*2.5. Macro–Micro Coupling of the Temperature Field*

The calculation of the welding heat transfer process is the basis of the microstructure simulation. However, the calculation of the heat transfer process of the weld pool is performed on a macro scale, while the calculation of the dendrite growth based on the CA method is carried out on a micro scale. Therefore, it is necessary to establish a macro-micro coupling model for temperature field calculation. The macroscopic temperature field was solved by the finite difference (FD) method using the ABAQUS finite element software. The CA model was built to simulate the microstructure evolution of the columnar to equiaxed transition (CET) process. The temperature of the CA element is affected by the temperature of the macro elements around it. It is related to the distance from the central node of the element to the surrounding macro elements as shown in Figure 5.

**Figure 5.** The macroscopic and microscopic coupling analysis.

The temperature value of the CA element can be expressed by the following formula [16]:

$$T\mathbf{o} = \sum\_{i=1}^{N} Li^{-1}Ti / \sum\_{i=1}^{N} Li^{-1} \tag{7}$$

where *T*o is the temperature of the micro-element O; *Ti* is the macro-element temperature around the point O; *Li* is the distance from the point O to the surrounding macro-element; and *N* is the number of macro-elements around the micro-element, the value of which is 8.
