Specific Domain Design

[engsbk]

Thanks @lululxvi for the amazing library!
I used it to model a 1D wave equation:

def pde(x, u):
    c = 1
    du_xx = dde.grad.hessian(u, x, i=0, j=0)
    du_tt = dde.grad.hessian(u, x, i=1, j=1)
    return   du_tt - (c ** 2)*(du_xx) 

I achieved very good results in this geomtime domain:

geom = dde.geometry.Interval(0,1)
timedomain = dde.geometry.TimeDomain(0,1)
geomtime = dde.geometry.GeometryXTime(geom, timedomain)

The results legend uniform1, uniform2, ... is only indicating the points distribution and number of experiment, so you can ignore them.

My question now is: how can I change the domain to model the wave travelling in a human tissue depth of 10cm and 10mm of bone for example? How can I have multiple units of space involved and have accurate results?

How can I move from my current setup to this multi-region domain?

[lululxvi]

Have no idea of what a human tissue depth of 10cm and 10mm of bone is. Your question seems a research problem.

[engsbk]

To simplify, you can imagine this example sketch in a 2D domain:

I would like to utilize PINN to model the wave equation with a source at the point indicated. My questions are:

1. How can I specify that the distance in the first region is 10 cm and 2cm in the second region?
2. In each region the wave speed (c) is different, how can I implement this in pde(x,u)?
3. If it is simpler to implement in 1D and then extend to 2D please advise on how to tackle this setup problem?

Thank you so much for your continuous support I really appreciate your efforts!

[ragusa]

I think the question could be about how to define heterogenous zones inside a domain. I would create a domain of size 12 and would use a mask
c = ctissue * (x<10) + cbone *(x>=10)
where ctissue and cbone are the material values you want.

[engsbk]

Thanks for the reply!
What about the units in centimeters or millimeters? How can you control it in the domain?

[ragusa]

as with all mathematical modeling, the units have to be consistent so that each term of your PDE has the same units: ddu/dtt should have the same units as c*ddu/ddx.