MANE 4240

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

• Numerical Integration in 1D

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Reading assignment Lecture notes, Logan 10.4

• Summary
• Newton-Cotes Integration Schemes
• Gaussian quadrature

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Axially loaded elastic bar
A(x) cross section at x b(x) body force
distribution (force per unit length) E(x)
Youngs modulus
x2
x1
For each element
Element stiffness matrix
where
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Only for a linear finite element
Element nodal load vector
Question How do we compute these integrals using
a computer?
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Any integral from x1 to x2 can be transformed to
the following integral on (-1, 1)
Use the following change of variables
Goal Obtain a good approximate value of this
integral 1. Newton-Cotes Schemes (trapezoidal
rule, Simpsons rule, etc) 2. Gauss Integration
Schemes NOTE Integration schemes in 1D are
referred to as quadrature rules
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Trapezoidal rule Approximate the function f(x)
by a straight line g(x) that passes through the
end points and integrate the straight line
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• Requires the function f(x) to be evaluated at 2
  points (-1, 1)
• Constants and linear functions are exactly
  integrated
• Not good for quadratic and higher order
  polynomials
• How can I make this better?

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Simpsons rule Approximate the function f(x) by
a parabola g(x) that passes through the end
points and through f(0) and integrate the parabola
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• Requires the function f(x) to be evaluated at 3
  points (-1,0, 1)
• Constants, linear functions and parabolas are
  exactly integrated
• Not good for cubic and higher order polynomials

How to generalize this formula?
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Notice that both the integration formulas had the
general form
Trapezoidal rule M2
Accurate for polynomial of degree at most 1 (M-1)
Simpsons rule M3
Accurate for polynomial of degree at most 2 (M-1)
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Generalization of these two integration rules
Newton-Cotes

• Divide the interval (-1,1) into M-1 equal
  intervals using M points
• Pass a polynomial of degree M-1 through these M
  points (the value of this polynomial will be
  equal to the value of the function at these M-1
  points)
• Integrate this polynomial to obtain an
  approximate value of the integral

f(1)
f(x)
f(-1)
g(x)
x
-1
1
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With M points we may integrate a polynomial of
degree M-1 exactly. Is this the best we can do
? With M integration points and M weights, I
should be able to integrate a polynomial of
degree 2M-1 exactly!! Gauss integration rule
See table 10-1 (p 405) of Logan
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Gauss quadrature
How can we choose the integration points and
weights to exactly integrate a polynomial of
degree 2M-1? Remember that now we do not know, a
priori, the location of the integration points.
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Example M1 (Midpoint qudrature)
How can we choose W1 and x1 so that we may
integrate a (2M-11) linear polynomial exactly?
But we want
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Hence, we obtain the identity
For this to hold for arbitrary a0 and a1 we need
to satisfy 2 conditions
i.e.,
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For M1
f(x)
f(0)
g(x)
x
-1
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• Midpoint quadrature rule
• Only one evaluation of f(x) is required at the
  midpoint of the interval.
• Scheme is accurate for constants and linear
  polynomials (compare with Trapezoidal rule)

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Example M2
How can we choose W1 ,W2 x1 and x2 so that we may
integrate a polynomial of degree (2M-14-13)
exactly?
But we want
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Hence, we obtain the 4 conditions to determine
the 4 unknowns (W1 ,W2 x1 and x2 )
Check that the following is the solution
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For M2
f(x)
x


-1
1

• Only two evaluations of f(x) is required.
• Scheme is accurate for polynomials of degree at
  most 3 (compare with Simpsons rule)

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Exercise Derive the 6 conditions required to
find the integration points and weights for a
3-point Gauss quadrature rule
Newton-Cotes
Gauss quadrature
1. M integration points are necessary to
exactly integrate a polynomial of degree 2M-1
2. Less expensive 3. Exponential convergence,
error proportional to
1. M integration points are necessary to
exactly integrate a polynomial of degree M-1
2. More expensive
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Example
Exact integration
Integrate and check!
Newton-Cotes
To exactly integrate this I need a 4-point
Newton-Cotes formula. Why?
Gauss
To exactly integrate this I need a 2-point Gauss
formula. Why?
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Gauss quadrature
Exact answer!
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Comparison of Gauss quadrature and Newton-Cotes
for the integral
Newton-Cotes
Gauss quadrature
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In FEM we ALWAYS use Gauss quadrature
Linear Element
Stiffness matrix
Nodal load vector
Usually a 2-point Gauss integration is used. Note
that if A, E and b are complex functions of x,
they will not be accurately integrated
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Quadratic Element
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Nodal shape functions
You should be able to derive these!
Stiffness matrix
Assuming E and A are constants
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Need to exactly integrate quadratic terms. Hence
we need a 2-point Gauss quadrature scheme..Why?