Complex materials such as soft tissues exhibit nonlinear anisotropic response and heterogeneous mechanical properties. Data-driven methods have been recently developed to capture the rich mechanical behaviour of these materials under extreme deformations. In particular, we have contributed to the field by leveraging neural ordinary differential equations (NODEs) as the building blocks of strain energy density functions that automatically satisfy polyconvexity, objectivity, material symmetry and positive energy dissipation requirements for realistic and physically plausible material models. However, these data-driven models have lacked consideration of uncertainty. This is particularly problematic for soft tissues which exhibit a large variation in mechanical properties from one individual to another. Here we establish a generative modelling framework based on stable diffusion to model distributions of materials while satisfying physics constraints. We use NODEs to describe the material response. Because the NODE framework automatically satisfies the desired physics, any samples of parameters of the NODE produces feasible materials. For a given material of interest e.g. skin, we assume that stress-strain curves from the population are available. Fitting a subset of the NODE parameters to the stress-strain data yields samples over the parameter space of the NODEs. Diffusion probabilistic models are then employed to learn that distribution over these NODE parameters and, implicitly, over the constitutive models. We showcase the ability of the framework to learn the distribution of material behaviour for both synthetic examples and murine skin data, outperforming standard density estimation techniques. We anticipate that this work will further establish the use of data-driven methods for materials that exhibit large variation across a population for which uncertainty quantification is essential.
This is joint work with Vahidullah Tac, Manuel Rausch, Ilias Bilionis, and Francisco Sahli Costabal.
This video was produced by the Isaac Newton Institute, as part of the workshop Mathematical mechanical biology: old school and new school, methods and applications, forming part of the programme Uncertainty quantification and stochastic modelling of materials.
