Quadrature System

The Quadrature is used to integrate numerically in the domain and on sides. Some examples of integrals performed by MOOSE are the integration of the residual over an element, or the postprocessing of spatial variable integrals / averages over the domain.

The Quadrature order is chosen by default so that the product of the test function and the shape functions is integrated exactly. When using Galerkin's method, this would mean the quadrature needs to integrate exactly polynomials of order twice the order of the variables. In order to be able to integrate exactly with a potential third multiplied term in the definition of the residual, we actually use a quadrature of order , with n the order of the variables.

In 1D, the following quadratures are known to integrate exactly:

  • GAUSS (-Legendre), the default rule, with n points integrates exactly 1D polynomials of order up to 2n-1

  • GAUSS_LOBATTO with n points integrates exactly 1D polynomials of order up to 2n-3

  • SIMPSON's rule is known to integrate exactly 1D polynomials of order up to 3

In higher dimensions, the quadratures are defined differently based on the element types. The user is referred to the libMesh doxygen for each quadrature rule object. They generally may be either:

  • tensor product of 1D quadratures to be able to able to integrate exactly terms with with a quadrature of order .

  • using Dunavant's rule for quadratures on triangle elements

  • extracted from the finite element literature

commentnote

Due to a current issue in the documentation system, the type parameter of the quadrature does not display in the list of parameters. The available options are CLOUGH CONICAL GAUSS GRID MONOMIAL SIMPSON TRAP GAUSS_LOBATTO with a default of GAUSS (-Legendre).

Lowering the quadrature order

The quadrature order may be lowered intentionally for various purposes. The side integration quadrature can be set at different order than the element integration one, using the "element_order" and "side_order" parameters. Custom quadrature orders may be set on selected blocks using the "custom_blocks" and "custom_orders" parameters

Example syntax

In this example, we specify a different quadrature rule using the [Quadrature] block nested in the [Executioner] block.

[Executioner]
  type = Steady

  # In 1D, 5th-order Gauss-Lobatto quadrature has 4 points, so in 2D
  # it should have 16.
  [./Quadrature]
    type = GAUSS_LOBATTO
    order = FIFTH
  [../]
[]
(moose/test/tests/quadrature/gauss_lobatto/gauss_lobatto.i)

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