A hybrid experimental–differential equation model for predicting mechanical properties of FDM-printed PLA using Lucas polynomial collocation

A hybrid experimental–mathematical framework is proposed for predicting the mechanical properties of fused deposition modeling–printed polylactic acid components. Previously reported Taguchi L9 orthogonal array experimental data are used to evaluate the effects of infill density, layer thickness, and filling structure on tensile and Charpy impact strengths, while analysis of variance identifies infill density as the dominant governing parameter. To transform the discrete experimental observations into a continuous predictive formulation, first-order saturation-type ordinary differential equation models are developed for both mechanical properties as functions of infill density. The model parameters are identified analytically using experimentally determined mean strength values. The governing equations are solved both analytically and numerically using the Lucas polynomial collocation method, allowing verification of the numerical approximation against the exact analytical solutions. Independent validation against individual Taguchi experiments demonstrates that the proposed framework captures the dominant density-dependent mechanical trends, while residual variability arises from additional process parameters not explicitly included in the single-variable model. A Pareto-based mechanical performance assessment shows that both tensile and impact strengths attain their highest values within the investigated range at 60% infill density. The proposed ordinary differential equation–Lucas framework provides a compact and physically interpretable proof-of-concept methodology that bridges statistical experimental analysis and continuous mathematical modeling for additive manufacturing applications.