INTRODUCTION TO FINITE ELEMENT METHOD
ME-411
FALL 2023
CREDIT HOURS: 2+1
INTRODUCTION
• Instructor: Dr. Sana Zulfiqar
• BE Mechanical Engineering, NED University, Karachi
• MS Aerospace Engineering, IST, Islamabad
• PhD Mechanical Engineering, Universiti Sains Malaysia (USM), Malaysia
• Office: 1st Floor, Faculty Cabins
• Consulting hours: Available during office hours or appointment
RESEARCH INTERESTS
• Structure Design Analysis
• Material and mechanical characterization of stretchable and flexible polymers and their
composites, particularly for stretchable electronics applications.
COURSE DESCRIPTION
• This course covers finite element method and its applications.
• Theoretical analysis and lab work will be covered in this semester.
COURSE OBJECTIVES
• The objective of the course is to develop the concept of simulation software for the
analysis of various parameters involve in designing a product.
• Concepts of nodes, elements, meshing and stiffness matrix.
• The convergence and source of errors will also be discussed.
• Students are also exposed to static structural, thermal and heat transfer, and dynamic
analysis.
COURSE LEARNING OUTCOMES
• Determine the field variable using one / two dimensional finite elements by applying
stiffness based finite element method.
• Examine the given problem and identify the forces / boundary conditions / appropriate
meshing / element type for a structure / component in order to perform finite element
analysis.
• Create a solvable simulation model using a commercial finite element code based on the
given problem statement.
COURSE OUTLINE
• Introduction to FEA and Element Performance
• Introduction to Finite Element Modelling and preliminary decisions
• Elements types and their properties
• Basic concepts of equilibrium & compatibility
• General factors affecting element performance – Sources of errors, Convergence, FE
Methods, Shape Functions, Stiffness Matrix and Transformation, Static Structural Analysis,
Heat Transfer and Thermal Stress Analysis, Dynamic Analysis
TENTATIVE COURSE PLAN
Week
No.
Topics
Contact
Hours
1
Introduction: Course Outline, objectives, teaching plan, assessment methods, brief introduction to topics
to be covered in the course, revision of prerequisite information.
2
2-3
Introduction to the Stiffness Method - Details of stiffness method used for finite element analysis
4
4-6
Development of Truss Equations, Equilibrium and Compatibility - Develop finite element equations for one
dimensional truss elements
6
7
Element Performance and types, convergence / Simulation Exercise (2D Elements) - Introduction to Finite
Element Modelling, elements types and their properties
2
8
Mid Semester Exam (Assessment of CLO 1, 2 and 3 through examination questions)
9-10
Preliminary Decisions, Sources of errors / Simulation Exercise (3D Elements) - Preliminary decisions in
static structural analysis and various errors
4
11
Development of Beam Equations - Formulation of beam elements
2
12-13
Heat Transfer and Thermal Stress Analysis - BCs in heat transfer problems and steady state analysis for
temperature field
4
14
Dynamic Analysis – Modal Analysis
2
15
Course Review
2
16
End Semester Examination (ESE) (Assessment of CLO 1 through examination questions)
TEXT BOOK
• A first course in the finite element method by Daryl L. Logan
REFERENCE BOOKS
• Finite Element Modelling for Stress Analysis by Robert D. Cook, John Wiley & Sons Inc.,
1995
• Others as needed
ASSESSMENT METHODS & MARKS DISTRIBUTION
• Quizzes (2 Quizzes/credit hour)
15%
• Projects / Assignments (1 Assignment/credit hour or individual project) 10%
• Midterm Exam
30%
• End Semester Exam
45%
GRADING CRITERIA
• The quizzes will be unannounced and normally last for ten minutes.
• The dates for MSE and ESE will be announced and will be held in week No 7 and 13.
• Short homeworks will be assigned and submitted individually. Direct copying is not
acceptable and in this case all students with the same work will be given zero and
appropriate disciplinary action may also be taken.
• All the grading will be relative.
METHOD OF TEACHING
• Detailed discussion on the topics with few examples will be done in the class.
• Questions can be asked during the lecture.
• Quizzes will be unannounced and depend on the previous lectures or some from the
current slides.
• Anything directly or indirectly related to a studied topic can be part of quizzes or exams.
• Bonus marks will be given to those students who actively participate in the class
discussions.
JOIN MICROSOFT TEAMS
• Teams can only be joined using official email id.
• All information, announcements, assignments, notes, slides, etc. will be
available on teams.
• Any communication about course can be done through teams.
GENERAL INSTRUCTIONS
• Lectures will start at 11.05 am sharp.
• Take your own notes, but lecture slides will be fairly complete.
• Ask questions during lecture.
• Use of mobile phones, laptops, etc are not allowed during class.
• Revise the previous lectures before coming to class.
• Plagiarism will not be tolerated.
INTRODUCTION TO FINITE ELEMENT
METHODS
OVERVIEW
• Many physical phenomena in engineering
and science can be described in terms of
partial differential equations.
• In general, solving these equations by
classical analytical methods for arbitrary
shapes is almost impossible.
• The finite element method (FEM) is a
numerical approach by which these
partial differential equations can be
solved approximately.
Model
Physical
Phenomenon
•
•
•
•
•
Solid Mechanics
Fluid Mechanics
Thermal Conduction
Diffusion
Electrical Conduction
Governing
Equations &
BCs
Finite
Element
Equations
Approximation
FINITE ELEMENT METHODS
INTRODUCTION
PURPOSE OF FEM
• The Finite Element Analysis (FEA) or Finite
Element Methods is a numerical method
for solving problems of engineering and
mathematical physics.
• Based on discretization, assembly and then
solution of system equation
• Useful for problems with complicated
geometries, loadings, and material
properties where analytical solutions can
not be obtained.
• While imposing kinematic constraint
(compatibility), material constitutive behavior
and equilibrium
• Complicated domains can be handled with ease
• Virtual investigations of design
• or to observe phenomena that cannot be
observed experimentally, e.g. stress field inside a
solid component
WHY FEM IS IMPORTANT?
• Knowledge of how the method works will enhance your analysis skills and provide better
understanding of numerical analysis in general
• The intelligent use of commercial FE software and correct interpretation of the output
requires basic knowledge of the method
• Possible to extend commercial software packages by implementing user defined subroutines.
• Widespread use in industry for virtual experiments during design stage. Extremely cost
effective and fast compared to physical testing.
ORIGINS OF FEM
• 1940s -discretising continuous systems into triangular elements
• 1950s -a team from Boeingdemonstrated that complex surfaces could be analysed with a matrix of triangular shapes
• The term “finite element” was coined by R. Clough in 1960s
• 1960s –Under a contract with NASA, a general purpose FEA code developed by MacNeal-SchwendleCorporation (MSC).
68000 degrees of freedom
• Released to the public. Around the same time, ANSYS, MARC and SAP were introduced
• 1970s+80s –CAD integrated with FEA
• 2000s –commercial FEA software rapidly developed.
• Current: multi physics capabilities, damage and failure modelling, complex nonlinear material models, XFEM, mesh free etc.
Millions of dof.
WHAT WENT WRONG?
• Sinking of the SleipnerA offshore platform
• Platform produces oil and gas in the Norwegian North Sea. Supported on the
seabed at a water depth of 82 m. Concrete gravity base structure consisting of
24 cells. Four cells are elongated to shafts supporting the platform deck
• First concrete base structure for SleipnerA sprang a leak and sank under a
controlled ballasting operation (August 1991). The failure involved a total
economic loss of about $700 million.
• Investigation traced the error to inaccurate finite element approximation of the
linear elastic model. Shear stresses were underestimated by 47%, leading to
insufficient design. In particular, certain concrete walls were not thick enough
WHAT WENT WRONG?
• Titan Incident 2023
• Went under catastrophic implosion. An implosion is the opposite of an
explosion. In an explosion, the force acts outwards, but in an implosion the
force acts inwards. When a submersible is deep in the ocean it experiences
the force on its surface due to water pressure. When this force becomes
larger than the force hull can withstand, the vessel implodes violently.
• The key is the design of the hull that protects the vessel against the large
external water pressure that is trying to crush the hull. Much of the existing
technology is based on steel, titanium and aluminum. The performance of
these materials under extreme stress is well understood.
• However, the Titan’s hull had an experimental design. It used mostly carbon
fibers, which have the advantage of being lighter than titanium or steel, so
Titan could have more space for passengers. Properties of carbon fibers for
deep sea applications are, however, not that well understood. It can crack and
break suddenly.
35 milliseconds
33 milliseconds
34 milliseconds
38 milliseconds
42 milliseconds
APPLICATIONS OF FEM
• Mechanical/Aerospace/Civil/Automotive
• Engineering
• Structural/Stress Analysis
• Static/Dynamic
• Linear/Nonlinear
• Fluid Flow
• Heat Transfer
• Electromagnetic Fields
• Soil Mechanics
• Acoustics
• Biomechanics
INGREDIENTS OF FEM
PHYSICAL
MATHEMATICAL
• Discretization of the physical domain
• Solution of the PDE
• Breakdown of structural system into
components (elements) and
reconstruction by the assembly process
• Numerical approximation of a boundary
value problem by Galerkindiscretization
with functions of local support
PHYSICAL FEM
• Discretization : Model body by dividing it into an equivalent system
of many smaller bodies or units (finite elements) interconnected at
points common to two or more elements (nodes or nodal points)
and/or boundary lines and/or surfaces.
• Use simple shapes (elements)
• All elements connected at nodes
• Solution at Element 1 is described using the values at Nodes 1, 2, 6,
and 5 (Interpolation).
• Elements 1 and 2 share Nodes 2 and 6
OBJECT, ELEMENTS, NODES