CE 620
FINITE ELEMENT METHOD
Yogesh M. Desai
Department of Civil Engineering
Indian Institute of Technology Bombay
Powai, Mumbai - 400076
Lecture 1: 02-01-2023 (c) Prof. Yogesh M Desai 1
Instructor Yogesh M. Desai
Office Room No. 126
Civil Engineering Dept.
Phone No 7333
E-mail desai@civil.iitb.ac.in
Lectures Monday 08:30 – 09:25
(SLOT 1) Tuesday 09:30 – 10:25
Thursday 10:35 – 11:30
Office Hours Wednesday 15:00 – 17:00
Extra Lectures Saturday 09:30 – 11:30
(Need Based)
Lecture 1: 02-01-2023 (c) Prof. Yogesh M Desai 2
Extra Lectures
Saturday 7th January 2023
09:30 – 11:30
Room No. 208, 2nd Floor, CED
Lecture 1: 02-01-2023 (c) Prof. Yogesh M Desai 3
Introduction
Many problems in engineering and applied
science are governed by differential or
integral equations.
Due to complexities in geometry,
properties and boundary conditions in
most real-world problems, an exact
solution cannot be obtained.
Lecture 1: 02-01-2023 (c) Prof. Yogesh M Desai 4
Introduction
Finite element method is an
approximate numerical method for
solving problems of engineering and
mathematical sciences.
Useful for problems with complicated
geometries, external influences and
properties for which analytical
solutions are not available.
Lecture 1: 02-01-2023 (c) Prof. Yogesh M Desai 5
Brief History
It is difficult to document the exact origin of the FEM,
because the basic concepts have evolved over a period
of 150 or more years.
Lecture 1: 02-01-2023 (c) Prof. Yogesh M Desai 6
Brief History
R. Courant,
Variational methods for the solution of problems of
equilibrium and vibration, Bull. Amer. Math. Soc.,49,1-23
(1943).
Presented an approximate solution to the St. Venant
torsion problem in which he approximated the warping
function linearly in each of an assemblage of triangular
elements and proceeded to formulate the problem using
the principle of minimum potential energy.
Courant’s piece-wise application of the Ritz method
involves all the basic concepts of the procedure now
known as the finite element method.
Lecture 1: 02-01-2023 (c) Prof. Yogesh M Desai 7
Brief History
Levy [1947, 1953] - Flexibility and Stiffness
Argyris [1955] - Energy Theorems and Structural
Analysis
Lecture 1: 02-01-2023 (c) Prof. Yogesh M Desai 8
Brief History
M. J. Turner, R. W. Clough, H. C. Martin and
L. J. Topp, Stiffness and deflection analysis of
complex structures, J. Aero. Sci., 23, 805-823
(1956).
showed that small portions or elements in a
continuum behave in a simplified manner.
The formal presentation of the finite element
method together with the direct stiffness
method for assembling elements is attributed
to Turner et al. (1956), who employed the
equations of classical elasticity to obtain
properties of a triangular element for use in the
analysis
Lecture of plane stress
1: 02-01-2023 problems.
(c) Prof. Yogesh M Desai 9
Brief History
Who Coined Term “Finite Elements”?
Lecture 1: 02-01-2023 (c) Prof. Yogesh M Desai 10
Brief History
R.W. Clough,
The finite element in plane stress analysis,
Proc. 2nd ASCE Conf. on Electronic
Computation, PA, Sept.1960
Termed “Finite Elements”
Lecture 1: 02-01-2023 (c) Prof. Yogesh M Desai 11
Brief History
In early 1960s, engineers used the method
for approximate solution of problems in stress
analysis, fluid flow, heat transfer, and other
areas.
The first book on the FEM by Zienkiewicz
and Chung was published in 1967.
Lecture 1: 02-01-2023 (c) Prof. Yogesh M Desai 12
How can FEM Help ?
• Can be applied to a variety of fields like
structural mechanics, aerospace
engineering, geotechnical engineering,
fluid mechanics, hydraulic and water
resource engineering, mechanical
engineering, nuclear engineering, electrical
and electronics engineering, metallurgical,
chemical and environmental engineering,
meteorology and bioengineering, etc.
Lecture 1: 02-01-2023 (c) Prof. Yogesh M Desai 13
• Easily applied to complex, irregular-shaped
objects composed of several different
properties and having complex boundary
conditions and external influences.
• Applicable to steady-state (static), time
dependent as well as characteristic value
problems.
• Applicable to linear as well as nonlinear
problems.
Lecture 1: 02-01-2023 (c) Prof. Yogesh M Desai 14
Finite Element Method is an Approximate Numerical
Method to Solve Problems of Engineering and
Mathematical Sciences.
Any given problem reduces to
Lecture 1: 02-01-2023 (c) Prof. Yogesh M Desai 15
OBJECTIVES / LEARNING OUTCOMES
• Understanding of different semi-analytical /
numerical methods to solve a variety of problems
• Understanding of general steps of FEM
• Understanding of various finite element formulations
• Ability to derive equations related to FE analysis of
various1-D , 2-D and 3-D problems
Lecture 1: 02-01-2023 (c) Prof. Yogesh M Desai 16
OBJECTIVES / LEARNING OUTCOMES
• Understanding of advantages and disadvantages of the
FEM
• Exposure to computer implementation of the FEM
• Ability to do FE analysis independently with proper
interpretation of results
Lecture 1: 02-01-2023 (c) Prof. Yogesh M Desai 17
Course Contents
Introduction
Review of Matrix Algebra 6–8 Lectures
Solution of Integral and Differential Equations
using Various Methods
Calculus of Variations
Basics of Finite Element Methods
Local and Global Finite Element Methods
Application of FEM to Solve Various 1-D, 2-D 20–24 Lectures
and 3-D Problems
C0 Continuum
C1 Continuum
Convergence and Error Estimation
Iso-Parametric Formulation
Numerical Integration
Lecture 1: 02-01-2023 (c) Prof. Yogesh M Desai 18
Course Contents
Concept of Sub-structuring 4 – 6 Lectures
Conditions of symmetry / anti-symmetry
Computer Implementation of Finite
Element Method
Application of Finite Element Method to 4 – 6 Lectures
Time Dependent Problems
Partial Finite Element Method
Exposure to Hybrid Finite Element
Methods
Total Lectures
~ 40
(~ 40 Hrs)
Lecture 1: 02-01-2023 (c) Prof. Yogesh M Desai 19
Assessment Scheme
Assignments and Term Projects :25 %
Mid - Term Exam :25 % (as per time table)
End - Term Exam :50 % (as per time table)
Notes: (1) Bring calculator to all the lecture sessions.
(2) 80% Attendance is required.
Lecture 1: 02-01-2023 (c) Prof. Yogesh M Desai 20
Lecture 1: 02-01-2023 (c) Prof. Yogesh M Desai 21
Reference Books
• Cook, R.D., Malkus, D. S., Plesha, M. E. and
Witt, R. J.: Concepts and Applications of
Finite Element Analysis, Wiley, 2013.
• Reddy, J. N.: An Introduction To The Finite
Element Method, McGraw Hill Education,
2005.
• Zienkiewicz, O. C. and Taylor, R. L.: The
Finite Element Method, Butterworth
Heinemann, 2000.
• Logan, D. L.: A First Course in the Finite
Element Method, Thomson, 2014.
