Quadrature schemes and extrapolation

OpenGeoSys has implemented certain numerical integration/quadrature rules for each mesh element type. Currently, all of the implemented quadrature rules are Gauss-Legendre quadrature schemes or related schemes in that spirit. I.e., integration points are located inside the elements, not at its faces, edges, or corners.

The table below gives an overview over the implemented schemes and which polynomials can be integrated exactly with the respective schemes. The data are unit tested.

Integration methods are implemented up to integration order 4. For each integration order n there is a maximum polynomial degree P that can be integrated exactly with the respective integration method. For the classical Gauss-Legendre integration the following holds: P = 2 · n - 1. Methods that fulfill this relation are marked bold in column P in the table, methods that are deficient are marked italic.

The columns contain the following data:

  • #IP: The number of integration points of the integration method.
  • P: The maximum polynomial degree that the integration method can integrate exactly. The monomials in the polynomial are xi yj zk with i + j + k ≤ P
  • Q: The maximum polynomial degree that the integration method can integrate exactly. The monomials in the polynomial are xi yj zk with i ≤ Q, j ≤ Q and k ≤ Q. I.e., maximum monomial degrees are higher in the Q column than in the P column.
Integration order → 1 1 1 2 2 2 3 3 3 4 4 4
Mesh element ↓ #IP Q P #IP Q P #IP Q P #IP Q P
Point 1 * * 1 * * 1 * * 1 * *
Line 1 1 1 2 3 3 3 5 5 4 7 7
Quad 1 1 1 4 3 3 9 5 5 16 7 7
Hexahedron 1 1 1 8 3 3 27 5 5 64 7 7
Triangle 1 0 1 3 1 3 4 1 3 7 2 5
Tetrahedron 1 0 1 5 1 3 14 1 5 20 1 5
3-sided prism 1 0 1 6 1 3 21 2 3 28 2 5
Pyramid 1 1 1 5 1 3 13 3 3 13 3 3

Note, that on pyramids the determinant of the Jacobian originating from mapping the unit element to the physical one, det($J$), varies over the mesh element, even for linear elements. This is because a pyramid is neither a simplex, nor a direct superposition of simplices, implying that it is deformed by construction. Therefore, for pyramids we are actually integrating a polynomial of higher degree than the degrees P and Q given in the table above. Moreover, please note, that det($J$) also varies for all deformed direct superpositions of linear elements. This implies that det($J$) will vary on deformed bi-linear elements like quadrilaterals (quads) and 3-sided prisms as well as on deformed hexahedra, which are tri-linear.

Extrapolation of integration point data to mesh nodes

OpenGeoSys can extrapolate integration point data to mesh nodes for easy post-processing. Note, however, that the extrapolation procedures is not exact and can lead to more or less subtle errors that are hard to find!

Since OpenGeoSys extrapolates integration point data element-wise, the number of integration points of each mesh element must be greater or equal to the number of nodes of the element. Therefore, the ability to extrapolate data is linked to the chosen integration order, i.e., the number of integration points must be greater or equal to the number of nodes. The relation is presented in the table below.

The columns contain the following data:

  • #IP: The number of integration points of the integration method.
  • L: Whether extrapolation can be performed on the (multi-)linear elements (e.g. Quad4) of the respective type with the given integration order.
  • Q: Whether extrapolation can be performed on the (multi-)quadratic elements (e.g. Hex20) of the respective type with the given integration order.

As you can see, for integration order ≥ 2 extrapolation works for all (multi-)linear elements and for integration order ≥ 3 extrapolation works for all element types implemented in OpenGeoSys.

Integration order → 1 1 1 2 2 2 3 3 3 4 4 4
Mesh element ↓ #IP L Q #IP L Q #IP L Q #IP L Q
Point 1 1 1 1
Line 1 2 3 4
Quad 1 4 9 16
Hexahedron 1 8 27 64
Triangle 1 3 4 7
Tetrahedron 1 5 14 20
3–sided prism 1 6 21 28
Pyramid 1 5 13 13

This article was written by Christoph Lehmann. If you are missing something or you find an error please let us know.
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