This course covers the classical partial differential equations of mathematical physics: the heat equation, the Laplace equation, and the wave equation. The primary technique covered in the course is separation of variables, which leads to solutions in the form of eigenfunction expansions. The topics include Fourier series, separation of variables, Sturm-Liouville theory, unbounded domains and the Fourier transform, spherical coordinates and Legendre’s equation, cylindrical coordinates and Bessel’s equation. The software package Mathematica will be used extensively. Prior knowledge of Mathematica is helpful but not essential. In the last two weeks of the course, there will be a project on an assigned topic. The course will include applications in heat conduction, electrostatics, fluid flow, and acoustics.
The course deals with computational methods to analytically intractable mathematical problems in biological research. For the first half of the course, general numerical analysis topics are reviewed such as linear algebra, ODE and PDE. Through homework assignments, students write their own computer code. Sufficient sample solutions are given to practice various numerical methods within limited time. The rest of the course is comprised of case studies and projects. Examples of computational analyses are drawn from life science problems such as biodynamics of human loco motion, ion channel kinetics, ionic diffusion, and finite element analysis of cells/tissues. For final project, students bring their own research problems, express them in mathematical equations, solve them using custom written computer programs and interpret the solutions.
Basic plasma parameters; quasi-neutrality, Debye length, plasma frequency, plasma parameter, Charged particle motion: orbit theory. Basic plasma equations; derivation of fluid equations from the Vlasov equation. Waves in plasmas. MHD theory. Energy balance.
The study of incompressible flow covers fluid motions which are gentle enough that the density of the fluid changes little or none. Topics: Conservation equations. Bernoulli’s equation, the Navier-Stokes equations. Inviscid flows; vorticity; potential flows; stream functions; complex potentials. Viscosity and Reynolds number; some exact solutions with viscosity; boundary layers; low Reynolds number flows. Waves.
This is an introduction to turbulence theory and modeling for graduate students in engineering and the physical sciences. This course stresses intuitive physical understanding, mathematical analysis techniques,and numerical methodologies. It will highlight applications in various disciplines, including aeronautics,fusion sciences, geophysics and astrophysics.
This course provides a thorough grounding on the theory and application of linear steady-state finite element method (FEM) applied to solid mechanics. Topics include: review of matrix algebra and solid mechanics, Principle of Minimum Potential Energy, Rayleigh Ritz Method, FEM computational procedures, isoparametric shape functions and numerical integration for 1D, 2D, and 3D elements, error estimation and convergence, and the demonstration of FEM best practices using a commercial FEM code. A semester project that involves coding FEM software in Matlab is required for graduate students.
This course focuses teaching the multidisciplinary aspects of designing complex, precise systems. In these systems, aspects from mechanics, optics, electronics, design for manufacturing/assembly, and metrology/qualification must all be considered to design, build, and demonstrate a successful precision system. The goal of this class is to develop a fundamental understanding of multidisciplinary design for designing the next generation of advanced instrumentation. This course is open to graduate students in engineering and physics backgrounds although it has a strong emphasis on mechanical engineering and systems engineering topics. This course is open to undergraduates who are in their senior year.
Review of basic thermodynamic quantities and laws; equations of state; statistical mechanics; heat capacity; relations between physical properties; Jacobian algebra; phase transformations, phase diagrams and chemical reactions; partial molal and excess quantities, phases of variable composition; free energy of binary and multicomponent systems; surfaces and interfaces. The emphasis is on the physical and chemical properties of micro and nano solids including stress and strain variables.
In this course, you will apply previously learned theoretical concepts to practical problems and applications. In addition, you will learn experimental techniques and enhance your technical writing skills. This course has two parts, a series of small laboratory exercises and a project. During the semester, students will work in groups of three to complete the assigned work, labs, and reports. The lab section of the course is designed to present basic applied concepts that will be useful to a broad base of engineering problems. The project portion is where you will work on a more specific idea, tailored around your desired future goals.
Description: The mechanical response of crystalline (metals, ceramics, semiconductors)and amorphous solids (glasses, polymers) and their composites in terms of the relationships between stress, strain, damage, fracture, strain-rate, temperature, and microstructure. Topics include: (1) Material structure and property overview. (2) Isotropic and anisotropic elasticity and viscoelasticity. (3) Properties of composites. (4) Plasticity. (5) Point and line defects. (6) Interfacial and volumetric defects. (7) Yield surfaces and flow rules in plasticity of polycrystals and single crystals. (8) Macro and micro aspects of fractures in metals, ceramics and polymers.(9) Creep and superplasticity. (10) Deformation and fracture mechanism maps. (11) Fatigue damage and failure; fracture and failure in composites (If time permits).
In this course, we will survey the role of mechanics in cells, tissues, organs and organisms. A particular emphasis will be placed on the mechanics of the musculoskeletal system, the circulatory system and the eye. Engineering concepts will be used to understand how physical forces contribute to biological processes, especially disease and healing. Experimental and modeling techniques for characterizing the complex mechanical response of biosolids will be discussed in detail, and the continuum mechanics approach will highlighted.
Fusion energy. Lawson criterion for thermonuclear ignition. Fundamentals of implosion hydrodynamics, temperature and density in spherical implosions. Laser light absorption. Implosion stability. Thermonuclear energy gain.