Uncertainty Analysis of Experimental Parameters for Reducing Warpage in Injection Molding

TL;DR

This work tackles warpage in injection molding by building polynomial regression surrogates for wall displacements and embedding them in a Bayesian framework to quantify uncertainty in the optimal process settings. By sampling posterior regression coefficients, it yields a distribution of optimal inputs $\hat{\mathbf{x}}^*$ and provides insight into robustness through marginal distributions and contour visuals. It further introduces a Monte Carlo boundary-analysis approach to construct confidence bands around zero-level sets $\partial\mathcal{X}_\ell$, enabling visualization of regions where warpage transitions between convex and concave profiles. Applied to a box-shaped part with four process parameters, the framework identifies a stable optimum and clearly delineates the boundaries of defect-prone regions, offering a practical tool for robust IM design under parameter uncertainty.

Abstract

Injection molding is a critical manufacturing process, but controlling warpage remains a major challenge due to complex thermomechanical interactions. Simulation-based optimization is widely used to address this, yet traditional methods often overlook the uncertainty in model parameters. In this paper, we propose a data-driven framework to minimize warpage and quantify the uncertainty of optimal process settings. We employ polynomial regression models as surrogates for the injection molding simulations of a box-shaped part. By adopting a Bayesian framework, we estimate the posterior distribution of the regression coefficients. This approach allows us to generate a distribution of optimal decisions rather than a single point estimate, providing a measure of solution robustness. Furthermore, we develop a Monte Carlo-based boundary analysis method. This method constructs confidence bands for the zero-level sets of the response surfaces, helping to visualize the regions where warpage transitions between convex and concave profiles. We apply this framework to optimize four key process parameters: mold temperature, injection speed, packing pressure, and packing time. The results show that our approach finds stable process settings and clearly marks the boundaries of defects in the parameter space.

Figures (8)

• Figure 1: Left: CAD model of the test geometry highlighting dimensional details, including asymmetric wall thicknesses, 1.25 mm and 2 mm. Right: warpage direction nomenclature and sign convention is categorized as positive (convex) or negative (concave) based on surface displacement.
• Figure 2: Multiple realizations of the quadratic function (blue lines) and their corresponding optimal decisions (red dots) for the polynomial model in \ref{['eq:cure']}.
• Figure 3: Empirical density of the uncertainty set in \ref{['eq:xuncertainty']} for the polynomial model in \ref{['eq:cure']}.
• Figure 4: Panel of Monte Carlo-based confidence bands for the quadratic surrogate model in Example 2 at four standardized tolerances. Each subplot shows the 95% boundary (blue dashed contour) approximately satisfying $\mathbb P\bigl(|y(x_1,x_2)/\sigma_y(x_1,x_2)| < \varepsilon\bigr)\ge0.95$ with $\varepsilon$ equal to 1.5, 2.0, 2.5, and 3.0 standard deviations, respectively. The purple solid curve is the posterior mean zero-level set, the green solid curve is the original true function zero-level set, and the gray curves are ten random posterior sample zero-level sets.
• Figure 5: Post-processed 2D warpage contour images from Moldex3D are analyzed to convert pixels into deformation units i.e. mm.