Lecture on Introduction to FEM (Lecture #1)

1. Finite Element Method (FEM)
Module Code:
Lecture on Introduction to FEM, 15 September 2019
By: Dr. Mesay Alemu Tolcha
Faculty of Mechanical Engineering, JiT, Ethiopia.
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2. Content I
1. Introduction
• What is the Finite Element Method?
• Common Application Areas
• Modeling and Finite Element Methods
• FEM Procedures
2. Direct approach to solution of one-dimensional problem
• Spring Element and Truss Element
• Analysis of Truss Structure
• Heat Transfer in a composite wall
• Pipe flow element
3. Finite Element Methods Formulation
• Introduction
• Weighted Residual Methods
• Galerkin’s Method
• Galerkin’s Finite Element Method/ Dicretization Principle
• Variational Method 2/21

3. Content II
4. Interpolation Functions and Isoparametric Elements
• One-Dimensional Elements
• Two-Dimensional Elements
• Triangular Elements
• Area Coordinates
• Rectangular Elements
• Natural Coordinates
• Three-Dimensional Elements
• Isoparametric Formulation
• Axisymmetric Elements
5. Numerical Integration
• Transformation of integral from global to local coordinates
• Jacobian Matrix
• Gaussian quadrature, Trapezoidal rule, Simpson’s rule
6. Plain Stress and Strain Analysis by FEM
• Constitutive Equations
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4. Outline III
• Plane stress and Plane Analysis using plate Elements
• Axis Symmetric problems
7. Plane Steady State heat Conduction Analysis
• Element Thermal Stiffness Matrix and Load Vector Derivation
• Assembly of Element Matrix
• Boundary Conditions
• Imposition of Prefixed Temperature
• Imposition of Heat Flux
• Imposition of Convective Heat Transfer
8. Transient Heat Conduction Analysis
• Element Thermal Stiffness and Capacitance Matrix Derivation
• Assembly of element matrix
• Imposition of boundary conditions
9. Dynamic Analysis of a Beam by FEM
• Element Mass, stiffness matrices and load vectors
• Modal analysis 4/21

5. ...Cont
10. Software Practice
• Python
• ABAQUS/ANSYS
Assessment Strategy:
• Exercises: 15%
• Project 1: 15%
• Project 2: 20%
• Final exam: 50%
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6. Application area of FEM
What do you expect from this course?
• The finite element method is now widely used for analysis of
structural engineering problems.
◦ In civil, aeronautical, mechanical, ocean, mining, nuclear,
biomechanical,... engineering
• Since the first applications two decades ago,
• Now we see applications in linear, nonlinear, static and
dynamic analysis
• Various computer programs are available and in significant use
• Therefore, how do we cover the entire course within the
stipulated time?
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7. Engineering Analyzing Tools
Theoretical justification
Constitutive equations
Math models
New Exist
Governing models
Physical problem
Establish FEM models
Solve the models/simulate
Interpret the results
Results OK?
Extract the results
Experimental validation
Analytical
methods
FEM
solution
process
Experimental
design &
test
No
Yes
Refine
the models
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8. Methods of Engineering Analysis...
There are three approaches usually followed to undertake any
engineering analysis:
• Experimental methods
◦ Accurate but it needs man power and materials. So, it is time
consuming and cost to process.
• Analytical methods
◦ Quick and closed form solution but for simple geometries and
simple loading conditions.
• FEM or approximate methods
◦ Approximate but acceptable solution for problems involving
complex material properties and loading.
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9. Numerical Methods
The common four numerical methods are:
• Finite Difference Method
◦ For heat transfer, fluid and structural mechanics. This method
is difficult to use when regions have curved or irregular
boundaries
• Boundary element method (BEM) and Finite volume method
(FVM)... to solve thermal and computational fluid dynamics
• Finite Element Method
◦ This method is a popular numerical technique that used to
determine the approximated solution for a partial differential
equation (PDE)
• Functional approximation/Method implemented in FEM
◦ Rayleigh-Ritz (variation approach) for complex structural
◦ Weighted residual method for solving non-structural
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10. What is the finite element method ?
Finite element method is a numerical method for solving
problems of Engineering and Mathematical Physics.
• So, Finite element method can be viewed simply as a method
of finding approximate solutions for partial differential
equations.
• Or as a tool to transform partial differential equations into
algebraic equations, which are then easily solved. In this
method the body is considered as an assemblage of elements
connected at a finite number of Nodes.
• On other the hand, the FEM reduces the degree of freedom
from infinite to finite with the help of discretion or meshing
(nodes and elements)
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11. Introduction to FEM more...
Structural/Stress Analysis
Figure 1: Application area of FEM 11/21

12. Introduction to FEM more...
ements
a)
b)
c) d)
Figure 2: Multiphyics application. a) Car crush, b)Thermal stress analysis, c)
Electromagnetic analysis, d) Aerodynamics analysis
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13. History of Finite Element Methods
• Hrenikoff (1941), proposed framework method
• Courant (1943), used principle of stationary potential energy
• 1953, -Stiffness equations were written and solved using
digital computers by several research
• Clough (1960), made up the name "finite element method"
• 1970s, -FEA carried on "mainframe" computers
• 1980, -FEM code run on PCs
• 2000s, -Parallel implementation of FEM (large-scale analysis,
virtual design
• There are improvement every times in feature also.....
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14. Major FEM Analysis Steps
• Discretization of the domain into a finite number of
subdomains (elements)
• Selection of interpolation functions
• Development of the element matrix for the subdomain
(element)
• Assembly of the element matrices of each subdomain to obtain
the global matrix for the entire domain,
• Imposition of the boundary conditions
• Solution of equations
• Additional computations (if desired).
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15. Simple Example,...Approximation the area of a circle
ements
Node
Edge
Element
Figure 3: Discretisation of a Continuum
Deiscretization modelling is dividing a body it into a equivalent
system of finite elements interconnected at finite number points on
each element called node.
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16. Finite Element Method Continued
• Assume a trial Solution that satisfies the boundary condition
• The domain residual or error is calculated while satisfying the
differential equation
• The weighted sum of the domain residual computed over the
entire domain is rendered zero
• The accuracy of the assumed trial solution can be improved by
taking additional higher order terms but computations
becomes tedious.
• Therefore, it is not a trivial task to choose a single trial
function over the entire domain satisfying the boundary
condition.
• It is preferred to discretize the domain in to several elements
and use several piece wise continuous trial functions, each
valid with in a segment. 16/21

17. Common Types of Elements
1. One-Dimensional Elements (1D)
• Line, Rods, Beams, Trusses, Frames
2. Two-Dimensional Elements (2D)
• Triangular, Quadrilateral Plates, Shells, 2D Continua
3. Three-Dimensional Elements (3D)
• Tetrahedral, Rectangular, Prism (Brick), 3-D Continua
If this the case, finite element method only makes
calculations at a limited (finite) number of points and then
interpolated the results for the entire domain
Therefore, any continuous solution field such as stress,
displacement, temperature, pressure, etc. can be approximated by
a discrete model composed of a set of piecewise continuous
functions defined over a finite number of sub-domains.
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18. Two-Dimensional Discretization Refinement
ements Node
Piecewise linear representation
With 228 elements
Triangular element
Figure 4: A finite element partition
To understand the physics of such representation, mathematical
equation must be developed. The finite element equation can be
derived by either of the following methods:
1. Direct equilibrium method
2. Variation method /or Rayleigh-Ritz method
3. Weighted Residual method /or Galerkin method
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19. FE Equation Derivation
• The direct method is easy to understand but difficult for
programming while the variation and weighted residual
methods are difficult to understand but easy from a
programming.
• The Galerkin Weighted Residual formulation is the most
popular from the finite element point of view.
Weighted Residual Method
• For any problem where the differential equation of the
phenomenon can be easily formulated, Weighted residual
becomes very useful. But,
• For structural problems, potential energy function can be easily
formed, Rayleigh-Ritz method is used.
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20. FE Equation Derivation...In Direct Method
Direct approach to solution of one-dimensional problem including
the following points:
• Spring Element
• Truss Element
• Analysis of Truss Structure
• Element Transformation
• Heat Transfer in a composite wall
• Pipe flow element
Direct method, also known as the matrix stiffness method, is
particularly suited for computer-automated analysis of complex
structures including the statically indeterminate type.
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21. FE Equation Derivation...
There are many types of weighted residual methods, among of
them, three are very popular.
1. Point collocation method
2. Least square method
3. Galerkin’s method
• Among these three, the Galerkin approach has the widest
choice and is used in FEMs.
• In Galerkin’s, the trial function itself is considered as the
weighting function.
• In point collocation, residuals are set to zero at n different
locations
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