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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. Complex Numbers.

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

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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.

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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.

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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.

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Application of principles of mechanics. Elements of statics in two and three dimensions, equivalent systems of forces. Equilibrium of rigid bodies, distributed forces, analysis of structures, forces in beams. Friction. Kinematics of particles, kinetics of particles, energy and momentum methods, kinematics of rigid bodies, plane motion of rigid bodies.

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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.

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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 central limit theorem. Regression analysis. Applications in engineering.

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

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Mathematical modeling of engineering problems, Numerical solution of nonlinear single variable equation and system of linear and nonlinear equations. Curve fitting and interpolating polynomials. Numerical differentiation and integration. Numerical solution of ordinary differential equations. Optimization.

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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.

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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.

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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.

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Engineering Antropometry. Fundamental of cell and tissue physiology. Gross anatomy and physiology of human skeletal, muscular, nervous, cardiovascular, respiratory, and urinary systems. Energy metabolism and requirements of human body. Body interactions with environment. Sleep and body rhytms.

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Structure and organization of tissues. Mechanics of Tissues. Cell-matrix interactions. Introduction to basic concepts of tissue engineering: Cell source, cellular therapy, scaffold guided tissue engineering, tissue dynamics and microenvironment. Design principles of biomimetic environments. Cartilage, skin, and bone tissue engineering. Standards and regulatory considerations of tissue engineered products.

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Introduction of the concept of bioengineering. Application of fluid mechanics, mass transfer, bioheat transfer, control theory to physiological systems and artificial organs. Structure-property relationships of biomedical materials. Problems associated with the
selection and function of biomedical materials. Design principles for biomaterials. The basis for the molecular therapeutics and drug delivery systems. Engineering of biomaterial structures and surfaces. Biomechanics of human musculoskeletal system. Devices for medical imaging. Prerequisite: None

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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.

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Ordinary differential equations. Series solutions of ordinary differential equations. Fourier series and Fourier integral. Partial differential equations. Separation of variables. Gamma, Bessel, Laguerre functions, Legendre, Chebyshev polynomials.

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Formulation of basic engineering problems. Sturm-Liouville theory. Fourier series. Complex calculus; Cauchy-Riemann equations, power series, Cauchy's integral formula, residue theorem, improper integrals. Laplace, Fourier, Hankel, Mellin transforms. Green's function method. Integral equations.

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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.

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

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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 problems into the problem of calculus of variation. Direct methods; Ritz method, Galerkin method, Kantorovich method, Weighted residual method.

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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.

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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.

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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.

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

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

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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.

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

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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 tensor. Balance laws of momenta and energy, entropy principle. Constitutive theory and its axioms, thermomechanical materials. Some illustrative applications.

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

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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.

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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.

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

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

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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 work and reciprocity.Application to elastic rods, structural systems, elastic plates and shells. Stability.

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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.

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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.

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

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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.

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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.

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

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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.

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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.

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Similar to ES 591 but open to doctoral students only.

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PSEUDOSPECTRAL METHODS

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