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**Advisors:**Phanish Suryanarayana, Edmond Chow, John E. Pask, Glaucio H. Paulino, Arash Yavari

Development of new materials needs better understanding of the behavior of materials at nanoscale which involves accurate simulation of atomic and electronic interactions. Electronic structure is especially important when the atomic interactions involve breaking or formation of chemical bonds. When such interactions are present, first principles based ab-initio electronic structure calculations of atoms, which do not involve any empirical potentials, would be a suitable choice to study the behavior of materials at nanoscale. Such simulations involving many thousands of atoms are intractable by current software (especially for metals) due to their cubic scaling with respect to the system size. In this dissertation, the cubic scaling bottleneck is overcome by developing a linear scaling method amenable to massive parallelization. A linear scaling Density Functional Theory (DFT) framework has been developed using Clenshaw-Curtis Spectral Quadrature (SQ) method and implemented on massively parallel computers to simulate the electronic structure of hundreds of thousands of atoms. Finite difference representation has been employed in order to exploit the locality of electronic interactions in real space, enable systematic convergence and facilitate large-scale parallel implementation. In combination with linear scaling electrostatics, the electron density, energy and atomic forces can be calculated with effort that scales linearly with the number of atoms for both insulating and metallic systems. This method allows computation of the $Gamma$-point and infinite-cell calculations without resorting to Brillouin zone integration or large supercells. The method is validated and systematic convergence of energy and forces to the exact diagonalization result is demonstrated. Convergence with respect to mesh size to established cubic scaling planewave results has also been shown. The efficiency and suitability of the method for high temperature calculations is also discussed. Energy and forces for systems with many thousands of atoms have been computed. The parallel scaling of the method to more than hundred thousand processors has been studied. The extreme parallelizability demonstrated by the method promises the potential to make use of the next generation exascale computer architectures for scientific simulations. In the spirit of massive parallelizability and efficiency, new extrapolation techniques have been developed to accelerate the convergence of fixed point iterations. These techniques when applied to basic iterative methods give rise to efficient solvers for linear systems of equations. Robust and efficient performance of these methods is demonstrated in acceleration of the non-linear fixed point iteration that is used to solve the electronic structure problem. The SQ method enables simulation of very large systems of both metals and insulators under a unified framework, at high temperatures. It also enables performing ab-initio molecular dynamics simulations at high temperatures which is impractical using cubic-scaling codes. This method also provides the basis on which an accurate simulation of the mechanics of materials at nanoscale can be performed in multi-scale modeling studies using coarse graining techniques.

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MATIN Development Team