Reliable Uncertainty Quantification for Fiber Orientation in Composite Molding Processes using Multilevel Polynomial Surrogates

Abstract

Fiber orientation is decisive for the mechanical performance of composite materials. During manufacturing, variations in material and process parameters can influence fiber orientation. We employ multilevel polynomial surrogates to model the propagation of uncertain material properties in the injection molding process. To ensure reliable uncertainty quantification, a key focus is deriving novel error bounds for statistical measures of a quantity of interest. Numerical experiments employ the Cross-WLF viscosity model and Hagen-Poiseuille flow to investigate the impact of uncertainties in fiber length and matrix temperature on the fractional anisotropy of fiber orientation. The Folgar-Tucker equation and the improved anisotropic rotary diffusion model, incorporating analytical solutions, are used for verification. Results show that the method improves significantly upon standard Monte Carlo estimation, while also providing error guarantees. These findings offer the first step toward a reliable and practical tool for optimizing fiber-reinforced polymer manufacturing processes in the future.

Figures (9)

• Figure 1: Illustration of model input and output relationships: (a) fixed parameter case $x \in \mathcal{X}$; (b) parameter augmented case, where both uncertainties $\omega \in \Omega$ and parameters $x \in \mathcal{X}$ are treated as random inputs.
• Figure 2: Hagen-Poiseuille flow $v(x, y)$, Eq. \ref{['eq:hp']}, with viscosity $\eta = \eta(T)$ for $T=565$ K. The red dot marks the arbitrary selected point in the channel $(x_0, y_0) := (2.0 ,\, 0.1)$ mm, where fiber orientation is investigated. The arrows show the velocity gradient.
• Figure 3: Response surface $Q = F\left(\mathbf{A}[t_\text{end}; \cdot]\right)$ showing the fractional anisotropy over the parameter domains $\Omega_T = [550, 580]$ K (temperature) and $\Omega_{L_f} = [0.38 \pm 50\%]$ mm (fiber length). Computations use the FTE and iARD model as FOM, respectively, with isotropic initial orientation tensor $\mathbf{A}(0) = \mathbf{I} / 3$.
• Figure 4: Input tolerance $\epsilon$ vs. the resulting error $\| Q - \hat{Q}_\epsilon\|$ of the polynomial approximation of fractional anisotropy $Q = F(\mathbf{A})$. Also shown are the errors for the expected value $\mathbb{E}[Q]$, standard deviation $\mathbb{V}[Q]^{1/2}$, CDF $F_Q$, and the $p$-quantile $q_Q$. The left plot presents results based on the FTE, while the right plot corresponds to the iARD model.
• Figure 5: Shown are the PDFs and histograms of the QoI, fractional anisotropy, for the FTE and the iARD model. They were computed using a KDE of evaluations of their polynomial surrogates $\hat{Q}_\epsilon$ with $\epsilon = 10^{-3}$ on the Sobol sample $\omega_S$, Eq. \ref{['eq:sobol']}.

Theorems & Definitions (4)

• Lemma 1
• Lemma 2
• Lemma 3
• Remark 1