Vector spaces, matrices, systems of linear equations, linear transformations, change of basis, eigenvalue problems, quadratic forms and diagonalization. Vector calculus, line, surface, and volume integrals. Gradient, divergence, curl. Green, Gauss and Stokes theorems.

Introduction to vector spaces and linear algebra. Vector differential calculus. Line, surface, volume integrals and integral theorems. Algebra of matrices. System equations and Gauss elimination. Linear transformations, change of basis. Characteris tic value problems, diagonalization and quadratic forms. Concept of probability, random variables, some useful distributions, estimation of parameters, confidence intervals and tests of hypothesis, linear regression.

Principles of mechanics. Elements of statics in two and three dimensions, centroids, analysis of structures and machines, friction. Internal force diagrams. Moment of inertia.

Kinematics of a particle. Dynamics of a particle. Kinematics of a rigid body in plane motion. Dynamics of a rigid body in translation. Dynamics of a rigid body in rotation. Dynamics of a rigid body in plane motion. Impulse and momentum.

Principles of mechanics. Elements of statics in two dimensions. Centroids and moments of inertia. Analysis of simple plane structures. Internal force diagrams. Concepts of stress and strain. Axially loaded members. Torsion. Laterally loaded members .

Prerequisite: MATH 156 or MATH 158.

State of stress and strain. Idealizations and principles in solving engineering problems. Axially loaded members. Torsion. Laterally loaded members. Thermal stress and strain. Indeterminate problems. Deflections. Failure theories.

Error analysis. Sources and propagation. Introductory linear algebra: review of systems of linear equations, matrix algebra and introductory vector differential calculus. Interpolation and extrapolation. Roots of polynomials. Data fitting and least squares problems. Numerical differentiation and integration. Numerical solution of ordinary differential equations.

Descriptive statistics, histograms, central tendency, dispersion and correlation measures. Basic probability concepts, random variables, probability density and mass function. Hypothesis testing, confidence intervals. Law of large numbers and centr al limit theorem. Regression analysis. Applications in engineering.

Numerical solution of linear and nonlinear systems of equations. Interpolating polynomials. Numerical differentiation and integration. Numerical solution of ordinary differential equations.

Analysis of error in numerical computations. Solution of linear algebraic system of equations. Eigenvalues. Roots of nonlinear equations. Interpolation and approximations. Numerical differentiation and integration. Difference equations. Solution of system of ordinary differential equations.

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

Mathematical modeling and reduction of engineering problems to ordinary or partial differential systems. Applications of Fourier series, separation of variables, Fourier and Laplace transforms, Bessel functions, Legendre polynomials to basic equati ons in engineering such as wave, continuity, heat conduction, beam and Navier equations.

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

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

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

Geometrical foundations. Analysis of stress and deformation. Balance laws. Constitutive equations. Finite and infinitesimal theories of elasticity. Applications in fluid mechanics and viscoelasticity.

Vibration and stability of systems with finite degrees of freedom. Rotation of rigid bodies about fixed and moving axes. Gyroscope. Impulsive motion. Nonholonomic systems. Selected problems of pursuit and orbital flight.

Fundamental concepts of rheology. A classification of material behavior. Linear viscoelasticity, creep, relaxation and complex modulus. Relaxation and retardation spectra, correspondence principle. Nonlinear viscoelasticity. Elasto-plastic and viscoplastic substances. Engineering application.

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.

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

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. Panhomechanics.

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

Perceptual, central and motor processes in man-machine systems. Human capabilities and limitations. Use of anthropometric data. Body mechanics and posture. Man-machine interface design. Physical work capacity. Thermal stress and comfort. Vision and illumination. Noise, vibrations. Fatigue, vigilance and accidents. Technological skills and training.

Measurement systems. Error analysis. Operational amplifiers. Force, pressure, temperature, flow, strain and other relevant measurements. Microprocessor applications in measurement and control. Manipulation, transmission and recording of data.

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

Review of rigid body dynamics. Generalized coordinates and forces. Lagrangian and Hamiltonian formulations. Small oscillations. Natural modes. Response of multi-degree-of-freedom systems. Vibration of continuous elastic systems. Introduction to non linear vibrations.

These code numbers will be used for technical elective courses which are not listed regularly in the catalog. The course contents will be announced before the semester commences.

• ES 500 M.S. Thesis NC
• ES 591 Seminar (0-2)NC
• 7 elective courses

Total minimum credit: 21

No of courses with credit (min) : 7

• ES 600 Ph.D. Thesis NC
• ES 691 Seminar (0-2)NC
• 7 elective courses

Total minimum credit: 21

No of courses with credit (min): 7

• ES 500 M.S. Thesis NC
• ES 501 Analytical Methods in Engineering I (3-0)3
• ES 502 Analytical Methods in Engineering II (3-0)3
• ES 503 Finite Element Method (3-0)3
• ES 504 Numerical Solution of Partial Differential Equations (3-0)3
• ES 505 Variational Methods in Engineering (3-0)3
• ES 506 Reliability (3-0)3
• ES 507 Boundary Element Method (3-0)3
• ES 508 Statistical Methods for Engineering Scientists
• ES 509 PDEs in Computer Vision and Image Processing
• ES 511 Basic Principles of Mechanics (3-0)3
• ES 512 Experimental Analysis (3-0)3
• ES 514 Mechanical Behaviour of Deformable Bodies (3-0)3
• ES 521 Theory of Elasticity (3-0)3
• ES 522 Advanced Theory of Elasticity (3-0)3
• ES 523 Advanced Mechanics (3-0)3
• ES 524 Thermal Stress Analysis (3-0)3
• ES 525 Theory of Continuous Media I (3-0)3
• ES 526 Theory of Continuous Media II (3-0)3
• ES 527 Fracture Mechanics (3-0)3
• ES 528 Wave Propagation in Solids (3-0)3
• ES 531 Mechanics of Composite Materials (3-0)3
• ES 532 Theory of Plasticity (3-0)3
• ES 534 Elastic Stability (3-0)3
• ES 536 Energy Methods (3-0)3
• ES 538 Soil-Structure Interaction Analysis (3-0)3
• ES 541 Introduction to Biomechanics (3-0)3
• ES 542 Advanced Biomechanics (3-0)3
• ES 551 Stochastic Methods in Engineering. Mechanics I (3-0)3
• ES 552 Stochactic Methods in Engineering. Mechanics II (3-0)3
• ES 554 Nonlinear Dynamics (3-0)3
• ES 571 Basic Principles of Fluid Mechanics (3-0)3
• ES 572 Advanced Fluid Mechanics (3-0)3
• ES 591 Seminar (0-2)NC
• ES 600 Ph.D. Thesis NC
• ES 691 Seminar (0-2)NC
• ES 7XX Special Topics in Engineering Sciences (3-0)3
• ES 8XX Special Studies (4-2)NC
• ES 9XX Advanced Studies (4-0)NC

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

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.

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.

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.

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.

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

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.

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 i ts application in engineering.

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.

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.

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.

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

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

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.

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

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.

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.

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.

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.

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.

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.

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.

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.

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

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.

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.

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.

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

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.

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

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.

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

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

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.

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.

Similar to ES 591 but open to doctoral students only.

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.

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

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.