OREGON STATE UNIVERSITY
Term assignment
Note: I simply want to evaluate how well you understood some of the main concepts we explored during the class. Please
iefly answer the following questions,
inging in figures and equations from course documents as needed. Simply put answers after the questions in this document. I have also collected documents on attached. These and the lab assignments will be useful for you.
1. The most common type of digital image co
elation uses a two-camera stereo-vision method. Describe
iefly the simplified mathematical basis for depth measurement that stereo-vision provides, and the role of camera cali
ation.
2. Describe the image co
elation steps that are required to analyze a set of stereo-image pairs, and how this generates “full-field” displacement data.
3. Describe how we used an Ordinary Least Squares method to calculate the displacement gradient tensor, and how the displacement gradient relates to strain. How is Lagrangian strain different from engineering strain, and what are the advantages of Lagrangian strain?
4. Both Ordinary and Nonlinear Least Squares methods determine the parameter values of a model. What distinguishes models suitable for Ordinary from those that require Nonlinear approaches? Give examples of each.
5. There are two key assumptions made in the development of the Nonlinear Least Squares method. What are they?
6. How is a “noise floor” established when conducting a digital image co
elation analysis?
7. How does su
egion size relate to the accuracy and spatial resolution of a digital image co
elation measurement?
8. Describe the objective function that we used to do parameter identification for the disc diametric compression analysis. What parameters did we identify?
9. Why do you have to have a good starting estimate of translation parameter values when using Nonlinear Least Squares for image co
elation? Use a figure to illustrate.
10. What is the role of image data interpolation in digital image co
elation, and why is it needed?
1
files/01-what-to-measure-and-why-3soxvdx2.pptx
Experimental Mechanics
What to measure, and why
Expanding roles of Digital Image Co
elation
Term Project: getting started
The term project (this year) is:
The development of a proposal for and experimental project
Focused on a well-defined objective (make product lighter <-> save the world)
Grounded in technical background research
Incorporating Digital Image Co
elation as a primary tool
Details justified through alignment with the iDICs Good Practices Guide
Objectives for today:
Introduction to companies that provide measurement products
Examples to motivate a project topic
Digital Image Co
elation “thinking”: understanding scope and possibilities
Digital Image Co
elation reference points (partial list)
https:
www.idics.org
https:
www.co
elatedsolutions.com
https:
trilion.com
https:
www.matchid.eu
http:
www.lavision.de/en
https:
www.gom.com
Professional Society:
General Metrology:
Digital Image Co
elation specialists:
This popped up in my
owser after lecture …
Digital Image Co
elation “thinking”
If you can cover the surface of an object with suitable speckle
Cali
ate the measurement space through measurement of target images
And collect images of suitable quality while the object is under motion/load
Then you can achieve dense point-based full-field measurements of surface:
Shape
With subsequent calculation of:
Motion
Deformation (strain)
Stress (if we have material property information)
Quantitative model comparisons
* Everything highlighted becomes a topic for technical understanding and assessment.
Discrete Surface Characterization
Shape
Motion
Deformation (strain)
Continuous Information
Much of basic Engineering Mechanics (and advanced topics such as Theory of Elasticity and Continuum Mechanics) are based on continuously defined functions
e.g. The basic beam bending stress equation, s = My/I, defines stress at all locations over the beam cross-section
Theory of Elasticity
Starts from the differential (infinitesimal) perspective
Define strain, stress, and material behavior at every point within an object …
Stress
Strain
Material Properties
… Generating Three Sets of Equations
These equations are simplified forms that embody the symmetry of the stress and strain tensors.
They are also a 2D versions of the full 3D equation set.
Body forces are assumed zero in the equili
ium equations.
Hooke’s Law
Engineering Strain
Stress Equili
ium
The displacement vector is symbolized as disp = {u,v,w} in 3D, {u,v} in 2d.
u = x component of displacement, v = the y component of displacement.
Strain is built from the spatial derivatives of displacement.
The Problem with Complex Shapes
But there is a problem in generalizing this approach to objects with complex shapes:
It may be difficult, or even impossible, to define equations representing the basic quantities of interest (displacements, strains, stresses) over a region of complex geometry
That is why traditional mechanics approaches are limited to objects of simple shape (prismatic bars, rectangular and circular plates, cylinders)
Discretization
The solution to the dilemma is discretization …
eak the object up into small chunks of simpler geometry … that is the basis of Finite Element Analysis
Nodes and Elements
Look closely and you can see the small pieces (elements) and connection points (nodes) that define the geometry
Each individual piece has a simple geometry with simple equations that describe problem variables (displacements, strains, stresses) within that region
The nodes connect the individual pieces together and provide “sharing” of information between the elements
Taken together, the nodes and elements define the behavior of the entire component
Displacement
Now the very important point that helps connect Finite Element Analysis with Digital Image Co
elation .. And by extension to solid mechanics in general
The information that is shared at the nodes (in most FEA formulations) is displacement
In fact, the FEA solution is the displacement of the nodes
Strain and stress are subsequently calculated from the the nodal displacements
Displacements Measured at Discrete Points
Wait … that looks familiar!
Biomedical example from Co
elated Solutions
Digital image co
elation was used to directly measure the displacements of discrete points over the surface of a test sample
This is analogous to the calculation of nodal displacements by finite element analysis
Simulation of artery deformation during stent installation … displacements shown as vectors, colors indicate the radial component of displacement.
Next time … Data Workflow
Shape
Motion (displacement)
Deformation (strain)
Stress
Analysis Comparisons (validation, identification)
σ xx
σ yy
τ xy
⎧
⎨
⎪⎪
⎩
⎪
⎪
⎫
⎬
⎪⎪
⎭
⎪
⎪
=
Exx Exy Exz
Exy Eyy Eyz
Exz Eyz Ezz
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
ε xx
ε yy
γ xy
⎧
⎨
⎪⎪
⎩
⎪
⎪
⎫
⎬
⎪⎪
⎭
⎪
⎪
s
xx
s
yy
t
xy
ì
í
ï
ï
î
ï
ï
ü
ý
ï
ï
þ
ï
ï
=
E
xx
E
xy
E
xz
E
xy
E
yy
E
yz
E
xz
E
yz
E
zz
é
ë
ê
ê
ê
ê
ù
û
ú
ú
ú
ú
e
xx
e
yy
g
xy
ì
í
ï
ï
î
ï
ï
ü
ý
ï
ï
þ
ï
ï
files/02-data-workflow-jeflhnzy.pptx
Data Workflow
Shape
Motion (displacement)
Deformation (strain)
Stress
Analysis Comparisons (validation, identification)
From Shape to Stress
Stress is often the goal of a mechanics analysis
Most closely related to common failure criteria (max normal stress, von Mises stress, etc.)
A common language for comparing results to prior work in a field
But fundamentally DIC just measures object shape
Points on the surface of an object located within a 3D coordinate system
How do we get from discrete shape information to full field stress patterns?
What is Stereo-Imaging?
Use two imaging system separated by a distance to interpret a scene
With knowledge of the system geometry you can calculate distances
The is an old method used for navigation and surveying
Digital Image Co
elation is the method pushed to the extremes:
Conduct high-precision surface shape measurements
Measure displacement of objects as they rotate and translate within 3D space
Measure strain on object surfaces as they deform under load
Triangulation: Distance to an object
Aim telescopes at locations A and B (known distance apart) at an object, measure a and b angles.
Obviously, both telescopes have to have line-of-sight to the object.
Basic Digital Image Co
elation set-up
Camera, Sensor, and World Coordinate Systems
Better to have cameras in a fixed position and identify points within each image, working out the distance through a more complex coordinate transformation process … that is image co
elation.
Aluminum beverage can
A standard Aluminum beverage can
Nearly 20 mm of depth
About 90o of the surface covered by the image pair
Out of plane distance (z-height)
Student Project 1 – Can Buckling
Example shows very precise 3D shape measurement over a highly curved region with very large depth excursion (~ 20 mm).
Project Workflow
Equipment and Imaging set-up
Target Images and Cali
ation
Test Images and Co
elation
Post-processing (Data Workflow)
1. Cali
ation
Prepares images and establishes the geometric context for stereo-imaging
Intrinsic camera parameters
Lens effective focal length
Lens distortion co
ection parameters
Options available for various distortion characteristics
Extrinsic stereo system parameters
Geometry of the stereo set-up
Distance between and orientation of cameras
Coordinate system of the virtual working volume
This information supports very precise triangulation
2. Co
elation
Convert a ROI point cloud into x,y,z point positions in the volume coordinate system
Point cloud is defined as a set of positions on the object surface in the Ref-Left camera image
Location of each point in the Ref-Right camera image allows calculation of 3D positions
Connect ROI point clouds between in situ experimental steps
Co
elation between Ref-Left and Step1-Left moves the points with the object (as opposed to a new set of locations on the surface)
Convert point cloud at the first load step into x,y,z point positions
Co
elation between Step1-Left and Step1-Right
This completes a load step and provides information for displacement and strain post-processing
Ref
Step1
Step2
• • •
(a)
(b)
(c)
Left Camera
Right Camera
Stereo
Pair 1
Stereo
Pair 2
Stereo
Pair 3
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
3. Post-Processing
Calculate displacements from point positions
This is simple.
For each of the n points in the cloud, at each load step, subtract the reference position: u = (x1 – x0), v = (y1 – yo), w = (z1 – z0)
This establishes a “total Lagrangian” perspective, with the initial configuration used as the basis for analysis
Calculate the displacement gradient tenso
Not so easy for i
egularly spaced points
Often involves a filtering or local fitting approach to control noise
Method influences the effective spatial resolution of the result
Calculate strain tensor from the displacement gradient
Interpret the strain tenso
We are now done with co
elation, no more image processing. Post-processing operates purely from the x,y,z point positions established during the co
elation step.
Step 1 displacements
Step 2 displacements
etc. for the remaining steps
Co
elated Solutions output file shows the progression
A flat surface seems relatively simple
X
Y
How do we manage a curved surface?
files/03-strain-vob35r3q.pptx
Data Workflow
Shape
Displacement
Strain
Stress
From Shape to Stress
Stress is often the goal of a mechanics analysis
Most closely related to common failure criteria (max normal stress, von Mises stress, etc.)
A common language for comparing results to prior work in a field
But fundamentally DIC just measures object shape
Points on the surface of an object located within a 3D coordinate system
How do we get from