I have the following Laplace's equation on rectangle with length a and width b (picture is attached):
ΔU(x,y)=0
Ux(0,y)=0 : Isolated boundary
Ux(a,y)=f(y) : Current source
U(x,b)=0 : Zero potential
The fourth boundary is quite complex : Mixed boundary condition (isolated except in two points)
if x=a/3 or 2a/3
U(x,0)=0
else
Uy(x,0)=0
Is there an analytical way to solve this kind of mixed boundary problem? can someone point me to the right direction? I'm a bit lost
thanks in advance
Hi Dor,
No turnkey solution, but rather a track to explore...
By converting your rectangle domain in an cartesian grid and implementing a Multigrid Solving Scheme such this one proposed by Guillet et al.: https://arxiv.org/pdf/1104.1703.pdf
Another approach facing with interface discontinuities: https://arxiv.org/pdf/1401.8084.pdf
Regards
Thank you Stéphane Breton for the quick answer,
I will look into it.
I was wondering whether there is an analytical approach to this problem such as separation of variables (fourier series expansion)? It will save me a lot of time because I need to run this laplace solution for many times.
I came across the robin boundary conditions. Will it help to write it in this way?
a*U+b*du/dy=0
a=δ(x-a/3)+δ(x-2a/3)
b=H(x)*H(a/3- x)+H(x-a/3)*H( 2a/3-x)+H(x-2a/3)* H(a-x)
H->Heaviside step function
δ->Dirac delta function
Hi Dor,
Yes, in order to address your issue analytically, variable separation jointly with sub-problems decomposition/superposition by means of series expansion appears an interesting track. To this end, combining d'Alembert's formula with Duhamel principle may be helpful ( http://www.math.ucla.edu/~yanovsky/handbooks/PDEs.pdf and http://mate.dm.uba.ar/~jfbonder/libro/(Universitext_)Sandro_Salsa-Partial_Differential_Equations_in_Action_from_Modelling_to_Theory-Springer(2008).pdf ).
Here are some valuable references:
- http://researchonline.jcu.edu.au/15878/1/JRC_WWR_10.pdf
- https://www.researchgate.net/profile/Wayne_Read/publication/222782264_Series_solutions_for_Laplace's_equation_with_nonhomogeneous_mixed_boundary_conditions_and_irregular_boundaries/links/540ee0660cf2d8daaaceb0a7.pdf
- http://ac.els-cdn.com/S0377042706006169/1-s2.0-S0377042706006169-main.pdf?_tid=42b51128-031a-11e7-879a-00000aab0f6b&acdnat=1488879878_24a5d129da328db256f56338870b165d
- http://ac.els-cdn.com/S0895717797000721/1-s2.0-S0895717797000721-main.pdf?_tid=74143246-032c-11e7-8371-00000aacb35e&acdnat=1488887692_caecdfd304f1df938347a50b1f8170d2
Hoping it will be helpful
Article Series solutions for Laplace's equation with nonhomogeneous ...
THE ANALYTICAL SOLUTION COULD BE QUITE COMPLEX. BUT TRY TO USE THE FINITE DIFFERENCE METHOD. IT SHOULD WORK QUITE EASILY.
REGARDS
JAN SIKORA
The problem as posed cannot be solved, or rather what you get is different from what you expect. The boundary conditions at (a/3,0) and (2a/3,0) will have no effect. If you insist on a boundary condition of that kind you should prescribe it on segments with nonzero length. If you doubt this compare (numerical) solutions with these bc's on nonzero segments and let the length of the segment go to 0.
On a different note: forget about an analytical solution of this problem if anything it leads to very slowly convergent Fourier expansions, that will take all eternity to calculate. So Breton's suggestion is an excellent one: discretize and use multigrid. Any old multigrid package will do.
Finite difference methods using a discretization on Cartesian Grid (you can choose which type A, B or C Arakawa Grid) -- I think there are few tutorials on youtube
Finite Element methods (weak form) -- https://www.youtube.com/watch?v=U65GK1vVw4o&list=PLJhG_d-Sp_JHKVRhfTgDqbic_4MHpltXZ (really good tutorial with few examples and coding tutorials)
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