I am analyzing journal bearings and trying to solve Reynold's equation using finite element analysis. I have discretized the equation and now i want to solve surface integrals and line integrals in that equations using Gauss quadrature. The details are in attached file. I need help in understanding the values of weighted functions which i should take for 6 point line integral, 6 x 6 area integral and 3 point integral. Kindly help.
Please visit the following site. All weight values and respective abscissae are given there for n=2 to n=64.
https://pomax.github.io/bezierinfo/legendre-gauss.html
the Gauss Quadrature is best Approximately according with the problems and the mesh partitions of it, so you can see the attach file be the simplest and observations of Gauss Quadrature
Hello Dinesh Dhande,
I will use the Fortran 90 language
REAL*8 p(6), w(6), a, b, c,d, x, Integral
INTEGER i, j,np
The Gauss points and weights for a 6-point integration for ξ = -1 to ξ = +1 are
p(1)= .9324695142031521d0; p(2) = .6612093864662645d0
p(3) = .2386191860831969d0
w(1) = .1713244923791627d0; w(2) = .3607615730481387d0
w(3) = .4679139345726893d0
p(4) = -p(3); p(5) = -p(2); p(6) = -p(1)
w(4) = w(3); w(5) = w(2); w(6) = w(1)
The integral of f(x) from x = a to x = b is given by the following Fortran routine
Integral = 0.d0
np = 6
h = b - a
DO i = 1, np
x = a + .5*d0*h * (p(i) + 1.0d0)
Integral = Integral + .5d0 * h * w(i) * f(x)
ENDDO!
The integral of f(x,y) from x = a to x = b and y = c to y= d is given by the following Fortran routine
Integral = 0.d0
np = 6
hx = b - a ; hy = d- c
DO i = 1, np
DO j = 1, np
x = a + .5*d0*hx * (p(i) + 1.0d0)
y = c + .5*d0*hy * (p(j) + 1.0d0)
Integral = Integral+ .25d0 * hx*hy * w(i) *w(j)* f(x, y)
ENDDO!
ENDDO
I hope that this answer is useful to you
In my researchgate section, page 6, you may find my contribution Numerical Methods with theory and examples, which will solve your problem. Good luck with your study.
The program to calculate ordinates and weights for Gauss Legendre integration is in the page
http://userpages.umbc.edu/~squire/download/gauleg.f90
This program is from the book “Numerical recipes”, chapter: Gaussian Quadratures and Orthogonal polynomials.
Good luck!
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