Solving PDEs: time dependent c coefficient

I am solving a PDE to simulate current flow through a conductive medium in 2D.

My geometry is as follows:

With Neumann boundary conditions applied to edges E2 and E4, like so:

applyBoundaryCondition(model,'neumann','Edge',1:model.Geometry.NumEdges,'q',0,'g',0);
applyBoundaryCondition(model,'neumann','Edge',2,'q',0,'g',1);
applyBoundaryCondition(model,'neuman','Edge',4,'q',0,'g',-1);

I am only interested in solving the PDE in the form of the Laplace equation:

,

where is electric potential (which we are solving for) and σ is the conductivity tensor, which is represented as the c coefficient in our PDE model.

I want to apply a time-dependent c coefficient to the subdomain, F2, whereby the conductivity changes in time such that c = t/(t+constant).

I have tried the following lines of code:

c1 = [1;0;0;1];
specifyCoefficients(model,'m',0,'d',0,'c',c1,'a',0,'f',0,'Face',1);

c2 = @(location,state)(repmat([1;0;0;1].*state.time./(state.time + 1),1,length(location.x))); % Conductivity tensor for Face 2.
specifyCoefficients(model,'m',0,'d',0,'c',c2,'a',0,'f',0,'Face',2);

However, I am unsure about how to specify the time scale as the rest of the PDE is time-independent. I have found that

tlist = 0:0.01:1;
results = solvepde(model,tlist);

does not work.

Any help is appreciated.

Additional, but less urgent:

Once this step is completed, I aim to have the conductivity tensor also non-constant in x and y, so that it has something resembling a normal distribution (smallest conductivity in the centre of Face 2 and parabolically tending towards equilibrium at the edges. If anyone has any ideas on how to tackle this, that would be great.

Answer 1

[Alan Weiss]

If I understand you correctly, you have a sequence of independent problems; the solution at one time t does not depend on or influence the solution at a different time.

As such, you do not have a time-dependent model. I believe that all you need to do is solve a set of time-independent models, where each model uses the same geometry but a different c coefficient. Solve the problems in a loop, something like

tspan = linspace(0,5,20); % 20 times from 0 through 5, just for example
for i = 1:length(tspan)
    c2 = tspan(i)/(1 + tspan(i));
    specifyCoefficients(model,'m',0,'d',0,'c',c2,'a',0,'f',0,'Face',2);
    results{i} = solvepde(model);
end

As for your second question, I am not sure exactly what you need, but if you can write out the equations for your conductivity, then I would imagine that the c coefficient documentation would lead you to the correct formulation.

Good luck,

Alan Weiss

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