## 1. Elementary Heat Conduction Test

These tests check 2D and 3D heat equation (heat conduction) elements with linear conductivity. Given a wall that extends infinitely in the y- and z-directions, the tests compute the temperature through the wall.

Case 1 specifies temperatures at the wall surfaces T(x=0)=T1 and T(x=L)=T2. For a temperature-invariant thermal conductivity the theoretical solution is

T(x) = T1+(T2-T1)x/L

where T1 is the wall temperature at x=0, T2 the wall temperature at x=L, and L the wall thickness. The constant heat flux is

q = -k(T2-T1)/L

T, Q, HE, TE, and PR regular and skew elements are tested, the problem being the same in case of 2D or 3D elements. The figure below shows the model and the temperature distribution for the H20 case (4 elements)

Case 2 specifies temperatures at the wall surfaces T(x=0)=T1 and T(x=L)=T2 and the wall is heated internally with a constant heat source w. For a temperature-invariant thermal conductivity the theoretical solution is

T(x) = T1 - 0.5 w x2/k + (T2-T1)x/L + 0.5 w L x / k

where T1 is the wall temperature at x=0, T2 the wall temperature at x=L, L the wall thickness, and k the thermal conductivity. The figure below shows the model and the temperature distribution for the H20 case (4 elements)

T, Q, HE, TE, and PR elements are tested, the problem being the same in case of 2D or 3D elements. A similar 1D test is performed with axisymmetric T and Q elements.