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These slides are a gentle introduction to mappings in continuum mechanics. Goals: Introduce students to the concept of mapping. Point out that y=f(x) is a mapping from x to y, and then slowly build to vector-vector mappings, y=f(x). Introduce the concept of reference and spatial configurations. Show some common deformation mappings in continuum mechanics. Explain the physical meaning of a derivative of a mapping. These slides should be viewed in slideshow mode. Press F5 to go into slideshow mode.

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

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

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

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EXAMPLE: Length of a vector: input is a vector x output is a scalar L Q: What shape is described by = constant? 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! Isosurfaces (contours) are lines of constant f(x) COPYRIGHT:

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Mapping from a scalar to a vector input is a scalar output is 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. COPYRIGHT:

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EXAMPLE 1: MAPPING TO POINTS ON A CIRCLE scalar vector COPYRIGHT:

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1-D reference configuration (single scalar ranging from 0 to ) Symbolically, we would write this mapping as Two outputs (components of vector x) determined from just one scalar input (the angle ). Each 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 Range: semi-circle Given the velocity of a point in the reference configuration, you can use the mapping to figure out velocity in the spatial configuration. COPYRIGHT:

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1-D reference configuration (single scalar ranging from -1 to ) 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 =0. Different domain! Same range (semicircle), but this is a different mapping is tangent to the curve. Its length is proportional to hash mark spacing. COPYRIGHT: As always, speed and direction come from the derivative.

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EXAMPLE 2: CUBIC SPLINES (start PowerPoint on your laptop and follow along) COPYRIGHT: 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.

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Cubic spline COPYRIGHT:

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

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

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Right-click an endpoint and select “smooth point.” 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) move this new handle Then move the new handle. COPYRIGHT:

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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. COPYRIGHT:

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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. COPYRIGHT:

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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. COPYRIGHT:

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

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control vector reference configuration A curve may be represented parametrically as Each “t” corresponds to a different x. Then… 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 t, so it makes sense to introduce a “stretch” vector: COPYRIGHT:

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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). COPYRIGHT:

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With the a-vectors determined, the entire line may be drawn parametrically by allowing t to vary from 0 to 1, and finding x by For example, the midpoint in the reference configuration (t=1/2) maps to in the spatial configuration. The midpoint in the reference configuration does not necessarily map to the midpoint of the spatial curve (recall our second example of a zoomed map to the hemisphere). Length of the curve: Integrals over the spatial configuration are easier when transformed to integrals over the reference configuration! COPYRIGHT:

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Mapping from a vector to a vector input is a vector output is 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). COPYRIGHT:

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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 COPYRIGHT:

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Everything is Mapping Suppose that this is a color plot of pressure (= the average of the diagonal components of stress). 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) 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. What additional mappings are needed to make this pressure plot during postprocessing of the FEM output? COPYRIGHT:

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TERMINOLOGY scalar-to-scalar function vector-to-vector function linear affine quadratic COPYRIGHT:

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

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A point originally at deforms to Example: homogeneous deformation: This mapping is “homogeneous” because each little square deforms the same as all the others initial location deformed location 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. COPYRIGHT:

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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. COPYRIGHT:

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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. COPYRIGHT:

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Here is a another quadratic mapping. Look at the formula for the mapping. It says that the vertical component remains unchanged after deformation (x 2 =X 2 ), and the horizontal displacement (u 1 =x 1 -X 1 ) increases quadratically with vertical distance from the base. COPYRIGHT:

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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 (u 1 =x 1 -X 1 =2X 1 +2) involves doubling the horizontal width and translating horizontally by a distance 2. The vertical displacement in the spatial configuration, u 2 =x 2 -X 2 =sin(3X 1 ), varies sinusoidally as you move horizontally in the reference configuration. COPYRIGHT:

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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. COPYRIGHT:

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SIMPLE SHEARPURE SHEAR COPYRIGHT:

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ROTATION COPYRIGHT:

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Vortex: The rotation angle increases with proximity to the origin. Nonlinear because varies with position COPYRIGHT:

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Torsion: The rotation angle increases with distance up the axis, X 3. Nonlinear because varies with position COPYRIGHT:

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A positive Jacobian is necessary, but not sufficient, for invertibility of the mapping. Nonphysical Material Interpenetration COPYRIGHT: 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!

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Deformation of a unit square (or unit cube in 3D) unit square 1 1 3/2 1/4 1/2 2 deformed size is times larger! COPYRIGHT: deformed parallelogram

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QUADRATIC FORM Visualization of a vector-to-scalar mapping describes an “isosurface” COPYRIGHT:

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Quadratic Forms COPYRIGHT: ellipsoid if >0 cylinder as disk as as hyperboloid if <0 These plots show the surface changing as changes!

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QUADRATIC FORM FOR TENSORS! Visualization of a tensor-to-scalar mapping describes an “isosurface” in 9D tensor space 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. COPYRIGHT:

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