CHANCE CONSTRAINED PDE-CONSTRAINED OPTIMAL DESIGN STRATEGIES UNDER HIGH-DIMENSIONAL UNCERTAINTY

This study focuses on developing a computational framework for model-based design of the thermal insulation elements of net-zero buildings based on silica aerogel porous materials, ensuring they provide superinsulation while upholding structural integrity. This approach employs a multiphase continuum model, capturing the thermomechanical properties of the insulation component through a set of partial differential equations (PDE). The framework considers the uncertainty associated with both the physical parameters like elasticity and thermal conductivity for the solid and fluid phases, as well as the design parameter, which is the spatial distribution of the aerogel porosity over the domain of the component. The combination of spatially varying design and uncertainty parameters, along with their finite element discretization, results in a high dimensional PDE-constrained optimal design problem. A mean cost functional is implemented to achieve both target insulation performance and uncertainty reduction during the design process. To avoid stress concentration in the component, chance constraints are included in the optimization formulation, which ensures that the probability of a function that measures the difference between evaluated stress from the multiphase model and a critical threshold value lies within tolerance. A scalable method is introduced for solving PDE-constrained optimization under uncertainty that is both efficient and dimension-independent. For efficiency, this method exploits a second-order Taylor approximation of the design objective and chance constraint function, which solves a generalized eigenvalue problem. Combined with a gradient-based optimization built on Lagrangian formulation, it results in dimension-independent (scalable) computational costs. The numerical experiments on the design of thermal breaks of the buildings demonstrate that the proposed framework leads to a significant reduction in computational cost while preserving thermal insulation performance and avoiding mechanical failure due to stress concentration.