Additive manufacturing with 3D printers is growing in popularity as more uses for printed products are invented. While multi-scale models exist for various aspects of additive manufacturing, none reproduce filament-based extrusion with fused deposition technology.

Xia et al. developed a mathematical model for the fused deposition process, which uses high temperatures to create complex geometric components.

“The lack of a multi-scale theoretical model poses significant challenges for prediction in 3D printing processes, which are driven by rapidly evolving temperature fields, including the nonuniformity of tracks, the spheroidization effect of materials, and inter-track voids,” said author Yibao Li. “However, we have established a significant mathematical model to describe the fused deposition process in the 3D printing process.”

The mathematical model works by coupling conductive and convective heat transfer with a dendritic solidification model, allowing the model to describe material relationships on multiple scales. The model describes physical phenomena in a material including cooling, solidification, crystallization, and deposition of the filament. To reduce computational costs, the model assumes the materials are homogenous across all scales.

The model was tested by calculating and simulating an additive manufacturing process. The results showed acceptable accuracy compared to the final manufactured product and the model was able to reproduce the temperature distribution during production.

In addition to the model for fused deposition modeling in this paper, the authors are planning to develop more general mathematical models for other additive manufacturing techniques, such as selective laser melting technology, and multi-material additive manufacturing techniques. They hope their work will improve computational efficiency in the manufacturing process.

Source: “Multi-scale modeling and simulation of additive manufacturing based on fused deposition technique,” by Qing Xia, Gangming Sun, Junseok Kim, and Yibao Li, Physics of Fluids (2023). The article can be accessed at