Ph.D. and M.S. in Computational Science and Engineering. Two different M.S. and Ph.D. degrees in Applied mathematics and Engineering

Transcription

1 Ph.D. and M.S. in Computational Science and Engineering Two different M.S. and Ph.D. degrees in Applied mathematics and Engineering MS in CSE - Mathematics MS in CSE Mechanical/Electrical Engineering PhD in CSE - Mathematics PhD in CSE Mechanical/Electrical Engineering YONSEI UNIVERSITY Graduate School Department of Computational Science and Engineering

2 Contents 1. Introduction Overview of Program 2.1 Aims and Objectives of the Program Teaching and Learning Major steps leading to the M.S. degree Assessment 4.1 Progression and Assessment Regulations Plagiarism Miscellaneous Matters 5.1 Seminars Communication Attendance Dissertations Advice and Guidance Course Descriptions 6.1 Preliminary Course Core Courses Syllabuses Academic Calendar

3 1. Introduction Welcome to the Department of Computational Science and Engineering at Yonsei University! We hope this program enables you to extend your interest and enjoyment of mathematics and computing and to enhance your knowledge of both theoretical and practical, industrially relevant, aspects of the subject. This program handbook contains details of the program in Computational Science and Engineering and is specific to this program. More general aspects of postgraduate study, including computer facilities, attendance requirements, staff contact details, and so on, are covered in the Postgraduate Handbook of the University, which should be read in conjunction with this handbook. Prof. Jin Keun Seo Head and Program Director 176 Engineering Research Park, Yonsei University 2. Overview of Program 2.1 Aims and Objectives of the Program Computational Science and Engineering (CSE) is a new graduate department at Yonsei university which focuses on a computation oriented multidisciplinary area combining the knowledge and techniques of the traditional engineering fields - electrical, mechanical, biomedical, chemical, aerospace, and materials science - with the knowledge and techniques of computational and applied mathematics. This department is based on World Class University Project funded by government. We provide a venue for innovative research and education in scientific computing. CSE involves the invention, implementation, testing, and application of algorithms and software used to solve large-scale scientific and engineering problems. And it focuses on the development of problem-solving methodologies and robust tools for the solution of scientific and engineering problems. Our faculty come from a wide variety of traditional mathematical, scientific and engineering disciplines who are primarily interested in developing knowledge and tools for computational science and engineering and applying those tools for the solution of problems in a variety of applications. Among the many research subjects involved in CSE, our research group focuses on numerical methodologies and tools, mechanical/environmental engineering and bio-medical engineering applications Details of module specific aims and learning outcomes can be found in the individual syllabuses. 2.2 Teaching and Learning The teaching methods employed in the program are small group lecturing, examples classes, supervised laboratory exercises, and written projects and a written dissertation. Examples classes are classes in which the students work through problem sheets with a member of staff in attendance and able to offer help and advice; these are used for the more mathematically oriented modules. Laboratory classes are supervised classes in a computer laboratory, in which students work on computer-based exercises. Students are expected to spend additional private study time on each course unit, which is essential for reviewing and understanding course material, consulting further sources of information, and preparing for the various types of class. Lecturers will include staff at the university, visiting academics, and also guest lecturers from industry who will impart knowledge of real-life applications and industrial numerical software needs. 3

4 3. Major steps leading to the M.S./Ph.D. degree There are two different Master's Programs in CSE: CSE-mathematics and CSE-Engineering. Within the first two years, the student is expected to demonstrate proficiency in three core areas of CSE or passing written examinations. We require a thesis, while others emphasize course work. Next, a student is expected, via the M.S./Ph.D. preliminary oral examination, to demonstrate proficiency in a major area. This examination should be taken before the first week of the final semester of the graduate study. All have a Final Oral Examination in some form. Program Provisions Required written exams and course work; The only course work requirements involve breadth in basic areas (3 core area: 9 credits) and the remaining supporting courses depending on specific programs (e.g., 21 credits for M.S. student). All these are described in detail in the next course section. The course work planning is done in consultation with the student's Graduate Advisor and/or Director of Graduate Studies (DGS) in CSE. The M.S./Ph.D. preliminary written examinations (qualifying exams) are given twice each year, once before the start of spring and fall semester. A student who wishes to become a candidate for the M.S. /Ph.D. degree must pass core course work requirements or the related written examinations. 4. Assessment 4.1 Progression and Assessment Regulations The assessment for the M.S. and Ph.D. consists of two parts: (1) an assessment of the taught modules undertaken, and (2) an assessment of the dissertation. Thesis The University regulation can be found at on thesis. A student should take qualifying exams for 2 modules of core courses and pass. This can be made after one semester of CSE study. A student who does not satisfy the CSE standards in qualifying exams will be asked to re-sit the exams; the chance is given one time only. A student is then expected to submit a proposal of a dissertation. Progress on the dissertation will be considered with the proposal at an examiner s meeting, where an approval of starting writing up is made for each student. A Student who does satisfy the examiners will work toward submission of an M.S. thesis or Ph.D. dissertation. M.S. student Each M.S. student is assessed on 30 credits worth of coursework and examination questions. Coursework will include practical computing exercises (individually or in groups), written work, and/or other forms appropriate to each individual module. Those who get grades over A0 in all the core courses can be exempted from their qualifying exams. A student then can work on dissertation. The work on dissertation should be published on an academic journal. Ph.D. student Each Ph.D. student is assessed on 60 credits worth of coursework and examination questions (this may include 30 credits of the M.S. course of a student). Coursework will include practical computing exercises (individually or in groups), written work, and/or other forms appropriate to each individual module. A student should take qualifying exams for 2 modules of core courses and pass. Those who get grades over A0 in all the core courses 4

5 or graduates of CSE MS program can be exempted from their qualifying exams. A student then can work on dissertation. At least two pieces of the work should be published on an international academic journal. M.S./Ph.D. Student Each M.S./Ph.D. student is assessed on 54 credits worth of coursework and examination questions. Coursework will include practical computing exercises (individually or in groups), written work, and/or other forms appropriate to each individual module. At least two pieces of the work should be published on an international academic journal. Students who meet all requirements of CSE will be awarded either a M.S., or Ph.D. The rules for award of each are specified in: See Plagiarism Plagiarism is the theft or expropriation of someone else's work without proper acknowledgement, presenting the material as if it were one's own. Plagiarism is a serious academic offence and the consequences are severe. The students are encouraged to take a research ethnic course. 5. Miscellaneous Matters 5.1 Seminars The department of Computational Science and Engineering encourage all postgraduate students to attend a selection of seminars, which enables them to hear leading researchers speaking about work in applied mathematics and computing-based engineering. Students should keep records of the seminars they have attended, with the title, name of speaker, and their comments/notes. This enables students to build a record of which areas of current research they find the most interesting, which will help with the selection of a dissertation topic. 5.2 Communication The principal means of written communication between staff and students is by . Students are therefore expected to check their on a daily basis. (They should also check their mailbox where hard copy circulars and mail is deposited.) 5.3 Attendance We stress here the importance of students discussing acceptable times for holidays, particularly those during the dissertation period, with their supervisors. Students should not expect to be able to complete their dissertations significantly in advance of the deadline for submission. 5.4 Dissertations The dissertation is started as early as Semester Two and the topic chosen from a range of available projects offered by members of staff. Some projects are on topics suggested by industrial partners. These projects have an academic supervisor and offer the possibility of spending time on-site with the industrial partner. The list of projects offered varies from year to year. 5

6 For the award of the M.S., candidates passing the examinations at M.S. level are required to submit a satisfactory dissertation by the 1st week of the final semester. The Examinations and Awards office requires advance notice ( notice of submission") to be given of the student's intention to submit a thesis on a particular date and it is the student's responsibility to ensure that this notice is given. Typesetting the Dissertation: LATEX is recommended as the typesetting of CSE dissertation. 5.5 Advice and Guidance The Program Director has general responsibility for the program arrangements and may be consulted at any time about problems. It is important to inform the Director promptly if any illness or personal circumstances affect a student's work, in particular at examination periods. Such factors can be taken into account provided the student supplies medical certificates or other supporting evidence. Specific questions on lecture courses should normally be addressed to the lecturer concerned. Once the dissertation is in progress the project supervisor will advise on all aspects of dissertation work. 6. Course descriptions The CSE course This innovative graduate program is a WCU-led initiative offered in collaboration with interdisciplinary subjects: Computation, Mathematics, Science and Engineering. The CSE is linked to the departments of College of Science and College of Engineering at Yonsei University Preliminary Course Basics of Computational Science and Engineering: CSE5001 This course is intended for graduate students (or undergraduate) who need a rapid and uncomplicated introductions to the field of applied mathematics involving computational linear algebra and differential equations. The lecture has two themes-how to understand equations, and how to solve them. This course includes numerical linear algebra(qr,svd, singular system), Newton s method for minimization, Equilibrium and stiffness matrix, Least squares, Nonlinear problems, Covariances and Recursive Least squares, Differential equations and finite elements, Finite Difference and Fast Poisson, Boundary value problems in Elasticity and Solid mechanics Basics of Computational Fluid Dynamics: CSE5002 This course introduces advanced undergraduate and the beginning graduate students to basic knowledge of computational fluid dynamics with an emphasis on finite-difference methods as means of solving different type of differential equations in fluid mechanics and heat transfer. This course includes fundamentals of fluid mechanics, numerical analysis, ordinary differential equations and partial differential equations related to fluid mechanics, projection method. Challenging projects involving the use of low-level programing language (e.g., Fortran or C++) offer a unique programing experience for the students to solve the Navier-Stokes equations. Prerequisites: Fluid mechanics, Advanced Calculus, and Engineering Math 6

7 6.2. Core Courses Numerical Analysis: CSE5810 This course is designed to acquaint students in mathematical and physical sciences and engineering with the fundamental theory of numerical analysis. This course is devoted to interpolation, spline, and the solution of nonlinear equations, and numerical linear algebra covering direct methods, error analysis, structured matrices, iterative methods, least squares, and parallel techniques. Prerequisites: Advanced Calculus and Linear Algebra (or Engineering Math) Partial Differential Equations for Science and Engineering: CSE5950 This is an introductory, graduate-level course on partial differential equations (PDE) for science and engineering. This course focuses on derivation, interpretation, and analysis for model equations including Laplace equation, heat, and wave equations. This course covers maximum principle and uniqueness results, variation principles, Lax-Milgram theorem and applications to boundary value problems. Prerequisites: Advanced Calculus and Linear Algebra (or Engineering Math) Numerical Partial Differential Equations: CSE5840 A presentation of the fundamentals of modern numerical techniques for a wide range of linear and nonlinear elliptic, parabolic and hyperbolic partial differential equations and integral equations central to a wide variety of applications in science, engineering, and other fields. Topics include: Mathematical Formulations; Finite Difference Method, Finite Volume Method, Collocation Method, Finite Element Discretizations. Prerequisites: Advanced Calculus and Linear Algebra (or Engineering Math) Viscous fluid flow: CSE6623 This course is designed to give graduate students in-depth understanding of viscous dominated flow dynamics. The course is concerned with the mathematical theory of viscous fluid flow, its applications to a variety of practical problems with appropriate simplifications and solution strategies, and a deep understanding of the character and properties of general flow fields. The course includes fluid properties, dimensional analysis, Cartesian tensors, kinematics, Navier-Stokes equations, vorticity dynamics, potential flows, laminar flows, boundary layers, etc. Prerequisites: Fluid mechanics and Engineering Math Students are expected to get the score of B0 or above for all the courses. They are not allowed to continue their degree program next semester when they have not met this requirement. 7. Syllabuses Basics of Computational Science and Engineering: CSE5001 This course is intended for graduate students (or undergraduate) who need a rapid and uncomplicated introductions to the field of computational science and engineering. This course includes the followings: Text book: Computational Science and Engineering by Gilbert Strang (Wellesley-Cambridge press) 7

8 0. Review:. Vector and tensor analysis. - Green's theorem, divergence theorem, Stokes' theorem, Taylor Expansion, Lagrange Multiplier - Linear independence, linear dependence, basis, dimension - Inner products, Schwarz's inequality, norms, Gram-Schmidt orthogonalization - Basic ideas of approximation in function spaces - convergence, sequences, Cauchy sequences, compactness 1. Applied Linear Algebra - Eigenvalues and Eigenvectors - Positive Definite Matrices - Numerical Linear Algebra: LU, QR, SVD - Best Basis from the SVD 2. A Framework for Applied Mathematics - Equilibrium and the Stiffness Matrix - Oscillation by Newton's Law - Least Squares for Rectangular Matrices - Structures in Equilibrium - Covariances and Recursive Least Squares 3. Boundary Value Problems - Differential Equations of Equilibrium - Gradient and Divergence - Laplace's Equation - Finite Differences and Fast Poisson Solvers - The Finite Element Method - Finite Difference and Fast Poisson, -Boundary value problems in Elasticity and Solid mechanics. Basics of Computational Fluid Dynamics: CSE5002 This course introduces advanced undergraduate and the beginning graduate students to basic knowledge of computational fluid dynamics with an emphasis on finite-difference methods as means of solving different type of differential equations in fluid mechanics and heat transfer. Fundamentals of fluid mechanics, numerical analysis, ordinary differential equations and partial differential equations related to fluid mechanics and heat transfer will be reviewed. Challenging projects involving the use of Fortran or C++ offer a unique programing experience for the students to solve the Navier-Stokes equations. This course includes the followings: 1. Fundamentals of fluid mechanics - Fluid property, flow variables 2. Navier-Stokes equations - Conservations laws - Similarity and scaling - Some exact solutions 3. Basic numerical analysis - Root finding - Interpolation, Integration, Least-square method - Ordinary differential equation 4. Finite difference method - One-dimensional heat equation, Stability - Two-dimensional Laplace equation, Poisson equation 5. Methods for incompressible Flows - Projection method - Staggered/Non-staggered grid - Explicit/Implicit method - Advanced CFD solvers - Visualization 8

9 Partial Differential Equations for Science and Engineering: CSE5950 This is an introductory, graduate-level course on partial differential equations (PDE) for science and engineering. This course focuses on derivation, interpretation, and analysis for model equations including Laplace equation, heat, and wave equations. This course covers maximum principle and uniqueness results, variation principles, Lax-Milgram theorem and applications to boundary value problems. 1. Basics in PDE - Laplace's Equation: Fundamental solution; mean-value formulas; properties of harmonic functions; Green's function for the half-space and ball; Dirichlet's principle. - Heat Equations: Fundamental solution; initial-value and nonhomogeneous problems; mean-value formula; regularity; energy methods. - Wave Equation: d 'Alembert 's formula; solution by spherical means (n = 2, 3); Huygen's principle; nonhomogeneous problem, energy methods. 2. First-Order Equations: Complete integrals; envelopes; characteristic ODE; local existence theorem. 3. Fourier Transform, Hilbert space theory, Sobolev space - Basic geometry, orthogonality, bases, projections, and examples; Bessel s inequality and the Parseval Theorem, - Fourier Transform, The Plancherel Theorem, Convolutions, Sobolev spaces. 4. Differential Calculus and Calculus of Variations (4 weeks) - The Frechet derivatives. - The Chain Rule and Mean Value Theorems. - Higher order derivatives and Taylor's Theorem. - The Euler-Lagrange equation. - Applications to classical mechanics and geometry. 5. Variational Boundary Value Problems (BVP) (3 weeks) - Weak solutions to elliptic BVP's. - Variational forms. - Lax-Milgram Theorem. - Galerkin approximations Numerical Analysis: CSE5810 This course is designed to acquaint students in sciences and engineering with the fundamental theory of numerical analysis. This course is devoted to nonlinear equations, optimization, approximation theory, numerical quadrature and numerical linear algebra (including linear systems, least squares problems and eigenvalue problems). The course will stress both on analytic and computational aspects of numerical methods. 1. Nonlinear equations and optimizations: Newton, Quasi-Newton, Line search 2. Approximation theory and numerical quadrature: Interpolation, Splines, Least square fitting, Simple integration rules, Gaussian quadratures 3. Linear systems and LSM : Matrix decomposition, Direct and iterative methods, Structured matrices 4. Eigenvalue problems: Power methods and QR algorithms Numerical Partial Differential Equations: CSE5840 This course focuses on the fundamentals of modern numerical techniques for a wide range of linear and nonlinear elliptic, parabolic and hyperbolic partial differential equations and integral equations central to a wide variety of applications in science, engineering, and other fields. Topics include: Mathematical Formulations; Finite Difference Method, Finite Volume Method, Collocation Method, Finite Element Discretizations 9

10 1. Finite difference methods - Stability of time stepping - Stability region in the complex plane - linear multistep time stepping. - Runge-Kutta methods - Stability/consistency and difference formula - Spectrum of some finite difference operators - Numerical dispersion and dissipation - Convergence and accuracy 2. Finite volume methods - Basic concept - Basic limiter 3. Discontinuous Galerkin (DG) Methods - Interpolation on the triangle - DG for advection equations - Basic finite element method 4. Spectral Methods and Collocation - Basic principles - Legendre and Chebyshev expansions Viscous fluid flow: CSE6623 This course is designed to give graduate students in-depth understanding of viscous dominated flow dynamics. The course is concerned with the mathematical theory of viscous fluid flow, its applications to a variety of practical problems with appropriate simplifications and solution strategies, and a deep understanding of the character and properties of general flow fields such as both high- and low-reynolds-number flows, laminar flows, boundary layer flows, potential flows, jet and wakes, etc. The course includes the followings: 1. Introduction to fluid mechanics - Continuum / Fluid properties /Dimensional analysis 2. Cartesian tensors - Scalars, vectors and tensors - Gradient, divergence, and curl - Gauss theorem / Stokes theorem 3. Flow kinematics - Fluid motions / Acceleration / Strain and rotation rates 4. Conservation laws - Mass conservation - Navier-Stokes equations - Special forms of the Navier-Stokes equations 5. Potential flows - Circulation theorem / Vortex theorem / Vorticity equation 6. Laminar flows - Exact solutions for steady incompressible viscous flow - Similarity solution for unsteady incompressible viscous flow - Low-Reynolds number viscous flow 7. Boundary layers - Boundary layer properties - Von Karman momentum integral equation - Blasius solution / Falkner-Skan similarity solution - Flow past a circular cylinder / sphere - Wakes, jets, secondary flows 10

11 8. Academic Calendar September 1 st week: First week for the Fall Semester Registration takes place this week, along with introductory talks and induction events. December : Last day of teaching in Fall Semester January: Period of exams for Semester One course units (to be confirmed) March 1 st week: First week of teaching in Spring Semester June 3rd week: Last week of teaching in Spring Semester September 1 st week: Deadline for submitting M.S. and Ph.D. dissertation 11