# Finite Element Methods

By Dr. TAMAL PRAMANICK   |   National Institute of Technology (NIT) Calicut
Learners enrolled: 101
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 Galerkinâ€™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.

Summary
 Course Status : Upcoming Course Type : Elective Duration : 16 weeks Category : Teacher Education Credit Points : 5 Level : Undergraduate Start Date : 01 Jul 2024 End Date : 30 Sep 2024 Enrollment Ends : 31 Aug 2024 Exam Date : 15 Dec 2024 IST EXAM SHIFT : SHIFT 2

Note: This exam date is subjected to change based on seat availability. You can check final exam date on your hall ticket.

### Course layout

week 1
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  With Finite Difference
Methods (FDMs)
Day 4 Interaction based on the three Modules covered
Day 5 Deadline for submitting assignments

week 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 assignments

week 3
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

week 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

week 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

week 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

week 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

week 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

week 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

week 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
Day 4 Interaction based on the three Modules covered
Day 5 Deadline for submitting assignments

week 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

week 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

week 13
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

week 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

week15
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

week 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

week 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

### Books and references

1. J. N. Reddy, Introduction to the Finite Element Methods, Tata McGraw-Hill, 2003.
2. K.J. Bathe, Finite Element Procedures, Prentice-Hall, 2001.
3. R.D. Cook, D.S. Malkus and M.E. Plesha, Concepts and Applications of Finite
Element Analysis, John Wiley and Sons, 2002.
4. Thomas J.R. Hughes, The Finite Element Method: Linear Static and Dynamic Finite
Element Analysis, Dover Publication, 2000.
5. George R. Buchanan, Finite Element Analysis, McGraw Hill, 1994.

### Dr. TAMAL PRAMANICK

National Institute of Technology (NIT) Calicut
The course coordinator completed his M.Sc and Ph.D from IIT Guwahati. After that he selected for NBHM Post-Doctoral Fellowship and continued his Post-doctoral research at IISc Bangalore. Next, he continued his academic journey as an Assistant professor at NIT Calicut from March, 2020 onwards.
The course coordinator published article in highly reputed journals, such as SIAM J. Numer. Anal., Appl. Numer. Math., J. Comp. Math, Numer. Methods of PDEs, Comput. Appl. Math. and others.
As an Assistant Professor the course coordinator taught various subjects to B.Tech., M.Sc., M.Tech. and Ph.D. students. He is a Faculty Advisor of M.Sc. students, and also Departmental Lab In-charge, Selection committee member of Ph.D. students.
He has delivered lectures as an Invited speaker at various Colleges and Universities. Also, he visited as an invited speaker at the Institute for Advanced Study in Mathematics (IASM) Hangzhou, China.  He has completed one FRG project, supported by NIT Calicut and currently one NBHM project is ongoing. Also, one internship project VRITIKA, sponsored by SERB has been completed as a coordinator of the project. Three students have completed their internship under this sponsored fund.

### Course certificate

Course Completion  will carry 70% weightage of end term exam and 30% weightage of internal assessments. A minimum 40% in each is required to qualify for the Course Completion Certificate.