This course provides a knowledge and understanding of a numerical approach for solving differential equations such as ODEs, PDEs and various mathematical models. This course starts with an introduction to the finite element methods, comparison of FEMs with the finite difference methods, Methods of weighted residuals, Least square Galerkin’s method, Variational formulations, Ritz method etc. This course also helps to understand solving procedure of simple problems of ODEs, construction of linear quadratic and higher order element in one dimension, construction of simplex element in two and three dimensions, constructions of quadratic triangular elements and rectangular elements etc.In this course we will introduce the domain discretization in one dimension, two dimension and discretization with the curved boundaries. Together with we discuss construction of Basis functions, Interpolation functions in order to solve various problems related to ODEs and PDEs. Finally, we discuss about the solving procedure of two dimensional partial differential equations under different geometric conditions.This course gives a complete knowledge for better understanding the subject Finite Element Methods. The learning outcome will be assessed through relevant questions on the assignments and comprehensive final. As an outcome of this course students will learn how to solve ODEs and PDEs by using the finite element methods. ODEs and PDEs have huge applications in various fields of science and engineering and hence, learning finite element methods in order to solve ODEs and PDEs numerically is highly beneficial.
Overview
Syllabus
Weeks Weekly Plan
1 Weekly Lecture Topics
Day 1 Module 1 : Introduction to Finite Element Methods Day 2 Module 2 : Consideration of Mathematical Models Appears in Science and Engineering Day 3 Module 3 : Comparison Between Finite Element Methods (FEMs) With Finite Difference Methods (FDMs) Day 4 Interaction based on the three Modules covered Day 5 Deadline for submitting assignments
2
Day 1 Module 4 : Methods of Weighted Residuals Day 2 Module 5 : Collocation Methods Day 3 Module 6 : Point and Subdomain Collocation Methods Day 4 Interaction based on the three Modules covered Day 5 Deadline for submitting assignments3
Day 1 Module 7 : Least Square Method Day 2 Module 8 : Galerkin Method Day 3 Module 9 : Ritz Method Day 4 Interaction based on the three Modules covered Day 5 Deadline for submitting assignments
4
Day 1 Module 10 : Applications of Rayleigh-Ritz Method Day 2 Module 11 : Variational Methods Day 3 Module 12 : Weak Derivatives Day 4 Interaction based on the three Modules covered Day 5 Deadline for submitting assignments
5
Day 1 Module 13 : Examples of Weak Derivatives Day 2 Module 14 : Types of Boundary Conditions Day 3 Module 15 : Linear and Bilinear Functionals Day 4 Interaction based on the three Modules covered Day 5 Deadline for submitting assignments
6
Day 1 Module 16 : Quadratic Functionals and Comparison Between Galerkin and Rayleigh-Ritz Method Day 2 Module 17 : Applications of FEM: Solving Nodal Displacement of the Rod Day 3 Module 18 : Constructions of Linear Elements in FEM Day 4 Interaction based on the three Modules covered Day 5 Deadline for submitting assignments
7
Day 1 Module 19 : One Dimensional Linear Shape Functions and Quadratic Elements Day 2 Module 20 : Calculating Temperature Distribution for a Model Poisson Problem Day 3 Module 21 : General Formulas for Solving One Dimensional Problems in FEM and Examples Day 4 Interaction based on the three Modules covered Day 5 Deadline for submitting assignments
8
Day 1 Module 22 : Calculation of Nodal Displacement, Element Stresses and Reaction Forces for a Steel Stepped Bar Day 2 Module 23: Calculation of Stiffness Matrix, Nodal Displacement for a Taper Steel Plate Problem Day 3 Module 24 : Solving Mathematical Model of a Tapered Bar Using FEM in 1D Day 4 Interaction based on the three Modules covered Day 5 Deadline for submitting assignments
9
Day 1 Module 25 : Finite Element Meshes in One and Two Dimension Day 2 Module 26 : Three Dimensional Finite Elements, Finite Element Models for Second Order Differential Equations Day 3 Module 27: Constructions of Weak or Variational Form for Model Second Order Differential Equations Day 4 Interaction based on the three Modules covered Day 5 Deadline for submitting assignments
10
Day 1 Module 28 : Linear and Quadratic Interpolation Functions Day 2 Module 29 : Finite Element Formulation for Second Order Differential Equations Day 3 Module 30 : Derivation of Stiffness Matrix and the Matrix Equation for Linear and Quadratic Elements Day 4 Interaction based on the three Modules covered Day 5 Deadline for submitting assignments
11
Day 1 Module 31 : Connectivity of Elements, Serendipity Elements in FEM Day 2 Module 32 : Shape Functions for Corner Nodes of Eight Noded Serendipity Elements Day 3 Module 33: Shape Functions for Middle Nodes of Eight Noded Serendipity Elements, Isoparametric Elements Day 4 Interaction based on the three Modules covered Day 5 Deadline for submitting assignments
12
Day 1 Module 34 : Types of Isoparametric Elements Day 2 Module 35 : Numerical Example on Coordinate Transformation in Isoparametric Formulation Day 3 Module 36 : Solving Second Order Differential Equations in One Dimension, Numerical Integration Day 4 Interaction based on the three Modules covered Day 5 Deadline for submitting assignments
13
Day 1 Module 37 : Gauss Quadrature Method, Newton-Cotes Quadrature Method Day 2 Module 38 : Two Dimensional Partial Differential Equations Under Different Geometric Conditions Day 3 Module 39 : Weak Form of Two Dimensional Partial Differential Equations Day 4 Interaction based on the three Modules covered Day 5 Deadline for submitting assignments
14
Day 1 Module 40 : Finite element model for Two-Dimensional PDE, Interpolation Functions for Triangular Element Day 2 Module 41 : Interpolation Functions for Linear Rectangular Element Day 3 Module 42 : Quadratic Elements, Evaluation of Element Matrices and Vectors Day 4 Interaction based on the three Modules covered Day 5 Deadline for submitting assignments
15
Day 1 Module 43 : Element Matrices of a Linear Triangular Element Day 2 Module 44 : Axisymmetric Problems, Various Types of Triangular Meshes Day 3 Module 45 : Evaluation of Shape Functions for Two-Dimensional Constant Strain Triangle (CST) Element Day 4 Interaction based on the three Modules covered Day 5 Deadline for submitting assignments
16
Day 1 Module 46 : Evaluation of Stiffness Matrix for Two-Dimensional CST element Day 2 Module 47 : Evaluation of Stiffness Matrix for an Axisymmetric Element Day 3 Module 48 : Calculation of Stress-Strain Relationship Matrix, Strain Displacement Matrix and Stiffness Matrix for CST Element Day 4 Interaction based on the three Modules covered Day 5 Deadline for submitting assignments
17
Day 1 Module 49 : Plane Strain Problem, Calculate the Element Stresses, Principal Stresses and Principal Angle for Triangular Element Day 2 Module 50 : Derivation of Stiffness Matrix for 2D Heat Conduction and Convection Day 3 Interaction based on the three Modules covered Day 4 Deadline for submitting assignments Day 5
1 Weekly Lecture Topics
Day 1 Module 1 : Introduction to Finite Element Methods Day 2 Module 2 : Consideration of Mathematical Models Appears in Science and Engineering Day 3 Module 3 : Comparison Between Finite Element Methods (FEMs) With Finite Difference Methods (FDMs) Day 4 Interaction based on the three Modules covered Day 5 Deadline for submitting assignments
2
Day 1 Module 4 : Methods of Weighted Residuals Day 2 Module 5 : Collocation Methods Day 3 Module 6 : Point and Subdomain Collocation Methods Day 4 Interaction based on the three Modules covered Day 5 Deadline for submitting assignments3
Day 1 Module 7 : Least Square Method Day 2 Module 8 : Galerkin Method Day 3 Module 9 : Ritz Method Day 4 Interaction based on the three Modules covered Day 5 Deadline for submitting assignments
4
Day 1 Module 10 : Applications of Rayleigh-Ritz Method Day 2 Module 11 : Variational Methods Day 3 Module 12 : Weak Derivatives Day 4 Interaction based on the three Modules covered Day 5 Deadline for submitting assignments
5
Day 1 Module 13 : Examples of Weak Derivatives Day 2 Module 14 : Types of Boundary Conditions Day 3 Module 15 : Linear and Bilinear Functionals Day 4 Interaction based on the three Modules covered Day 5 Deadline for submitting assignments
6
Day 1 Module 16 : Quadratic Functionals and Comparison Between Galerkin and Rayleigh-Ritz Method Day 2 Module 17 : Applications of FEM: Solving Nodal Displacement of the Rod Day 3 Module 18 : Constructions of Linear Elements in FEM Day 4 Interaction based on the three Modules covered Day 5 Deadline for submitting assignments
7
Day 1 Module 19 : One Dimensional Linear Shape Functions and Quadratic Elements Day 2 Module 20 : Calculating Temperature Distribution for a Model Poisson Problem Day 3 Module 21 : General Formulas for Solving One Dimensional Problems in FEM and Examples Day 4 Interaction based on the three Modules covered Day 5 Deadline for submitting assignments
8
Day 1 Module 22 : Calculation of Nodal Displacement, Element Stresses and Reaction Forces for a Steel Stepped Bar Day 2 Module 23: Calculation of Stiffness Matrix, Nodal Displacement for a Taper Steel Plate Problem Day 3 Module 24 : Solving Mathematical Model of a Tapered Bar Using FEM in 1D Day 4 Interaction based on the three Modules covered Day 5 Deadline for submitting assignments
9
Day 1 Module 25 : Finite Element Meshes in One and Two Dimension Day 2 Module 26 : Three Dimensional Finite Elements, Finite Element Models for Second Order Differential Equations Day 3 Module 27: Constructions of Weak or Variational Form for Model Second Order Differential Equations Day 4 Interaction based on the three Modules covered Day 5 Deadline for submitting assignments
10
Day 1 Module 28 : Linear and Quadratic Interpolation Functions Day 2 Module 29 : Finite Element Formulation for Second Order Differential Equations Day 3 Module 30 : Derivation of Stiffness Matrix and the Matrix Equation for Linear and Quadratic Elements Day 4 Interaction based on the three Modules covered Day 5 Deadline for submitting assignments
11
Day 1 Module 31 : Connectivity of Elements, Serendipity Elements in FEM Day 2 Module 32 : Shape Functions for Corner Nodes of Eight Noded Serendipity Elements Day 3 Module 33: Shape Functions for Middle Nodes of Eight Noded Serendipity Elements, Isoparametric Elements Day 4 Interaction based on the three Modules covered Day 5 Deadline for submitting assignments
12
Day 1 Module 34 : Types of Isoparametric Elements Day 2 Module 35 : Numerical Example on Coordinate Transformation in Isoparametric Formulation Day 3 Module 36 : Solving Second Order Differential Equations in One Dimension, Numerical Integration Day 4 Interaction based on the three Modules covered Day 5 Deadline for submitting assignments
13
Day 1 Module 37 : Gauss Quadrature Method, Newton-Cotes Quadrature Method Day 2 Module 38 : Two Dimensional Partial Differential Equations Under Different Geometric Conditions Day 3 Module 39 : Weak Form of Two Dimensional Partial Differential Equations Day 4 Interaction based on the three Modules covered Day 5 Deadline for submitting assignments
14
Day 1 Module 40 : Finite element model for Two-Dimensional PDE, Interpolation Functions for Triangular Element Day 2 Module 41 : Interpolation Functions for Linear Rectangular Element Day 3 Module 42 : Quadratic Elements, Evaluation of Element Matrices and Vectors Day 4 Interaction based on the three Modules covered Day 5 Deadline for submitting assignments
15
Day 1 Module 43 : Element Matrices of a Linear Triangular Element Day 2 Module 44 : Axisymmetric Problems, Various Types of Triangular Meshes Day 3 Module 45 : Evaluation of Shape Functions for Two-Dimensional Constant Strain Triangle (CST) Element Day 4 Interaction based on the three Modules covered Day 5 Deadline for submitting assignments
16
Day 1 Module 46 : Evaluation of Stiffness Matrix for Two-Dimensional CST element Day 2 Module 47 : Evaluation of Stiffness Matrix for an Axisymmetric Element Day 3 Module 48 : Calculation of Stress-Strain Relationship Matrix, Strain Displacement Matrix and Stiffness Matrix for CST Element Day 4 Interaction based on the three Modules covered Day 5 Deadline for submitting assignments
17
Day 1 Module 49 : Plane Strain Problem, Calculate the Element Stresses, Principal Stresses and Principal Angle for Triangular Element Day 2 Module 50 : Derivation of Stiffness Matrix for 2D Heat Conduction and Convection Day 3 Interaction based on the three Modules covered Day 4 Deadline for submitting assignments Day 5
Taught by
Dr. TTHAMAL PRAMANICK
