Continuum Mechanics and the Finite Element...
Transcript of Continuum Mechanics and the Finite Element...
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Continuum Mechanics and
the Finite Element Method
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Assignment 2
Due on March 2nd @ midnight
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Suppose you want to simulate
this…
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The familiar mass-spring
system
Deformation Measure Elastic Energy Forces
fint(x) =
X xyi yi
l = |x - yi|
l0 = |X - yi|
ll0
Spring length
before/after
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Mass Spring Systems
Can be used to model arbitrary elastic/plastic objects, but…
• Behavior depends on tessellation Find good spring layout
Find good spring constants
Different types of springs interfere
No direct map to measurable
material properties
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Alternative…
• Start from continuum mechanics
• Discretize with Finite Elements
– Decompose model into simple elements
– Setup & solve system of algebraic equations
• Advantages
– Accurate and controllable material behavior
– Largely independent of mesh structure
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Mass Spring vs Continuum
MechanicsMass spring systems require:
1. Measure of Deformation
2. Material Model
3. Deformation Energy
4. Internal Forces
We need to derive the same types of concepts using continuum mechanics principles
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Continuum Mechanics: 3D
Deformations• For a deformable body, identify:
– undeformed state described by positions
– deformed state described by positions
• Displacement field u describes in terms of
x
y
z
u(x)
': u )(XX ux
3' R
3R
x
X
'
'
xX )(Xu
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Continuum Mechanics: 3D
Deformations• Consider material points and such that
is infinitesimal, where
x
y
z
1X
|| d2X
12' xxd
• Now consider deformed vector 'd
1X2X
1x2x
'dd
d)( uI
12 XX d
'
Deformation gradient F=
u
www
vvv
uuu
zyx
zyx
zyx
X
x
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So…
• Displacement field transforms points
FdduId )('
)(XX ux
• Jacobian of displacement field (deformation
gradient) transforms differentials (infinitesimal
vectors) from undeformed to deformed
F=X
x
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Displacement Field and
Deformation Gradient
In general, displacement field is not explicitly described. Nevertheless, toy examples:
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Displacement Field and
Deformation Gradient
In general, displacement field is not explicitly described. Nevertheless, toy examples:
![Page 13: Continuum Mechanics and the Finite Element Methodscoros/cs15467-s16/lectures/12-FEM.pdfAlternative… •Start from continuum mechanics •Discretize with Finite Elements –Decompose](https://reader031.fdocuments.us/reader031/viewer/2022030719/5b046ebf7f8b9a8c688da674/html5/thumbnails/13.jpg)
Displacement Field and
Deformation Gradient
In general, displacement field is not explicitly described. Nevertheless, toy examples:
![Page 14: Continuum Mechanics and the Finite Element Methodscoros/cs15467-s16/lectures/12-FEM.pdfAlternative… •Start from continuum mechanics •Discretize with Finite Elements –Decompose](https://reader031.fdocuments.us/reader031/viewer/2022030719/5b046ebf7f8b9a8c688da674/html5/thumbnails/14.jpg)
Displacement Field and
Deformation Gradient
In general, displacement field is not explicitly described. Nevertheless, toy examples:
![Page 15: Continuum Mechanics and the Finite Element Methodscoros/cs15467-s16/lectures/12-FEM.pdfAlternative… •Start from continuum mechanics •Discretize with Finite Elements –Decompose](https://reader031.fdocuments.us/reader031/viewer/2022030719/5b046ebf7f8b9a8c688da674/html5/thumbnails/15.jpg)
Displacement Field and
Deformation Gradient
In general, displacement field is not explicitly described. Nevertheless, toy examples:
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Measure of deformations
How can we describe deformations?
• Displacement field transforms points
FdduId )('
)(XX ux
• Jacobian of displacement field (deformation
gradient) transforms vectors
F=X
x
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Deformation measure (strain):
Undeformed spring:
Undeformed (infinitesimal) continuum volume:
Back to spring deformation
Similar to F
= 1
= I ?
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Strain (description of deformation
in terms of relative displacement)
Deformation measure (strain):
Desirable property: if spring is undeformed, strain is 0 (no change in shape)
Can we find a similar measure that would work for infinitesimal volumes?
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3D Nonlinear Strain
Idea: to quantify change in shape, measure change in
squared length for any arbitrary vector
dIFFddddddd )(''|||'| 22 TTTT
Green strain )(21 IFFE T
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Strain (description of deformation
in terms of relative displacement)
Deformation measure (strain):
Desirable property: if spring is undeformed, strain is 0 (no change in shape)
Can we find a similar measure that would work for infinitesimal volumes?
Green strain )(21 IFFE T
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3D Linear Strain
• Green strain is quadratic in displacements
21
wvwuw
wvvuv
wuvuu
zzyzx
yzyyx
xzxyx
2
2
2
)()(21
21 uuuuIFFE TTT
• Neglecting quadratic term (small deformation
assumption) leads to the linear
Cauchy strain
(small strain)
• Written out:
Notation
𝜺 =1
2𝛻𝐮 + 𝛻𝐮t =
1
2𝐅 + 𝐅𝑡 − 𝐈
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3D Linear Strain
• Linear Cauchy strain
• Geometric interpretation
y,v
x,u
i : normal strains
i : shear strains
21
wvwuw
wvvuv
wuvuu
zzyzx
yzyyx
xzxyx
2
2
2
x yxy
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Cauchy vs. Green strain
Nonlinear Green strain is rotation-invariant
• Apply incremental rotation 𝐑 to given deformation 𝐅 to obtain 𝐅′ = 𝐑𝐅
• Then
Linear Cauchy strain
is not rotation-invariant
→ artifacts for larger rotations
)''('21 IFFE T E
휀′ =1
2𝐅′ + 𝐅′
𝑡≠ 휀
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Mass spring systems:
1. Measure of Deformation
2. Material Model
3. Deformation Energy
4. Internal Forces
Continuum Mechanics:
1. Measure of Deformation: Green or Cauchy strain
2. Material Model
Mass Spring vs Continuum
Mechanics
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Material Model: linear isotropic
material
• Linear isotropic material (generalized Hooke’s law)
– Energy density Ψ =1
2𝜆tr(𝜺)2+𝜇tr(𝜺2)
– Lame parameters 𝜆 and 𝜇 are material constants related to
Poisson Ratio and Young’s modulus
• Interpretation
– tr 𝜺2 = 휀 𝐹2 penalizes all strain components equally
– tr 𝜺 2 penalizes dilatations, i.e., volume changes
• Material model links strain to energy (and stress)
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Volumetric Strain (dilatation,
hydrostatic strain)Consider a cube with side length 𝑎
For a given deformation 𝜺, the volumetric strain is
Δ𝑉/𝑉0 = (𝑎 1 + 휀11 ⋅ 𝑎 1 + 휀22 ⋅ 𝑎 1 + 휀33 − 𝑎3)/𝑎3
= 휀11 + 휀22 + 휀33 + 𝑂 𝜺2 ≈ tr 𝜺
𝑎
http://en.wikipedia.org/wiki/Infinitesimal_strain_theory
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Linear isotropic material
Energy density: Ψ =1
2𝜆tr(𝜺)2+𝜇tr(𝜺2)
Problem: Cauchy strain is not invariant under rotations
→ artifacts for rotations
Solutions:
• Corotational elasticity
• Nonlinear elasticity
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Material Model: non-linear
isotropic model
Replace Cauchy strain with Green strain → St. Venant-Kirchhoff material (StVK)
Energy density: Ψ𝑆𝑡𝑉𝐾 =1
2𝜆tr(𝐄)2+𝜇tr(𝐄2)
Rotation invariant!
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Problems with StVK
StVK softens under compression
Ψ𝑆𝑡𝑉𝐾 =1
2𝜆tr(𝐄)2+𝜇tr 𝐄2
Energy
Deformation
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Advanced nonlinear materials
Green Strain 𝐄 =1
2𝐅𝑡𝐅 − 𝐈 =
1
2(𝐂 − 𝐈)
Split into deviatoric (i.e. shape changing/distortion) and volumetric (dilation, volume changing) deformations
Volumetric: 𝐽 = det 𝐅
Neo-Hookean material:
Ψ𝑁𝐻 =𝜇
2tr 𝐂 − 𝐈 − 𝜇ln(𝐽) +
𝜆
2ln 𝐽 2
Energy
Deviatoric: 𝐂 = det 𝐅 −2/3 𝐂
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Different Models
St. Venant-Kirchoff
Neo-Hookean
Linear
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Mass Spring vs Continuum
MechanicsMass spring systems:
1. Measure of Deformation
2. Material Model
3. Deformation Energy
4. Internal Forces
Continuum Mechanics:
1. Measure of Deformation: Green or Cauchy strain
2. Material Model: linear, StVK, Neo-Hookean, etc
3. From Energy Density to Deformation Energy:
Finite Element Discretization
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Finite Element Discretization
Divide domain into discrete elements, e.g., tetrahedra
• Explicitly store displacement values at nodes (𝐱𝑖).
• Displacement field everywhere else obtained
through interpolation: 𝐱 𝑋 = ∑𝑁𝑖 𝑋 𝐱𝑖
• Deformation Gradient: 𝐅 =𝜕𝐱 𝑋
𝜕𝑋= ∑𝑖 𝐱𝑖
𝜕𝑁𝑖
𝜕𝑋
𝑡
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• Basis functions (aka shape functions) 𝑁𝑖 𝑋𝑗 : 𝐑3 → 𝐑
• Satisfy delta-property: 𝑁𝑖 𝑋𝑗 = 𝛿𝑖𝑗
• Simplest choice: linear basis functions
𝑁𝑖 ҧ𝑥, ത𝑦, ҧ𝑧 = 𝑎𝑖 ҧ𝑥 + 𝑏𝑖 ത𝑦 + 𝑐𝑖 ҧ𝑧 + 𝑑𝑖
• Compute 𝑁𝑖 (and 𝜕𝑁𝑖
𝜕𝑋) through
Basis Functions
Ω𝑒
𝑿1
𝑿2
𝑿3
𝑿4
ഥV𝑒
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• Displacement field is continuous in space
• Deformation Gradient, strain, stress are not
– Constant Strain per element
• Deformation Gradient can be computed as
– 𝐅 = 𝒆𝑬-1 where e and E are matrices whose
columns are edge vectors in undeformed and
deformed configurations
Constant Strain Elements
(Linear Basis Functions)
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Constant Strain Elements: From
energy density to deformation energy
• Integrate energy density over the entire element:
𝑊𝑒 = නΩ𝑒
Ψ 𝐅
• If basis functions are linear:
− 𝐅 is linear in 𝐱𝑖
− 𝐅 is constant throughout element:
𝑊𝑒 = නΩ𝑒
Ψ 𝐅 = Ψ 𝐅 ⋅ ത𝑉𝑒
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Mass Spring vs Finite Element
MethodMass spring systems:
1. Measure of Deformation
2. Material Model
3. Deformation Energy
4. Internal Forces
Continuum Mechanics:
1. Measure of Deformation: Green or Cauchy strain
2. Material Model: linear, StVK, Neo-Hookean, etc
3. Deformation Energy: integrate over elements
4. Internal Forces:
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Discretize into elements (triangles/tetraderons, etc)
For each element
• Compute deformation gradient 𝐅 = 𝒆𝑬-1
• Use material model to define energy density Ψ(𝐅)
• Integrate over elements to compute energy: W
• Compute nodal forces as:
FEM recipe
![Page 39: Continuum Mechanics and the Finite Element Methodscoros/cs15467-s16/lectures/12-FEM.pdfAlternative… •Start from continuum mechanics •Discretize with Finite Elements –Decompose](https://reader031.fdocuments.us/reader031/viewer/2022030719/5b046ebf7f8b9a8c688da674/html5/thumbnails/39.jpg)
FEM recipe
𝒇 = −𝜕𝑊
𝜕𝒙= −𝑉
𝜕Ψ
𝜕𝑭
𝜕𝑭
𝜕𝒙
First Piola-Kirchhoff stress tensor P
Area/volume of element
![Page 40: Continuum Mechanics and the Finite Element Methodscoros/cs15467-s16/lectures/12-FEM.pdfAlternative… •Start from continuum mechanics •Discretize with Finite Elements –Decompose](https://reader031.fdocuments.us/reader031/viewer/2022030719/5b046ebf7f8b9a8c688da674/html5/thumbnails/40.jpg)
FEM recipe
𝒇 = −𝜕𝑊
𝜕𝒙= −𝑉
𝜕Ψ
𝜕𝑭
𝜕𝑭
𝜕𝒙
First Piola-Kirchhoff stress tensor P
Area/volume of element
![Page 41: Continuum Mechanics and the Finite Element Methodscoros/cs15467-s16/lectures/12-FEM.pdfAlternative… •Start from continuum mechanics •Discretize with Finite Elements –Decompose](https://reader031.fdocuments.us/reader031/viewer/2022030719/5b046ebf7f8b9a8c688da674/html5/thumbnails/41.jpg)
FEM recipe
𝒇𝟏 𝒇𝟐 𝒇𝟑 = −𝑉𝑷𝑬−𝑻; 𝒇𝟒 = −𝒇𝟏 − 𝒇𝟐− 𝒇𝟑
Additional reading: http://www.femdefo.org/
For a tetrahedron, this works out to:
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Material Parameters
Lame parameters 𝜆 and 𝜇 are material constants related to the
fundamental physical parameters: Poisson’s Ratio and Young’s
modulus (http://en.wikipedia.org/wiki/Lamé_parameters)
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Young’s Modulus and Poisson
RatioLame parameters 𝜆 and 𝜇 are material constants related to the
fundamental physical parameters: Poisson’s Ratio and Young’s
modulus (http://en.wikipedia.org/wiki/Lamé_parameters)
Young’s modulus (E),
measure of stiffness
Poisson’s ratio (ν), relative
transverse to axial deformation
10
Kg
10
Kg
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What Do These Parameters
Mean
Stiffness is pretty intuitive
OBJECT
E=10000000
GPa
10
Kg
OBJECT
E=60 KPa
10
Kg
Pascals = force/area
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What Do These Parameters
Mean
Poisson’s Ratio controls volume preservation
OBJECT
OBJECT
1
1
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What Do These Parameters
Mean
Poisson’s Ratio controls volume preservation
OBJECT
OBJECT1/2
2
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What Do These Parameters
Mean
Poisson’s Ratio controls volume preservation
OBJECT
OBJECT1/2
2
Poisson’s Ratio ( ) = ?
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What Do These Parameters
Mean
Poisson’s Ratio controls volume preservation
OBJECT
PR = 0.5
OBJECT
PR = 0.5 1/2
2
Poisson’s Ratio ( ) = 0.5
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What Do These Parameters
Mean
Poisson’s ratio is between -1 and 0.5
OBJECT
PR = 0.5
OBJECT
PR = 0.5 1.5
2
Poisson’s Ratio ( ) = -0.5
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Negative Poisson’s Ratio
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Measurement
Where do material parameters come from?
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Simple Measurement: Stiffness
OBJECT
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Simple Measurement
OBJECT
1kg
What’s the Force (Stress)?
What’s the Deformation (Strain)?
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Simple Measurement
OBJECT
1kg
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Simple Measurement
OBJECT
2kg
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Simple Measurement
OBJECT
3kg
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Simple Measurement
OBJECT
7kg
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Simple Measurement
OBJECT
7kg
How do we get the stiffness ?
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Simple Measurement
OBJECT
7kg
How do we get the stiffness ?
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Simple Measurement
OBJECT
7kg
How do we get the stiffness ?
Stiffness
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Simple Measurement: Poisson’s
Ratio
OBJECT
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Simple Measurement: Poisson’s
Ratio
OBJECT
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Simple Measurement: Poisson’s
Ratio
OBJECT
Compute changes in width and height
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Simple Measurement: Poisson’s
Ratio
OBJECT
Poisson’s Ratio
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Measurement Devices
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Simulating Elastic Materials
with CM+FEM
You now have all the mathematical tools you need
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Suppose you want to simulate
this…
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Plastic and Elastic Materials
Elastic Materials
• Objects return to their original shape in the absence of other forces
Plastic Deformations:
• Object does not always return to its original shape
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Example: Crushing a Coke Can
Old Reference State New Reference State
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Example: Crushing a van
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A Simple Model For Plasticity
Recall our model for strain:
Let’s consider how to encode a change of reference shape into this metric
• Changing undeformed mesh is not easy!
We want to exchange F with , a deformation gradient that takes into account the new shape of our object
New Reference State
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Continuum Mechanics:
Deformation
deformation gradient maps undeformedvectors (local) to deformed (world) vectors
Reference Space World Space
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Continuum Mechanics:
Deformation
deformation gradient maps undeformedvectors (reference) to deformed (world) vectors
Reference Space World Space
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Continuum Mechanics:
Deformation
F transforms a vector from Reference space to World Space
Reference SpaceWorld Space
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Continuum Mechanics:
Deformation
Introduce a new space
Reference SpaceWorld SpacePlastic Space
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Continuum Mechanics:
Deformation
Our goal is to use but we only have access to
Reference SpaceWorld SpacePlastic Space
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Continuum Mechanics:
Deformation
Our goal is to use but we only have access to
Reference SpaceWorld SpacePlastic Space
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Continuum Mechanics:
Deformation
Our goal is to use but we only have access to
Keep an estimate of per element, built incrementally
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How to Compute the Plastic
Deformation Gradient• We compute the strain/stress for each element during
simulation
• When it gets above a certain threshold store F as
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How to Compute the Plastic
Deformation Gradient• We compute the strain/stress for each element during
simulation
• When it gets above a certain threshold store F as
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How to Compute the Plastic
Deformation Gradient• Each subsequent simulation step uses
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So now you too can simulate
this…
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Questions?
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93