Core Course Structure and Syllabus

Dept. of Mechanical Engineering

M Tech (Computational Mechanics)

SEMESTER-I

SEMESTER-II

SEMESTER-III

SEMESTER-IV

ME 541 Continuum Mechanics

Introduction to Tensors: Vectors and second order tensors; Tensor operation; Properties of tensors; Invariants, eigenvalues and eigenvectors of second order tensors; Tensor fields; Differentiation of tensors; Divergence, Stokes and Localization theorems.; Kinematics of Deformation: Continuum hypothesis; Deformation mapping; Material (Lagrangian) and Spatial (Eulerian) field descriptions; Length, area and volume elements in deformed configuration; Material and spatial time derivatives - velocity and acceleration; Linearized kinematics; Balance Laws: Conservation of mass; Balance of linear and angular momentum - Cauchy stress tensor, state of stress; Spatial and material forms of balance laws - concept of first and second Piola-Kirchoff stress tensors; Conservation of energy; Continuum Thermodynamics: Basic laws of thermodynamics; Energy equation; Entropy; Clausius-Duhem inequality. Constitutive Equations: Material frame-indifference; Objective stress and stress-rates; Material symmetry; Constitutive relations for Hyperelastic Solids, Generalized Hooke's law; Simple fluids; Navier-Stokes equation.

Texts/References:

1. Jog, C. S., Continuum Mechanics: Foundations and Applications of Mechanics, Volume-I, Third edition, Cambridge-IISc Series, Cambridge university press, 2015.
2. Tadmor, E. B., Miller, R. E., and Elliot, R. S., Continuum Mechanics and Thermodynamics: From Fundamental Concepts to Governing Equations, Cambridge University Press, 2012.
3. Lai, W. M., Rubin, D., and Krempl, E., Introduction to Continuum Mechanics, Butterworth-Heinemann, 4th edition, 2015.
4. Bower, A. F., Applied Mechanics of Solids, CRC Press, 2010. Website: http://solidmechanics.org/
5. Rudnicki, J. W., Fundamentals of Continuum Mechanics, John Wiley & Sons, 2015.
6. Heinbockel, J. H., Introduction to Tensor Calculus and Continuum Mechanics, Trafford Publishing, 2001.
7. Gurtin, M., Fried, E. and Anand, L., The Mechanics and Thermodynamics of Continua, Cambridge University Press, 2013.
8. Mase, G. T., and Mase, G. E., Continuum Mechanics for Engineers, CRC Press, 2nd Edition, 1999.
9. Malvern, L. E., Introduction to the Mechanics of A Continuous Medium, Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1969.
10. Jaunzemis, W., Continuum Mechanics, The Macmillan Company, New York, 1967.
11. Chadwick, P., Continuum Mechanics: Concise Theory and Problems, Dover Publications Inc., New York, 1999.
12. Chandrasekharaiah D. S., Debnath L., Continuum Mechanics, Academic press, 1994..

ME 542 Numerical Analysis

Introduction to numerical analysis, Significant digits, Types of errors; Stability; Accuracy; Solutions of Linear Algebraic Equations: Direct elimination methods, Pitfalls of elimination methods, Norm and condition number; Iterative methods, Accuracy and convergence of iterative methods; Solution of Eigenvalue Problems; Solutions of Nonlinear Equations: Newton’s method, System of nonlinear equations, Convergence and Error analysis; Interpolation: Lagrange polynomials, Divided difference polynomials, Hermite and cubic spline interpolation, Least square approximation; Numerical Differentiation – Unequally spaced data and Equally spaced data, Error estimation and extrapolation; Numerical quadrature – Newton-Cotes, Gauss quadrature, Multiple integrals; Initial and boundary value problems – Classification of ODEs, One step methods, Convergence and numerical stability analysis, Solution of higher order equations, Multistep methods, Convergence and stability analysis.

Laboratory component : The lab is intended to be a platform for students to get used to scientific computing. Strong emphasis is laid on computer programming and the student is expected to write his own programs/codes for prototypical mathematical problems which will have real--life applications in the area of computational mechanics.

Texts/References:

1. M. T. Heath, Scientific Computing - An Introductory Survey, Revised Second Edition, SIAM, 2018
2. S. D. Conte and C. de Boor, Elementary Numerical Analysis, Third Edition, Tata McGraw-Hill Education, 2005.
3. F.B. Hildebrand, Introduction to Numerical Analysis, Second (Revised) Edition, Courier Dover Publications, 1987.
4. E. Kreyszig, Advanced Engineering Mathematics, Tenth Ed., John Wiley and Sons, 2010.
5. R. L. Burden and J. D. Faires, Numerical Analysis, 9th Edition (second Indian Reprint 2012), Brooks/Cole, 2011.
6. L.N. Trefethen, David Bau III, Numerical Linear Algebra, SIAM, 1997.
7. A.Quarteroni, R. Sacco, and F. Saleri. Numerical Mathematics, Springer-Verlag, New York, 2000.
8. G. M. Phillips and P. J. Taylor, Theory and Applications of Numerical Analysis, Second Edition, Academic Press, 1996.
9. J. D. Hoffman, Numerical Methods for Engineers and Scientists, Second Edition (Special Indian Edition), CRC Press, 2001.
10. K. E. Atkinson. An Introduction to Numerical Analysis, Second Edition, Wiley, 2004.
11. R. W. Hamming, Numerical Methods for Scientists and Engineers, Second Edition, Dover, 1986.

ME 543 Computational Fluid Dynamics

Basic equations of Fluid Dynamics: General form of a conservation law; Equation of mass conservation; Conservation law of momentum; Conservation equation of energy. The dynamic levels of approximation. Mathematical nature of PDEs and flow equations. Basic Discretization techniques: Finite Difference Method (FDM); Analysis and Application of Numerical Schemes: Consistency; Stability; Convergence; Fourier or von Neumann stability analysis; Modified equation; Application of FDM to wave, Heat, Laplace and Burgers equations. Integration methods for systems of ODEs: Linear multi-step methods; Predictor-corrector schemes; ADI methods; The Runge-Kutta schemes. Vorticity-stream function formulation. Solution of Navier-Stokes equations using MAC algorithm. The Finite Volume Method (FVM) and conservative discretization. Numerical solution of the incompressible Navier-Stokes equations: Primitive variable formulation; Pressure correction techniques like SIMPLE, SIMPLER and SIMPLEC; Brief introduction to compressible flows and numerical schemes – quick idea of Euler equations, homogenity and flux jacobian. Introduction to upwind schemes.

Texts/References:

1. J. C. Tannehill, D. A. Anderson, and R. H. Pletcher, Computational Fluid Mechanics and Heat Transfer, CRC Press, 2012.
2. J. D. Anderson Jr., Computational Fluid Dynamics, McGraw-Hill International Edition, 2017.
3. S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corporation, 2017.
4. J. H. Ferziger, and M. Peric, Computational Methods for Fluid Dynamics, Springer, 2001.
5. T. J. Chung, Computational Fluid Dynamics, Cambridge University Press, 2010.
6. C. A. J. Fletcher, Computational Techniques for Fluid Dynamics, Vol. 1 and 2, Springer, 1998.
7. H. K. Versteeg and W. Malalasekera, An introduction to computational fluid dynamics: The finite volume method 3e, Pearson Education, 2007.
8. C. Hirsch, Numerical Computation of Internal and External Flows, Vol.1 and 2, John Wiley & Sons, 2007.

ME 544 Computational Mechanics Lab

Introduction to Ansys modules including general steps for solving a problem. Simple mesh generation for one-dimensional or two-dimensional domain; Using appropriate commercial package: A simple 2-D heat conduction problem. Heat convection in a heated cavity. A problem from bluff body flows/shear flows; Using appropriate commercial package: One dimensional problem: A cantilevered beam/simply supported beam with distributed load/concentrated load. An I-beam under distributed/concentrated load. A two dimensional problem: plate under tensile loading. Truss, closed cylinder under pressure.

ME 532 Finite Element Methods in Engineering

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