A variational multiscale computational framework for nonlinear interfacial solid mechanics

Abstract

This dissertation develops a robust computational framework for solving solid mechanics problems containing strong or weak discontinuities arising from interfaces. The framework consists of a stabilized version of the Discontinuous Galerkin method to handle interfaces combined with a stabilized mixed method for elasticity with a built-in error estimation module. The unifying approach for deriving these components is the Variational Multiscale (VMS) method, a guiding philosophy for recovering stability through the modeling of features that are filtered out by discretization. The enhanced stability of the framework enables the treatment of various interface kinematics, such as nonmatching meshes in domain decomposition or substructure modeling, contact and friction in mechanical systems, and delamination at bi-material interfaces in composites.

The common launching point for developing the components of this framework according to the VMS approach is an additive decomposition of the solution field into coarse scales and fine scales. This separation into numerical scales is an artifact of the discretization process whereby the coarse scales represented on the finite element mesh are unable to resolve the fine-scale features of the solution, and failure to account for the fine scales manifests instabilities in the computed results. In the case of interfaces, the source of instability is the discontinuity and/or nonconformity of the primary field along with the inf-sup condition governing the nontrivial selection of Lagrange multiplier interpolations. Likewise, the choice of displacement-pressure interpolations for mixed elasticity leads to instability in the discrete setting. In this work, models for these fine scales in the interior and along the interface are constructed using simple polynomial bubble functions. The variational embedding of these models serves as a vehicle for systematically deriving a robust interface framework that is mathematically consistent, admits common element types, and is free from user-defined tuning parameters. Additionally, the fine-scale models serve as a natural mechanism for estimating the numerical error, thereby providing built-in feedback on assessing confidence in the computed results. The underlying mathematical structure of the framework enables ready extension of the stabilization and error estimation methods to problems involving nonlinearity in the material and interface responses.