Review of FE techniques for linear elasticity; Review of continuum mechanics—kinematics, balance laws, stress measures, Clausius Duhem inequality, frame indifference, stress rates and constitutive equations; Introduction to directional derivatives, formulation of variational principles for nonlinear problems and linearisation; Linearisation of variational principles for nonlinear problems; Generalised Newton Raphson scheme; Applications to hyperelasticity, metal plasticity and crystal plasticity; Issues of convergence rates, measures and patch tests; Techniques for dealing with locking issues; Using and incorporating UMAT and UEL subroutines in ABAQUS.

Reference Books:

1. Ted Belytschko, Nonlinear Finite Elements for Continua and Structures. John Wiley & Sons, Ltd.

2. K. J. Bathe, Finite Element Procedures. Prentice – Hall Ltd.

3. M. A. Crisfield, Non-linear Finite Element Analysis: Essentials (Volume 1), John Wiley & Sons, Ltd.

4. M. A. Crisfield, Non-linear Finite Element Analysis: Advanced topics (Volume 2), John Wiley & Sons, Ltd.

Pre-requisites:

An introductory course on linear FEM (ME623 or equivalent) and a course on Continuum Mechanics (ME621 or equivalent). Familiarity with tensor operations will be assumed.

Introduction to the Course and Some Applications of Difference Equations in Engineering, Preliminaries in linear algebra and analysis, Analogies between differential and difference equations, Elementary Difference Operations: the Difference and the Shift operators, The Difference and Summation Calculus, Linear difference equations, First order equations, Higher Order Difference Equations, Linear difference equations with constant coefficients, Linear difference equations with variable coefficients, Method of undetermined coefficients and variation of parameters, Generating functions, The z-transform and its applications, Systems of Linear difference equations and applications, The Sturmian theory and Fourier techniques, Asymptotic methods, Limiting behavior of solutions, Nonlinear difference equations and boundary value problems, Stability theory and relevance to dynamical systems, Partial Difference Equations, Differential-difference equations, Discrete Mechanics, Open problems.

Lecture wise Breakup

I. Lecture 1 (1 Lecture):

• Introduction and Some Applications of Difference Equations in Engineering (1 lecture): Overview of the course. Discussion on some applications, to mechanical systems, of finite difference schemes, Some applications of finite difference schemes in understanding dynamics of discrete mechanical systems as well as discretized systems.

II. Lecture 2–4 (3 Lectures): (Preliminaries in linear algebra and analysis)

• Sets, Mappings and some Sequence Spaces with useful Structure, `p spaces on finite and infinite grids (1 lecture)

• Review of linear algebra and vector spaces, Definition of Banach space and Hilbert space, Discretespaces, Some motivating examples from discrete mechanics, particle mechanics etc, involving such spaces (1 lecture)

• Analogies between: differential and difference equations, integration and summation, Interpolation,Extrapolation (1 lecture)

III. Lecture 5–12 (8 Lectures): (The Difference and Summation Calculus)

• The Difference operator, combinatorial identities (1 lecture)

• The Shift operator, Symbolic calculus (2 lectures)

• Relations between Shift and Difference Operators, Stirling numbers, Factorial polynomial, Taylor’sTheorem and Lagrange identities (3 lectures)

• The Summation operator and applications, Bernoulli polynomials, summation by parts, symbolic operators (2 lectures)

IV. Lecture 13–22 (10 Lectures): (Linear difference equations)

• First order equations, the simplest difference equation (2 lectures)

• Higher Order Difference Equations, existence and uniqueness theorem (1 lecture)

• Linear difference equations with constant coefficients, characteristic roots, fundamental set of solutions(1 lectures)

• Method of undetermined coefficients and variation of parameters, application of symbolic calculus (2 lectures)

• Method of Generating functions (1 lecture)

• Method of z-transform and its applications (1 lecture)

• Linear difference equations with variable coefficients, Gamma function, factorial series, inverse factorialseries (2 lectures)

V. Lecture 23–27 (5 Lectures): (Systems of Linear difference equations)

• As a Generalization of high order difference equations (1 lecture)

• Matrix based methods for constant coefficients (2 lecture)

• Matrix based methods for variable coefficients, applications to mechanics (1 lecture) 4. Method of variation of parameters (1 lecture)

VI. Lecture 28–32 (5 Lectures): (The Sturmian theory and Fourier analysis)

• Adjoint problem, Self-adjoint operator (1 lecture)

• discrete Sturm-Liuoville problems, Comparison and Separation Theorems (2 lectures)

• Eigenvalues and Eigenfunctions, separated boundary conditions and periodic boundary conditions,Spectral Theorem (1 lecture)

• applications to mechanics (1 lecture)

VII. Lecture 33–36 (4 Lectures): (Asymptotics)

• Limiting behavior of solutions (2 lecture): Poincar´e Theorem, Levinson Theorem

• Stability theory and relevance to dynamical systems (1 lecture)

• Nonlinear difference equations and boundary value problems (1 lecture)

VIII. Lecture 37–38 (2 Lectures): (Partial Difference Equations)

• Taylor’s theorem, Partial Difference Equations in two dimensions (1 lecture)

• Partial Difference Equations in arbitrary dimensions with Constant coefficients (1 lecture)

IX. Lecture 39–40 (2 Lectures): (Special Topics)

• Difference Equations with Continuous Time, Differential-difference equations (1 lecture)

• Lagrangian and Hamiltonian Formalism for Difference Equations. Discrete Symmetry, Discrete Mechanics, Open problems in difference equations (1 lecture)

References:

1. An Intoduction to Difference Equations, S. N. Elaydi (Springer) (Textbook)

2. Difference Equation, W. G. Kelley and A. C. Peterson (Academic Press)

3. Finite Difference Equations. H. Levy and F. Lessman (MacMillan)

4. A treatise on the Calculus of Finite Differences. G. Boole. (MacMillan)

Syllabus:

Introduction; Channel Flow; Dissipation effect, Compliance of channel wall. Transport Laws; Boundary slip, Momentum accommodation coefficient, Thermal accommodation coefficient, Diffusion, Dispersion and Mixing; Surface Tension Dominated Flows; Thermo-capillary flows, Diffuso-capillary flows, Electrowetting, Charged Species Flow; Electro-osmosis, Electrophoresis, Dielectrophoresis, Magnetism and Microfluidics;. Microscale Heat Conduction; Energy Carriers, Scale effects, Kinetic theory, Boltzmann transport theory, Microscale Convection; Flow boiling, Condensation, Micro heat pipes, Micro Fabrication; Measurements.

Lecture-wise Breakup:

I. Introduction:

• Historical Perspectives, Definition, Biological Systems, Analogy with computational platforms, Benefits, Application Examples: Micro Electro-Mechanical Systems (MEMS), Lab on a Chip, Micro reactor, Micro heat pipes, Micro sensors, Micro actuators,  Micro Pumps, Drug delivery systems. (1 lecture)

II. Scaling Analysis:

• Natural systems, Parallel plate capacitor for sensor, Micro droplets, Micro resonator, Micro reactor, Micro heat exchangers (1 lecture)

III. Channel Flow:

• N-S equations, Dimensional Analysis, Hydraulic resistance, Arbitrary shaped channel flow, Elliptic, Equilateral and Rectangular channel flow, Dissipation effect, Compliance of channel wall. (3 lectures)

IV. Transport Laws:

• Boundary slip, Momentum accommodation coefficient,  Thermal accommodation coefficient, Thermal creep, Knudsen Compressor, Slip flow boundary condition in liquids and gases, Physical parameters affecting Slip, Slip Model Derivation, Compressibility effect, Slip flow between parallel plates and Couette  flow, Introduction to molecular modeling, Deterministic molecular modeling, Statistical molecular modeling, Boltzmann Equation, Direct Simulation Monte-Carlo (DSMC) Method. (4 lectures)

V. Diffusion, Dispersion and Mixing:

• Random walk model of diffusion, Stokes-Einstein Law, Fick's law, Governing equation of multicomponent system, Characteristic nondimensional parameters, Fixed planar source diffusion, Constant planar source diffusion,  Convection-diffusion equation, Taylor dispersion, Micromixer examples, Soluble or rapidly reacting wall, Reverse osmosis channel flow. (4 lectures)

VI. Surface Tension Dominated Flows:

• Microscopic model of surface tension, Gibbs free energy, Young-Laplace equation, Contact angle (Static and Dynamic), Wetting, Super hydrophobicity and hydrophilicity, Coating flows, Thermo-capillary flows, Thermo capillary pump, Diffuso-capillary flows, Electro-wetting, Taylor flows, Two-phase liquidliquid flows, Clogging pressure, Digital microfluidics, Marangoni convection and instability. (4 lectures)

VII. Charged Species Flow:

• Electrical conductivity and charge transport, Electrohydrodynamic transport theory, Transport equation of dilute binary electrolyte, Electrolytic cell, Electric double layer or Debye sheath, Electro-kinetic phenomena, Electro-osmosis, Electro-osmotic micro-channel systems, Electro-osmotic Pumps, Electrophoresis, Dielectrophoresis, Particle trapping. (4 lectures)

VIII. Magnetism and Microfluidics:

• Introduction to magnetism theory, Magnetic beads, Magnetic force, Motion of magnetic particles, Magnetophoresis, Magnetic fluid flow fractionation, Magnetic sorting, Magnetic separation, Ferrofluidic pump, Heat transfer enhancement using ferrofluid, Magneto-hydrodynamics, MHD based micro-pump. (3 lectures)

IX. Microscale Heat Conduction:

• Energy Carriers, Time and length scales, Scale effects, Fourier's law, Hyperbolic heat conduction, Kinetic theory, Electron thermal conductivity in metals, Lattice thermal conductivity, Scale effects of thermal conductivity, Boltzmann transport theory, Heat transport in thin films and at solid-solid interfaces, Heat conduction in semiconductor devices and interconnects, Laser heating. (4 lectures)

X. Microscale Convection:

• Scaling laws, Temperature jump boundary condition, Convection in parallel plate channel flow and Couette flow with and without viscous dissipation, Similarity and dimensionless parameters, Flow boiling in micro channels, Mini-channel versus micro-channel, Nucleate and convective boiling, Dryout incipience quality, Saturated and sub-cooled flow boiling, Condensation heat transfer in mini-micro channels, Micro heat pipes (4 lectures)

XI. Micro Fabrication:

• Functional materials, Lithography, Subtractive technique, Etching, Wet etching, Dry etching, Deep reactive ion etching, Additive techniques, Physical vapor deposition, Chemical vapor deposition, PDMS based molding, Bonding, Laser micro fabrication technique. (4 lectures)

XII. Measurements:

• Micro scale velocimetry,  Microscale thermometry (3 lectures)

Reference Books:

1. Introduction to Microfluidics, P. Tabeling, Oxford University Press, 2005

2. Microflows & Nanoflows: Fundamental and Simulation, G. Karniadakis, A. Beskok, N. Aluru, Springer Publication, 2005

3. Microfluidics for Biotechnology, J. Berthier and P. Silberzan, Artech House, 2006.

4. Theoretical Microfluidics, H. Bruus, Oxford University Press, 2008.

5. Fundamentals and Applications of Microfluidic 2nd edition, N.T. Nguyen and S.T. Wereley, Artech House, 2006 .