model module¶

The heat equation model.

This model solves the 1D transient equation, a.k.a. heat equation. It describes the temperature evolution $u$ in a $L$-length rod from the initial time 0 to the final time $T$ with a thermal diffusivity $nu(mathbf{X})$ depending on a random vector $mathbf{X}$. The heat equation is

$$frac{partial u(x,t;mathbf{X})}{partial t}

nu(mathbf{X})frac{partial^2 u(x,t;mathbf{X})}{partial x^2} = 0$$

with the boundary condition $u(0,t;mathbf{X})=u(L,t;mathbf{X})=0$ where $xinmathcal{D}=[0,L]$ and $tin[0,T]$.

To obtain an analytical solution, Geraci et al. (2015) chose $L=1$ and the uncertain initial condition:

$$u(x,0;mathbf{X}) =: mathcal{G}(mathbf{X})mathcal{F}_1(x) +mathcal{I}(mathbf{X})mathcal{F}_2(x)

where

$mathcal{F}_1(x)=sin(pi x)$,
$mathcal{F}_2(x)=sin(2pi x)+sin(3pi x)
+50left(sin(9pi x)+sin(21pi x)right)$,
$mathcal{I}(mathbf{X})=3.5
left(sin(X_1)+7sin^2(X_2)+0.1X_3^4sin(X_1)right)$,
$mathcal{G}(mathbf{X})=50prod_{i=5}^7(4|X_i|-1)$.

This uncertainty on the initial condition is modelled by the random variables $X_1,ldots X_7$ that are independent and distributed as:

$X_isimmathcal{U}(-pi,pi)$, for $iin{1,2,3}$,
$nu(mathbf{X})=X_4simmathcal{U}(nu_{min},nu_{max})$,
$X_isimmathcal{U}(-1,1)$, for $iin{5,6,7}$.

Then, Geraci et al. (2015) consider the integral of the temperature along the rod

$$mathcal{M}(mathbf{X}) = int_{mathcal{D}}u(x,T;mathbf{X})dx$$

and are interested in the estimation of its [HeatEquationConfiguration][gemseo_umdo.use_cases.heat_equation.configuration.HeatEquationConfiguration.expectation]:

$$mathbb{E}[mathcal{M}(mathbf{X})] = 50H_1+frac{49}{4}(H_3+50H_9+50H_{21})$$

where $H_k=frac{2}{k^3pi^3T} frac{exp(-nu_{min}k^2pi^2T)-exp(-nu_{max}k^2pi^2T)}{nu_{max}-nu_{min}}$.

The [HeatEquationModel][gemseo_umdo.use_cases.heat_equation.model.HeatEquationModel] computes the temperature at final time from instances of the random variables "X_1", …, "X_7" defined over the [HeatEquationUncertainSpace] [gemseo_umdo.use_cases.heat_equation.uncertain_space.HeatEquationUncertainSpace]. The temperature "u_mesh" is computed at each mesh node while the temperature "u" is an integral over the rod.