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SUMMARYA multiscale computational framework is presented that provides a coupled self-consistent system of equations involving molecular mechanics at small scales and quasi-continuum mechanics at large scales. The proposed method permits simultaneous resolution of quasi-continuum and atomistic length scales and the associated displacement fields in a unified manner. Interatomic interactions are incorporated into the method through a set of analytical equations that contain nanoscale-based material moduli. These material moduli are defined via internal variables that are functions of the local atomic configuration parameters. Point defects like vacancy defects in nanomaterials perturb the atomic structure locally and generate localized force fields. Formation energy of vacancy is evaluated via interatomic potentials and minimization of this energy leads to nanoscale force fields around defects. These nanoscale force fields are then employed in the multiscale method to solve for the localized displacement fields in the vicinity of vacancies and defects. The finite element method that is developed based on the hierarchical multiscale framework furnishes a two-level statement of the problem. It concurrently feeds information at the molecular scale, formulated in terms of the nanoscale material moduli, into the quasi-continuum equations. Representative numerical examples are shown to validate the model and demonstrate its range of applicability.

This paper presents B-splines and nonuniform rational B-splines (NURBS)-based finite element method for self-consistent solution of the Schrödinger wave equation (SWE). The new equilibrium position of the atoms is determined as a function of evolving stretching of the underlying primitive lattice vectors and it gets reflected via the evolving effective potential that is employed in the SWE. The nonlinear SWE is solved in a self-consistent fashion (SCF) wherein a Poisson problem that models the Hartree and local potentials is solved as a function of the electron charge density. The complex-valued generalized eigenvalue problem arising from SWE yields evolving band gaps that result in changing electronic properties of the semiconductor materials. The method is applied to indium, silicon, and germanium that are commonly used semiconductor materials. It is then applied to the material system comprised of silicon layer on silicon–germanium buffer to show the range of application of the method.

This paper presents two stabilized formulations for the Schrödinger wave equation. First formulation is based on the Galerkin/least-squares (GLS) method, and it sets the stage for exploring variational multiscale ideas for developing the second stabilized formulation. These formulations provide improved accuracy on cruder meshes as compared with the standard Galerkin formulation. Based on the proposed formulations a family of tetrahedral and hexahedral elements is developed. Numerical convergence studies are presented to demonstrate the accuracy and convergence properties of the two methods for a model electronic potential for which analytical results are available.