solid_.NumericalDampingLeftCauchy(...) * inverse_F.transpose()?

[DongWuTUM]

@DongWuTUM Could you explain the left Cauchy damping to me again? Is the damping stress elastic_solid_.NumericalDampingLeftCauchy(...) * inverse_F.transpose()?

If you use the Green-Lagrangian strain tensor $\mathbb{E}$ to calculate stress, then you should use the 'NumericalDampingRightCauchy(...)' which can be added directly to the second Piola-Kirchhoff stress $\mathbb{S}$.

If you use left Cauchy-Green deformation gradient tensor $\mathbb{b}$, then you should use the 'NumericalDampingLeftCauchy(...)' which can be added directly to the Kirchhoff stress $\mathbb{\tau}$.

[WeiyiVirtonomy]

@DongWuTUM Could you explain the left Cauchy damping to me again? Is the damping stress elastic_solid_.NumericalDampingLeftCauchy(...) * inverse_F.transpose()?

If you use the Green-Lagrangian strain tensor E to calculate stress, then you should use the 'NumericalDampingRightCauchy(...)' which can be added directly to the second Piola-Kirchhoff stress S .

If you use left Cauchy-Green deformation gradient tensor b , then you should use the 'NumericalDampingLeftCauchy(...)' which can be added directly to the Kirchhoff stress τ .

Thanks a lot!

[WeiyiVirtonomy]

@DongWuTUM @Xiangyu-Hu The dual loop for contact problem works after we modify the contact force formulation by using the harmonic average of the contact stiffness.

Test: a stiff catheter (represented by a rectangular with a round tip) moves upward and collides with the soft aorta (represented by a quarter circle).
Sound speed ratio between two materials: 191

Left: dual loop, the contact force acting on the stiff material is assumed to be constant during the inner loop
Right: single-time-stepping

Single-time stepping computation time: 134s
Dual-time stepping computation time: 40s

The instability observed before was caused by the fact that we were using $p^{*} = \sigma_i K_i + \sigma_j K_j $ to compute contact force. As the square of sound speed of the stiff material is included in this formula, it would be too large for the soft material acoustic time step size.

By using the harmonic average, the smaller contact stiffness dominates and reduces the instability.

I can go on to clean up this test case if you think it would be good to include it in SPHinXsys

[Xiangyu-Hu]

Sure. Please go ahead.