# 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

1. at x=-0.5, a non-homogeneous Neumann condition k \frac{\partial T}{\partial x} = \delta;

2. 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.