Chance Constrained PDE-Constrained Optimal Design Strategies Under High-Dimensional Uncertainty

2025-01-03

AI Summary

Silicon aerogel porous insulation materials face challenges in thermal-bridge design for buildings due to high-dimensional uncertainties—particularly in porosity fields—making it difficult to simultaneously ensure thermal insulation performance and structural safety. Method: This paper proposes a robust thermo-mechanical coupled optimization framework. It innovatively integrates second-order Taylor expansion with Hessian low-rank approximation to achieve dimension-independent computation for chance-constrained PDE optimization. A risk-averse objective function based on statistical moments is coupled with probabilistic stress constraints. The approach employs multiphase continuum modeling within a gradient-based optimization framework. Results: Validated on 2D and 3D benchmarks, the method demonstrates high accuracy, strong scalability, and a key computational advantage: the number of PDE solves remains constant regardless of parameter dimensionality. It significantly enhances design reliability and engineering applicability under uncertainty.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search
• Constraint Satisfaction and Optimization: Constraint Optimization

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Energy management for devices in mobile Web and WoT environments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

This study presents an advanced computational framework for the optimal design of thermal insulation components in buildings, utilizing silica aerogel porous materials. The framework aims to achieve superior thermal insulation while maintaining structural integrity of the component under stress concentrations. A multiphase continuum model is employed to simulate the thermomechanical behavior of the insulation system, governed by a set of coupled partial differential equations (PDEs). The design process explicitly accounts for spatially correlated uncertainties in the design parameters, particularly the spatially varying aerogel porosity, resulting in a high-dimensional, PDE-constrained optimization under uncertainty. The optimization problem is formulated using a risk-averse cost functional to balance insulation performance with uncertainty mitigation, incorporating statistical moments of the design objective. Additionally, chance constraints are imposed to limit the probability of stress exceeding critical thresholds. To address the challenges arising from high-dimensional parameters, the optimization leverages a second-order Taylor expansion of both the design objective and the chance constraint functions, combined with a low-rank approximation of the Hessian matrix for efficient evaluation of the generalized eigenvalue problem. This approach supports scalable computations, either directly or as a variance-reduction control variate for Monte Carlo estimations. Combined with a gradient-based optimization approach, we achieve a scalable solution algorithm with dimension-independent computational costs in terms of number of PDE solved. Two- and three-dimensional numerical experiments on the design of thermal breaks in buildings showcase the features of the proposed design under uncertainty framework with respect to accuracy, scalability, and computational cost.

Problem

• Silica Aerogel
• Optimal Thermal Insulation
• Mechanical Durability

Innovation

• Advanced Mathematical Modeling
• Silica Aerogel in Building Design
• Performance Optimization