How to calculate the energy reflection coefficients in LAMB wave edge reflection problem?

01 September 2020 2 4K Report

Hello everyone,

Using the spectral collocation method (SCM), I have successfully obtained the complex wavenumber k, displacement U and stress field S of different LAMB modes.

However, when I used the obtained k/U/S form the SCM method to solve the LAMB wave edge reflection problem, I had trouble with the calculation of energy reflection coefficients.

The sum of all the non-zero energy reflection coefficients (corresponding to the propagating modes) should be equal to 1, but I failed to get that right and the sum of those coefficients turned to be variable.

Power flux has been calculated as described in Mode-exciting method for Lamb wave-scattering analysis (JASA, 2004) by Arief Gunawan, and Sohichi Hirose in the form:

= Real ( -iw/2*intgrate(sigma*·(du/dt), -h,h) )

If anyone has ever met this problem or knows how to solve it? Or, if there is any program available online for us to use and study?

I have been checking my program over and over again for half a month, but I still can't find the causes. Thank you so much for spending your time reading my question. I would really appreciate it if you can help me with this problem.

My program was written with reference to the article of Prof. Pagneux and has been attached as NromalIncident-Vincent Pagneux way.zip, in which the LambDispersionValidation.m can be run to check the basic results of my program (compared with the online program from https://www.mathworks.com/matlabcentral/fileexchange/73050-lamb-wave-dispersion-curve, as shown in the figure below).

To help you understand my program and save your time, I want to tell you some details about my codes:

I can get the dispersion curves (as shown in the figure below). So I think the results of complex wavenumber k are right.

For LAMB waves propagating in +x direction, the displacement field was assumed as u = U*exp(i(kx-wt)). The numerically obtained k may have the form a + bi, a - bi, -a + bi, -a - bi (a and b are positive real numbers). In the GetkInOrder.m, I have kept only those k with positive real part (+x propagating) and positive imaginary part (physically decaying with propagation).

For LAMB waves propagating in -x direction, the displacement field was assumed as u = U*exp(i(-kx-wt)), so the k will be still kept k = a+bi.

Best regards,

Hao Qiu

Steffen Tai

Hello, it looks like I couldn't download your "Program.zip." In any case, I have some questions that could help us to investigate the issue in the energy balance.

Assume the roots of the dispersion equation and the modal functions are correct, at which x location (where x is the wave propagation direction) you are calculating the power flow? I have the impression that the power flow associated to the complex roots near the free edge are not insignificant and should be accounted for. Sounds like you have the complex roots as well, which is very good news, but you have only sum the powerflow associated to only the propagating mode (real roots)?

I have seen the discussion in the following publication that seems to refer to the same problem:

"C. Schaal and A. Mal, "Lamb Wave Propagation in a Plate with Step Discontinuities," Wave Motion, vol. 66, pp. 177-189, 2016"

The discussion is in section 3.1 and partly in section 2 as well.

Hao Qiu

Steffen Tai

Hi Steffen,

Thank you for your reply.

About the link error:

I think there may be some error with the Program.zip link. I have re-edited my question and attached my program as NromalIncident-Vincent Pagneux way.zip since I have been programming with reference to the article of Prof. Pagneux.

About the power flow

You are right. I have read that article before. The power flow associated with the complex roots near the free edge is not insignificant as shown in Eq. (11) in the paper you mentioned. However, while considering the area far away from the edge (several times the wavelength will be enough, I set x1 = 100m in my program), the power flux of the non-propagating modes can be neglected. This can also been validated with Eq. (11).

Which is suitable for use with Python? MySQL or SQL Server?

How to remove the residual PDMS on the silicon mold?

What are the important topics in the field of Artificial Intelligence in Business world?

Why CNN algorithm performs well in Network Intrusion Detection based on some recent resources?

What is the difference between the supplementary cementitious materials and alkali-activated concrete?

Formulate model for one integer optimization?

Do anyone have a full MATLAB code for structural topology optimization using Genetic Algorithm? Such as a cantilever beam?

How is it possible to rank variables using Likert scale?

Can anyone provide me with the scoring method and the questionnaire of Work life balance scale by Hayman (2005)?

What are the pretrained-language-model that is obviously better than BERT and RoBERTa?

Why the results when using wall thickness and when using shell conduction are different in ansys fluent?

How much pressure we can reduce at the outlet to avoid Reverse Flow?

Timeout error!! Processor in the loop (PIL) using STM32F4 Matlab 2018a?

Could anyone tell me what is the range of submillimeter wave?

How to define tie constraint between rigid body and another part in dynamic explicit model of Abaqus?

Can polymer concentrations be calculated based on the number average molecular weight?

How to eliminate Steady state error from Reinforcement learning genetic algorithm code?

Is there a deep reason for using the squared modulus of the wave function as the probability density of the existence of the particle?

How to prevent ANFIS Model from negative prediction values in MATLAB?

Complex wave form in ANSYS?