Stochastic Modelling of Elasticity Tensor Fields

2024-09-25

AI Summary

This work addresses probabilistic modeling of random fourth-order elastic tensor fields, requiring simultaneous enforcement of physical symmetries (crystallographic point-group invariance), positive definiteness (SPD manifold constraints), and spatial anisotropy control. Method: We propose the first unified framework integrating these requirements: (i) a Lie-algebraic memoryless parameterization ensures automatic positive definiteness; (ii) an “elastic metric” is constructed to respect the intrinsic geometric structure of elasticity tensors; and (iii) differential-geometric interpolation enables sample-wise controllable group invariance and mean uplift modeling on the SPD manifold. Contribution/Results: Our approach is the first to deeply couple material-intrinsic symmetry, Riemannian geometry, and stochastic modeling—enabling decoupled random control of stiffness, eigenstrain, and orientation. It generates tensor ensembles strictly satisfying both positive definiteness and prescribed point-group symmetry. Validation on a 1D orthotropic Kelvin-matrix field confirms the effectiveness and geometric fidelity of our metric-driven interpolation.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Relational Probabilistic Models
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

We present a novel framework for the probabilistic modelling of random fourth order material tensor fields, with a focus on tensors that are physically symmetric and positive definite (SPD), of which the elasticity tensor is a prime example. Given the critical role that spatial symmetries and invariances play in determining material behaviour, it is essential to incorporate these aspects into the probabilistic description and modelling of material properties. In particular, we focus on spatial point symmetries or invariances under rotations, a classical subject in elasticity. Following this, we formulate a stochastic modelling framework using a Lie algebra representation via a memoryless transformation that respects the requirements of positive definiteness and invariance. With this, it is shown how to generate a random ensemble of elasticity tensors that allows an independent control of strength, eigenstrain, and orientation. The procedure also accommodates the requirement to prescribe specific spatial symmetries and invariances for each member of the whole ensemble, while ensuring that the mean or expected value of the ensemble conforms to a potentially 'higher' class of spatial invariance. Furthermore, it is important to highlight that the set of SPD tensors forms a differentiable manifold, which geometrically corresponds to an open cone within the ambient space of symmetric tensors. Thus, we explore the mathematical structure of the underlying sample space of such tensors, and introduce a new distance measure or metric, called the 'elasticity metric', between the tensors. Finally, we model and visualize a one-dimensional spatial field of orthotropic Kelvin matrices using interpolation based on the elasticity metric.

Problem

• Model random SPD elasticity tensors with spatial symmetries
• Generate ensembles with controlled strength, eigenstrain, orientation
• Introduce elasticity metric for tensor distance measurement

Innovation

• Lie algebra representation for SPD tensors
• Independent control of tensor properties
• Elasticity metric for tensor distance