# Courses

## MAE Graduate Courses

**501 (APC 501) Mathematical Methods of Engineering Analysis I** A complementary presentation of theory, analytical methods, and numerical methods for the solution of problems in physics and engineering. Topics include an introduction to functional analysis, linear spaces and linear operators, including matrices, eigenproblems and Sturm-Liouville theory; basic ordinary differential equation (ODE) theory, Green’s functions for the solution of linear ODEs and Poisson’s equation, and the calculus of variations.

**502 Mathematical Methods of Engineering Analysis II** A continuation of MAE 501. Complex variables, including contour integration using residues and conformal mapping; Fourier and Laplace transforms; linear partial differential equations (hyperbolic, parabolic and elliptic PDEs) and solution methods; traveling waves for nonlinear PDE; an introduction to numerical methods for ODEs and PDEs, and regular perturbation methods.

**506 (APC 524) Software Engineering for Scientific Computing** The goal of this course is to teach basic tools and principles of writing good code, in the context of scientific computing. Specific topics include an overview of relevant compiled and interpreted languages, build tools and source managers, design patterns, design of interfaces, debugging and testing, profiling and improving performance, portability, and an introduction to parallel computing in both shared memory and distributed memory environments. The focus is on writing code that is easy to maintain and share with others. Students will develop these skills through a series of programming assignments and a group project.

**507 (APC 523) Numerical Algorithms for Scientific Computing** A broad introduction to numerical algorithms used in scientific computing. The course will begin with a review of the basic principles of numerical analysis, including sources of error, stability and convergence of algorithms. The theory and implementation of techniques for linear and nonlinear systems of equations, and ordinary and partial differential equations will be covered in detail. Examples of the application of these methods to problems in physics, astrophysics and other disciplines will be given. Issues related to the implementation of efficient algorithms on modern high-performance computing systems will be discussed.

**509, 510 Advanced Topics in Engineering Mathematics I, II** Selected topics in mathematical methods, with an emphasis on advances relevant to research activities represented in the department. Possible topics include analytical methods for differential equations, numerical solution of hyperbolic equations, and statistical methods.

**511 Experimental Methods: Introduction to Electronics for Engineering and Science** A laboratory course that focuses on basic electronics techniques, digital electronics, and data acquisition and analysis. Topics include introduction to digital and analog electronics, digital-to-analog and analog-to-digital conversion, microcomputer sampling, and data analysis. There are four laboratory hours and two lecture hours per week. There is one project. Enrollment is limited.

**512 Experimental Methods II** An exploration of experimental techniques in fluid mechanics and combustion. The course introduces experimentation, error analysis, and technical communication. Methods covered include pressure and temperature probes, flow visualization, hot-wire and laser anemometry, line reversal, Raman techniques, fluorescence, absorption, gas chromatography, and mass spectroscopy. There are three lecture hours and laboratory time per week.

**513, 514 Independent Project I, II** Directed study for Master of Engineering students. The topic is proposed by the student and must be approved by the student’s research advisor and received approval from the MAE Graduate Committee

**519, 520 Advanced Topics in Experimental Methods I, II** Selected topics in experimental methods, with an emphasis on advances relevant to research activities represented in the department. Possible topics include dynamic data analysis; instrumentation and systems analysis, scanning probe techniques, and nanoscale materials property measurements.

**521 Optics and Lasers** An introduction to principles of lasers. Topics include a review of propagation theory, interaction of light and matter, Fourier optics, a survey and description of operational characteristics of lasers, light scattering, and nonlinear optics. Some introductory quantum mechanics will be covered to give students an appreciation of the basic tools for the interaction of light with matter and nonlinear optical phenomena.

**522 (AST564) Applications of Quantum Mechanics to Spectroscopy and Lasers** An intermediate-level course in applications of quantum mechanics to modern spectroscopy. The course begins with an introduction to quantum mechanics as a “tool” for atomic and molecular spectroscopy, followed by a study of atomic and molecular spectra, radiative, and collisional transitions, with the final chapters dedicated to plasma and flame spectroscopic and laser diagnostics. Prerequisite: one semester of quantum mechanics. (Offered in alternate years)

**523 Electric Propulsion** Based on a review of pertinent atomic physics and electromagnetic theory, the particle and continuum representations of ionized gas dynamics are developed and applied to various electro-thermal, electrostatic, and electromagnetic acceleration mechanisms, each illustrated by various thruster designs, contemporary applications, and performances. (Offered in alternate years)

**524 Plasma Engineering** The purpose of this course is to expose interested graduate and undergraduate students in engineering and the natural sciences to basic aspects of plasma physics and chemistry applicable to a variety of technologies, such as plasma propulsion, lasers, and materials processing. It involves extension of classical fluid mechanics, kinetic theory, statistical thermodynamics, and reaction engineering methods to relatively-low-temperature plasmas in electric and magnetic fields. (Offered in alternate years)

**525 (AST 551) General Plasma Physics I** Characterization of the plasma state, Debye length, plasma and cyclotron frequencies, collision rates and mean free paths, atomic processes, adiabatic invariance, orbit theory, magnetic confinement of single charged particles, two-fluid description, magnetohydrodynamic waves and instabilities, heat flow, diffusion, finite-pressure effects, kinetic description, and Landau damping.

**527 Physics of Gases** Physical and chemical topics of basic importance in modern fluid mechanics, plasma dynamics, and combustion science: statistical calculations of thermodynamic properties of gases; chemical and physical equilibria; adiabatic temperatures of complex reacting systems; quantum mechanical analysis of atomic and molecular structure and atomic-scale collision phenomena; transport properties; reaction kinetics, including chemical, vibrational, and ionization phenomena; and propagation, emission, and absorption of radiation.

**528 (AST 566) Physics of Plasma Propulsion** Focus of this course is on fundamental processes in plasma thrusters for spacecraft propulsion with emphasis on recent research findings. Start with a review of the fundamentals of mass, momentum & energy transport in collisional plasmas, wall effects, & collective (wave) effects, & derive a generalized Ohm’s law useful for discussing various plasma thruster concepts. Move to detailed discussions of the acceleration & dissipation mechanisms in Hall thrusters, mangetoplasmadynamic thrusters, pulsed plasma thrusters, & inductive plasma thrusters, & derive expressions for the propulsive efficiencies of each of these concepts.

**529,530 Advanced Topics in Applied Physics I, II** Selected topics in applied physics, with an emphasis on advances relevant to research activities represented in the department. Possible topics include advanced plasma propulsion, linear and nonlinear wave phenomena, and x-ray lasers in biological investigations.

**531 Combustion** Fundamentals of combustion: thermodynamics; chemical kinetics; explosive and general oxidative characteristics of fuels; premixed and diffusion flames; laminar and turbulent flame phenomena; ignition and flame stabilization; detonation, environmental combustion considerations; and coal combustion.

**532 Combustion Theory** Theoretical aspects of combustion: the conservation equations of chemically-reacting flows; activation energy asymptotics; chemical and dynamic structures of laminar premixed and non-premixed flames; aerodynamics and stabilization of flames; pattern formation and geometry of flame surfaces; ignition, extinction, and flammability phenomena; turbulent combustion; boundary layer combustion; droplet, particle, and spray combustion; and detonation and flame stabilization in supersonic flows.

**534 Energy Storage Systems** This is a survey course on energy storage systems with a focus on electrochemical energy storage. Fundamentals of thermodynamics will be reviewed and fundamentals of electrochemistry introduced. These fundamentals will then be applied to de-vices such as batteries, flywheels and compressed air storage. Device optimization with respect to energy density, power density, cycle life and capital cost will be considered.

**536 (MSE 586) Synchrotron and Neutron Techniques for Energy Materials **Topics include an introduction to radiation generation at synchrotron and neutron facilities, elastic scattering techniques, inelastic scattering techniques, imaging and spectroscopy. Specific techniques include X-ray and neutron diffraction, small-angle scattering, inelastic neutron scattering, reflectometry, tomography, microscopy, and X-ray absorption spectroscopy. Emphasis is placed on data analysis and use of Fourier transforms to relate structure/dynamics to experiment data. Example materials covered include energy storage devices, sustainable concrete, CO2 storage, magnetic materials, mesostructured materials and nanoparticles.

**539, 540 Advanced Topics in Combustion I, II** Selected topics in theoretical and experimental combustion, with an emphasis on advances relevant to research activities represented in the department. Possible topics include turbulent combustion, theoretical calculations of rate constants, plasma fuels and natural resources, and nuclear propulsion power plants.

**541 (APC 571) Applied Dynamical Systems** Phase-plane methods and single-degree-of-freedom nonlinear oscillators; invariant manifolds, local and global analysis, structural stability and bifurcation, center manifolds, and normal forms; averaging and perturbation methods, forced oscillations, homoclinic orbits, and chaos; and Melnikov’s method, the Smale horseshoe, symbolic dynamics, and strange attractors. (Offered in alternate years)

**542 Advanced Dynamics** Principles and methods for formulating and analyzing mathematical models of physical systems; Newtonian, Lagrangian, and Hamiltonian formulations of particle and rigid and elastic body dynamics; canonical transformations, Hamilton-Jacob-Jacobi Theory; and integrable and non-integrable systems. Additional topics are explored at the discretion of the instructor.

**543 Advanced Orbital Mechanisms** An advanced course in orbital motion of earth satellites, interplanetary probes, and celestial mechanics. Topics include orbit specification, orbit determination, Lambert’s problem, Hill’s equations, intercept and rendezvous, air-drag and radiation pressure, Lagrange points, numerical methods, general perturbations and variation of parameters, earth-shape effects on orbits, Hamiltonian treatment of orbits, Lagrange's planetary equations, orbit resonances, and higher-order perturbation effects. (Offered in alternate years)

**544 Nonlinear Control** Nonlinear control of dynamical systems, with an emphasis on the geometric approach. The course gives an introduction to differential geometry, nonlinear controllability and constructive controllability, nonlinear observability, state-space transformations and stability, followed by study of a selection of nonlinear control design methods, including techniques motivated by geometric mechanics.

**545 Special Topics (Fall 2015)- Lessons from Biology for Engineering Tiny Devices -**Over millions of years of evolution nature invented many tiny sensors, machines and structures that are important for functions of cells and organisms. In this course we present a survey of problems at the interface of statistical mechanics, biology and engineering to discuss how cells move around, transport cargo and separate their genome during division, how microorganisms sense and swim in response to changing environment, how viruses assemble and infect other cells, how organs, such as brain and gut, obtain their shape, how animals obtain structural colors and how they camouflage, etc. Using this knowledge we study how to engineer and self-assemble tiny devices with DNA origami, how to design thin structures that can transform into specific shapes in response to external stimulus, how to make metamaterials with unusual properties, etc. (Offered in alternate years)

**546 Optimal Control and Estimation** An introduction to stochastic optimal control theory and application. It reviews mathematical foundations and explores parametric optimization, conditions for optimality, constraints and singular control, numerical optimization, and neighboring-optimal solutions. Least-squares estimates, propagation of state estimates and uncertainty, and optimal filters and predictors; optimal control in the presence of uncertainty; certainty equivalence and the linear-quadratic-Gaussian regulator problem; frequency-domain solutions for linear multivariable systems and robustness of closed-loop control are all studied.

**547 (ELE521) Linear System Theory** Advanced topics in linear system analysis. The course gives a review of linear vector spaces and differential equations. It covers characterization of continuous and discrete time linear systems, transfer functions and state-space representations, properties of transition matrices, observability and controllability, minimal realizations, stability, feedback, and pole assignment.

**548 (ELE 523) Nonlinear System Theory** Mathematical techniques useful in the analysis and design of nonlinear systems. This course covers topics in nonlinear dynamical systems including qualitative behavior, Lyapunov stability, input-output stability, passivity, averaging and singular perturbations. (offered in alternate years)

**549, 550 Advanced Topics in Dynamics and Control I, II** Selected topics in dynamics and control, with an emphasis on advances relevant to research activities represented in the department. Possible topics include bifurcation theory, nonlinear mechanics, system identification, intelligent control, learning control, and applied aerodynamics.

**551 Fluid Mechanics** An introduction to fluid mechanics. The course explores the development of basic conservation laws in integral and differential forms: one-dimensional compressible flows, shocks and expansion waves; effects of energy addition and friction; unsteady and two-dimensional flows and method of characteristics. Reviews classical incompressible flow concepts, including vorticity, circulation, and potential flows. Introduces viscous and diffusive phenomena.

**552 Viscous Flows and Boundary Layers** The mechanics of viscous flows. The course explores the kinematics and dynamics of viscous flows; solution of the Navier Stokes equations; the behavior of vorticity; the boundary layer approximation; laminar boundary layer with and without pressure gradient; separation; integral relations and approximate methods; compressible laminar boundary layers; instability and transition; and turbulent boundary layers and self-preserving turbulent shear flows.

**553 Turbulent Flow** Physical and statistical descriptions of turbulence; and a critical review of phenomenological theories for turbulent flows. The course examines scales of motion; correlations and spectra; homogeneous turbulent flows; inhomogeneous shear flows; turbulent flows in pipes and channels; turbulent boundary layers; calculation methods for turbulent flows (Reynolds stress equations, LES, DNS); and current directions in turbulence research.

**555 Non-Equilibrium Gasdynamics** Noncontinuum description of fluid flow and Liouville and Boltzmann equations. The course examines molecular collisions; detailed balancing; Chapman-Enskog expansion for near-equilibrium flows; transport phenomena; flows with translational, vibrational and chemical non-equilibrium; shock structure; and shear and mixing layers with chemical reactions.

**557 Simulation and Modeling of Fluid Flows** Numerical methods are applied to solve the equations that govern fluid motion. Fluid flow problems involve convection, diffusion, and source terms. The governing equations are non-linear and coupled. Finite-difference and finite volume methods are considered, together with concepts of accuracy, consistency, stability, convergence, conservation, and shock capturing. A range of current methods is reviewed with emphasis on multidimensional steady and unsteady compressible flows. Homework topics include writing codes to solve the conservation equation for a scalar, boundary layer flow, shock tube flow, application to curvilinear coordinates.

**559 Advanced Topics in Fluid Mechanics** Selected topics in fluid mechanics, with an emphasis on advances relevant to research activities represented in the department. Possible topics include advanced computational fluid dynamics, turbulence in fluids and plasmas, hydrodynamic stability, low Reynolds number hydrodynamics, and capillary phenomena.

**561 (MSE 501) Introduction to Materials** Emphasizes the connection between microstructural features of materials (e.g., grain size, boundary regions between grains, defects) and their properties, and how processing conditions control structure. Topics include thermodynamics and phase equilibria, microstructure, diffusion, kinetics of phase transitions, nucleation and crystal growth, phase separation, spinodal decomposition, glass formation, and the glass transition.

**562 (MSE 540) Fracture Mechanics** Fracture involves processes at multiple time and length scales. This course covers the basic topics, including energy balance, crack tip fields, toughness, dissipation processes, and subcritical cracking. Fracture processes are then examined as they occur in some modern technologies, such as advanced ceramics, coatings, composites, and integrated circuits. The course also explores fracture at high temperatures and crack nucleation processes. (Offered in alternate years)

**563 Instabilities in Fluids: Linear and Non-Linear Analysis of Waves and Patterns in the Environmen**t This course describes natural patterns arising from instabilities in nature, and discusses their importance in the environment. We analyze phenomena at various scales, as diverse as wave breaking at the ocean surface, internal mixing in the atmosphere and the ocean, volcanic plumes, convection cells in the atmosphere, the break-up of fluid ligaments or bubble bursting at an interface. The course details mathematical tools (linear and non-linear stability analysis, symmetry arguments, solutions to non-linear equations such as shocks and solitons), as well as present laboratory and numerical demonstration of the instabilities.

**564 (MSE 512) Structural Materials** Stress/strain behavior of materials; dislocation theory and strengthening mechanisms; yield strength; materials selection. Fundamentals of plasticity, Tresca and Von Mises yield criteria. Case study on forging: upper and lower bounds. Basic elements of fracture. Fracture mechanics. Mechanisms of fracture. The fracture toughness. Case studies and design. Fatigue mechanisms and life-prediction methodologies. (Offered in alternate years)

**MSE 452 Phase Transformations and Evolving Microstructures in Hard and Soft Matter Systems** This course covers the fundamental principles of thermodynamics and phase transformation kinetics in hard and soft matter systems, such as metals and alloys, semiconductors, polymers, and lipid bilayer membranes. The course synthesizes descriptive observations, principles of statistical thermodynamics, and mathematical theories to address emergent physical, chemical, mechanical, and biological properties of multi-component, multiphase materials systems.

**569, 570 Advanced Topics in Materials and Mechanical Systems I, II** Selected topics in materials and mechanical systems, with an emphasis on advances relevant to research activities represented in the department. Possible topics include high temperature protective coatings, multifunctional materials, MEMS, advanced computational methods in materials engineering.

**571 Inspiring Young Engineers through Outreach** We study effective ways to inspire young students to think about science and engineering. Four concepts of modern engineering are identified and demonstration labs are built around them. The setups are built using modern yet simple tools and are accompanied by a video explaining how the concepts fit together in a larger picture. A field trip is made to a science exhibition to study methods to inspire and teach science to young people. At the end of the course the students perform demonstrations to students from Harlem Prep Elementary who will visit the MAE department.

**574 Unmaking the Bomb, not offered every year **This course covers the science and technology underlying existing and emerging nuclear security issues. It introduces the principles of nuclear fission, nuclear radiation, and nuclear weapons (and their effects) and develops the concepts required to model and analyze nuclear systems. The second half of the semester is centered around a hands-on team project.

**579, 580 Advanced Topics in Energy and Environment I, II** Selected topics in energy and the environment, with an emphasis on advances relevant to research activities represented in the department. Possible topics include combustion control and emissions, economic development and energy resources, and energy efficiency.

**597, 598 Graduate Seminar in Mechanical and Aerospace Engineering** A seminar of graduate students and staff presenting the results of their research and recent advances in flights, space, and surface transportation; fluid mechanics; energy conversion; propulsion; combustion; environmental studies; applied physics; and materials sciences. There is one seminar per week and participation at presentations by distinguished outside speakers.