Thesis title: Direct and Inverse Problems in Mathematics and Mechanics of Metamaterials: Numerical algorithms and their computational implementation for determining model parameters and solving generalized continuum mechanics problems
Technological advances in the past decade in additive manufacturing has provided the opportunity to fabricate metamaterials which usually have detailed and complex microstructures. This has given rise to numerous studies towards investigating the metamaterials. In this work, direct and inverse mathematical problems in mechanics of metamaterials are studied. Utilizing generalized continuum mechanics for modeling metamaterials is an efficient approach which is capable of modeling the material with higher accuracy and lower computational cost. In generalized continuum mechanics, additional constitutive parameters are present which need to be identified. Herein, an automatized optimization algorithm is presented for parameter identification of metamaterials. The parameters are validated for the dynamic regime. Moreover, the strain gradient elasticity theory is numerically implemented using a mixed formulation, and it is verified by analytical solutions.
The thesis is comprised of two parts. In Part I, a summary of the fundamentals, mathematical approaches, theories of mechanics, and numerical methods and their implementation are discussed. The Part II, contains the original results of the present thesis, including two published journal articles, one accepted book chapter, and one manuscript submitted for publication.