Computers%20in%20Civil%20Engineering

1
Computers in Civil Engineering53081 Spring 2003
Lecture 16
Numerical Integration
2
Numerical Integration

• Important in numerous engineering applications
• Analogy between integration and summation
• Closely related to areas/volumes
• Strong coupling of integration and differentiation

3
Scope
Evaluation of definite integrals, e.g.
Evaluation of indefinite integrals, where the
result is a function whose derivative is given,
is outside of the scope of the course.
4
Integration in Engineering
A definite integral can be interpreted as the
area under a function
Mass transfer values
Total force
Kinetics
5
Examples Integration in Engineering
rainrate (mm/h)
time (s)
a
b
w(z)
z
D
front view
side view
6
Integration Summation
Analogy between Integration and Summation
7
Numerical Integration
Basic Strategy

• Approximate the function with something more
  easily integrable, and
• Integrate the approximation

where
The integral can also be approximated using a
series of piecewise polynomials, i.e. the same
polynomial does not necessarily have to pass
through all the points.
8
Example
Approximate the functions with several piecewise
first-order polynomials
approximation
function
9
The Trapezoidal Rule
Truncation error
10
Potential Pitfalls
11
Multiple-Application of The Trapezoidal Rule
12
Mathematical Formulation
width
average height
13
Example
Problem Use the two-segment trapezoidal rule to
estimate the integral of from a 0, to b 0.8.
Solution
14
Truncation Error
Approximate truncation error
where average 2nd derivative over the
whole interval a to b
Problem Compute for the function
Solution
15
Pseudocode

• function trapez(f,n,a,b)
• dimension f(n1)
• sum0.0
• do i2,n
• sumsumf(i)
• enddo
• trapez(f(1)2Sumf(n1))(b-a)/(2n)
• return
• end

h(b-a)/n
16
Do you have to use constant intervals?

• No it is not hard to develop a scheme using
  irregular intervals.

But then you cant interpret the method as just
taking the average of the function values, right?
17
Simpsons Rules
Concept instead of finer segmentation, use
higher-order polynomials to fit the data, and
calculate the integrals based on these
approximations. For example
The following analyses assume equal-sized
intervals.
18
Simpsons 1/3 Rule
Notice even number of segments
parabola
width
average height
19
Simpsons 3/8 Rule
width
average height
20
Other Issues

• Extension of Simpsons 1/3 Rule to
  Multiple-Application situations
• Integration with unequal segments
• Integration limits extended beyond the data
  points (open Newton-Coates) formulas
• Adaptive selection of the number of points and/or
  interval length

21
Multiple Application of Simpsons 1/3 Rule
f(x)
Notice Even number of segments
a
b
width
average height
22
Next Overview and Tips