How to implement irregular, time-dependent boundary condition in PDEPE function?

Hi

I am trying to use the MATLAB Partial Differential Equation solver, pdepe, for a simple 1D (one-dimensional) heat transfer in a space shuttle tile using Fourier's equation for heat transfer. All parameters are known, and I have organised the equation according to the MATLAB format.

However, the problem I have encountered is in the Boundary Conditions. For the shuttle tile, the outer surface (right hand boundary condition) varies in a time-dependent manner according to a predefined pattern that cannot be described mathematically. I can store the data in a vector, but I have thus far been unsuccessful in implementing this in the RHS boundary (pr).

Any help at all would be very much appreciated, a sample of my current code can be viewed below.

James

function pde1 % Function to perform parabolic PDE solver on Fourier's Heat transfer % equation in one dimension %

global rho cp k tinitial global n tempdata deltaT timedata

% tile properties

k = 0.141; % W/(m K)

rho = 352; % 22 lb/ft^3

cp = 1255; % 0.252 Btu/lb/F at 500F

n = 500;

tinitial = (60-32)*5/9; % intial temp throughout the tile

% Loads the outer surface temperature data in an array. % This data can then be extracted into a vector. load temp597.mat timedata tempdata

% properties required for Partial Differential Equation input

m = 0; % assume 'slab' shape

xmesh = linspace(0, 0.05, 21); % 21 spatial steps, up to 5cm

t = linspace(0, 4000, 500); % 250 time-steps over 2000 seconds

sol = pdepe(m, @pdef, @pdeic, @pdebc, xmesh, t);

u = sol(:,:,1);

% ========================================================================= % defines the properties of the PDE function

function[c,f,s] = pdef(x, t, u, DuDx)

c = rho*cp;

f = k*DuDx;

s = 0;

end % ========================================================================= % defines the Initial Conditions

function[u0] = pdeic(x)

u0 = tinitial;

end % ========================================================================= % defines the Boundary Conditions

function[pl,ql,pr,qr] = pdebc(xl, ul, xr, ur, t)

pl = 0;

ql = 1;

pr = ???;

qr = 1;

end end

end

Answer 1

[Bill Greene]

I think that

pr = interp1(timedata, tempdata, t) - ur;

may be what you need. Assuming the lengths of timedata and tempdata are the same, this will do a piecewise linear interpolation for the temperature at time t.

Beyond that, if you want to prescribe the temperatures at the right and left ends to a prescribed value, you want ql and qr equal zero rather than 1.

Bill

[James Fraser]

Thankyou very much for your help, the code does seem to at least run now, but I am met with the following warnings.

Warning: Failure at t=0.000000e+00.  Unable to meet integration tolerances
without reducing the step size below the smallest value allowed (7.905050e-323)
at time t. 

Warning: Time integration has failed. Solution is available at requested time
points up to t=0.000000e+00.

It tells me the problem lies with

sol = pdepe(m, @pdef, @pdeic, @pdebc, xmesh, t);

in my pde1 function. And lines 317 and 323 of the pdepe function.

Do you know how I can prevent this from happening?

[James Fraser]

On further checking through my function, when I output the variable pr, it informs me that, at all points,

pr = NaN

[James Fraser]

got it! Just needed

interp1(timedata, tempdata, t, 'linear', 'extrap')

many thanks!