Course Title: 
Introduction to Finite Element Methods
Course Code: 
Year Taught: 
Subject Core
Undergraduate (UG)
School of Engineering

'Introduction to Finite Element Methods' is a course offered in the sixth semester of B. Tech. in Mechanical Engineering program at School of Engineering, Amrita Vishwa Vidyapeetham.

Unit 1

Introduction: Equilibrium equations in elasticity subjected to body force, traction forces, stress strain relations for plane stress and plane strain, Boundary conditions, Initial conditions, Euler’s Lagrange’s equations of bar, beams, Principal of a minimum potential energy, principle of virtual work, Rayleigh-Ritz method, Galerkins method, Guass elimination method, Numerical integration.

Basic Procedure: General description of Finite Element Method, Engineering applications of finite element method, Discretization process; types of elements 1D, 2D and 3D elements, size of the elements, location of nodes, node numbering scheme, half Bandwidth, Stiffness matrix of bar element by direct method, Properties of stiffness matrix, Preprocessing, post processing.

Interpolation Models: Polynomial form of interpolation functions - linear, quadratic and cubic, Simplex, Complex, Multiplex elements, Selection of the order of the interpolation polynomial, Convergence requirements, 2D Pascal triangle, Linear interpolation polynomials in terms of global coordinates of bar, triangular (2D simplex) elements, Linear interpolation polynomials in terms of local coordinates of bar, triangular (2D simplex) elements, CST element.

Higher Order and Isoparametric Elements: Lagrangian interpolation, Higher order one dimensional elements - quadratic, Cubic element and their shape functions, properties of shape functions, Truss element, Shape functions of 2D quadratic triangular element in natural coordinates, 2D quadrilateral element shape functions – linear, quadratic, Biquadric rectangular element (Noded quadrilateral element), Shape function of beam element. Hermite shape function of beam element, Numerical integration.

Unit 2

Solid Mechanics Applications: Direct method for bar element under axial loading, trusses, beam element with concentrated and distributed loads, matrices, Jacobian, Jacobian of 2D triangular element, quadrilateral, Consistent load vector.

Solution of bars, stepped bars, plane trusses, space truss, beams and frames by direct stiffness method. Solution for displacements, reactions and stresses by using elimination approach, penalty approach. Plane stress, plane strain and Axisymmetric problems. Dynamic Analysis.

Unit 3

Heat Transfer and Fluid Flow Problems: Steady state heat transfer, 1D and 2D heat conduction governing equation, boundary conditions, One dimensional element, Functional approach for heat conduction, Galerkin approach for heat conduction, heat flux boundary condition, 1D heat transfer in thin fins, heat transfer 1D and 2D problems with conduction and convection.

Fluid flow problems and Introduction to Finite Element Packages and its application to solid mechanics, fluid and heat transfer problems

  • S. S. Rao - ‘The Finite Element Method in Engineering’ Butterworth-Heinemann Ltd. - 2010 - 5th Edition
  • K. J. Bathe - ‘Finite Element Procedures’ - Prentice-Hall of India - 2014 - 2nd Edition
  • O. C. Zienkiewicz - ‘The FEM its basics and fundamentals’ - Elsevier - 2013 - 7th Edition
  • J. N. Reddy - ‘Finite Element Method’ – McGraw Hill International Edition - 2005 - 3rd Edition
  • K. H. Huebner, D. L. Dewhirst, D. E. Smith and T. G. Byron - ‘The Finite Element Method for Engineers’ - John Wiley & Sons Inc., New York - 2001 - 4th Edition
  • Chandrupatla T. R. - ‘Finite Elements in Engineering’ - Pearson Edition - 2011 - 4th Edition