Chapter 1, page 1

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Chapter 1, Introduction

• FEM numerical procedure for obtaining solutions
  to many engineering problems
• Uses discrete elements for joint displacements
  and member forces
• Uses continuum elements for field problems
  heat transfer, fluid mechanics, and solid
  mechanics problems.

2
An analytical solution best

• Numerical methods must be used to obtain an
  approximate solution when an analytical solution
  can not be found, due to
• Irregular shape
• Impossible to describe the boundary
  mathematically
• Composite materials
• Anisotropic materials
• Etc.

3
SOLUTION OF BOUNDARY VALUE PROBLEMS (Solving
physical problems governed by a differential
equation to obtain an approximate solution.)

• Finite difference method approximates the
  derivatives in the differential equation.
• Variational method integral of a function
  produces a number find the function that
  produces the lowest number (min.)
• Weighted residual methods approximate solution
  is substituted into the differential equation.

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Common Weighted Residual Methods

• Collocation method impulse function
• Subdomain method weighting functions 1
• Galerkins method the approximating function
  is used for the weighting functions
• Least squares method - uses the residuals as the
  weighting function you get a new error term,
  which must then be minimized
• What we will use for the field problems

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INTEGRAL FORMULATION FOR NUMERICAL SOLUTIONS

• Variational Method
• It is not applicable to a differential equation
  containing a first derivative
• Based on Calculus of Variation
• A functional appropriate for the differential
  equation is minimized with respect to
  undetermined coefficients in the approximate
  solution.

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Governing differential equation
The calculus of variations shows that the
equation giving the lowest value for p is the
solution of the differential equation
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Weighted Residual Methods
An approximate solution is substituted into the
governing differential equation for p.
THE RESIDUAL
So we require that
Where wi(x) is the weighting function an R(x) is
the residual. Note Write one residual equation
for every unknown
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• Common choices for the weighting functions

Method
Collocation - residual vanishes at points(Impulse
function)
Subdomain - residual vanishes over interval
Approx. function
Galerkins method - similar results to
variational method
Least Squares Method - error minimized with
respect to unknown coefficients in the
approximating solution
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For a simply supported beam with concentrated end
moments
EI
H
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Exact Solution
11
Method
Collocation
Subdomain
Galerkin
Least squares
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In the example, the Galerkins method uses the
same function for wi(x) that was used in the
approximate solution.
.
13
Integrating yields
Solving gives
and the approximate solution is
This solution is identical to the solution
obtained using the variational method.
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• Comparison of Errors

30
?
?
?
20
Subdomain method
?
10
?
0.125
0.25
0.375
0.50
0
Percent error in deflection
?
?
?
Variational method Galerkins method Least
squares method
?
10
?
?
20
Collocation method
?
?
30
?
?
Collocation and subdomain method errors depend on
choice of collocation point/subdomain.
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Structural and Solid Mechanics
Potential Energy Formulation
The displacements at the equilibrium position
occur such that the potential energy of a stable
system is a minimum value.
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• The Finite Element Method (FEM) is a numerical
  technique for obtaining approximate solutions to
  engineering problems.
• Subdivisions
• Discrete element formulation (Matrix Analysis of
  Structures) Utilizes discrete elements to
  obtain the joint displacements and member forces
  of a structural framework.
• Continuum element formulation yields
  approximate values of the unknowns at nodes.
• The FEM produce a system of linear or nonlinear
  equations.

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THE FINITE ELEMENT METHOD

• Basic Steps

1. Discretize the region gt nodes 2. Specify
the approximation equation (linear, quadratic) 3.
Develop the system of equations 4. Solve the
system 5. Calculate quantities of interest
derivative of the parameter
Galerkin - 1/node Potential Energy - 1/displ.