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

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.