Unit - 1 Fundamental Concepts of FEA
Introduction - Brief history of FEM, Finite element terminology (Nodes, Elements, Domain, Continuum, Degrees of freedom, Loads and constraints) General FEM procedure, Applications of FEM in various fields, P and h formulation, Advantages and disadvantages of FEM. Consistent units system. Review of solid mechanics stress equilibrium equations, Strain-displacement equations, Stress-Strain-Temperature relations, Plane stress, Plane strain and Axi-symmetric problems, Strain energy, Total potential energy. Essential and natural boundary conditions. Review of matrix algebra (Vectors, Matrices, Symmetric banded matrix, Determinants, Inverses), Banded skyline solutions. Introduction to solvers (Sparse solver, Iterative solver, PCG, Block lanczos). Introduction to different approaches used in FEA such as direct approach, Variational approach, Weighted residual, Energy approach, Galerkin and raleigh ritz approach.
Unit - 2 1D Elements
Types of 1D elements. Displacement function, Global and local coordinate systems, Order of element, Primary and secondary variables, Shape functions and its properties. Formulation of elemental stiffness matrix and load vector for spring, Bar, Beam, Truss and Plane frame. Transformation matrix for truss and plane frame, Assembly of global stiffness matrix and load vector, Properties of stiffness matrix, Half bandwidth, Boundary conditions elimination method and penalty approach, Symmetric boundary conditions, Stress calculations.
Unit - 3 2D Elements
Types of 2D elements, Formulation of elemental stiffness matrix and load vector for plane stress/strain such as Linear Strain Rectangle (LSR), Constant Strain Triangles (CST), Pascal‘s triangle, Primary and secondary variables, Properties of shape functions. Assembly of global stiffness matrix and load vector, Boundary conditions, Solving for primary variables (displacement), Overview of axi-symmetric elements.
Unit - 4 Isoparametric Elements
Concept of isoparametric elements, Terms isoparametric, Super parametric and subparametric. Isoparmetric formulation of bar element. Coordinate mapping - Natural coordinates, Area coordinates (for triangular elements), Higher order elements (Lagrangean and serendipity elements). Convergence requirements - Patch test, Uniqueness of mapping - Jacobian matrix. Numerical integration - 2 and 3 point Gauss quadrature, Full and reduced integration. Sub-modeling, Substructuring.
Unit - 5 1D Steady State Heat Transfer Problems
Introduction, Governing differential equation, Steady-state heat transfer formulation of 1D element for conduction and convection problem, Boundary conditions and solving for temperature distribution.
Unit - 6 Dynamic Analysis
Types of dynamic analysis, General dynamic equation of motion, Point and distributed mass, lumped and Consistent mass, Mass matrices formulation of bar and beam element. Undamped-free vibration- Eigenvalue problem, Evaluation of eigenvalues and eigenvectors (natural frequencies and mode shape