*Note: Duke CEE graduate students may also take courses from other engineering departments within Duke's Pratt School of Engineering and from other professional schools at Duke with the permission of their advisor and the Director of Graduate Studies.*

**CEE 307(207). Transport Phenomena in Biological Systems.** An introduction to the modeling of complex biological systems using principles of transport phenomena and biochemical kinetics. topics include the conservation of mass and momentum using differential and integral balances; rheology of Newtonian and non-Newtonian fluids; steady and transient diffusion in reacting systems; dimensional analysis; homogeneous versus heterogeneous reaction systems. Biomedical and Biotechnological applications are discussed. Instructor: Katz, Truskey, or Yuan. 3 units. C-L: see Biomedical Engineering 307(207); also C-L: Mechanical Engineering and Materials Science 307(207).

### Registration Links

**CEE 520(201). Continuum Mechanics.** Tensor fields and index notation. Analysis of states of stress and strain. Conservation laws and field equations. Constitutive equations for elastic, viscoelastic, and elastic-plastic solids. Formulation and solution of simple problems in elasticity, viscoelasticity, and plasticity. Instructors: Hueckel, or Nadeau. 3 units. **CEE 521(206). Elasticity.** Linear elasticity will be emphasized including concepts of stress and strain as second order tensors, equilibrium at the boundary and within the body, and compatibility of strains. Generalized solution to two and three dimensional problems will be derived and applied to classical problems including torsion of noncircular sections, bending of curved beams, stress concentrations and contact problems. Applications of elasticity solutions to contemporary problem in civil and biomedical engineering will be discussed. 3 units. C-L: see Biomedical Engineering 526(206). ** **

**CEE 525(272). Wave Propagation in Elastic and Poroelastic Media.** Basic theory, methods of solution, and applications involving wave propagation in elastic and poroelastic media. Analytical and numerical solution of corresponding equations of motion. Linear elasticity and viscoelasticity as applied to porous media. Effective medium, soil/rock materials as composite materials. Gassmann's equations and Biot's theory for poroelastic media. Stiffness and damping characteristics of poroelastic materials. Review of engineering applications that include NDT, geotechnical and geophysical case histories. Prerequisite: Mathematics 353(108) or consent of instructor. Instructor: Boadu. 3 units.

**CEE 530(254). Introduction to the Finite Element Method.** Investigation of the finite element method as a numerical technique for solving linear ordinary and partial differential equations, using rod and beam theory, heat conduction, elastostatics and dynamics, and advective/diffusive transport as sample systems. Emphasis placed on formulation and programming of finite element models, along with critical evaluation of results. Topics include: Galerkin and weighted residual approaches, virtual work principles, discretization, element design and evaluation, mixed formulations, and transient analysis. Prerequisites: a working knowledge of ordinary and partial differential equations, numerical methods, and programming in FORTRAN. Instructor: Dolbow and Laursen. 3 units.

**CEE 541(283). Structural Dynamics.** Formulation of dynamic models for discrete and continuous structures; normal mode analysis, deterministic and stochastic responses to shocks and environmental loading (earthquakes, winds, and waves); introduction to nonlinear dynamic systems, analysis and stability of structural components (beams and cables and large systems such as offshore towers, moored ships, and floating platforms). Instructor: Gavin. One course.

**CEE 560(208). Environmental Transport Phenomena.** Introduction to environmental modeling, fluid flow and mass and heat transfer. Conservation principles in the atmosphere and bodies of water, fundamental equations for transport in the atmosphere and bodies of water, scaling principles, simplification, turbulence, turbulent transport, Lagrangian transport, applications to transport of particles in water, from volcanoes and stacks, case studies: volcanic eruption, Chernobyl accident, forest fires and Toms River power plant emission. Instructor: Wiesner. 3 units.**CEE 561(242). Environmental Aquatic Chemistry.** Principles of chemical equilibria and kinetics applied to quantitative description of the chemistry of lakes, rivers, oceans, groundwater, and selected treatment processes. Equilibrium and steady state models applied to processes such as acid-base chemistry, the carbonate system, coordination chemistry, precipitation and dissolution, oxidation-reduction, and adsorption. Instructor: Hsu-Kim. 3 units.**CEE 562(244). Biological Processes in Environmental Engineering.** Biological processes as they relate to environmental systems, including wastewater treatment and bioremediation. Concepts of microbiology, chemical engineering, stoichemistry, and kinetics of complex microbial metabolism, and process analyses. Specific processes discussed include carbon oxidation, nitrification/denitrification, phosphorus removal, methane production, and fermentation. Consent of instructor required. Instructor: Deshusses. 3 units.

**CEE 563(240). Chemical Fate of Organic Compounds.** Equilibrium, kinetic and analytical approaches applied to quantitative description of processes affecting the distribution and fate of anthropogenic and natural organic compounds in surface and groundwater, including chemical transfers between air, water, soils/ sediments, and biota; and thermochemical and photochemical transformations. The relationships between organic compound structure and environmental behavior will be emphasized. Sampling, detection, identification and quantification of organic compounds in the environment. Prerequisites: university-level general chemistry and organic chemistry within last four years. Instructors: Stapleton. 3 units. C-L: see Environment 540(240)**CEE 564(241). Physical-Chemical Processes In Environmental Engineering.** Principles of surface chemistry, particle and solute separation, and oxidation/ disinfection, gas tranfer, precipitation, adsorption, membrane processes. Applications to potable water treatment, fuel cells, photovoltaics, treatment of aqueous streams in energy production, hazardous waste treatment and ground water remediation. Prerequisites: Environmental Transport Phenomena recommended, but not required. introductory environmental engineering, chemistry, or permission of instructor. Instructor: Wiesner. 3 units.

**CEE 566(250). Environmental Microbiology.** Fundamentals of microbiology and biochemistry as they apply to environmental engineering. General topics include cell chemistry, microbial metabolism, bioenergetics, microbial ecology and pollutant biodegradation. Prerequisites: CE 462L(124L) or graduate standing or consent of the instructor. Instructor: Gunsch. 3 units.

**CEE 575(247). Air Pollution Control Engineering.** The problems of air pollution with reference to public health and environmental effects. Measurement and meteorology. Air pollution control engineering: mechanical, chemical, and biological processes and technologies. Instructor: Khlystov. 3 units.

**CEE 581(245). Pollutant Transport Systems.** Distribution of pollutants in natural waters and the atmosphere; diffusive and advective transport phenomena within the natural environment and through artificial conduits and storage/treatment systems. Analytical and numerical prediction methods. Prerequisites: Civil Engineering 301L(122L) and Mathematics 353(108) or equivalents. Instructor: Medina. 3 units.

**CEE 585(260). Vadose Zone Hydrology.** Transport of fluids, heat, and contaminants through unsaturated porous media. Understanding the physical laws and mathematical modeling of relevant processes. Field and laboratory measurements of moisture content and matric potential. Prerequisites: Civil Engineering 301L(122L) and Mathematics 353(108), or consent of instructor. Instructor: Kabala. 3 units.

**CEE 621(203). Plasticity.** Inelastic behavior of soils and engineering materials. Yield criteria. Flow rules. Concepts of perfect plasticity and plastic hardening. Methods of rigid-plasticity. Limit analysis. Isotropic and kinematic hardening. Plastic softening. Diffused damage. Thermo-plasticity. Visco-plasticity. Prerequisite: Civil Engineering 520(201) or consent of instructor. Instructor: Hueckel. 3 units.**CEE 622(212). Fracture Mechanics.** Theoretical concepts concerning the fracture and failure of brittle and ductile materials. Orowan and Griffith approaches to strength. Determination of stress intensity factors using compliance method, weight function method, and numerical methods with conservation laws. Cohesive zone models, fracture toughness, crack growth stability, and plasticity. Prerequisites: CE 520(201) or instructor consent. Instructor: Dolbow. 3 units.**CEE 623(205). Mechanics of Composite Materials.** Theory and application of effective medium, or homogenization, theories to predict macroscopic properties of composite materials based on microstructural characterizations. Effective elasticity, thermal expansion, moisture swelling, and transport properties, among others, are presented along with associated bounds such as Voigt/Reuss and Hashin-Shtrikman. Specific theories include Eshelby, Mori-Tanaka, Kuster-Toksoz, self-consistent, generalized self-consistent, differential method, and composite sphere and cylinder assemblages. Tensor-to-matrix mappings, orientational averaging, and texture analysis. Composite laminated plates, environmentally induced stresses, and failure theories. Prerequisite: Civil Engineering 520(201) or consent of instructor. Instructor: Nadeau. 3 units.

**CEE 625(210). Intermediate Dynamics.** Comprehensive treatment of the dynamic motion of particles and rigid bodies with an introduction to nonlinear dynamics and the vibration of continuous systems. Topics include: conservation of linear and angular momentum, superposition applied to linear systems, motion in inertial and noninertial frames of reference, Hamilton's principle and Langrange's equations, and generalized coordinates. Instructor: Hall or Knight. 3 units. C-L: see Mechanical Engineering and Materials Science 541(210).

**CEE 626(211). Energy Flow and Wave Propagation in Elastic Solids.** Derivation of equations for wave motion in simple structural shapes: strings, longitudinal rods, beams and membranes, plates and shells. Solution techniques, analysis of systems behavior. Topics covered include: nondispersive and dispersive waves, multiple wave types (dilational, distortion), group velocity, impedance concepts including driving point impedances and moment impedances. Power and energy for different cases of wave propagation. Prerequisites: Engineering 244L(123L) and Mathematics 353(108) or consent of instructor. Instructor: Franzoni. 3 units. C-L: Mechanical Engineering and Materials Science 543(234)**.**

**CEE 630(255). Nonlinear Finite Element Analysis.** Formulation and solution of nonlinear initial/boundary value problems using the finite element method. Systems include nonlinear heat conduction/diffusion, geometrically nonlinear solid and structural mechanics applications, and materially nonlinear systems (for example, elastoplasticity). Emphasis on development of variational principles for nonlinear problems, finite element discretization, and equation-solving strategies for discrete nonlinear equation systems. Topics include: Newton-Raphson techniques, quasi-Newton iteration schemes, solution of nonlinear transient problems, and treatment of constraints in a nonlinear framework. An independent project, proposed by the student, is required. Prerequisite: Civil Engineering 530(254) or consent of instructor. Instructor: Laursen. 3 units.

**CEE 635(256). Computational Methods for Evolving Discontinuities.** Presents an overview of advanced nomenical methods for the treatment of engineering problems such as brittle and ductile failure and solid-liquid phase transformations in pure substances. Analytical methods for arbitrary discontinuities and interfaces are reviewed, with particular attention to the derivation of jump conditions. Partition of unity and level set methods. Prerequisites: CE 530(254), CE 630(255), or instructor consent. Instructor: Dolbow. 3 units.

**CEE 641(237). Advanced Soil Mechanics.** Characterization of behavior of geomaterials. Stress strain incremental laws. Nonlinear elasticity, hypo-elasticity, plasticity and viscoplasticity of geomaterials; approximated laws of soil mechanics; fluid-saturated soil behavior; cyclic behavior of soils; liquefaction and cyclic mobility; elements of soil dynamics; thermal effects on soils. Prerequisite: Civil Engineering 302L(139L) or equivalent. Instructor: Hueckel. 3 units.**CEE 642(238). Environmental Geomechanics.** The course addresses engineered and natural situations, where mechanical and hydraulic properties of soils and rocks depend on environmental (thermal chemical, biological) processes. Experimental findings are reviewed, and modeling of coupled thermo-mechanical, chemo-mechanical technologies are reviewed. Instructor: Hueckel. 3 units.**CEE 643(270). Environmental and Engineering Geophysics.** Use of geophysical methods for solving engineering and environmental problems. Theoretical frameworks, techniques, and relevant case histories as applied to engineering and environmental problems (including groundwater evaluation and protection, siting of landfills, chemical waste disposals, roads assessments, foundations investigations for structures, liquefaction and earthquake risk assessment). Introduction to theory of elasticity and wave propagation in elastic and poroelastic media, electrical and electromagnetic methods, and ground penetrating radar technology. Prerequisite: Mathematics 353(108) or Physics 152L(62L) or consent of instructor. Instructor: Boadu. 3 units.**CEE 644(271). Inverse Problems in Geosciences and Engineering.** Basic concepts, theory, methods of solution, and application of inverse problems in engineering, groundwater modeling, and applied geophysics. Deterministic and statistical frameworks for solving inverse problems. Strategies for solving linear and nonlinear inverse problems. Bayesian approach to nonlinear inverse problems. Emphasis on the ill-posed problem of inverse solutions. Data collection strategies in relation to solution of inverse problems. Model structure identification and parameter estimation procedures. Prerequisite: Mathematics 353(108) or consent of instructor. Instructor: Boadu. 3 units.

**CEE 647(252). Buckling of Engineering Structures.** An introduction to the underlying concepts of elastic stability and buckling, development of differential equation and energy approaches, buckling of common engineering components including link models, struts, frames, plates, and shells. Consideration will also be given to inelastic behavior, postbuckling, and design implications. Prerequisite: Civil Engineering 421L(131L) or consent of instructor. Instructor: Virgin. 3 units. C-L: Mechanical Engineering and Materials Science 527(252).

**CEE 661L(239L). Environmental Molecular Biotechnology (GE, MC).** Principles of genetics and recombinant DNA for environmental systems. Applications to include genetic engineering for bioremediation, DGGE, FISH, micro-arrays and biosensors. Laboratory exercises to include DNA isolation, amplification, manipulation and analysis. Prerequisites: CE 462L(124L)/BIO 201L(25) or consent of the instructor. Instructor: Gunsch. 3 units. C-L: Biomedical Engineering 565L(240L).**CEE 676(269). Fundamentals and Applications of Advanced Physical-Chemical Processes in Environmental Systems.** Fundamental basis for design of membranes systems, applications of environmental nanotechnology, advanced oxidation, principles of surface chemistry and photcatalysis. Prerequisites: CE 564(241) or consent of instructor. Instructor: Wiesner. One course.

**CEE 683(227). Groundwater Hydrology and Contaminant Transport.** Review of surface hydrology and its interaction with groundwater. The nature of porous media, hydraulic conductivity, and permeability. General hydrodynamic equations of flow in isotropic and anisotropic media. Water quality standards and contaminant transport processes: advective-dispersive equation for solute transport in saturated porous media. Analytical and numerical methods, selected computer applications. Deterministic versus stochastic models. Applications: leachate from sanitary landfills, industrial lagoons and ponds, subsurface wastewater injection, monitoring of groundwater contamination. Conjunctive surface-subsurface models. Prerequisite: Civil Engineering 463L(123L) or consent of instructor. Instructor: Medina. 3 units.

**CEE 684(224). Physical Hydrology and Hydrometeorology.** The objective of this course is to introduce and familiarize graduate students with the fundamental physical processes in Hydrology and Hydrometeorology that control and modulate the pathways and transformations of water in the environment. The content of the course will be strongly oriented toward providing students with a specific basis for quantitative analysis of the terrestrial water cycle including land-atmosphere interactions and clouds and precipitation (rain and snow) processes. The course should be of interest to undergraduate and graduate students interested in Environmental Science and Engineering, and Atmospheric and Earth Sciences.**CEE 686(220). Ecohydrology.** Provides the theoretical basis for understanding the interaction between hydrologic cycle, vegetation, and soil biogeochemistry. This is key for a proper management of soil, and water resources and terrestrial ecosystems especially in view of the possible intensification and alteration of the hydrologic regime due to climate change. The course begins with a review of probability and stochastic processes, with special attention to marked-Poisson processes for daily rainfall modeling, followed by an analysis of the soil water balance and the probabilistic theory of soil moisture dynamics. Finally, the main hydrologic controls on soil biogeochemical cycles and issues related to the ecohydrology of managed ecosystems are discussed. Prerequisites: Civil Engineering 463L(123L) and Engineering 305(115) or equivalent. Instructor: Porporato. 3 units.**CEE 690(265). Advanced Topics in Civil and Environmental Engineering.** Opportunity for study of advanced subjects relating to programs within the civil and environmental engineering department tailored to fit the requirements of individuals or small groups. Instructor: Staff. Variable credit.

**CEE 701(301). Graduate Colloquium. **Current topics in civil and environmental engineering theory and practice. Weekly seminar series. Instructor: Staff. 0 units.**CEE 702(302). Graduate Colloquium.** Current topics in civil and environmental engineering theory and practice. Weekly seminar series. Instructor: Staff. 0 units.

**CEE 890(399). Special Readings in Civil and Environmental Engineering.** Special individual readings in a specific area of study in civil and environmental engineering. Approval of director of graduate studies required. 1 to 3 units. Instructor: Graduate faculty. Variable credit.

## Other Courses Accepted by the Department

**STA 611(213). Introduction to Statistical Methods.** Emphasis on classical techniques of hypothesis testing and point and interval estimation, using the binomial, normal, t, F, and chi-square distributions. Not open to students who have had Statistical Science 250(114) or Mathematics 342(136). Prerequisite: Mathematics 212(103) (may be taken concurrently) or equivalent, or consent of instructor.

**MATH 551(211). Applied Partial Differential Equations and Complex Variables.** Initial and boundary value problems for the heat and wave equations in one and several dimensions. Fourier series and integrals, eigenvalue problems. Laplace transforms, solutions via contour integration, and elementary complex variables. Solutions via Green's functions. Intended for applied math students and students in science and engineering. Prerequisite: Mathematics 216(107) and 353(108) or the equivalent.

**MATH 541(216). Applied Stochastic Processes.** An introduction to stochastic processes without measure theory. Topics selected from: Markov chains in discrete and continuous time, queuing theory, branching processes, martingales, Brownian motion, stochastic calculus. Prerequisite: Mathematics 230(135) or equivalent.

**MATH 561(224). Scientific Computing.** Structured scientific programming in C/C++ and FORTRAN. Floating point arithmetic and interactive graphics for data visualization. Numerical linear algebra, direct and iterative methods for solving linear systems, matrix factorizations, least squares problems and eigenvalue problems. Iterative methods for nonlinear equations and nonlinear systems, Newton's method. Prerequisite: Mathematics 212(103) and 230(104).

**MATH 577(229). Mathematical Modeling. **Formulation and analysis of mathematical models in science and engineering. Emphasis on case studies; may include individual or team research projects.

**ME 631(226). Intermediate Fluid Mechanics.** A survey of the principal concepts and equations of fluid mechanics, fluid statics, surface tension, the Eulerian and Lagrangian description, kinematics, Reynolds transport theorem, the differential and integral equations of motion, constitutive equations for a Newtonian fluid, the Navier-Stokes equations, and boundary conditions on velocity and stress at material interfaces.

**ME 632(227). Advanced Fluid Mechanics.** Flow of a uniform incompressible viscous fluid. Exact solutions to the Navier-Stokes equation. Similarity methods. Irrotational flow theory and its applications. Elements of boundary layer theory. Prerequisite: Mechanical Engineering 631(226) or consent of instructor.

**ENVIRON 564(282). Biogeochemistry.** Processes controlling the circulation of carbon and biochemical elements in natural ecosystems and at the global level, with emphasis on soil and surficial processes. Topics include human impact on and social consequences of greenhouse gases, ozone, and heavy metals in the environment. Prerequisite: Chemistry 101DL(31L) or equivalent.

**ENVIRON 710(210). Applied Data Analysis for Environmental Sciences.** Graphical and exploratory data analysis; modeling, estimation, and hypothesis testing; analysis of variance; random effect models; nested models; regression and scatterplot smoothing; resampling and randomization methods. Concepts and tools involved in data analysis. Special emphasis on examples drawn from the biological and environmental sciences. Students to be involved in applied work through statistical computing using software, often S-plus, which will highlight the usefulness of exploratory methods of data analysis. Other software, such as SAS, may be introduced.