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This paper introduces a computationally efficient framework for the optimal design of engineering systems governed by multiphysics, nonlinear partial differential equations (PDEs) and subject to high-dimensional spatial uncertainty. The focus is on 3D printed silica aerogel-based thermal break components in building envelopes, where the objective is to maximize thermal insulation performance while ensuring mechanical reliability by mitigating stress concentrations. Material porosity is modeled as a spatially correlated Gaussian random field, yielding a high-dimensional stochastic design space whose dimensionality corresponds to the mesh resolution after finite element discretization. A robust design objective is employed, incorporating statistical moments of the thermal performance metric and in conjunction with a probabilistic (chance) constraint that restricts the p-norm of the von Mises stress field below a critical threshold, effectively controlling stress concentrations across the domain. To alleviate the substantial computational burden associated with Monte Carlo estimation of statistical moments, a second-order Taylor series approximation is introduced as a control variate, significantly accelerating convergence. Furthermore, a continuation-based strategy is developed to regularize the non-differentiable chance constraints, enabling the use of an efficient gradient-based Newton–Conjugate Gradient optimization algorithm. The proposed framework achieves computational scalability that is effectively independent of the stochastic design space dimensionality. Numerical experiments on two- and three-dimensional thermal breaks in building insulation demonstrate the method’s efficacy in solving large-scale, PDE-constrained, chance-constrained optimization problems with uncertain parameter spaces reaching dimensions in the hundreds of thousands.
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A scalable framework for multi-objective PDE-constrained design of building insulation under uncertainty
This paper introduces a scalable computational framework for optimal design under high-dimensional uncertainty, with application to thermal insulation components. The thermal and mechanical behaviors are described by continuum multi-phase models of porous materials governed by partial differential equations (PDEs), and the design parameter, material porosity, is an uncertain and spatially correlated field. After finite element discretization, these factors lead to a high-dimensional PDE-constrained optimization problem. The framework employs a risk-averse formulation that accounts for both the mean and variance of the design objectives. It incorporates two regularization techniques, the L0-norm and phase field functionals, implemented using continuation numerical schemes to promote spatial sparsity in the design parameters. To ensure efficiency, the framework utilizes a second-order Taylor approximation for the mean and variance and exploits the low-rank structure of the preconditioned Hessian of the design objective. This results in computational costs that are determined by the rank of preconditioned Hessian, remaining independent of the number of uncertain parameters. The accuracy, scalability with respect to the parameter dimension, and sparsity-promoting abilities of the framework are assessed through numerical examples involving various building insulation components.
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- Award ID(s):
- 2143662
- PAR ID:
- 10484116
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Computer Methods in Applied Mechanics and Engineering
- Volume:
- 419
- Issue:
- C
- ISSN:
- 0045-7825
- Page Range / eLocation ID:
- 116628
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1016/j.cma.2023.116628
