MANE 4240

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MANE 4240 CIVL 4240Introduction to Finite
Elements
Prof. Suvranu De

• Mapped element geometries and shape functions
  the isoparametric formulation

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Reading assignment Chapter 10.1-10.3, 10.6
Lecture notes

• Summary
• Concept of isoparametric mapping
• 1D isoparametric mapping
• Element matrices and vectors in 1D
• 2D isoparametric mapping rectangular parent
  elements
• 2D isoparametric mapping triangular parent
  elements
• Element matrices and vectors in 2D

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For complex geometries
General quadrilateral elements
Elements with curved sides
4
Consider a special 4-noded rectangle in its local
coordinate system (s,t)
t
Displacement interpolation
2
1
1
1
1
s
1
Shape functions in local coord system
4
3
5
Recall that
Rigid body modes
Constant strain states
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Goal is to map this element from local coords to
a general quadrilateral element in global coord
system
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In the mapped coordinates, the shape functions
need to satisfy
1. Kronecker delta property
Then
2. Polynomial completeness
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The relationship
Provides the required mapping from the local
coordinate system To the global coordinate system
and is known as isoparametric mapping
(s,t) isoparametric coordinates (x,y) global
coordinates
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Examples
t
t
1
1
1
s
1
y
s
x
t
t
1
s
s
y
1
x
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1D isoparametric mapping
3 noded (quadratic) element
x2
x1
x3
1
2
3
x
s
1
2
3
1
1
Isoparametric mapping
Local (isoparametric) coordinates
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NOTES 1. Given a point in the isoparametric
coordinates, I can obtain the corresponding
mapped point in the global coordinates using the
isoparametric mapping equation
Question x? at s0.5?
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2. The shape functions themselves get mapped In
the isoparametric coordinates (s) they are
polynomials. In the global coordinates (x) they
are in general nonpolynomials Lets consider the
following numerical example
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2
x
1
2
3
Isoparametric mapping x(s)
Simple polynomial
Inverse mapping s(x)
Complicated function
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Now lets compute the shape functions in the
global coordinates
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Now lets compute the shape functions in the
global coordinates
N2(x)
N2(s)
1
1
2
3
1
4
2
x
s
1
2
3
1
1
N2(x) is a complicated function
N2(s) is a simple polynomial
However, thanks to isoparametric mapping, we
always ensure 1. Knonecker delta property 2.
Rigid body and constant strain states
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Element matrices and vectors for a mapped 1D bar
element
1
2
3
x
s
1
2
3
1
1
Displacement interpolation
Strain-displacement relation
Stress
The strain-displacement matrix
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The only difference from before is that the shape
functions are in the isoparametric coordinates
We know the isoparametric mapping
And we will not try to obtain explicitly the
inverse map. How to compute the B matrix?
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(No Transcript)
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What does the Jacobian do?
Maps a differential element from the
isoparametric coordinates to the global
coordinates
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The strain-displacement matrix
For the 3-noded element
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The element stiffness matrix
NOTES 1. The integral on ANY element in the
global coordinates in now an integral from -1 to
1 in the local coodinates 2. The jacobian is a
function of s in general and enters the
integral. The specific form of J is determined
by the values of x1, x2 and x3. Gaussian
quadrature is used to evaluate the stiffness
matrix 3. In general B is a vector of rational
functions in s
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Isoparametric mapping in 2D Rectangular parent
elements
Parent element
Mapped element in global coordinates
Isoparametric mapping
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Shape functions of parent element in
isoparametric coordinates
Isoparametric mapping
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• NOTES
• The isoparametric mapping provides the map (s,t)
  to (x,y) , i.e., if you are given a point (s,t)
  in isoparametric coordinates, then you can
  compute the coordinates of the point in the (x,y)
  coordinate system using the equations
• 2. The inverse map will never be explicitly
  computed.

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8-noded Serendipity element
t
7
4
3
1
1
1
6
8
s
1
5
2
1
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8-noded Serendipity element element shape
functions in isoparametric coordinates
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NOTES1. Ni(s,t) is a simple polynomial in s and
t. But Ni(x,y) is a complex function of x and
y. 2. The element edges can be curved in the
mapped coordinates 3. A midside node in the
parent element may not remain as a midside node
in the mapped element. An extreme example
t
5
y
1
1
2
1
1
5
1
8
2,6,3
8
6
s
x
1
7
4
3
7
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4. Care must be taken to ensure UNIQUENESS of
mapping
2
y
t
2
1
1
1
1
3
1
x
s
1
4
3
4
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Isoparametric mapping in 2D Triangular parent
elements
t
Parent element a right angled triangles with
arms of unit length Key is to link the
isoparametric coordinates with the area
coordinates
2
1
P(s,t)
s
t
s
3
1
1
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Now replace L1, L2, L3 in the formulas for the
shape functions of triangular elements to obtain
the shape functions in terms of (s,t) Example
3-noded triangle
2 (x2,y2)
t
2
1
1
3 (x3,y3)
s
1 (x1,y1)
3
1
Isoparametric mapping
Parent shape functions
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Element matrices and vectors for a mapped 2D
element
Recall For each element
Displacement approximation
Strain approximation
Stress approximation
Element stiffness matrix
Element nodal load vector
STe
e
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• In isoparametric formulation
• Shape functions first expressed in (s,t)
  coordinate system
• i.e., Ni(s,t)
• 2. The isoparamtric mapping relates the (s,t)
  coordinates with the global coordinates (x,y)
• 3. It is laborious to find the inverse map s(x,y)
  and t(x,y) and we do not do that. Instead we
  compute the integrals in the domain of the parent
  element.

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NOTE 1. Ni(s,t) s are already available as simple
polynomial functions 2. The first task is to find
and
Use chain rule
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In matrix form
This is known as the Jacobian matrix (J) for the
mapping (s,t) ? (x,y)
We want to compute these for the B matrix
Can be computed
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How to compute the Jacobian matrix? Start from
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Need to ensure that det(J) gt 0 for one-to-one
mapping
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3. Now we need to transform the integrals from
(x,y) to (s,t)
Case 1. Volume integrals
hthickness of element
This depends on the key result
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Proof
y
db
da
j
x
i
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Problem Consider the following isoparamteric map
y
ISOPARAMETRIC COORDINATES
GLOBAL COORDINATES
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Displacement interpolation
Shape functions in isoparametric coord system
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The isoparamtric map
In this case, we may compute the inverse map, but
we will NOT do that!
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The Jacobian matrix
since
NOTE The diagonal terms are due to stretching of
the sides along the x-and y-directions. The
off-diagonal terms are zero because the element
does not shear.
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Hence, if I were to compute the first column of
the B matrix along the positive x-direction
I would use
Hence
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The element stiffness matrix
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Case 2. Surface integrals For dST we consider 2
cases Case (a)
dST
2
t
y
ds
1
1
2
3
s
dST
4
x
3
4
ds
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Case (b)
dST
2
t
y
1
1
2
dt
3
s
dST
4
dt
x
3
4
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Summary of element matrices in 2D plane
stress/strain
Quadrilateral element
t
s
1
1
Suppose s-1 gets mapped to ST
1
1
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Summary of element matrices in 2D plane
stress/strain
Triangular element
t
1
t
Suppose t0 gets mapped to ST
s
s1-t
t
1