1D Heat Transfer Benchmark
Note This benchmark applies to both Heat Transfer and Reduced Basis.
Description
We consider a 1D heat diffusion problem. It is described by two geometrical domains in \mathbb{R} : \Omega_1 =[0.5~,~0] and \Omega_2=[0~,~0.5]. The temperature T of the domain \Omega = \cup_{i=1}^2 \Omega_i is then solution of the following steady heat diffusion equation :  k ~ \Delta T = \varphi where \varphi is the source term and k is thermal conductivity defined as k=0.1+k_1 if x \leq 0 and by k=0.1+k_2 if x>0.
This problem has been proposed to validate the implementation of CRB methods and especially the Successive Constraint Method (SCM) algorithm.
Boundary Conditions
We set

at x=0.5, a nonhomogeneous Neumann condition k \frac{\partial T}{\partial x} = \delta;

at x=0.5, a Robin condition k \ \frac{\partial T}{\partial x} = T .
Inputs
As inputs, We define the parameter set \mu by \mu=(k_1,k_2,\delta,\varphi)~\in~[0.2~,~50]^2 \times [1~,~5] \times [0.1~,~5]
Outputs
The output s(\mu) is the mean temperature on the central part of domain \Omega, defined by s(\mu) = \frac{1}{0.2} \int_{0.1}^{0.1} T This output depends on the solution of the equation above and is dependent on the parameter set \mu.