In this paper, we present a Bayesian framework for the identification of the parameters of nonlinear constitutive material laws using full-field displacement measurements. The concept of force-based Finite Element Model Updating (FEMU-F) is employed, which relies on the availability of measurable quantities such as displacements and external forces. The proposed approach particularly unfolds the advantage of FEMU-F, as opposed to the conventional FEMU, by directly incorporating information from full-field measured displacements into the model. This feature is well-suited for heterogeneous materials with softening, where the localization zone depends on the random microstructure. Besides, to account for uncertainties in the measured displacements, we treat displacements as additional unknown variables to be identified, alongside the constitutive parameters. A variational Bayesian scheme is then employed to identify these unknowns via approximate posteriors under the assumption of multivariate normal distributions. An optimization problem is then formulated and solved iteratively, aiming to minimize the discrepancy between true and approximate posteriors. The benefit of the proposed approach lies in the stochastic nature of the formulation, which allows to tackle uncertainties related to model parameters and measurement noise. We verify the efficacy of our proposed framework on two simulated examples using gradient damage model with a path-dependent nonlinear constitutive law. Based on a nonlocal equivalent strain norm, this constitutive model can simulate a localized damage zone representing softening and cracking. The first example illustrates an application of the FEMU-F approach to cracked structures including sensitivity studies related to measurement noise and parameters of the prior distributions. In this example, the variational Bayesian solver demonstrates a sizable advantage in terms of computational efficiency compared to a traditional least-square optimizer. The second example demonstrates a sub-domain analysis to tackle challenges associated with limited domain knowledge such as uncertain boundary conditions.

中文翻译：

---

一个贝叶斯框架，用于通过使用全场测量进行本构模型识别，并应用于异质材料


在本文中，我们提出了一个贝叶斯框架，用于使用全场位移测量来识别非线性本构材料定律的参数。采用了基于力的有限元模型更新 （FEMU-F） 的概念，它依赖于可测量量（如位移和外力）的可用性。与传统的 FEMU 相比，所提出的方法通过将全场测量位移的信息直接整合到模型中，特别展示了 FEMU-F 的优势。这一特征非常适合软化的异质材料，其中定位区取决于随机微观结构。此外，为了解决测量位移的不确定性，我们将位移与本构参数一起视为需要识别的附加未知变量。然后在多变量正态分布的假设下，采用变分贝叶斯方案通过近似后验来识别这些未知数。然后制定优化问题并迭代求解，旨在最小化真实和近似后验之间的差异。所提出的方法的优点在于公式的随机性，它允许解决与模型参数和测量噪声相关的不确定性。我们使用具有路径依赖非线性本构律的梯度损伤模型在两个模拟实例上验证了我们提出的框架的有效性。基于非局部等效应变模，该本构模型可以模拟表示软化和开裂的局部损伤区。 第一个例子说明了 FEMU-F 方法在裂纹结构中的应用，包括与测量噪声和先验分布参数相关的敏感性研究。在此示例中，与传统的最小二乘优化器相比，变分贝叶斯求解器在计算效率方面表现出相当大的优势。第二个示例演示了子域分析，以解决与有限域知识相关的挑战，例如不确定的边界条件。