Chapter 6, page 1

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Chapter 6 Local and Natural Coordinate Systems

• Why?
• Because changing the coordinate system can
  decrease some of the difficulties associated with
  evaluating an integral.

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Local Coordinate Systems A local coordinate
system has its origin located on the element.
Recall the 1-D linear shape functions
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The integrations can be simplified by developing
new shape functions that are defined relative to
a coordinate system with the origin located on
the element - called the local coordinate system.
Some problems require the evaluation of integrals
with products of the shape functions
and
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• 1. If the origin is at node i

1) Replace x with
?
?
Note shape function properties still hold 1 at
own node 0 at the other node ? to 1.
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2. Origin at the center of the element

• 1) Replace x with

?
?
Note shape function properties hold and q ranges
from L/2 to L/2
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• You will need to change the integration variables

where p is your new coordinate variable and g(p)
is equation relating x and p (i.e., xg(p))
For a local coordinate system, s, with the origin
at node i and s defined by x Xi s
ds
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For a local coordinate system q, with the origin
at the center of the element where,
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Example of how this simplifies things Look at

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Similarly, for q coordinates
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Natural Coordinate Systems A local coordinate
system that uses dimensionless numbers varying
from -1 to 1. Well use the symbol,

• System suitable for programs which use
  numerical integration schemes such as the
  Gauss-Legendre method.

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• Converting local coordinate systems to natural
  coordinate systems. For the q system (origin at
  center of the element) divide q by L/2

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• Natural coordinate systems for the
  one-dimensional element.

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New shape functions

• Ni(q) (1/2 q/L)
• Nj(q) (1/2 q/L)

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We must also change the variables for integration.
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The natural coordinate system uses length ratios,
It is often used for evaluating integrals that
involve a product. For example, let s distance
from node i and define

Note
are the shape function we found for
the local coordinate system, s, with origin at
node i.
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Since
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Applying change of variable rule changing
But
This relationship was shown by Abramowitz and
Stegun in 1964.
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• In general,

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Therefore,
Try

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Coordinate Systems and Limits of Integration for
the One-Dimensional Element
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2-D ElementsLocal Coordinate System

• -independent of orientation?

s
t
y
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• What is the coordinate range for each element?

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Natural Coordinate Systems

• Using natural coordinate systems are more
  convenient for both numerical and analytical
  integration.
• A local system that permits the specification of
  a point within the element by a dimensionless
  number whose absolute magnitude never exceeds
  unity (1).

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Rectangular Element. Define a coordinate system
located at the center of the element
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Converting the shape functions to the natural
coordinate system gives
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• Triangular Element
• Define 3 length ratios,

S perpendicular distance from one side
h altitude of that side
Area coordinates
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For some point, B
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So L1 Ni Similarly,
So?

• Well (again) we now have a much easier way to
  evaluate the integrals. Eisenberg and Malvern
  showed (1973)

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• For example

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• Area coordinates on the boundary of a triangle

Along side i-j, L3 0
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• The area for the triangle coordinates reduce to
  the 1-D shape functions for the local coordinate
  system in s.

?
?
?
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Using the 1-D natural coordinate notation we can
write
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Therefore, any integral on the edge of a
triangular element can be replaced by a line
integral written in terms of s or ?2
Example along of triangle
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but
so
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CONTINUITY
The function for approx. ?(x,y) consists of a set
of continuous piecewise smooth equations, each
defined over a single element. The need to
integrate this piecewise smooth function places a
requirement on the order of continuity between
elements. If the integral contains 1st
derivative terms, ? must be continuous between
elements but or need not
to be. For 1-D elements continuity is assured
since we have a common node.
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Show continuity along a common boundary of 2
triangular elements
Given
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But,

• Then,

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Homework

• Due next class period.