Optimization Techniques

1
University of Illinois-Chicago
Chapter 7 Introduction to Finite Element Method
Principles of Computer-Aided Design and
Manufacturing Second Edition 2004 ISBN
0-13-064631-8 Author Prof. Farid.
Amirouche University of Illinois-Chicago
2
CHAPTER 7
7.1 Introduction
7.1 INTRODUCTION
FEM is a technique that discretizes a given
physical or mathematical problem into smaller
fundamental parts, called elements . Then an
analysis of the element is conducted using the
required mathematics.
Figure 7.1 A finite element mesh of hip prosthesis
Principles of Computer-Aided Design and
Manufacturing Second Edition 2004 ISBN
0-13-064631-8 Author Prof. Farid. Amirouche,
University of Illinois-Chicago
3
CHAPTER 7
7.1 Introduction
Figure 7.2 A output of stress distribution after
simulation
Principles of Computer-Aided Design and
Manufacturing Second Edition 2004 ISBN
0-13-064631-8 Author Prof. Farid. Amirouche,
University of Illinois-Chicago
4
7.2 BASIC CONCEPTS IN THE FINITE-ELEMENT METHOD
CHAPTER 7
7.2 Basic Concepts in FEM
The basic steps in FEA (Finite Element Analysis)
consists of 3 phases

• Preprocessing Phase
• 1) Create and discretize the solution domain into
  finite elements that is ,subdivide the problem
  into nodes and elements.
• 2) Assume a shape function to represent the
  physical behavior of an element that is, an
  approximate continuous function
• is assumed to represent the solution of an
  element.
• 3) Develop equations for an element.
• 4) Assemble the elements to present the entire
  problem. Construct the global stiffness matrix.
• 5) Apply boundary conditions, initial conditions,
  and loading.

Principles of Computer-Aided Design and
Manufacturing Second Edition 2004 ISBN
0-13-064631-8 Author Prof. Farid. Amirouche,
University of Illinois-Chicago
5
CHAPTER 7
7.2 Basic Concepts in FEM
b) Solution Phase 6) Solve a set of linear or
nonlinear algebraic equations simultaneously to
obtain nodal results, such as displacement
values at different nodes or temperature values
at different nodes in a heat transfer
problem. c) Post processor Phase 7) Obtain
other important information. At this point, you
may be interested in values of principal
stresses, heat fluxes, etc.
Principles of Computer-Aided Design and
Manufacturing Second Edition 2004 ISBN
0-13-064631-8 Author Prof. Farid. Amirouche,
University of Illinois-Chicago
6
CHAPTER 7
7.2 Basic Concepts in FEM
Figure 7.3 A general discretization of a body
into finite elements
Principles of Computer-Aided Design and
Manufacturing Second Edition 2004 ISBN
0-13-064631-8 Author Prof. Farid. Amirouche,
University of Illinois-Chicago
7
CHAPTER 7
7.2 Basic Concepts in FEM
where ke (e being the element) is the local
element stiffness matrix, f?e is the external
forces applied at each element, ue represents the
nodal displacements for the element, and f?eadd
represents the additional forces. (7.1)
Assuming
Equation 7.1 reduces to
(7.2)
In terms of stresses and strains equation (7.2)
becomes
?e seue
(7.3)
where ?e is the element stress matrix, and se is
the constitutive relationship connecting ue and
?e .
Principles of Computer-Aided Design and
Manufacturing Second Edition 2004 ISBN
0-13-064631-8 Author Prof. Farid. Amirouche,
University of Illinois-Chicago
8
CHAPTER 7
7.2 Basic Concepts in FEM
Example 7.1
A load of P 180 lbs act on the variable cross
sectional bar in which one end is fixed and a
compressive load act at the other end. Dimensions
of the bar are shown in the figure 7.4. Calculate
the deflection at various points along its
length. Calculate the reaction force at the fixed
end and compute stresses in each element.
Figure 7.4(a) A variable cross-sectional bar
subject to a compressive load
Principles of Computer-Aided Design and
Manufacturing Second Edition 2004 ISBN
0-13-064631-8 Author Prof. Farid. Amirouche,
University of Illinois-Chicago
9
CHAPTER 7
7.2 Basic Concepts in FEM
Solution
From figure 7.4 we compute the cross-sectional
areas , and assign values of the Youngs modulus
of elasticity to each of the segment bars.
A12.3248 in2. E1669400.12 lb/in2 . A20.7750
in2. E2458 lb/in2 . A32.0922
in2. E3669400.12 lb/in2 . P 180 lb.
The stiffness of each bar segment is derived from
Hookes Law where
(7.4)
(7.5)
(7.6)
Principles of Computer-Aided Design and
Manufacturing Second Edition 2004 ISBN
0-13-064631-8 Author Prof. Farid. Amirouche,
University of Illinois-Chicago
10
CHAPTER 7
7.2 Basic Concepts in FEM
Figure 7.4(b) Free body diagram of the forces
Principles of Computer-Aided Design and
Manufacturing Second Edition 2004 ISBN
0-13-064631-8 Author Prof. Farid. Amirouche,
University of Illinois-Chicago
11
CHAPTER 7
7.2 Basic Concepts in FEM
Applying the I law of mechanics where ?F0, at
each node
At node 1
(7.7)
At node 2
(7.8)
At node 3
(7.9)
At node 4
(7.10)
The above equations can be written in matrix form
as
(7.11)
Principles of Computer-Aided Design and
Manufacturing Second Edition 2004 ISBN
0-13-064631-8 Author Prof. Farid. Amirouche,
University of Illinois-Chicago
12
CHAPTER 7
7.2 Basic Concepts in FEM
Since u1 0 , this is equivalent to eliminating
row 1 and column 4 of the stiffness matrix.
Column 1
The matrix equation (7.11) reduces to
(7.12)
Plugging in the values for all the Ks and
substituting the value of P
(7.13)
Principles of Computer-Aided Design and
Manufacturing Second Edition 2004 ISBN
0-13-064631-8 Author Prof. Farid. Amirouche,
University of Illinois-Chicago
13
CHAPTER 7
7.2 Basic Concepts in FEM
From the Equation (7.13) we get
Principles of Computer-Aided Design and
Manufacturing Second Edition 2004 ISBN
0-13-064631-8 Author Prof. Farid. Amirouche,
University of Illinois-Chicago
14
CHAPTER 7
7.2 Basic Concepts in FEM
Example 7.2 This is a similar example to 7.1,
however element z is replaced with two identical
supports which forms element 2 and 3. A load of P
180 lbs acts on the variable cross sectional
bar in which one end is fixed and a compressive
load acts at the other end. Dimensions of the bar
are indicated in the figure 7.5. Calculate the
deflection at various points along its length.
Calculate the reaction force at the fixed end and
the corresponding stresses in each element.
Figure 7.6 A variable cross-section bar with
identical supports in the middle.
Principles of Computer-Aided Design and
Manufacturing Second Edition 2004 ISBN
0-13-064631-8 Author Prof. Farid. Amirouche,
University of Illinois-Chicago
15
CHAPTER 7
7.2 Basic Concepts in FEM
The cross sectional areas are found to be
The Youngs modulus of elasticity for each segment
 
       
The load P is
The element stiffness is found as
Principles of Computer-Aided Design and
Manufacturing Second Edition 2004 ISBN
0-13-064631-8 Author Prof. Farid. Amirouche,
University of Illinois-Chicago
16
CHAPTER 7
7.2 Basic Concepts in FEM
At node 1
(7.14)
At node 2
(7.15)
At node 3
(7.16)
At node 4
(7.17)
The above equations can be written in matrix form
as
(7.18)
Principles of Computer-Aided Design and
Manufacturing Second Edition 2004 ISBN
0-13-064631-8 Author Prof. Farid. Amirouche,
University of Illinois-Chicago
17
CHAPTER 7
7.2 Basic Concepts in FEM
Since u1 0, Equation ( 7.18 ) can be reduced to
(7.19)
Substituting the values of Ks and P we obtain
 
The solution of which is found to be
Principles of Computer-Aided Design and
Manufacturing Second Edition 2004 ISBN
0-13-064631-8 Author Prof. Farid. Amirouche,
University of Illinois-Chicago
18
CHAPTER 7
7.2 Basic Concepts in FEM
Stresses for each element are obtained as
lb/in2
lb/in2
lb/in2
Principles of Computer-Aided Design and
Manufacturing Second Edition 2004 ISBN
0-13-064631-8 Author Prof. Farid. Amirouche,
University of Illinois-Chicago
19
CHAPTER 7
7.2 Basic Concepts in FEM
Example 7.3 A tapering round bar is fixed at one
end and a tensile load P180 lbs is applied at
the other end. The maximum and minimum radius of
the bar is 20 in and 10 in respectively. The
bars modulus of elasticity E 669400.12 lb/in2.
Consider the bar as a set of 4 elements of equal
length and uniformly increasing diameter. Find
the global stiffness matrix and displacements at
each node and reaction force.
Figure 7.6 A Tapered round bar subject to a
tensile load P
Principles of Computer-Aided Design and
Manufacturing Second Edition 2004 ISBN
0-13-064631-8 Author Prof. Farid. Amirouche,
University of Illinois-Chicago
20
CHAPTER 7
7.2 Basic Concepts in FEM
Figure 7.7 A four element representation of the
tapered round bar
Principles of Computer-Aided Design and
Manufacturing Second Edition 2004 ISBN
0-13-064631-8 Author Prof. Farid. Amirouche,
University of Illinois-Chicago
21
CHAPTER 7
7.2 Basic Concepts in FEM
The geometrical data and stiffness properties for
each element are
Since ?F0 at each node therefore,
(7.20)
At node 1
(7.21)
At node 2
(7.22)
At node 3
(7.23)
At node 4
(7.24)
At node 5
(7.25)
Principles of Computer-Aided Design and
Manufacturing Second Edition 2004 ISBN
0-13-064631-8 Author Prof. Farid. Amirouche,
University of Illinois-Chicago
22
CHAPTER 7
7.2 Basic Concepts in FEM
Applying the boundary conditions, that is u5 0
we get
(7.26)
       
A solution of which is found to be
(7.27)
The reaction force is obtained by writing the
equation form, (7.25) where
(7.28)
Substituting the values of u5 and u4 and k4, we
obtain
(7.29)
Principles of Computer-Aided Design and
Manufacturing Second Edition 2004 ISBN
0-13-064631-8 Author Prof. Farid. Amirouche,
University of Illinois-Chicago
23
7.3 POTENTIAL ENERGY FORMULATION
CHAPTER 7
7.3 Potential Energy Formulation

• Finite element method is based on the
  minimization of the total potential energy
  formulation.
• When close form solution is not possible
  approximation methods such as Finite Element are
  the most commonly used in solid mechanics.
• To illustrate how finite element is formulated,
  we will consider an elastic body such as the one
  shown in Figure 7.9 where the body is
  subjected to a loading force which causes it to
  deform.

Principles of Computer-Aided Design and
Manufacturing Second Edition 2004 ISBN
0-13-064631-8 Author Prof. Farid. Amirouche,
University of Illinois-Chicago
24
CHAPTER 7
7.3 Potential Energy Formulation
Figure 7.9 Elastic body
Principles of Computer-Aided Design and
Manufacturing Second Edition 2004 ISBN
0-13-064631-8 Author Prof. Farid. Amirouche,
University of Illinois-Chicago
25
CHAPTER 7
7.3 Potential Energy Formulation
From Hookes Law, we write

(7.30)

where
F represents the compressive
force A represents area of the elastic
body E represents the youngs modulus
of elasticity
be the displacement in the y-direction,
and L is the length of the segment. From Hookes
Law the force displacement relationship is

where represents the energy
(7.32)
Principles of Computer-Aided Design and
Manufacturing Second Edition 2004 ISBN
0-13-064631-8 Author Prof. Farid. Amirouche,
University of Illinois-Chicago
26
CHAPTER 7
7.3 Potential Energy Formulation
We can express the energy equation in terms of
the stress-strain as
Where represents the element strain
energy and the volume of element is
we can then write the strain energy for an
element introducing the superscript e
(7.35)
The stress can be substituted for by
(7.36)
Using the above relation in Equation (7.32 ), we
obtain
(7.37)
Principles of Computer-Aided Design and
Manufacturing Second Edition 2004 ISBN
0-13-064631-8 Author Prof. Farid. Amirouche,
University of Illinois-Chicago
27
CHAPTER 7
7.3 Potential Energy Formulation
The total energy, which consists of the strain
energy due to deformation of the body and the
work performed by the external forces can be
expressed as function of combined energy as
Where denotes displacement along the force
direction Fi .
(7.38)
It follows that a stable system requires that the
potential energy be minimum at equilibrium
(7.39)
We know that strain is defined making use of the
relative displacement between adjacent elements
(7.40)
Principles of Computer-Aided Design and
Manufacturing Second Edition 2004 ISBN
0-13-064631-8 Author Prof. Farid. Amirouche,
University of Illinois-Chicago
28
CHAPTER 7
7.3 Potential Energy Formulation
Equation (7.39) has two components. Expression
for the first term is
(7.41)
and
(7.42)
In matrix form
(7.43)
where
Principles of Computer-Aided Design and
Manufacturing Second Edition 2004 ISBN
0-13-064631-8 Author Prof. Farid. Amirouche,
University of Illinois-Chicago
29
CHAPTER 7
7.3 Potential Energy Formulation
Similarly , we take the second term in (7.39)
(7.44)
and
(7.45)
Combining Equations (7.43),(7.44), and (7.45)
leads to elemental force-displacement relation
(7.46)
(7.47)
or
where k(e) is the element stiffness matrix, u(e)
the element displacement associated with node i
or i1 , and F(e) denotes the external forces
acting at the nodes.
Principles of Computer-Aided Design and
Manufacturing Second Edition 2004 ISBN
0-13-064631-8 Author Prof. Farid. Amirouche,
University of Illinois-Chicago
30
7.4 CLOSED FORM SOLUTION
CHAPTER 7
7.4 Closed Loop Formulation

• The closed form solution is used when all the
  variables
• have explicit mathematical forms that can be
  dealt with
• in terms of extracting a solution.
• Consider a continuous body subject to a
  compressive
• load P (Figure 7.10).The objective is to
  determine the
• displacement or deformation at any point.
• Let the continuous body have a variable
  cross-sectional area .

Principles of Computer-Aided Design and
Manufacturing Second Edition 2004 ISBN
0-13-064631-8 Author Prof. Farid. Amirouche,
University of Illinois-Chicago
31
CHAPTER 7
7.4 Closed Loop Formulation
Figure 7.9 A non-uniform bar subjected to
compressive load P.
Principles of Computer-Aided Design and
Manufacturing Second Edition 2004 ISBN
0-13-064631-8 Author Prof. Farid. Amirouche,
University of Illinois-Chicago
32
CHAPTER 7
7.4 Closed Loop Formulation
The equilibrium equation at any cross sectional
area cab be written as
where
(7.48)
(7.49)
Substituting the above equation into (7.48) we get
(7.50)
Recall that strain is defined as
(7.51)
Substituting the above equation into (7.50) we get
(7.52)
(7.53)
Principles of Computer-Aided Design and
Manufacturing Second Edition 2004 ISBN
0-13-064631-8 Author Prof. Farid. Amirouche,
University of Illinois-Chicago
33
CHAPTER 7
7.4 Closed Loop Formulation
Integrating the above equation will lead to exact
solution of displacement.
(7.54)
If we assume the load is constant and that
(7.55)
Equation (7.48) becomes
( 7.54 )
(7.56)
or
(7.57)
where
(7.58)
Principles of Computer-Aided Design and
Manufacturing Second Edition 2004 ISBN
0-13-064631-8 Author Prof. Farid. Amirouche,
University of Illinois-Chicago
34
CHAPTER 7
7.4 Closed Loop Formulation
Let
Hence,
(7.59)
, then
Let
(7.60)
or
(7.61)
Principles of Computer-Aided Design and
Manufacturing Second Edition 2004 ISBN
0-13-064631-8 Author Prof. Farid. Amirouche,
University of Illinois-Chicago
35
CHAPTER 7
7.4 Closed Loop Formulation
The above solution can be used to find
displacement at various points Along the length
. From (7.58) we have
And the displacement is
(7.62)
and
Thus, the solution depends upon
which we should know first hand.
Principles of Computer-Aided Design and
Manufacturing Second Edition 2004 ISBN
0-13-064631-8 Author Prof. Farid. Amirouche,
University of Illinois-Chicago
36
7.5 WEIGHTED RESIDUAL METHOD
CHAPTER 7
7.5 Weighted Residual Method

• The WRM assumes an approximate solution to the
  governing differential equations.
• The solution criterion is one where the boundary
  conditions and initial conditions of the problem
  are satisfied.
• It is evident that the approximate solution
  leads to some marginal errors.
• If we require that the errors vanish over a
  given interval or at some given points then we
  will force the approximate solution to converge
  to an accurate solution.

Principles of Computer-Aided Design and
Manufacturing Second Edition 2004 ISBN
0-13-064631-8 Author Prof. Farid. Amirouche,
University of Illinois-Chicago
37
CHAPTER 7
7.5 Weighted Residual Method
Consider the differential equation discussed in
previous example where
(7.63)
Let us choose a displacement field u to
approximate the solution, that is let
(7.64)
are unknown coefficients
where
if we substitute u(y) A(y) into the
differential equation we get
(7.65)
stands for residual.
where
Principles of Computer-Aided Design and
Manufacturing Second Edition 2004 ISBN
0-13-064631-8 Author Prof. Farid. Amirouche,
University of Illinois-Chicago
38
CHAPTER 7
7.5 Weighted Residual Method
The equation above has two constants C1 C2. If
we require that e vanishes at two points we will
get two equation which can be used to solve for
C1 C2. Solving Equation (7.66) for these
conditions , we get
(7.65)
and
(7.66)
for
The final solution for the displacement field is
(7.67)
Principles of Computer-Aided Design and
Manufacturing Second Edition 2004 ISBN
0-13-064631-8 Author Prof. Farid. Amirouche,
University of Illinois-Chicago
39
CHAPTER 7
7.6 Galerkin Method
7.6 GALERKIN METHOD
The Galerkin method requires the integer of the
error function over some selected interval to be
forced to zero and that the error be orthogonal
to some weighting functions ?i, according to the
integral
(7.68)
Where 0 , L for i 1,.., n.
?i s are selected as part of the approximate
solution. This is simply done by assigning the ?
function to the terms to multiply the
coefficients.
Because we assume
as defined in (7.63) then
?1 y and ?2 y2.
(7.64)
Now we use (7.67) and substitute ?1 y and the
residual e from (7.65). Furthermore let the
values of r1 and r2 be given from Example 7.3
then we obtain,
Principles of Computer-Aided Design and
Manufacturing Second Edition 2004 ISBN
0-13-064631-8 Author Prof. Farid. Amirouche,
University of Illinois-Chicago
40
CHAPTER 7
7.6 Galerkin Method
(7.69)
P 180 lb from Example 7.3
(7.70)
Solving the above two equations we get

C1yC2y2
(7.71)
The solution is then approximated by
Principles of Computer-Aided Design and
Manufacturing Second Edition 2004 ISBN
0-13-064631-8 Author Prof. Farid. Amirouche,
University of Illinois-Chicago
41
CHAPTER 7
7.6 Galerkin Method
Table 7.1 Comparison of displacement values by
different methods
Principles of Computer-Aided Design and
Manufacturing Second Edition 2004 ISBN
0-13-064631-8 Author Prof. Farid. Amirouche,
University of Illinois-Chicago