Multi-scale processes governed on each scale by separate principles for evolution or equilibrium are coupled by matching the stored energy and dissipation in line with the Hill-Mandel principle. We are interested in cementitious materials, and consider here the macro- and meso-scale behaviour of such a material. The accurate representations of stored energy and dissipation are essential for the depiction of irreversible material behaviour, and here a Bayesian approach is used to match these quantities on different scales. This is a probabilistic upscaling and as such allows to capture, among other things, the loss of resolution due to scale coarsening, possible model errors, localisation effects, and the geometric and material randomness of the meso-scale constituents in the upscaling. On the coarser (macro) scale, optimal material parameters are estimated probabilistically for certain possible behaviours from the class of generalised standard material models by employing a nonlinear approximation of Bayes’s rule. To reduce the overall computational cost, a model reduction of the meso-scale simulation is achieved by combining unsupervised learning techniques based on a Bayesian copula variational inference with functional approximation forms.

中文翻译：

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基于能量考虑的贝叶斯随机多尺度分析

通过按照Hill-Mandel原理匹配存储的能量和耗散，将通过单独的演化或平衡原理在各个范围上进行控制的多尺度过程耦合在一起。我们对胶结材料感兴趣，在这里考虑这种材料的宏观和中观行为。储能和耗散的准确表示对于描述不可逆的材料行为至关重要，这里使用贝叶斯方法在不同尺度上匹配这些数量。这是一个概率放大，因此可以捕捉到由于缩放粗化导致的分辨率损失，可能的模型误差，定位效应以及放大中的中尺度成分的几何和材料随机性。在较粗的（宏观）范围内，通过使用贝叶斯规则的非线性逼近，从广义标准材料模型的类别中概率地估计某些特定行为的最佳材料参数。为了降低总体计算成本，通过将基于贝叶斯copula变分推论的无监督学习技术与函数逼近形式相结合，实现了中尺度模拟的模型简化。