# 3D Benchmark - Moving heat source

Simulations of welding process require accurate and reliable heat repartition evaluations.

In the welding process, the heat source is moving.

Fachinotti & al. proposed an analytical solution of the non steady linear heat equation in a semi infinite body with the input source given by Goldak et al..

## Physical parameters

Symbol |
Name |
Unit |

\rho |
density |
\frac{[kg]}{[m^3]} |

c |
heat capacity |
\frac{[J]}{[K]} |

q |
heat source |
x |

k |
thermal conductivity |
\frac{[W]}{[m.K]} |

K=\frac{k}{\rho c} |
thermal diffusivity |
\frac{[m^2]}{[s]} |

T=T(x,y,z,t) |
Temperature at time t and position (x,y,z) |
[K] |

T_0=T_0(x,y,z) |
Initial temperature at position (x,y,z) |
[K] |

## Temperature field

Heat conduction in a homogeneous solid is governed by the linear partial differential equation: \rho c \partial_t T - k \Delta T = q endowed with initial and boundary condition.

In the following, we will considere infinite body. That is if the computational domain is big enough, we will set T_0 as a boundary condition.

## Moving double ellipsoidal heat source

Consider a fixed Cartesian reference frame (x,y,z), in which a heat source located initially at z=0 and at time t=0, moves with constant velocity, v, along the z-axis. In the case of welding applications, Goldak et al. defined the heat source at a position (x,y,z) and time, t, by means of the following double-ellipsoidal distribution:

q(x,y,z,t) = \frac{6 \sqrt{3}Q}{\pi \sqrt{\pi}ab} \times \begin{cases} \frac{f_f}{c_f} \exp\left[-3 \frac{x^2}{a^2}-3 \frac{y^2}{b^2}-3 \frac{\left(z-vt\right)^2}{c_f^2}\right], & \text{for $z>vt$}.\\ \frac{f_r}{c_r} \exp\left[-3 \frac{x^2}{a^2}-3 \frac{y^2}{b^2}-3 \frac{\left(z-vt\right)^2}{c_r^2}\right], & \text{for $z<vt$}. \end{cases}

where ._f stands for **front** and ._r stands for **rear**.

## Analytical solution

Given the linear equation and the heat source, we have (see Fachinotti & al.) the following analytical solution:

\begin{aligned} T\left(x,y,z,t\right) &= T0 + \frac{3\sqrt{3} Q }{\pi\sqrt{\pi} \rho c} \times \int_{0}^{t} \frac{\exp\left[-3 \frac{x^2}{12 K \left(t-t'\right) + a^2}-3 \frac{y^2}{12 K \left(t-t'\right) + b^2}\right]}{\sqrt{12 K \left(t-t'\right) + a^2}\sqrt{12 K \left(t-t'\right) + b^2}} \\ &= \times \left[ f_r A_r \left(1-B_r\right) + f_f A_f \left(1-B_f\right)\right] \mathrm{d}t' \end{aligned} with - using i=r or f: A_i = A(z,t,t'; c_i) = \frac{\exp\left[-3 \frac{\left(z-vt'\right)^2}{12 K \left(t-t'\right) + c_i^2}\right]}{\sqrt{12 K \left(t-t'\right) + c_i^2}} and B_i = B(z,t,t'; c_i) = \text{erf}\left[\frac{c_i}{2}\frac{z-vt'}{\sqrt{K\left(t-t'\right)}\sqrt{12 K \left(t-t'\right) + c_i^2}}\right]

## Comparison

We evaluate the temperature field at (0,0,0.05) over the time and compare it with the analytical solution. That is actually the reproduction (3D only) of the figure 7 in Fachinotti & al.

We also evaluate the temperature profile along the welding path, 10s after the start of the simulation. That is actually the reproduction (3D only) of the figure 8 in Fachinotti & al.

{% youtube %}https://www.youtube.com/watch?v=31nQtOMwDLQ{% endyoutube %}