Variational Multiscale Discontinuous Galerkin Interface Method for High-Performance Homogenization and Uncertainty Quantification of Microstructures

In this dissertation, initially we studied the sources of nonlinearity and reliability of the previously developed combined crystal plasticity and cohesive zone (CZ) physics-based grain boundary model. For this purpose, we performed a comprehensive sensitivity analysis and uncertainty quantification of the model through advanced machine learned (ML) models and generalized polynomial chaos expansion method (gPCE). This study showed that the grain boundary parameters are the most sensitive ones in the model. This demonstrates a need for more sophisticated grain boundary models.

Next, we derived a thermodynamically consistent nonlinear crystal plasticity variational multiscale discontinuous Galerkin (VMDG) interface. Beginning with the Hu-Washizu principle and the generalized principle of maximum plastic dissipation, we showcase its thermodynamic consistency for unified elastoplastic behavior. Introducing an inner interface, we propose a Lagrange multiplier term augmented to the Hu-Washizu potential using a trace operator, preserving displacement continuity. Additionally, we included a Galerkin least square term (GLS) to recover the interface traction average and interface traction jump terms for stabilizing the finite element solution. These augmentations reveal that the Hu-Washizu variational principle can derive all governing equations for an interface problem as seen in Nitche and VMS methods. We then implemented these weak forms in MOOSE framework. To validate the robustness of the implementation, we performed various crystal plasticity VMDG interface problems.

The final chapter conducted a detailed comparative analysis to assess the effectiveness and computational efficiency of periodic boundary conditions and homogenization methods in nonlinear macro strain analysis. Using fluctuation fields and total displacement fields as field variables, the analyses employed volumetric integrals and interface integrals, respectively, to calculate homogenized macro stress and strains, yielding varied accuracies and high-performance efficiencies. Four methods Multipoint Constraint (MPC), Lagrange Multipliers (LM), Penalty Method, and Discontinuous Galerkin (DG) are employed, revealing that the DG method excels in accuracy for macro strain values. Additionally, the MPC method exhibits the lowest expected runtime. A crucial comparison involves iterative and direct solvers, highlighting that while iterative solvers are notably faster upon convergence, they face convergence challenges in methods with Lagrange multipliers. Conversely, direct solvers, though time-consuming, demonstrate consistent and robust convergence across different methods.