Chance-Constrained Optimal Design of Porous Thermal Insulation Systems Under Spatially Correlated Uncertainty

TL;DR

Addresses robust design of silica aerogel thermal breaks under spatially correlated porosity uncertainty using PDE-constrained optimization. The objective combines mean and variance of the thermal compliance $\mathcal{J}(d) = \mathbb{E}[Q(d,m)] + \beta_V \mathbb{V}[Q(d,m)] + \beta_{RG} R_p(d)$, while a chance constraint $P(f(m,d) \ge 0) \le \alpha_c$ enforces mechanical stability with $f = T_{cr} - T_{pn}$ and $T_{pn} = \left( \int_{\Omega} T_{VM}^p \, d\Omega \right)^{1/p}$. Scalability is achieved via quadratic Taylor approximations and Monte Carlo with control variates, solved by an Inexact Newton-Conjugate-Gradient method with a continuous surrogate for the indicator. The framework is demonstrated in 2D and 3D beam-insulator problems with design spaces reaching hundreds of thousands of parameters, highlighting its potential for large-scale additive manufacturing of robust aerogel insulation components.

Abstract

This paper presents a computationally efficient method for the optimal design of silica aerogel porous material systems, balancing thermal insulation performance with mechanical stability under stress concentrations. The proposed approach explicitly accounts for additive manufacturing uncertainties by modeling material porosity as a spatially correlated stochastic field within a multiphase finite element formulation. A risk-averse objective function, incorporating statistical moments of the design objective, is employed in conjunction with chance constraints that enforce mechanical stability by restricting the probability of exceeding critical stress thresholds. To mitigate the prohibitively high computational cost associated with the large-dimensional uncertainty space and Monte Carlo estimations of the objective function's statistical moments, a second-order Taylor expansion is utilized as a control variate. Furthermore, a continuation-based smoothing strategy is introduced to address the non-differentiability of the chance constraints, ensuring compatibility with gradient-based optimization. The resulting framework achieves computational scalability, remaining agnostic to the dimensionality of the stochastic design space. The effectiveness of the method is demonstrated through numerical experiments on two- and three-dimensional thermal break systems for building insulation. The results highlight the framework's capability to solve large-scale, chance-constrained optimal design problems governed by finite element models with uncertain design parameter spaces reaching dimensions in the hundreds of thousands.

Figures (13)

• Figure 1: The domain of a beam-insulator system used for numerical experiments, indicating both the mechanical (left) and thermal (right) boundary conditions. The solid displacement are fixed at the boundaries $\Gamma_1$ and $\Gamma_3$ and roller boundary conditions are implemented at $\Gamma_2$. At $\Gamma_1$ and $\Gamma_4$, we consider convective heat transfer cold ambient temperature in the building exterior and hot ambient temperature in the building interior. no heat flux boundary is implemented at $\Gamma_2$.
• Figure 2: Finite element solution of the thermomechanical (forward) model for the beam-insulator system with constant porosity $\phi_f=0.7$, depicting the solid temperature $\theta_s$, fluid temperature $\theta_f$, displacement magnitude $||\mathbf{u}_s||$, and von Mises stress $T_{\text{VM}}$.
• Figure 3: Approximation of the discontinuous indicator function $\chi_{[0,\infty)}(x)$ with a sigmoid function as given in \ref{['eq:smooth_approximation']} and its convergence with increasing $\omega$
• Figure 4: Samples of uncertain parameter $m$ and corresponding aerogel porosity field $\phi_f(\boldsymbol{x})$ for correlation length $L_{CR} = 0.25$ and variance of $\sigma^2 = {0.5}^2$. The mean of the uncertain parameter is $\Tilde{m}=0$ and anisotropy is $\upvartheta_x = 1, \upvartheta_y = 1 \times 10^{-4}$ for both cases.
• Figure 5: Convergence plots corresponding to scenario in Figure \ref{['fig:LU']} for the mean and variance of the design objective $Q$ : (a) standard Monte Carlo method (b) Monte Carlo with control variate and (c) Quadratic approximation. The uncertain parameter $m$ has the correlation length $L_{CR} =0.25$ and a variance of $\sigma^2 ={0.5}^2$.