The integration methods are of two kinds. Exact integrations of polynomials and approximated integrations (cubature formulas) of any function. The exact integration can only be used if all the elements are polynomial and if the geometric transformation is linear.
A descriptor on an integration method is given by the function:
ppi = getfem::int_method_descriptor("name of method");
where "name of method" is a string to be choosen among the existing methods.
The program integration located in the tests directory lists and checks the degree of each integration method.
The list of available exact integration methods is the following
Exact Integration Methods "IM_NONE()" Dummy integration method. "IM_EXACT_SIMPLEX(n)" Description of the exact integration of polynomials on the simplex of reference of dimension n. "IM_PRODUCT(a, b)" Description of the exact integration on the convex which is the direct product of the convex in a and in b. "IM_EXACT_PARALLELEPIPED(n)" Description of the exact integration of polynomials on the parallelepiped of reference of dimension n. "IM_EXACT_PRISM(n)" Description of the exact integration of polynomials on the prism of reference of dimension n
Even though a description of exact integration method exists on parallelepipeds or prisms, most of the time the geometric transformations on such elements are nonlinear and the exact integration cannot be used.
Beware: In fact a lot of computation cannot be done with exact integration methods. So, it is recommended to use cubature formulas instead.
Newton cotes integration of order K on simplices, parallelepipeds and prisms are denoted by "IM_NC(N,K)", "IM_NC_PARALLELEPIPED(N,K)" and "IM_NC_PRISM(N,K)" respectively.
Gauss-Legendre integration on the segment of order K (with K/2+1 points) are denoted by "IM_GAUSS1D(K)". Gauss-Lobatto-Legendre integration on the segment of order K (with K/2+1 points) are denoted by "IM_GAUSSLOBATTO1D(K)". It is only available for odd values of K. The Gauss-Lobatto integration method can be used in conjunction with "FEM_PK_GAUSSLOBATTO1D(K/2)" to perform mass-lumping.
Integration methods on dimension 2 graphic coordinates (x, y) weights function to call / order ![]()
(1/3, 1/3) 1/2 "IM_TRIANGLE(1)"
1 point, order 1.
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(1/6, 1/6)
(2/3, 1/6)
(1/6, 2/3)
1/6
1/6
1/6
"IM_TRIANGLE(2)"
3 points, order 2.
Integration methods on dimension 2 graphic coordinates (x, y) weights function to call / order ![]()
(1/3, 1/3)
(1/5, 1/5)
(3/5, 1/5)
(1/5, 3/5)
-27/96
25/96
25/96
25/96
"IM_TRIANGLE(3)"
4 points, order 3.
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(a, a)
(1-2a, a)
(a, 1-2a)
(b, b)
(1-2b, b)
(b, 1-2b)
c
c
c
d
d
d
"IM_TRIANGLE(4)"
6 points, order 4
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(1/3, 1/3)
(a, a)
(1-2a, a)
(a, 1-2a)
(b, b)
(1-2b, b)
(b, 1-2b)
9/80
c
c
c
d
d
d
"IM_TRIANGLE(5)"
7 points, order 5
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(a, a)
(1-2a, a)
(a, 1-2a)
(b, b)
(1-2b, b)
(b, 1-2b)
(c, d)
(d, c)
(1-c-d, c)
(1-c-d, d)
(c, 1-c-d)
(d, 1-c-d)
e
e
e
f
f
f
g
g
g
g
g
g
"IM_TRIANGLE(6)"
12 points, order 6
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Integration methods on dimension 2 graphic coordinates (x, y) weights function to call / order ![]()
(a, a)
(b, a)
(a, b)
(c, e)
(d, c)
(e, d)
(d, e)
(c, d)
(e, c)
(f, f)
(g, f)
(f, g)
(1/3, 1/3)
h
h
h
i
i
i
i
i
i
j
j
j
k
"IM_TRIANGLE(7)"
13 points, order 7
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"IM_TRIANGLE(8)"
(see [EncyclopCubature])
"IM_TRIANGLE(9)"
(see [EncyclopCubature])
"IM_TRIANGLE(10)"
(see [EncyclopCubature])
"IM_TRIANGLE(13)"
(see [EncyclopCubature])
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(
)
(
)
1/3
1/3
"IM_QUAD(2)"
3 points, order 2
Integration methods on dimension 2 graphic coordinates (x, y) weights function to call / order ![]()
(
)
(
)
1/4
1/4
"IM_QUAD(3)"
4 points, order 3
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(
)
(
)
(
)
2/7
5/63
5/36
"IM_QUAD(5)"
7 points, order 5
"IM_QUAD(7)"
12 points, order 7
"IM_QUAD(9)"
20 points, order 9
"IM_QUAD(17)"
70 points, order 17
There is also the "IM_GAUSS_PARALLELEPIPED(n,k)" which is a direct product of 1D gauss integrations.
Important note: do not forget that IM_QUAD(k) is exact for
polynomials up to degree
, and that a
polynomial has a degree
of
. For example, IM_QUAD(7) cannot integrate exactly the product
of two
polynomials. On the other hand,
IM_GAUSS_PARALLELEPIPED(2,4) can integrate exactly that product ...
Integration methods on dimension 3 graphic coordinates (x, y) weights function to call / order ![]()
(1/4, 1/4, 1/4) 1/6 "IM_TETRAHEDRON(1)"
1 point, order 1
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1/24
1/24
1/24
1/24
"IM_TETRAHEDRON(2)"
4 points, order 2} hspace{7em}
Integration methods on dimension 3 graphic coordinates (x, y) weights function to call / order ![]()
(1/4, 1/4, 1/4)
(1/6, 1/6, 1/6)
(1/6, 1/2, 1/6)
(1/6, 1/6, 1/2)
(1/2, 1/6, 1/6)
-2/15
3/40
3/40
3/40
3/40
"IM_TETRAHEDRON(3)"
5 points, order 3
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8/405
5/567
5/567
5/567
5/567
5/567
5/567
"IM_TETRAHEDRON(5)"
15 points, order 5
Others methods are:
name element type number of points "IM_TETRAHEDRON(6)" tetrahedron 24 "IM_TETRAHEDRON(8)" tetrahedron 43 "IM_SIMPLEX4D(3)" 4D simplex 6 "IM_HEXAHEDRON(5)" 3D hexahedron 14 "IM_HEXAHEDRON(9)" 3D hexahedron 58 "IM_HEXAHEDRON(11)" 3D hexahedron 90 "IM_CUBE4D(5)" 4D parallelepipeded 24 "IM_CUBE4D(9)" 4D parallelepipeded 145
You can use "IM_PRODUCT(IM1, IM2)" to produce integration methods on quadrilateral or prisms. It gives the direct product of two integration mathods. For instance "IM_GAUSS_PARALLELEPIPED(2,k)" is an alias for "IM_PRODUCT(IM_GAUSS1D(2,k),IM_GAUSS1D(2,k))" and can be use instead of the "IM_QUAD" integrations.
Use "IM_STRUCTURED_COMPOSITE(IM1, S)" to copy IM1 on an element with S subdivisions. The resulting integration method has the same order but with more points. It could be more stable to use a composite method rather than to improve the order of the method. Those methods have to be used also with composite elements. Most of the time for composite element, it is preferable to choose the basic method IM1 with no points on the boundary (because the gradient could be not defined on the boundary of sub-elements).
For the HCT element, it is advised to use the "IM_HCT_COMPOSITE(im)" composite integration (which split the original triangle into 3 sub-triangles).