Extrusion force prediction in FDM: A semi-empirical model with temperature- and geometry-dependent flow exponent

Fused deposition modeling process optimization confronts a fundamental dilemma: maximizing printing speed to improve productivity while avoiding excessive forces that cause filament buckling, feeding failures, or material damage. Extrusion force prediction resolves this trade-off by enabling quantitative assessment of mechanical loads at the operating point, allowing systems to run at maximum safe speeds rather than conservative safety margins. However, existing models either rely on computationally expensive first-principles simulations unsuitable for real-time control or employ purely empirical correlations lacking physical interpretability and transferability across operating conditions. A semi-empirical log-log quadratic model with continuously varying effective flow exponent was developed and calibrated on 267 experimental conditions spanning three nozzle diameters (0.4, 0.6, 0.8 mm), five temperatures (170–250 °C), and flow rates from 1.64 to 25.8 mm³/s using PLA filament. A per-nozzle five-parameter variant was additionally fitted to assess the trade-off between generality and accuracy. The global seven-parameter model achieved R² = 0.948 and RMSE = 2.83 N; the per-nozzle model improved this to R² = 0.962 and RMSE = 2.43 N, with the largest gains for the 0.8 mm nozzle (R² from 0.815 to 0.983 at 170 °C). Both models require only setpoint temperature, nozzle diameter, and flow rate as inputs, executing in under 0.1 ms, making them directly suitable for real-time embedded control and quantitative definition of safe operating envelopes. The continuously varying effective exponent framework provides physical interpretability while maintaining computational simplicity, offering a practical foundation for adaptive extrusion control in high-throughput FDM systems.