Quadrature Rule

Background

A QuadratureRule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration. The most common domain of integration for such a rule is taken as [−1, 1], so the rule is stated as:

• One-dimensional case:

  \[ \displaystyle{ \int _{-1}^{1}f(x)\,dx \approx \sum_{i=1}^{n} w_{i} \, f(x_{i})} \]

• Two-dimensional case:

  \[ \displaystyle{ \int_{-1}^{1} \int_{-1}^{1} f(x,y) \,dx\,dy \approx \sum_{i=1}^{n} \sum_{j=1}^{n} w_{i} \, w_{j} \, f(x_{i},y_{j}) } \]

• Three-dimensional case:

  \[ \displaystyle{ \int_{-1}^{1} \int_{-1}^{1} \int_{-1}^{1}f(x,y,z)\,dx\,dy\,dz \approx \sum_{i=1}^{n} \sum_{j=1}^{n} \sum_{k=1}^{n} w_{i} \, w_{j} \, w_{k} \, f(x_{i},y_{j},z_{k})} \]

where \(xi\) are the quadrature points and \(w_i\) for \(i = 1, \ldots, n\) the weights.

REFERENCES:

• Bathe, K. Jurgen, "Finite Element Procedures", Chapter 5: pages 465-468. Prentice-Hall, 1996.
• R. Lobatto, "Lessen over de differentiaal- en integraalrekening" , 1–2 , 's Gravenhage (1851–1852)

Gauss Quadrature

Gaussian quadrature is optimal because it fits all polynomials up to degree \(2n - 1\) exactly, where \(n\) is the number of integration points.

• One integration point:

  wi	xi
  2.000000000000000	0.000000000000000

• Two integration point:

  wi	xi
  1.000000000000000	-0.577350269189626
  1.000000000000000	0.577350269189626

• Three integration point:

  wi	xi
  0.555555555555556	-0.774596669241483
  0.888888888888889	0.000000000000000
  0.555555555555556	0.774596669241483

• Four integration point:

  wi	xi
  0.347854845137454	-0.861136311594053
  0.652145154862546	-0.339981043584856
  0.652145154862546	0.339981043584856
  0.347854845137454	0.861136311594053

• Five integration point:

  wi	xi
  0.236926885056189	-0.906179845938664
  0.478628670499365	-0.538469310105683
  0.568888888888889	0.000000000000000
  0.478628670499365	0.538469310105683
  0.236926885056189	0.906179845938664

• Six integration point:

  wi	xi
  0.171324492379170	-0.932469514203152
  0.360761573048139	-0.661209386466265
  0.467913934572691	-0.238619186083197
  0.467913934572691	0.238619186083197
  0.360761573048139	0.661209386466265
  0.171324492379170	0.932469514203152

• Seven integration point:

  wi	xi
  0.129484966168870	-0.949107912342759
  0.279705391489277	-0.741531185599395
  0.381830050505119	-0.405845151377397
  0.417959183673469	0.000000000000000
  0.381830050505119	0.405845151377397
  0.279705391489277	0.741531185599395
  0.129484966168870	0.949107912342759

  
  

  Note

  Integration in higher dimensions are performed using the the same information provided above.

Lobatto Quadrature

It is similar to Gaussian quadrature with the following differences: (1) The integration points include the end points of the integration interval. (2) It is accurate for polynomials up to degree \(2n - 3\), where \(n\) is the number of integration points.

• Two integration point:

  wi	xi
  1.000000000000000	-1.000000000000000
  1.000000000000000	1.000000000000000

• Three integration point:

  wi	xi
  0.333333333333333	-1.000000000000000
  1.333333333333333	0.000000000000000
  0.333333333333333	1.000000000000000

• Four integration point:

  wi	xi
  0.166666666666667	-1.000000000000000
  0.833333333333333	-0.447213595499958
  0.833333333333333	0.447213595499958
  0.166666666666667	1.000000000000000

• Five integration point:

  wi	xi
  0.100000000000000	-1.000000000000000
  0.544444444444444	-0.654653670707977
  0.711111111111111	0.000000000000000
  0.544444444444444	0.654653670707977
  0.100000000000000	1.000000000000000

  
  

  Note

  Integration in higher dimensions are performed using the the same information provided above.

  Currently integration for TRIA and TETRA are performed using Gauss quadrature.