1. NUMERICAL INTEGRATION
Stiffness matrix and distributed load calculations involve integration over the domain
In many cases, analytical integration is very difficult
Numerical integration based on Gauss Quadrature is commonly used in finite element programs
Gauss Quadrature:
Integral is evaluated using function values and weights.
si: Gauss integration points, wi: integration weights
f(si): function value at the Gauss point
n: number of integration points.

2. ONE INTEGRATION POINT Constant Function: f(s) = 4
Use one integration point s1 = 0 and weight w1 = 2 Why?
The numerical integration is exact.
Linear Function: f(s) = 2s + 1
Use one integration point s1 = 0 and weight w1 = 2
One-point Gauss Quadrature can integrate constant and linear functions exactly.

3. TWO POINTS AND MORE Quadratic Function: f(s) = 3s2 + 2s + 1
Let’s use one-point Gauss Quadrature
One-point integration is not accurate for quadratic function
Let’s use two-point integration with w1 = w2 = 1 and -s1 = s2 =
Gauss Quadrature points and weights are selected such that n integration points can integrate (2n – 1)-order polynomial exactly.

4. GAUSS QUADRATURE POINTS AND WEIGHTS
What properties do positions and weights have? n
Integration Points (si)
Weights (wi)
Exact for polynomial of degree 1
0.0
2.0 2 
1.0 3  5 4   7   9

5. General interval
In general we will have
From Wikipedia

6. TWO DIMENSIONAL INTEGRATION
multiplying two one-dimensional Gauss integration formulas
Total number of integration points = m×n. s t
(a) 11
(b) 22
(c) 33

7. EXAMPLE 6.9 Integrate the following polynomial: One-point formula
Two-point formula

8. EXAMPLE 6.9 3-point formula 4-point formula
4-point formula yields the exact solution. Why?

9. Quiz-like problems
When can’t you use Gaussian quadrature and must fall back on methods such as Simpson’s rule?
Estimate the integral with one and two points and compare to the exact value
What are the coordinates of the integration point at the bottom right corner of the 3x3 Gaussian quadrature points shown in the figure
Solution on the notes page
When the function is not available to compute at every value of x, such as when it is in a table, we have to fall back on methods like Simpson’s rule (if the function is given at constant intervals) or trapezoidal integration.
We transform from x to s
From the table on Slide 6 we can deduce that the coordinates are s=0.7746, t=

10. APPLICATION TO STIFFNESS MATRIX
Application to Stiffness Matrix Integral