Forward and inverse problems in mechanics: from a single to thousands of interacting bodies

Abstract

Mechanics is the branch of physics that characterizes how bodies deform in response to forces, which can involve two categories of problems. In forward problems, one seeks to predict the response of the system given full knowledge of its physical and geometric properties. In inverse problems, some of the physical or geometric properties of the system are unknown and are either to be identified through experiments, or to be designed to optimize a desired objective. Although the physical laws governing the deformation of single elastic bodies have been known for over a century, forward problems involving thousands of interacting elastic bodies still elude simple and accurate models, while inverse problems involving even a single elastic body lack effective solution methods.

In this thesis, we investigate forward and inverse problems in systems ranging from a single elastic body to thousands of interacting ones. In a first part, we derive analytically a model for the contact force between elastically anisotropic bodies. We then implement this contact model into a computational framework for the forward dynamics of systems composed of hundreds of interacting bodies, which we leverage to showcase examples where the elastic anisotropy of each body affects the macroscopic behavior of the system. In a second part, we derive a homogenized continuum model to predict the forward dynamics of granular materials consisting of millions of interacting elastic particles, such as sand, with a particular focus on the accurate description of the onset and arrest of flow in response to external loading variations. Besides its predictive abilities, this model also sheds light on the physical mechanisms responsible for various unique features of avalanches and landslides such as their large initial acceleration. In a last part, we propose a topology optimization framework for the inverse problem of identifying hidden voids or rigid inclusions in an elastic body using measurements of the surface deformation in response to a prescribed surface loading. This framework combines recent advances in machine learning with level-set methods and the equations governing the deformation of single elastic bodies. We demonstrate the effectiveness of our method in identifying the number, locations, and shapes of hidden voids and rigid inclusions in elastic and hyperelastic materials.