## Graduate Courses

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**Graduate Courses**

(Common prerequisite for all following courses: Graduate standing and departmental consent)

**ES 500 M.S. Thesis NC**

Program of research leading to M.S. degree arranged between the student and a faculty member. Students register to this course in all semesters starting from the beginning of their second semester.

**ES 501 Analytical Methods in Engineering I (3-0)3**

Ordinary differential equations. Series solutions of linear ordinary differential equations. Sturm-Liouville theory. Gamma, Bessel, Hermite, Laguerre functions, Legendre, Chebyshev polynomials. Laplace transform. Complex calculus; Cauchy-Riemann e quations, power series, Cauchy integral formula, residue theorem, improper integrals. Calculus of variations.

**ES 502 Analytical Methods in Engineering II (3-0)3**

Differential equations, Formulation of basic engineering problems, Fourier series and Fourier integral. Separation of variables. Laplace, Fourier, Hankel and Mellin transforms. Green's function method. Integral equations.

*Prerequisite: ES 501 or equivalent.*

**ES 503 Finite Element Method (3-0)3**

Introduction to calculus of variations, weighted residuals method. Properties of finite elements. Ritz and Galerkin methods. Applications in bondary value problems. Two dimensional and time dependent problems.

**ES 504 Numerical Solution of Partial Differential Equations (3-0)3**

Solution of systems of equations. Initial and boundary-value problems. Parabolic, elliptic and hyperbolic equations. Selected topics from solid and fluid mechanics.

**ES 505 Variational Methods in Engineering (3-0)3**

Problems of minimization and maximization. Functionals. Classical problems in calculus of variations, Euler equations, Variational notation, Natural boundary conditions, Hamilton's principle, Lagrange equations. Transformation of boundary value pro blems into the problem of calculus of variation. Direct methods; Ritz method, Galerkin method, Kantorovich method, Weighted residual method.

**ES 506 Reliability (3-0)3**

Brief review of applied probability. Distributions of sum and quotient of two random variables. Topics in risk-based engineering design. Methods available, advantages and disadvantages. System reliability concepts. Statistical decision theory and its application in engineering.

**ES 507 Boundary Element Method (3-0)3**

Gradient and directional derivative of position vector. Numerical evaluation of surface and line integrals, review of the equations of elasto dynamics, acoustics and heat conduction. Formulation of boundary element method: basic integral equation, fundamental solutions. Boundary element equation. Numerical implementation of boundary element method. Codes based on boundary element method. Numerical applications.

**ES 508 Statistical Methods for Engineering Scientists**

Advanced statistical techniques in the solutions of real life engineering problems. Analysis of experimental data, ANOVA, k-variable analysis. Statistical Modeling. Regression Analysis. Experimental design. Topics in time series, Bayesian analysis, discriminant analysis and clustering and their application to engineering problems.

**ES 509 PDEs in Computer Vision and Image Processing**

Axiomatic approach in Computer Vision. Nonlinear evolution equations. Representation of shape. Energy functionals. Heat equation. Multi resolution.

**ES 510 Numerical Solution of ODE (3-0)3**

Numerical solution of initial value problems: multi-step methods and Runge-Kutta methods, stability and convergence. Numerical solution of boundary value problems: shooting methods, finite difference and collocation methods, Green's function methods, transform methods, introduction to the finite element method. Nonlinear boundary problems.

**ES 511 Basic Principles of Mechanics (3-0)3**

Fundamentals of mechanics. Equivalent force systems. Equations of equilibrium. Internal forces. Introduction to continuum mechanics. Mechanical behaviour of Hookean materials. Stress-strain transformations. Strain energy. Introduction to viscoelast ic materials.

**ES 512 Experimental Analysis (3-0)3**

General concepts. Measuring devices. Manipulation, transmission and recording of data.

**ES 513 Effects of Noise and Vibration in Working Environment (3-0)3**

Noise and vibration at work, their main sources; principal risk groups; Effects of noise and vibration (whole body and hand arm vibration) on workers; measurement methods, evaluation and management of workplace noise and vibration; assessment of legislations and directives and their implementation in practice; combined effect of noise, vibration and other workplace environmental effects including temperature extremes; Preventive measures

**ES 514 Mechanical Behavior of Deformable Bodies (3-0)3**

Materials properties; structure of materials; stress and strain concepts; stress and strain tensors; elastic behavior; three dimensional analysis; plastic behavior; fracture; viscoelastic behaviour.

**ES 516 Spectral Methods (3-0)3**

Introduction to the concept of spectral methods. Fourier-collocation spectral methods. Chebyshev-collocation spectral methods. Smoothness and accuracy. Boundary Value Problems. Polar coordinates. Time stepping. Initial value problems. Introduction to spectral element method.

**ES 521 Theory of Elasticity (3-0)3**

Stress and strain tensors. Strain-displacement relations. Compatibility equations. Constitutive equtions. Plane strain, plane stress. Biharmonic equation, polynomial solutions, Fourier series solutions. Axisymmetric problems. Torsion, bending.

**ES 522 Advanced Theory of Elasticity (3-0)3**

Indicial notation, Cartesian tensors and field equations of elasticity theory. Integral transform solutions. Complex variable formulation. Nonlinear elasticity.

*Prerequisite: ES 521 or equivalent.*

**ES 523 Advanced Mechanics (3-0)3**

Axioms of mechanics. Mechanics of a particle. Mechanics of a system of particles. The virtual work principle. General survey of further variational principles of mechanics. Conservation theorems. Hamilton's equations of motion. Canonical transforma tions.The Hamilton-Jacobi theory.

**ES 524 Thermal Stress Analysis (3-0)3**

Theory of heat conduction in solids and review of basic principles in thermal stress analysis. Various formulations of thermo-elastic problems, uncoupled and coupled theories. Some three-and two-dimensional problems in thermoelasticity. Formulation of thermal stresses in thermo-viscoelastic and thermo-elastoplastic media. Some illustrative examples.

**ES 525 Theory of Continuous Media I (3-0)3**

Review of tensor analysis and integral theorems. Kinematics of deformation, strain tensor, compatibility condition. Material derivative of tensors, deformation rate, spin and vorticity. External and internal loads, Cauchy principle and stress tenso r. Balance laws of momenta and energy, entropy principle, jump conditions. Constitutive theory and its axioms, thermomechanical materials. Some illustrative applications.

**ES 526 Theory of Continuous Media II (3-0)3**

Theory of elasticity: General approach, linear constitutive equations, material symmetries, isotropic materials.Wave propagation in isotropic elastic solids. Thermoelasticity: General approach, linear constitutive equations, isotropic materials. Th ermoelastic waves. Fluid dynamics: Incompressible and compressible fluids, propagation of shock waves. Viscoelasticity: Mechanical models, linear theories.

*Prerequisite: ES 525 or equivalent.*

**ES 527 Fracture Mechanics (3-0)3**

Mechanisms of failure for brittle and ductile materials. Stress concentration. Elastic stress fields around cracks. Plasticity effect. Fracture criteria.Crack propagation and methods of crack arrest. Fatigue. Fracture testing.

**ES 528 Wave Propagation in Solids (3-0)3**

Elements of wave motion. Wave propagation in unbounded elastic media. Plane, cylindrical and spherical waves. Harmonic and transient waves in half-space. Surface waves. Waves in layered media. Waves in rods. Method of characteristics.

**ES 531 Mechanics of Composite Materials (3-0)3**

The nature and scope of composite materials. Fundamental aspects of the theory of the linear anisotropic elasticity. Prediction of macroscopic mechanical properties of composite materials. Analysis of internal fields in heterogeneous medium. Wave propagation and dynamic effects in composites. Effective stiffness theory considerations, lattice model representations.

**ES 532 Theory of Plasticity (3-0)3**

Physical background. Idealizations, yield criteria. Plastic-stress strain relations. Two measures of workhardening. Extremum principles, the plastic potential and uniqueness. Elastoplastic problems. Plane stress and plane strain (theory of slip-lin e field with some applications). Geometric effects. Plastic anisotropy.

**ES 534 Elastic Stability (3-0)3**

Various stability methods. Buckling of beams, columns, beams on elastic foundation. Bifurcation and snap through buckling. Plate and shell buckling. Introduction to dynamic buckling.

**ES 536 Energy Methods (3-0)3**

Force Fields. Work. Principles of dynamics. Elements of calculus of variations, variational principles for discrete systems. Elements of the mechanics of continua. Hellinger, Reissner and Hamilton principles, Castigliano's theorem, theorems of w ork and reciprocity.Application to elastic rods, structural systems, elastic plates and shells. Stability.

**ES 538 Soil-Structure Interaction Analysis (3-0)3**

Discrete Fourier transform. Soil-structure interaction analysis: direct and substructure methods, free field system, impedance relation, scattering analysis. Artificial boundary conditions: viscous boundary conditions in the absence and presence of free field. Description of seismic environment: types of control points, free displacements and forces in terms of control point motion.

**ES 541 Introduction to Biomechanics (3-0)3**

Structural and physical properties of bone, muscle, tendon and cartilage. Mechanics of joint and muscle action. Body equilibrium. Mechanics of the spinal column, of the pelvis and of the hip joint. Pathomechanics.

**ES 542 Advanced Biomechanics (3-0)3**

The knee joint, foot and ankle, shoulder-arm complex, the elbow joint. Pathomechanics. Gait analysis.

**ES 545 Tissue Engineering (3-0)3**

Principles of Tissue Engineering. Engineering of Functional Tissues. Scaffold Design and Processing Techniques. Multifunctional Scaffolds. Vascularization Strategies in Tissue Engineering. Clinical Translational Research in Tissue Engineering.

**ES 551 Stochastic Methods in Engineering Mechanics I (3-0)3**

Brief review of probability theory. Random processes. Random vibrations of linear single degree of freedom systems. Analysis of random response in the time and frequency domains. Statistical analysis of failure mechanisms.

**ES 552 Stochastic Methods in Engineering Mechanics II (3-0)3**

Review of the deterministic multi-degree-of-freedom vibratory systems. Random vibration of multi-degree-of-freedom and continuous systems. Markov processes, random walk problems, Fokker-Planck equation. Introduction to random vibration of nonlinear systems, stability of systems subjected to stochastic excitations and introduction to chaotic dynamics.

*Prerequisite : ES 551 or equivalent*

**ES 554 Nonlinear Dynamics (3-0)3**

Fundamentals of nonlinear dynamics and brief review of random vibrations. Periodic and chaotic attractors, stability and bifurcations of equilibria and cycles. Iterated maps as dynamical systems. Criteria for the onset of chaos. Applications and cu rrent literature.

**ES 570 Research Methods and Ethics in Engineering Sciences(0-0)0**

Definition of ethic and ethical principles in science and engineering. Scientific research methods. Evaluation and representation of research results. How to prepare and publish a scientific document from research outputs.

**ES 571 Basic Principles of Fluid Mechanics (3-0)3**

Fluid statics. Transport mechanisms. Compressible flow. Boundary layer. Introduction to unsteady flows.

**ES 572 Advanced Fluid Mechanics (3-0)3**

Development of the governing equations. Grid generation. Inviscid flows. Boundary layer type equations. Parabolized Navier-Stokes equations. Incompressible and compressible Navier-Stokes equations.

*Prerequisite: ES 571 or equivalent.*

**ES 591 Seminar (0-2)NC**

Students prepare and pesent a progress report or literature review on their thesis topic. The course is normally taken by students in their third semester.

**ES 600 Ph.D. Thesis NC**

Program of research leading to M.S. degree arranged between the student and a faculty member. Students register to this course in all semesters starting from the beginning of their second semester.

**ES 691 Seminar (0-2)NC**

Similar to ES 591 but open to doctoral students only.

**ES 7XX Special Topics in Engineering Sciences (3-0)3**

Courses not listed in the catalogue. Contents vary from year to year according to interest of students and instructor in charge. Typical contents include waves in viscoelastic media, mathematical simulation of engineering problems, cell biomechanic s.

**ES 8XX Special Studies (4-2) NC**

M.S. Students choose and study a topic under the guidance of a faculty member, normally his/her advisor.

**ES 9XX Advanced Studies (4-0) NC**

Graduate students as a group or a Ph.D. student choose and study advanced topics under the guidance of a faculty member, normally his/her advisor.