We use a direct boundary element algorithm in time-domain (Mansur, 1983) to investigate the seismic behavior of non-homogeneous geologic features. This algorithm uses fundamental solution of the displacement (as well as traction) to form the stiffness matrices. The employed code (Hybrid, Kamalian, 2005) does not consider attenuation in the analysis.
The problem is that in most cases despite stability of the results of homogeneous analysis, the non-homogeneous results becomes unstable rapidly, in a way that the displacements start oscillating around the correct values, although we examined almost all possible optimum time-step range.
we know that considering attenuation could decrease divergence, but is there any other way (regarding ease of use) to deal with divergence problem in displacement matrices ?
Hi Shahram.
I had such a problem in one of my projects. What I finally did was to calculate the maximum value of the terms that I had to multiply them by the time-step. Then at each iteration, I put the time step lower than the inverse of that value. It worked for me at that time, however, after that I noted that my algorithm was not programmed well for local differentiations. Correction of that bug, made me need-free of the mentioned calculation.
Dear Somayeh, Thank you for your response.
Actually that was not completely clear for me ! Lets say, we have some thing like : H*U=G*P+Z
H and U are determined in first time step and U and P are unknowns, need to be calculated in every time step. Z is the history term which is related to the time steps earlier than current step.
Now U and P could become unstable during analysis. when instability begins somewhere, it affects later time steps because of the history term.
Now, as you say the value of the terms I need to multiply by the time step are H*U and G*P
H and G are constant matrices, I need to extract the maximum values of U and P and make them inverse. you mean defining time step duration according to the inverse of U and P ?
I would be grateful if you make it a bit more clear for me.
Hi Shahram,
My PDE problem was something like the following,
dv/dt=(Constant matrix)*dv/ds+f(s,t,v).
You see that all the terms on the right side of the equation are multiplied by "dt",in each step, I found maximum element of it and then chose "dt" as dt*maximum_element be less than a desired value. Also, I knew that f function has the largest values then I just checked that matrix to find the maximum.
Besides, did you try normalizing of the equations? Sometimes it works.
Hi Somayeh,
actually I prefer to find another way to solve the problem because normalizing may affect the results.
I will try to check your solution. I hope it work.
Thank you.
OK. Please let me know if it works. Also, by normalizing, I meant making the equations dimensionless, my apologizes for not being accurate with my previous response.
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