Finite Element Method

Syllabus

1. Introduction (4 hours)
   1. Definition and terminologies
   2. Mathematical modeling of physical systems
   3. Steps of computational method
   4. Basic steps of finite element method
   5. Use of computer for finite element method implementation
   6. Applications of finite element method
   7. Advantages of finite element method
   8. Examples to demonstrate finite element method philosophy
2. Direct Stiffness Method for Discrete Elements (9 hours)
   1. Direct stiffness method for spring, bar and shaft elements: stiffness matrix for a spring element, assembly of element equations, applications of boundary conditions and solution; bar element; shaft element
   2. Direct stiffness method for truss elements: introduction to truss structures, stiffness matrix for plane and space truss elements, finite element solution for a truss structure, strain and stress on a truss element
   3. Direct stiffness method for beam and frame elements: stiffness matrix for beam element (with axial load, arbitrarily oriented, general three dimensional), finite element solutions of beam and frame elements
3. Finite Element Formulation of Continuum Problems (8 hours)
   1. Introduction to continuum
   2. Forms of continuum problems
   3. Need of weighted integral method
   4. Fundamental of variational calculus: independent variable, function and functional; differentiation and variation; fundamental lemma; Euler Lagrange equation; essential and natural boundary conditions
   5. Method of weighted residual
   6. Ritz method
   7. Strong and weak formulations
4. Interpolation Functions (6 hours)
   1. Introduction
   2. Types of interpolation functions
   3. Requirements of a polynomial interpolation function
   4. Selection of the order of a polynomial interpolation function
   5. Polynomial interpolation function in terms of global coordinates (one, two and three-dimensional problems)
   6. Numerical integration in one, two and three-dimensions
5. Applications in General One-dimensional Problems (4 hours)
   1. Finite element formulation of general one-dimensional problems
   2. Derivation of stiffness matrix and force vector
   3. Assembly and solution of general one-dimensional problems
6. Applications in Heat Transfer Problems (6 hours)
   1. Finite element formulation for one-dimensional plane wall
   2. Finite element formulation for one-dimensional fin
   3. Finite element formulation for two-dimensional heat transfer problem (weighted residual method and functional)
   4. Derivation of stiffness matrix and force vector using linear triangular element and bilinear rectangular element
   5. Assembly and solution of heat transfer problems
7. Applications in Elasticity Problems (6 hours)
   1. Basic equations of elasticity: equilibrium equations, stress-displacement relations, stress-strain relations
   2. Basic equations of plane elasticity: equilibrium equations and stress-strain relations for plane stress and plane strain problems
   3. Finite element formulation for plane elasticity problem (weighted residual method and functional)
   4. Derivation of stiffness matrix and force vector using linear triangular element and bilinear rectangular element
   5. Assembly and solution of plane elasticity problems
8. Higher Order Elements (2 hours)
   1. Higher order elements: quadratic and cubic interpolation functions for one, two and three-dimensional problems
   2. General interpolation functions and elements: Lagrange and serendipity elements
   3. Parametric mapping and parametric elements

Practicals

1. Analysis of nodal displacements and reaction forces in spring, bar and shaft elements
2. Computation of axial stresses and deformations in plane and space trusses
3. Evaluation of bending moments, shear forces and deflections in beam and frame elements
4. Numerical solution of field variables for general one-dimensional problems
5. Modeling of temperature distribution and heat flux in heat transfer problems
6. Determination of stress-strain fields and displacement vectors in elasticity problems

Evaluation

Final exam questions cover all chapters. Approximate marks distribution (total 60 marks over 45 hours):

1. Introduction: 8 marks
2. Direct Stiffness Method for Discrete Elements: 10 marks
3. Finite Element Formulation of Continuum Problems: 10 marks
4. Interpolation Functions: 8 marks
5. Applications in General One-dimensional Problems and Higher Order Elements: 8 marks
6. Applications in Heat Transfer Problems: 8 marks
7. Applications in Elasticity Problems: 8 marks