paraheat_functional, a MATLAB code which sets up and solves a parameterized steady heat equation in a 2D spatial domain, using diffusivity parameterized by vc, and reporting selected solution values vs, using a functional framework "vs=f(vc)".
This function formalizes a functional input/output structure for a problem involving the steady heat equation in a 2D domain with a diffusivity function, for which values of the solution are sampled at specified sensor locations.
In this functional framework, the diffusivity parameters VC are considered the input, and the sensor readings VS the output. Thus, at the simplest level, we have
vs = paraheat_functional ( vc );
which allows us to concentrate on the relationship between diffusivity
parameters and resulting sensor readings.
An interesting goal is to characterize the relationship of VS as a function of VC, including the sensitivity and the possibility of inverting the relationship, that is, to approximate VC given values of VS.
The steady state heat equation to be solved is:
- del ( k(x,y) * grad u ) = f(x,y)
over the unit square 0 < x, y < 1.
Zero Dirichlet boundary conditions are applied. The right hand side function is set as:
f(x,y) = 1000 * x * ( 1 - x ) * y * ( 1 - y );
The diffusivity is represented by k(x,y). For this problem, k(x,y) is a piecewise constant function. The region is divided into a 4x4 grid of rectangles, to each of which a constant value VC is assigned. These values are regarded as the parameters of the problem.
After the solution is computed, the value VS of the solution is determined at a certain number of sensor locations.
The information on this web page is distributed under the MIT license.
paraheat_functional is available in a MATLAB version and an Octave version.
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