Vector Space, norm, and angle – linear independence and orthonormal sets – row reduction and echelon forms, matrix operations, including inverses – effect of round-off error, operation counts – block/banded matrices arising from discretization of differential equations – linear dependence and independence – subspaces and bases and dimensions – orthogonal bases and orthogonal projections – Gram-Schmidt process – linear models and least-squares problems – eigenvalues and eigenvectors – diagonalization of a matrix – symmetric matrices – positive definite matrices – similar matrices – linear tr

Basic acoustic principles – acoustic terminology and definitions – plane and spherical wave prop- agation – theories of monopole, dipole and quadrapole sound sources – sound transmission and absorption – sound transmission through ducts – structure borne sound – sound radiation and structural response – introduction to noise control.

Introduction – linear programming – revised simplex method – duality and sensitivity analysis – dual simplex method – goal programming – integer programming – network optimization models – dynamic programming – nonlinear programming – unconstrained and constrained optimization – nontraditional optimization algorithms.

Properties of materials: strength, hardness, fatigue, and creep – Ferrous alloys: stainless steels, maraging steel, aging treatments – Aluminum alloys: alloy designation and tempers, Al-Cu alloys, principles of age hardening, hardening mechanisms, Al-Li alloys, Al-Mg alloys, nanocrystalline alu- minum alloys – Titanium alloys: α-β alloys, superplasticity, structural titanium alloys, intermetallics– Magnesium alloys: Mg-Al and Mg-Al-Zn alloys – Superalloys: processing and properties of super- alloys, single-crystal superalloys, environmental degradation and protective coatings – Composites:

Review of vibration of structural elements – one-dimensional motion in elastic media – discrete Fourier transform – spectral finite element method – standing waves – flexural waves in beams and plates – torsional waves in shafts – guided waves – structural health monitoring using wave propagation.

Basics of probability theory: axioms, definitions, random variable – probability structure of random variable – joint distributions – functions of random variables – some common random variables – random processes/random fields.

Structural reliability – fundamental concepts – first order reliability methods – second order reliability methods – probabilistic sensitivity – system reliability – simulation techniques – high dimensional model representation techniques for reliability analysis.

Linear elastic fracture mechanics; energy release rate, stress intensity factor (SIF), relation between SIF and energy release rate, anelastic deformation at the crack tip – J-integral, CTOD, test meth- ods for fracture toughness – crack growth and fracture mechanisms, mixed-mode fracture, fracture at nanoscale – numerical methods for analysing fracture, applications – fatigue and design against fatigue failure – prediction of fatigue life.

Introduction – materials deformation and fracture phenomena – strength of materials: flaws, defects, and a perfect material, brittle vs. ductile material behavior, the need for atomistic simulations – ap- plications basic atomistic modeling – classical molecular dynamics – interatomic potential-numerical implementation – visualisation – atomistic elasticity, the virial stress and strain – multiscale model- ing and simulation methods – deformation and dynamical failure of brittle and ductile materials – applications.

Finite element formulations for beam, plate, shell (Kirchhoff and Mindlin-Reissner), and solid ele- ments – large deformation nonlinearity – nonlinear bending of beams and plates – stress and strain measures – total Lagrangian and updated Lagrangian formulations – material nonlinearity – ideal and strain hardening plasticity – elastoplastic analysis – boundary nonlinearity – general contact formulations – solution procedures for nonlinear analysis, Newton-Raphson iteration method.

The variational principle and the derivation of the governing equations of static and dynamic systems – different energy methods: Rayleigh-Ritz, Galerkin etc. – applications: problems of stress analysis, determination of deflection in determinate and indeterminate structures, stability and vibrations of beams, columns and plates of constant and varying cross-sectional area.

Review of planar motion of rigid bodies and Newton-Euler equations of motion; constraints – holo- nomic and non-holonomic constraints, Newton-Euler equations for planar inter connected rigid bod- ies; D’Alembert’s principle, generalized coordinates; alternative formulations of analytical mechanics and applications to planar dynamics – Euler-Lagrange equations, Hamilton’s equations and ignorable coordinates, Gibbs-Appel and Kane’s equations; numerical solution of differential and differential al- gebraic equations; spatial motion of a rigid body – Euler angles, rotation matrices, quaternions

Review of tensor algebra – tensor analysis – concept of continuum – kinematics of a deformable body – deformation and strain – motion and flow – analysis of stress-stress tensors – conservation laws, mass and momentum conservation – continuum thermodynamics – first and second laws applied to a continuum – Clausius-Duhem inequality – constitutive relations – applications.