基于可微分有限元的梯度驱动贝叶斯模型更新框架
GRADIENT-DRIVEN BAYESIAN MODEL UPDATING FRAMEWORK BASED ON DIFFERENTIABLE FINITE-ELEMENT METHOD
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摘要: 采用贝叶斯模型更新进行材料参数反演与不确定性量化, 在航空航天、土木、水利等领域中得到广泛应用. 然而, 传统有限元方法难以获取梯度值, 只能采用无梯度的贝叶斯方法, 在复杂问题中其推断效率与收敛性显著低于梯度驱动的贝叶斯方法. 为此, 本文提出一种基于可微分有限元的贝叶斯模型更新框架, 该框架采用自动微分和伴随法的混合策略获取梯度, 具有低内存占用、高通用性等优势. 在进行正向有限元计算的同时高效获取有限元响应对模型参数的梯度值, 并与No-U-Turn Sampler (NUTS) 耦合, 实现了有限元正演与梯度驱动贝叶斯推断的一体化结合. 分别以位移和内力响应为观测数据构造似然函数进行参数反演, 并通过两个经典力学算例对所提框架进行了验证. 结果表明, 该框架具有高效的参数推断能力和强鲁棒性, 其采样效率与可处理的参数维度均显著优于无梯度方法. 该框架通过将可微分有限元与梯度驱动推断深度结合, 在保留有限元高精度求解优势的同时, 显著提升了复杂问题下的模型更新效率与收敛性.Abstract: Bayesian model updating has been widely applied for material parameter inversion and uncertainty quantification in aerospace, civil, and hydraulic engineering. However, gradient information was prohibitively expensive to obtain from conventional finite-element method. As a result, the reliance on gradient-free Bayesian methods was necessitated. The inference efficiency and convergence of these methods were known to degrade substantially in complex problems when compared with gradient-driven alternatives. To address this limitation, a Bayesian model updating framework based on the differentiable finite-element method was developed. A hybrid automatic-differentiation–adjoint strategy was adopted to obtain gradients, offering low memory consumption and high generality. This strategy enabled the differentiable finite-element method to compute gradients of structural responses with respect to model parameters efficiently alongside forward solutions. These exact gradients were supplied to the No-U-Turn Sampler (NUTS), by which an integrated coupling between gradient-driven Bayesian inference and finite-element forward modelling was established. Likelihood functions were constructed using displacement and internal force responses, respectively. The proposed framework was validated through two classical mechanics examples, where high accuracy in parameter inference and strong robustness were observed. Both the sampling efficiency and the number of parameters that could be handled simultaneously were found to substantially exceed those of gradient-free methods. Through the deep integration of gradient-driven inference with the differentiable finite-element method, the proposed framework substantially improved the efficiency and convergence of model updating in complex problems, while preserving the high-fidelity solution capability of the finite-element method.
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