Spatial symmetries and invariances play an important role in the description of materials. When modelling material properties, it is important to be able to respect such invariances. Here we discuss how to model and generate random ensembles of tensors where one wants to be able to prescribe certain classes of spatial symmetries and invariances for the whole ensemble, while at the same time demanding that the mean or expected value of the ensemble be subject to a possibly 'higher' spatial invariance class. Our special interest is in the class of physically symmetric and positive definite tensors, as they appear often in the description of materials. As the set of positive definite tensors is not a linear space, but rather an open convex cone in the linear vector space of physically symmetric tensors, it may be advantageous to widen the notion of mean to the so-called Fr\'echet mean, which is based on distance measures between positive definite tensors other than the usual Euclidean one. For the sake of simplicity, as well as to expose the main idea as clearly as possible, we limit ourselves here to second order tensors. It is shown how the random ensemble can be modelled and generated, with fine control of the spatial symmetry or invariance of the whole ensemble, as well as its Fr\'echet mean, independently in its scaling and directional aspects. As an example, a 2D and a 3D model of steady-state heat conduction in a human proximal femur, a bone with high material anisotropy, is explored. It is modelled with a random thermal conductivity tensor, and the numerical results show the distinct impact of incorporating into the constitutive model different material uncertainties$-$scaling, orientation, and prescribed material symmetry$-$on the desired quantities of interest, such as temperature distribution and heat flux.

中文翻译：

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对称正定材料张量的随机建模

空间对称性和不变性在材料描述中起着重要作用。在对材料属性进行建模时，重要的是能够遵守此类不变性。在这里，我们讨论如何建模和生成张量的随机集合，我们希望能够为整个集合规定某些类别的空间对称性和不变性，同时要求集合的均值或期望值服从一个可能“更高”的空间不变性类。我们特别感兴趣的是物理对称和正定张量类，因为它们经常出现在材料描述中。由于正定张量集合不是线性空间，而是物理对称张量的线性向量空间中的开凸锥，将均值的概念扩展到所谓的 Fr\'echet 均值可能是有利的，它基于正定张量之间的距离度量，而不是通常的欧几里得张量。为了简单起见，并尽可能清楚地展示主要思想，我们将自己限制在二阶张量上。它展示了如何对随机集合进行建模和生成，并在其缩放和方向方面独立地对整个集合的空间对称性或不变性以及其 Fr\'echet 均值进行精细控制。例如，我们探索了人类股骨近端（具有高材料各向异性的骨骼）中稳态热传导的 2D 和 3D 模型。它是用随机热导率张量建模的，