Stochastic Modelling of Material Tensor Fields

COLLOQUIUM TALK

Abstract: The computational modelling of heterogeneous materials often has to resort to a statistical or probabilistic description, as the precise details are unknown or uncertain. 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. As the material properties are usually collected in tensor quantities, the focus here is the class of even-order physically symmetric and positive definite (SPD) tensors. This talk introduces procedures for modelling, especially second and fourth-order, tensor-valued material tensor fields using a Lie algebra representation via a memoryless transformation. Well known examples of such SPD tensors are e.g., thermal conductivity, a 2nd order tensor mapping temperature gradients to thermal fluxes, and the elasticity tensor of a linearly elastic material as a 4th order tensor, mapping strains to stresses. With this, it is shown how to generate a random ensemble of material tensors that allows an independent control of strength and spatial orientation in second-order tensors and, similarly, allows a separation of strength, eigen-strain, and orientation in fourth-order tensors. Furthermore, as the set of positive definite tensors is not a linear space but an open convex cone in the linear space of symmetric tensors, the notion of mean to the so-called Fréchet mean on a metric space is explored.

Speaker Bio: Sharana Kumar Shivanand is working at the Karlsruhe Institute of Technology (KIT), Germany. Before joining KIT, he worked at The Alan Turing Institute and the University of Cambridge, United Kingdom. He earned his Ph.D. in Computational Sciences in Engineering from the Technical University of Braunschweig, Germany. His research focuses on statistical and probabilistic methods for physics-informed and data-driven numerical modeling. He works at the intersection of uncertainty quantification, computational physics, and machine learning, addressing challenges in materials science, mechanics, and related engineering domains.