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 ID: 555612
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Slide1
NUMERICAL INTEGRATION
Stiffness matrix and distributed load calculations involve integration over the
domainIn many cases, analytical integration is very difficultNumerical integration based on Gauss Quadrature is commonly used in finite element programsGauss Quadrature:Integral is evaluated using function values and weights.si: Gauss integration points, wi: integration weightsf(si): function value at the Gauss pointn: number of integration points.Slide2
ONE INTEGRATION POINT
Constant Function: f(
s) = 4Use one integration point s1 = 0 and weight w1 = 2 Why?The numerical integration is exact.Linear Function: f(s) = 2s + 1Use one integration point s1 = 0 and weight w1 = 2The numerical integration is exact.
One-point Gauss Quadrature can integrate constant and linear functions exactly.Slide3
TWO POINTS AND MORE
Quadratic Function: f(
s) = 3s2 + 2s + 1Let’s use one-point Gauss QuadratureOne-point integration is not accurate for quadratic functionLet’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 (2
n – 1)-order polynomial exactly.Slide4
GAUSS QUADRATURE POINTS AND WEIGHTS
What properties do positions and weights have?
n
Integration Points (
si
)
Weights (
w
i
)
Exact for polynomial of degree
1
0.0
2.0
1
2
.5773502692
1.0
3
3
.7745966692
0.0
.5555555556
.888888888954.8611363116.3399810436.3478546451.652145154975.9061798459.53846931010.0.2369268851.4786286705.56888888899Slide5
General intervalIn general we will have
From WikipediaSlide6
TWO DIMENSIONAL INTEGRATION
multiplying two one-dimensional Gauss integration formulas
Total number of integration points = m×n.
s
t
s
t
s
t
(a) 1
1
(b) 2
2
(c) 3
3Slide7
EXAMPLE 6.9
Integrate the following polynomial:One-point formula
Two-point formulaSlide8
EXAMPLE 6.9
3-point formula
4-point formula4-point formula yields the exact solution. Why?Slide9
Quiz-like problemsWhen 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 figureSolution on the notes page Slide10
APPLICATION TO STIFFNESS MATRIX
Application to Stiffness Matrix Integral