The Statistical Finite Element Method (statFEM) offers a Bayesian framework for integrating computational models with observational data, thus providing improved predictions for structural health monitoring and digital twinning. This paper presents an efficient sampling-free statFEM tailored for non-conjugate, non-Gaussian prior probability densities. We assume that constitutive parameters, modeled as weakly stationary random fields, are the primary source of uncertainty and approximate them using Karhunen-Lo\`eve (KL) expansion. The resulting stochastic solution field, i.e., the displacement field, is a non-stationary, non-Gaussian random field, which we approximate via Polynomial Chaos (PC) expansion. The PC coefficients are determined through projection using Smolyak sparse grids. Additionally, we model the measurement noise as a stationary Gaussian random field and the model misspecification as a mean-free, non-stationary Gaussian random field, which is also approximated using KL expansion. The coefficients of the KL expansion are treated as hyperparameters. The PC coefficients of the stochastic posterior displacement field are computed using the Gauss-Markov-K\'alm\'an filter, while the hyperparameters are determined by maximizing the marginal likelihood. We demonstrate the efficiency and convergence of the proposed method through one- and two-dimensional elastostatic problems.

翻译：统计有限元法（statFEM）提供了一个贝叶斯框架，用于整合计算模型与观测数据，从而为结构健康监测和数字孪生提供改进的预测。本文提出一种针对非共轭、非高斯先验概率密度的高效免采样统计有限元法。我们假设本构参数（建模为弱平稳随机场）是主要的不确定性来源，并使用Karhunen-Loève（KL）展开对其进行近似。由此得到的随机解场（即位移场）是一个非平稳、非高斯随机场，我们通过多项式混沌（PC）展开对其进行近似。PC系数通过Smolyak稀疏网格的投影方法确定。此外，我们将测量噪声建模为平稳高斯随机场，将模型误设建模为零均值的非平稳高斯随机场（同样使用KL展开近似）。KL展开的系数被视为超参数。随机后验位移场的PC系数通过高斯-马尔可夫-卡尔曼滤波器计算，而超参数则通过最大化边缘似然函数确定。我们通过一维和二维弹性静力学问题验证了所提方法的效率与收敛性。