对称正定材料张量的随机建模
Stochastic Modelling of Symmetric Positive Definite Material Tensors
AI总结：
本文提出一个随机建模框架，用于生成尊重空间对称性和正定性的材料张量系综，通过李代数表示独立控制缩放与方向，并示例于股骨热传导模型，展示不同不确定性对结果的影响。
摘要：
空间对称性和不变性在材料行为中扮演着重要角色，在描述和建模材料属性时应予以尊重。本文关注的是物理上对称且正定的张量类别，因为它们经常出现在材料描述中，并且人们希望能够为整个系综中的每个成员指定特定类别的空间对称性和不变性，同时要求系综的均值或期望值可能服从于“更高”的空间不变性类别。我们构建了一个建模框架，该框架不仅尊重这两个要求——正定性和不变性——而且允许对方向进行精细控制，同时对强度/大小进行精细控制。由于正定张量的集合不是线性空间，而是物理对称张量线性空间中的一个开放凸锥，我们认为将均值的概念扩展为度量空间上的所谓Fréchet均值是有利的，该均值基于正定张量之间除通常欧几里得度量之外的距离度量或度量。我们展示了如何独立地在其缩放和方向或定向方面，通过无记忆变换的李代数表示来建模和生成随机系综。然后，描述该李代数中元素的参数被视为感兴趣域上的随机场。作为示例，在人类近端股骨（一种具有高材料各向异性的骨骼）中，使用随机热导率张量对二维和三维稳态热传导模型进行建模，数值结果表明，将不同的材料不确定性——缩放、方向和规定的材料对称性——纳入本构模型对所需感兴趣量的显著影响。
英文摘要：
Spatial symmetries and invariances play an important role in the behaviour of materials and should be respected in the description and modelling of material properties. The focus here is the class of physically symmetric and positive definite tensors, as they appear often in the description of materials, and one wants to be able to prescribe certain classes of spatial symmetries and invariances for each member of 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. We formulate a modelling framework which not only respects these two requirements$-$positive definiteness and invariance$-$but also allows a fine control over orientation on one hand, and strength/size on the other. As the set of positive definite tensors is not a linear space, but rather an open convex cone in the linear space of physically symmetric tensors, we consider it advantageous to widen the notion of mean to the so-called Fréchet mean on a metric space, which is based on distance measures or metrics between positive definite tensors other than the usual Euclidean one. It is shown how the random ensemble can be modelled and generated, independently in its scaling and orientational or directional aspects, with a Lie algebra representation via a memoryless transformation. The parameters which describe the elements in this Lie algebra are then to be considered as random fields on the domain of interest. 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 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.
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