UNIT 1 : Fundamental Concepts of FEA Introduction : Solution methodologies to solve engineering problems, governing equations, mathematical modelling of field problems in engineering, discrete and continuous models. Brief history of FEM, Finite Element terminology (nodes, elements, domain, continuum, degrees of freedom, loads & constraints), general steps involved in FEM, applications of FEM in various fields, advantages and disadvantages of FEM, consistent units system, essential and natural boundary conditions, symmetric boundary conditions. Introduction to different approaches used in FEA : Direct approach, Variational formulation- Principal of Minimum Potential Energy (PMPE), Galerkin weighted residual method, Principle of Virtual Work, Rayleigh-Ritz method, relation between FEM and Rayleigh-Ritz method. Types of Analysis (Introduction) : Linear static analysis, Non-linear analysis, Dynamic analysis, Linear buckling analysis, Thermal analysis, Fatigue analysis, Crash analysis. (Chapter - 1) Unit 2 : 1D Elements Types of 1D elements, displacement function, global and local coordinate systems, polynomial form of interpolation functions - linear, quadratic and cubic, properties of shape function, primary and secondary variables. Formulation of elemental stiffness matrix and load vector for bar, truss and beam using any approach, Formulation of load vector due to uniform temperature change (only for bar). Assembly of global stiffness matrix and load vector, properties of stiffness matrix, half bandwidth, treatment of boundary conditions - elimination approach, stress and reaction forces calculations. (Chapter - 2) Unit 3 : 2D Elements Two-Dimensional Stress Analysis: Plane Stress/Strain problems in 2D elasticity, constitutive relations Constant Strain Triangle (CST), Liner Strain Rectangle (LSR), displacement function, Pascal‘s triangle, compatibility and completeness requirement, geometric isotropy, convergence requirements, strain filed, stress filed, Formulation of element stiffness matrix and load vector for Plane Stress/Strain problems. Assembly of global stiffness matrix and load vector, Boundary conditions, solving for primary variables (displacement), stress calculations. (Chapter - 3) Unit 4 : Isoparametric Elements and Numerical Integration Concept of isoparametric elements, Terms isoparametric, super parametric and subparametric. Coordinate mapping : Natural coordinates, Area coordinates (for triangular elements), higher order triangular and quadrilateral elements (Lagrangean and serendipity elements), geometry associative mesh, quality checks, mesh refinement - p vs h refinements, Uniqueness of mapping - Jacobian matrix. Numerical integration : Gauss Quadrature in one and two dimension, Order of Gauss integration, full and reduced integration, sub-modeling, substructuring. (Chapter - 4) Unit 5 : 1D Steady State Heat Transfer Problems Introduction, One dimensional steady-state heat transfer problem - Governing differential equation, Finite Element formulation using Galerkin’s approach for composite wall and thin Fin , essential and natural boundary conditions and solving for temperature distribution. (Chapter - 5) Unit 6 : Dynamic Analysis Types of dynamic analysis, general dynamic equation of motion, lumped and consistent mass, Mass matrices formulation of bar, truss and beam element. Undamped-free vibration : Eigen value problem, evaluation of eigen values and eigen vectors (characteristic polynomial technique). (Chapter - 6)
