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Define boundary condition between PDE and Solid Mechanics (Elastic wave)
Posted 11 dic 2020, 11:09 GMT-5 Structural & Acoustics, Acoustics & Vibrations Version 5.5 10 Replies
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I have two adjacent elastic wave regions respectively solved by self-defined Weak Form PDE and Solid Mechanics (Elastic Wave) Module. The details are shown in the attached image. My question is: How to define the continuity condition on the boundary to avoid reflection of incident wave .
The Solid Mechanics is used for the solution of elastic wave with implicit solver.
The PDE is used for the Perfect Match Layer in time-domain for implicit solver.
Both regions have their own dependent variables of displacement (u,v) and (u2, v2). The PDE has 4 more variables in (Pa * s) coupled with traditional wave equation. The reference can be found here: reference
Insdead of PDE+Solid, I have achieved no reflection with two Solid Mechanics Modules just using displacement contiuity (u=u2, v=v2) but this is not enough in the PDE+Solid version.
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Just so I understand, you want to define your own perfectly matched layer because the built-in one isn't good enough?
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Just so I understand, you want to define your own perfectly matched layer because the built-in one isn't good enough?
Yes, the built-in one for elastic wave module is either for frequency domain or explicity solver. The time-domain PML seems to work only for acoustic module. For elastic wave module I got the same results with or without the PML. It was actually the Low-refecltion boundary that worked and the infinite layer also somewhat benefited with its streching effect but no aborbing effect.
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Just so I understand, you want to define your own perfectly matched layer because the built-in one isn't good enough?
The frequency PML I also tried, but found that through built-in freq2time FFT, the signal response (v(freq) -> v(t)) was not right while the loading (force(freq) -> force(t)) was correctly reconstructed. If you are interested, I can also share this problem with you.
The explicit absorber layer works well, but the explicity solver cannot support multiphysics coupling especially when heat-elastic wave or thermo-acoustic coupling is needed. It does neither support nolinear geometry or materials for pre-stress analysis.
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OK I see the issue. I can't help you with implementing a different PML. If you haven't done it already you might try optimizing the low-reflecting boundary by using a non-isotropic wave velocity. I have done this in the past to (mostly) absorb both transverse and longitudinal waves.
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OK I see the issue. I can't help you with implementing a different PML. If you haven't done it already you might try optimizing the low-reflecting boundary by using a non-isotropic wave velocity. I have done this in the past to (mostly) absorb both transverse and longitudinal waves.
Yeah, I haven't succeeded by now in finding the proper formula to combine the regions. Low-reflection is always a much easier solution and I used it a lot after the explicit absorber is implemented in COMSOL 5.5. Could you share some expieriences and guidance about adjusting the isotropic boundary condition? I have beening using it by the default average of P and S wave impedance. Shall I add more weight on the shear wave part when it is to solve guided wave (lamb or rayleigh wave) problem?
I heard that the time-domain PML used here was from an invited talk on COMSOL confrerence and had been implemented (or something similar) in the acoustic module. I am not sure why it was not in the Solid mechanics module but it is indeed not so generalized based on small-deformation elasticity presumption. I hope it could be accessed in future, at least in linear elastic wave problem simulation.
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Hi Ernest,
The easiest way to set up the continuity is to give the displacement variables in the Solid Mechanics and the PDE interfaces the same name. Otherwise, you can impose two Constraint boundary conditions on the interface boundary in the PDE interface with the expressions "u - u2" and "v - v2". Either way should do the trick.
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y
. Could you share some expieriences and guidance about adjusting the isotropic boundary condition?
A simple custom boundary condition works best when longintudinal and transverse waves are incident on a boundary. Then you can write an expression for a boundary force which is proportional to velocity (basically a termination in the acoustic impedance). Since the boundary force has vector components, you can use a different proportionality factor for normal and tangential velocity. This would perfectly absorb normally incident longitudinal and transverse waves.
For Lamb waves it's not so simple, because Lamb waves can be thought of as a linear combination of longitudinal and transverse waves that are reflected (at an angle) from top and bottom boundaries. It may be possible to find proportionality factors that reflect less than the default factor (an average of cs and cp).
I have generally either (1) used the default PML or (2) used an approximate impedance boundary condition. In the second case the nonzero reflected waves are not very large and in any case they are usually identifiable by the delay time.
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Hi Ernest,
The easiest way to set up the continuity is to give the displacement variables in the Solid Mechanics and the PDE interfaces the same name. Otherwise, you can impose two Constraint boundary conditions on the interface boundary in the PDE interface with the expressions "u - u2" and "v - v2". Either way should do the trick.
Hi, Kiril,
The way you mentioned applies only when PDE (or Solid Module #2) has the same formulation as the Solid Module. However, my PDE is for a self-defined PML, which does not only have displacement as dependent variable but also 4 more stress-related variables (in terms of time integration of stress sx,sxy and sy for 2D). I tried only using u-u2 and v-v2 constraint but it didn't work.
Ernest
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y
. Could you share some expieriences and guidance about adjusting the isotropic boundary condition?
A simple custom boundary condition works best when longintudinal and transverse waves are incident on a boundary. Then you can write an expression for a boundary force which is proportional to velocity (basically a termination in the acoustic impedance). Since the boundary force has vector components, you can use a different proportionality factor for normal and tangential velocity. This would perfectly absorb normally incident longitudinal and transverse waves.
For Lamb waves it's not so simple, because Lamb waves can be thought of as a linear combination of longitudinal and transverse waves that are reflected (at an angle) from top and bottom boundaries. It may be possible to find proportionality factors that reflect less than the default factor (an average of cs and cp).
I have generally either (1) used the default PML or (2) used an approximate impedance boundary condition. In the second case the nonzero reflected waves are not very large and in any case they are usually identifiable by the delay time.
Hi, Dave,
Thanks for your guidance.
I kept failing these days trying different continuity condtions as mentioned.
In fact, the PDE is for lamb wave simulation just because the low-reflection boundary did not work that well as you said. The reflected wave has not enough time delay to screen out. I tried different weighing factors to combine cs and cp but got tired of adjusting it case by case, especially when using parametric sweep.
I guess your case uses Acoustic Module more often where the default PML works well, but mine only needs the Solid Module. Now I guess the best solution is still using explicit solver with absorbing layer. Really hope Comsol can either develop a PML for its implicit solver or enable the explicit solver to support more multiphysics couplings other than acoustic-structure coupling.
Ernest
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Hi Ernest,
I recommend you to contact COMSOL support team and start a support case with your modle attached. I think it would be easier to figure out how to make the two fields continuous.
Regarding the implementation of a PML for the implicit Solid Mechanics interface, I may assume the demand for such a functionality is not that high. Moreover, as you mentioned, there will be need for auxiliary variables in the PML domains. The number of those will be quite large in the 3D case - for the scalar wave equation for pressure acoustics there have to be 4 auxiliary variables in 3D. This will make the PML formulation computationally heavy. Let alone it will only be valid for a limited number of material models - I can only think of the linear elastic model.
The time-explicit interfaces implement the discontinuous Galerkin (dG) FEM: the PDEs are formulated as the 1st order conservation laws. The coupling between dG interfaces or even discontinuity of the material data in a single dG interface is not straightforward to implement. It is not the question of the solver support, but rather a meticulous mathematical work. The Absorbing Layers present for all dG interfaces are indeed very useful for solving open doman problems, even though the ALs are actually not perfectly matched.
Best regards Kirill