Variational Multiscale Based Multiresolution Discontinuous Galerkin Framework for Interface Problems

The dissertation presents a novel homogenization-based finite element framework, Multiresolution Discontinuous Galerkin (MRDG), for modeling microstructural material environments in the presence of externally applied loads and internally applied loading conditions, through the inclusion of traction jumps and displacement jumps along material interfaces. The MRDG framework offers variationally consistent and stable solutions to the proposed boundary value problem (BVP) using the Lagrange Multiplier method, Discontinuous Galerkin method, and Variational Multiscale Analysis-based Nitsche method. The main objective of the framework is to provide a stabilized solution for a wider range of interface problems that need accommodation of external loading from sources similar to the presence of multiple electromagnetic fields, thermomechanical, or similar fields working on an object simultaneously in the presence of heterogeneity. This novel capability of the framework enabled the numerical implementation of different types of kinematically admissible boundary conditions in a micromechanical environment in a unified manner and facilitates the estimation of mesoscale grain-level response measurement in a computationally cost-effective manner. The framework is then further extended to nonlinear problems through the derivation of two novel BVPs using variational methods – this extension opens up promises of using the framework in wider ranges of interface problems, including nondifferential interfacial stress-strain relations in the presence of externally applied traction and displacement jumps. The framework for the linear and nonlinear aspects of MRDG has been developed, and the algorithmic aspects of the developed models are studied through a series of numerical examples. The developed BVPs and their numerical studies demonstrate that the framework fills a critical gap in understanding the driving forces behind crack nucleation and prediction of fatigue behavior in structural materials, and it also opens its prospective usage in analyzing metallic responses in the presence of multiple electromagnetic fields, robotic surgery, optical trapping, or similar multifield loading situations.

Researchers are continually developing finite element methods to solve new types of solid mechanics problems with lower computational costs. The dissertation presents a novel computational framework, Multi-Resolution Discontinuous Galerkin (MRDG), for modeling microstructural material environments in the presence of externally applied loads and internally applied loading conditions, including traction jumps and displacement jumps along material interfaces. This framework is designed to solve new solid mechanics-related problems with reduced computational costs. The MRDG framework is a homogenization-based finite element framework that offers the features of both the discontinuous Galerkin and variational multiscale approaches (VMS) through decomposing the balance of force and displacement jumps along grain boundaries into contributions from the granular uniform field and fluctuation field within a small material region called a representative volume element (RVE). The framework for the linear aspects of MRDG has been developed, and the algorithmic aspects of the developed models are studied through a series of numerical examples. The framework is then further extended to nonlinear problems through the development of numerical schemes, element routines, and the testing of these numerical methods. The successful implementation of this MRDG framework fills a critical gap in understanding the driving forces behind crack nucleation, enabling the prediction of fatigue behavior in these structural materials.