1. COPYRIGHT: Rebecca.Brannon@utah.edu
A “mapping” is…
1. Input 2. Output 3. Set of rules giving output from input

2. Mapping from a scalar to a scalar
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Mapping from a scalar to a scalar
output is a scalar
input is a scalar
Example1:
Example 2:
Example 3:

3. Mapping from a vector to a scalar
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Mapping from a vector to a scalar
input is a vector
Can you think of a function that takes a vector as input and returns a scalar as output?
output is a scalar

4. EXAMPLE: Length of a vector:
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input is a vector x
output is a scalar L
Q: What shape is described by = constant?
Isosurfaces (contours)
are lines of constant f(x)
Hint: this says “length is constant”.
A: circle in 2D, sphere in 3D!
Q: Which way does the gradient point?
Hint: perpendicular to isosurfaces
A: Radially!

5. Mapping from a scalar to a vector
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Mapping from a scalar to a vector
Your physical location in space is a vector
(quantified by latitude, longitude and altitude)
When you walk to class, your location (a vector) changes with time (a scalar).
The time rate of your position is velocity, which is tangent to your path.
input is a scalar
output is a vector

6. EXAMPLE 1: MAPPING TO POINTS ON A CIRCLE
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scalar  vector
EXAMPLE 1: MAPPING TO POINTS ON A CIRCLE

7. 2-D Spatial configuration
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1-D reference configuration
(single scalar ranging from 0 to p)
Symbolically, we would write this mapping as
Two outputs (components of vector x) determined from just one scalar input (the angle q).
Each q maps to a unique x.
Moving from left to right on the domain (reference line segment) moves from right to left, or counterclockwise, on the range (spatial configuration, semicircle).
Domain: line segment
2-D Spatial configuration
Given the velocity of a point in the reference configuration, you can use the mapping to figure out velocity in the spatial configuration.
Range: semi-circle

8. 2-D Spatial configuration
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1-D reference configuration
(single scalar ranging from -1 to 1)
The “phase plot” is the trace mapped out by the x vectors as the parameter t is varied.
Here, the phase plot is the same as before (a semi-circle).
The distinction is non-uniform mapping of the hash marks.
A finite element code developer might use a mapping like this to generate a finer mesh near q=0.
Different domain!
2-D Spatial configuration
As always, speed and direction come from the derivative.
Same range (semicircle), but this is a different mapping
is tangent to the curve. Its length is proportional to hash mark spacing.

9. COPYRIGHT: Rebecca.Brannon@utah.edu
EXAMPLE 2: CUBIC SPLINES (start PowerPoint on your laptop and follow along)
This is a “hands-on” exercise for the students who have brought a laptop to class. Otherwise, download this presentation and play with it on your own.

10. Powerpoint exercise to show how to play with a cubic spline…
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Powerpoint exercise to show how to play
with a cubic spline…
From drawing tools make a straight line

11. Right-click the line and select “edit points”.
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Right-click the line and select “edit points”.
The endpoints, initially hollow circles,
will now be small squares.

12. Right-click an endpoint and select “smooth point.”
move
this new
handle
Then move the new handle.
new
“control
bar”
appears
Do the same with the other endpoint.
(if a handle doesn’t appear, turn it into a smooth point as done above)
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13. You now have a cubic spline! Try changing the curvature with the
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You now have a cubic spline! Try changing the curvature with the
left handle. (Below is a bitmap image – do it on the next page)
drag this
point…
down here.
If blue the control bar handle is not visible, right-click the curve and select “edit points” again. The handle re-appears when you single click an endpoint.

14. COPYRIGHT: Rebecca.Brannon@utah.edu
Play with it here: Escape out of slideshow mode so that you can see PowerPoint drawing tools. Right-click the curve, select “edit points.” Move the endpoints and the handles to change the shape of the curve.

15. The control bar may be regarded as a vector that
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control bar
control bar
The control bar may be regarded as a vector that
controls curvature and “stretch.”

16. spatial configuration
A curve may be represented parametrically as
reference configuration
Each “t” corresponds to a different x.
spatial configuration
control vector
control vector
The control vectors are tangent to the curve. They point in the direction that x would move if t is increased. A longer control vector corresponds to more distance covered by x for a given increment Dt, so it makes sense to introduce a “stretch” vector:
Then…
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17. Input is the scalar parameter t. Output is the x position vector.
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A cubic is the highest order polynomial capable of fitting the prescribed data.
Input is the scalar parameter t. Output is the x position vector.
That’s why the a-coefficients must be vectors.
The a-coefficient vectors are found by enforcing the following requirements:
This is a system of four equations for the four unknown coefficient vectors! (Method of solution is no different from what it would be if the a’s were scalars).

18. Mapping from a vector to a vector
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Mapping from a vector to a vector
Example 1: v=3w (output is 3 times longer than the input)
Example 2: rotate w by 30 degrees to get v
Example 3: v=w (identity operator! the output is the same as the input).
input is a vector
output is a vector

19. Continuum deformation is a mapping from a vector to a vector
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Continuum deformation is a mapping from a vector to a vector
input is initial location of a point
output is deformed location of the same point

20. COPYRIGHT: Rebecca.Brannon@utah.edu
Everything is Mapping
What additional mappings are needed to make this pressure plot during postprocessing of the FEM output?
Suppose that this is a color plot of pressure (= the average of the diagonal components of stress).
3D VECTOR (physical position) maps to 2D VECTOR (screen position)
This tells us which pixels need to be lit.
But in what color?
We need to map pressure to color.
Input to the FEM code (impact speeds, mesh parameters, material properties, etc.) is ultimately…
mapped to output from the FEM code (e.g., deformed position vectors in 3D space, stress tensor).
TENSOR (stress)
maps to SCALAR (pressure)
maps to 1D line segment (the legend)
maps to 3D array (RGB color)
maps to voltage (pixel illumination)

21. scalar-to-scalar function vector-to-vector function
TERMINOLOGY
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scalar-to-scalar function
vector-to-vector function
linear
linear
affine
affine
quadratic
quadratic

22. How do “beginners” plot the function
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Let’s plot an affine vector-to-vector mapping the same way

23. Example: homogeneous deformation:
This is the same as the matrix equation in the previous slide with X and x as the variable names instead of x and y.
Here is how to plot this function using Mathematica. Note that it is a parametric plot. As X varies, the mapping tells how x varies.
A point originally at
deforms to -1 1 2 3 4 5
initial
location
deformed
This mapping is “homogeneous”
because each little square deforms the same as all the others
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24. Distinctive features of homogeneous mapping:
The previous slide showed a square grid. This picture shows additional (circle and diagonal) “paint lines” that flow with the material.
Distinctive features of homogeneous mapping:
All squares (big or little) deform to self-similar parallelograms, circles deform to ellipses, and straight lines deform to rotated and stretched straight lines.
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25. Here is a quadratic mapping.
The little squares don’t all deform in the same way.
Some straight lines deform to straight lines, but others (the diagonals) don’t.
Circles don’t
deform to ellipses.
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26. Here is a another quadratic mapping.
Look at the formula for the mapping.
It says that the vertical component remains unchanged after deformation (x2=X2), and the horizontal displacement (u1=x1-X1) increases quadratically with vertical distance from the base.
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27. Here is a generally nonlinear mapping.
Each little square deforms differently  not homogeneous.
Look at the formula for the mapping. It says that the horizontal displacement (u1=x1-X1=2X1+2) involves doubling the horizontal width and translating horizontally by a distance 2. The vertical displacement in the spatial configuration, u2=x2-X2=sin(3X1), varies sinusoidally as you move horizontally in the reference configuration.
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28. COPYRIGHT: Rebecca.Brannon@utah.edu

29. COPYRIGHT: Rebecca.Brannon@utah.edu

30. Affine (homogeneous) tangent mapping.
A tangent mapping (colored part of the figure) is a homogeneous mapping (self-similar parallelograms) that coincides with a nonlinear mapping at a particular location.
This is like the local straight line that is tangent to a nonlinear curve at a point.
nonlinear
mapping
Affine (homogeneous) tangent
mapping.
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31. SIMPLE SHEAR
PURE SHEAR
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32. ROTATION
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33. Vortex: Nonlinear because a varies with position
The rotation angle increases with proximity to the origin.
Nonlinear because a varies with position
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34. Torsion: Nonlinear because a varies with position
The rotation angle increases with distance up the axis, X3.
Nonlinear because a varies with position
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35. That’s why FEM codes have contact algorithms!
A positive Jacobian is necessary, but not sufficient, for invertibility of the mapping.
This massive bending of a square into a big ring-like shape is locally invertible (positive Jacobian) everywhere on the domain, but material interpenetration makes it not globally invertible.
That’s why FEM codes have contact algorithms!
Nonphysical
Material
Interpenetration
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36. Deformation of a unit square (or unit cube in 3D)
1/2 2
unit
square 1 1
deformed parallelogram
3/2
1/4
 deformed size is times larger!
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37. QUADRATIC FORM describes an “isosurface”
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Visualization of a vector-to-scalar mapping
describes an “isosurface”
QUADRATIC FORM
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38. Quadratic Forms These plots show the surface changing as h changes!
ellipsoid if h>0
cylinder as h®¥
disk as as h®0
hyperboloid if h<0
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39. QUADRATIC FORM FOR TENSORS!
Visualization of a tensor-to-scalar mapping
describes an “isosurface” in 9D tensor space
QUADRATIC FORM FOR TENSORS!
EXAMPLE: Plasticity yield criteria say that yield occurs when the stress is on the zero-isosurface of the yield function The yield surface is embedded in 9D tensor space, and it can be regarded as being in 6D space since stress is symmetric, and it can be visualized in 3D principal space when the yield function depends only on the stress invariants.
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40. COPYRIGHT: Rebecca.Brannon@utah.edu