Chapter 1, page 1

1
Chapter 1 Introduction

• Solution of boundary value problems
• Integral formulations for numerical solutions
• Potential energy formulation (structural and
  solid mechanics, 2 3-D elasticity, plate shell
  structures
• The finite element method a numerical method
  used to obtain an approximate solution at
  discrete points.

2
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.

3
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

4
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.

5

• Weighted Residual Methods

THE RESIDUAL
(1.3)
Require that
(1.4)
Where is the weighting
function Note one residual equation for every
unknown
6

• 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
Least Squares Method - error minimized with
respect to unknown coefficients in the
approximating solution
7
For a simply supported beam with concentrated end
moments
EI
H
8
Exact Solution
9

• For the example problem
• Approximating function

10
Method
Collocation
Subdomain
Galerkin
Least squares
11

• Galerkins Method - used to develop the finite
  element equations for the field problems (not
  structural nor solid mechanics).

12
Integrating yields
Solving gives
and the approximate solution is
This solution is identical to the solution
obtained using the variational method.
13

• 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.
14
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.
15

• 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.