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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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HIGH-DIMENSIONAL STOCHASTIC DESIGN OPTIMIZATION UNDER DEPENDENT RANDOM VARIABLES BY A DIMENSIONALLY DECOMPOSED GENERALIZED POLYNOMIAL CHAOS EXPANSION
Newly restructured generalized polynomial chaos expansion (GPCE) methods for high-dimensional design optimization in the presence of input random variables with arbitrary, dependent probability distributions are reported. The methods feature a dimensionally decomposed GPCE (DD-GPCE) for statistical moment and reliability analyses associated with a high-dimensional stochastic response; a novel synthesis between the DD-GPCE approximation and score functions for estimating the first-order design sensitivities of the statistical moments and failure probability; and a standard gradient-based optimization algorithm, constructing the single-step DD-GPCE and multipoint single-step DD-GPCE (MPSS-DD-GPCE) methods. In these new design methods, the multivariate orthonormal basis functions are assembled consistent with the chosen degree of interaction between input variables and the polynomial order, thus facilitating to deflate the curse of dimensionality to the extent possible. In addition, when coupled with score functions, the DD-GPCE approximation leads to analytical formulae for calculating the design sensitivities. More importantly, the statistical moments, failure probability, and their design sensitivities are determined concurrently from a single stochastic analysis or simulation. Numerical results affirm that the proposed methods yield accurate and computationally efficient optimal solutions of mathematical problems and design solutions for simple mechanical systems. Finally, the success in conducting stochastic shape optimization of a bogie side frame with 41 random variables demonstrates the power of the MPSS-DD-GPCE method in solving industrial-scale engineering design problems.
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- Award ID(s):
- 2317172
- PAR ID:
- 10628786
- Publisher / Repository:
- Begell House
- Date Published:
- Journal Name:
- International Journal for Uncertainty Quantification
- Volume:
- 13
- Issue:
- 4
- ISSN:
- 2152-5080
- Page Range / eLocation ID:
- 23 to 59
- 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/10.1615/Int.J.UncertaintyQuantification.2023043457
