I'm not sure; Daniel might have a better idea. One thing you can do to manage this (if the anisotropic diffusion is necessary) is change 2c) to:
c) eq1 = fp.DiffusionTerm(coeff=S * [[[1., 0], [0., 0.]]]) == f(w, a) # matrix co-oeff has now been simplified by using a single facevariable whose values are appropriately zeroed out in the 2D grid > On Sep 12, 2016, at 4:34 PM, Gopalakrishnan, Krishnakumar > <krishnaku...@imperial.ac.uk> wrote: > > Dear all, > > We have a question regarding the original posting to this list (below) > > To recap and quickly summarise, we have > 1. a 1D elliptic PDE which is defined along the x-axis (at the top of a 2D > grid) and > 2. a standard diffusion PDE, which is restricted to diffuse only along the > y-axis. > > The BC of the "vertically diffusing PDE" depends on the values of the axial > PDE solution along the top of the mesh. > > This question is based on Jon's gist code 'crazycoupling2.ipynb' here, > https://gist.github.com/guyer/bb199559c00f6047d466daa18554d83d . > > For the axial (elliptic) PDE, to confine the direction of solution, in Input > cell 33 of the jupyter notebook, we can see that the following has been > defined. > > eq1 = fp.DiffusionTerm(coeff=[[[S, 0.], [0., 0.]]]) == f(w, a) > > > Is defining coeff=[[[S, 0.], [0., 0.]]]) the only way to achieve this effect > ? I am asking because, we have a high-order tensor coefficient which is > dependent spatially upon the mesh's x- co-ordinates (as x**2.0), and thus the > value S is not a simple scalar anymore. > > Does the following (simpler) approach work and produce an identical result ? > > 1. declare S as a rank-1 faceVariable in the 2D mesh, initialised to zero in > all faces > 2. Selectively set S only at the vertical faces bordering the top-cells of > the grid using the construct, > a) x = mesh.cellCenters[0] , Y = mesh.faceCenters[1] > b) S.setValue(x**2.0, where = (Y == 1.0 - 0.5*dy)) , where 1.0 is the > height of the mesh, and dy is the constant inter-nodal spacing. > c) eq1 = fp.DiffusionTerm(coeff=S) == f(w, a) # matrix co-oeff has now > been simplified by using a single facevariable whose values are appropriately > zeroed out in the 2D grid > > > In effect, my question is that, if the diffusion coefficient along all other > faces is set to zero, do we still need the coeff=[[[S, 0.], [0., 0.]]]) in > order to restrict the PDE to the axial direction ? > > > Best Regards > > Ian and Krishna > > > > > > -----Original Message----- > From: fipy-boun...@nist.gov [mailto:fipy-boun...@nist.gov] On Behalf Of > Guyer, Jonathan E. Dr. (Fed) > Sent: Thursday, July 14, 2016 5:26 PM > To: FIPY <FIPY@nist.gov> > Subject: Re: Coupling Boundary Condition of one PDE with source term of > another PDE > > My gut reaction is that the second implementation is probably better, as it > should be possible to make things more implicit and coupled, even though > you're "wasting" calculation of PDE1 over most of the 2D domain where you > don't care about it. > >> On Jul 14, 2016, at 11:18 AM, Guyer, Jonathan E. Dr. (Fed) >> <jonathan.gu...@nist.gov> wrote: >> >>> What’s the best way to implement this problem in FiPy? >> >> Don't! >> >> >> Assuming you won't take that advise, I've posted a couple of attempts at >> this problem at: >> >> https://gist.github.com/guyer/bb199559c00f6047d466daa18554d83d >> >> >> >>> On Jul 9, 2016, at 1:37 PM, Gopalakrishnan, Krishnakumar >>> <k.gopalakrishna...@imperial.ac.uk> wrote: >>> >>> Hi, >>> >>> **We’re sorry, the previous posting omitted the complete definition of the >>> boundary condition for the 2nd PDE, and also a couple of typos are fixed >>> below. >>> >>> We have a system of equations, wherein the BC of one PDE couples with the >>> source term of another PDE. >>> >>> We have a regular 2D unit grid in x and y. >>> >>> There are two PDEs to be solved: >>> a) The first PDE (elliptic diffusion problem) is defined only at y = 1, >>> acting along the x-axis (i.e. it acts in the x-direction and only along the >>> top of the cartesian grid). This x-axis is discretized with a fixed >>> grid-spacing, generating a finite number of nodes. Let this set of >>> nodes/co-ordinates be represented by ‘X’. >>> b) The second PDE is a time-varying diffusion problem. This is defined >>> only along the y-axis, but for all x-nodes (i.e. for all ‘X’), where the >>> 1st PDE is being solved. >>> >>> Mathematically (plain-text version) : >>> >>> PDE1: divergence(S * grad(a)) = f(w,a) >>> BC1: a = 0 , at x = 0 (dirichelet) >>> BC2: d_a/dx = 1 at x = 1 (Neumann) >>> >>> D = faceVariable(rank=2, value = 1.0) # Rank 2 tensor for anisotropic >>> diffusion (i.e. allowed only along the y-axis) >>> >>> PDE2: partial_B/partial_t = divergence( [[[0, 0], [0, D]]] * grad(B) ) >>> BC1: B = 0, at y = 0 at all ‘X’ (dirichelet), i.e. along the bottom face >>> BC2: d_B/dy = g(w,a) (Neumann) at y =1, i.e. along the top face of the grid >>> >>> f and g are linear functions in ‘w(x,y,t)’ and ‘a(x,y,t)’. B is defined in >>> the 2D grid as ‘B(x,y,t)’. >>> >>> Importantly, w = B at y = 1, for all ‘X’; i.e., the BC2 of the 2nd PDE >>> couples with the Implicit Source Term of the 1st PDE along the top face of >>> the Cartesian mesh. >>> >>> What’s the best way to implement this problem in FiPy? >>> >>> Best Regards, >>> >>> Krishna > > _______________________________________________ > fipy mailing list > fipy@nist.gov > http://www.ctcms.nist.gov/fipy > [ NIST internal ONLY: https://email.nist.gov/mailman/listinfo/fipy ] _______________________________________________ fipy mailing list fipy@nist.gov http://www.ctcms.nist.gov/fipy [ NIST internal ONLY: https://email.nist.gov/mailman/listinfo/fipy ]
