Appendix B. Cubature method list
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
| "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 not linear 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.
use "IM_NC(N,K)", "IM_NC_PARALLELEPIPED(N,K)" and "IM_NC_PRISM(N,K)" to have the Newton cotes
integration of order K on simplices,
parallelepipeds and prisms respectively.
use "IM_GAUSS1D(K)" to have the
Gauss-Legendre integration on the segment of order K (with K/2 + 1
points), and "IM_GAUSSLOBATTO1D(K)"
to have the Gauss-Lobatto-Legendre integration on the segment of
order K (with K/2 + 1 points). The latter integration method
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.
| graphic |
coordinates
|
weights |
function to call / order |
|
|
|
|
 |
|
|
"IM_TRIANGLE(1)" 1
point, order 1. |
|
|
|
|
 |
|
|
"IM_TRIANGLE(2)" 3
points, order 2. |
|
|
|
|
 |
| 1/3 |
1/3 |
|
| 1/5 |
1/5 |
|
| 3/5 |
1/5 |
|
| 1/5 |
3/5 |
|
|
"IM_TRIANGLE(3)" 4
points, order 3. |
|
|
|
|
 |
| a |
a |
| 1-2a |
a |
| a |
1-2a |
| b |
b |
| 1-2b |
b |
| b |
1-2b |
|
|
"IM_TRIANGLE(4)"
6 points, order 4, a = 0.445948490915965,
b = 0.091576213509771, c = 0.111690794839005,
d = 0.054975871827661. |
|
|
|
|
 |
| 1/3 |
1/3 |
| a |
a |
| 1-2a |
a |
| a |
1-2a |
| b |
b |
| 1-2b |
b |
| b |
1-2b |
|
|
"IM_TRIANGLE(5)"
7 points, order 5,
a = (6+sqrt(15))/(21),
b = 4/7 - a,
c = (155+sqrt(15))/(2400),
d = 31/240 - c. |
|
|
|
|
 |
| 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 |
|
|
"IM_TRIANGLE(6)"
12 points, order 6,
a = 0.063089104491502,
b = 0.249286745170910,
c = 0.310352451033785,
d = 0.053145049844816,
e = 0.025422453185103,
f = 0.058393137863189,
g = 0.041425537809187. |
|
|
|
|
 |
| 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,
a = 0.0651301029022,
b = 0.8697397941956,
c = 0.3128654960049,
d = 0.6384441885698,
e = 0.0486903154253,
f = 0.2603459660790,
g = 0.4793080678419,
h = 0.0266736178044,
i = 0.0385568804451,
j = 0.0878076287166,
k = -0.0747850222338. |
|
|
|
"IM_TRIANGLE(8)" (see [3])
16 points, order 8 |
|
|
|
"IM_TRIANGLE(9)" (see [3])
19 points, order 9 |
|
|
|
"IM_TRIANGLE(10)" (see [3])
25 points, order 10 |
|
|
|
"IM_TRIANGLE(13)" (see [3])
37 points, order 13 |
|
|
|
|
 |
| 1/2+sqrt(1/6) |
1/2 |
|
| 1/2-sqrt(1/24) |
1/2±sqrt(1/8) |
|
|
"IM_QUAD(2)" 3 points,
order 2. |
|
|
|
|
 |
| 1/2±sqrt(1/6) |
1/2 |
|
| 1/2 |
1/2±sqrt(1/6) |
|
|
"IM_QUAD(3)" 4 points,
order 3. |
|
|
|
|
 |
| 1/2 |
1/2 |
|
| 1/2 ±sqrt(7/30) |
1/2 |
|
| 1/2±sqrt(1/12) |
1/2±sqrt(3/20) |
|
|
"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 k, and that a Qk polynomial has a
degree of 2*k. For example, IM_QUAD(7) cannot integrate exactly the
product of two Q2 polynomials. On the other hand,
IM_GAUSS_PARALLELEPIPED(2,4) can
integrate exactly that product...
| graphic |
coordinates
|
weights |
function to call / order |
|
|
|
|
 |
|
|
"IM_TETRAHEDRON(1)" 1
point, order 1. |
|
|
|
|
 |
|
|
"IM_TETRAHEDRON(2)"
4 points, order 2
a = (5 - sqrt(5))/(20),
b = (5 + 3sqrt(5))/(20). |
|
|
|
|
 |
| 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 |
|
|
"IM_TETRAHEDRON(3)"
5 points, order 3 |
|
|
|
|
 |
| 1/4 |
1/4 |
1/4 |
| a |
a |
a |
| a |
a |
c |
| a |
c |
a |
| c |
a |
a |
| b |
b |
b |
| b |
b |
d |
| b |
d |
b |
| d |
b |
b |
| e |
e |
f |
| e |
f |
e |
| f |
e |
e |
| e |
f |
f |
| f |
e |
f |
| f |
f |
e |
|
| 8/405 |
| h |
| h |
| h |
| h |
| i |
| i |
| i |
| i |
| 5/567 |
| 5/567 |
| 5/567 |
| 5/567 |
| 5/567 |
| 5/567 |
|
"IM_TETRAHEDRON(5)"
15 points, order 5
a = (7 + sqrt(15))/(34),
b = (7 - sqrt(15))/(34),
c = (13 + 3sqrt(15))/(34),
d = (13 - 3sqrt(15))/(34),
e = (5 - sqrt(15))/(20),
f = (5 + sqrt(15))/(20),
h = (2665 - 14sqrt(15))/(226800),
i = (2665 + 14sqrt(15))/(226800), |
|
Others methods are:
| name |
convex type |
nb of points |
| IM_TETRAHEDRON(6) |
3D simplex |
24 |
| IM_TETRAHEDRON(8) |
3D simplex |
43 |
| IM_SIMPLEX4D(3) |
4D simplex |
6 |
| IM_HEXAHEDRON(5) |
3D parallelepipeded |
14 |
| IM_HEXAHEDRON(9) |
3D parallelepipeded |
58 |
| IM_HEXAHEDRON(11) |
3D parallelepipeded |
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 ca be use instead of the IM_QUAD
integrations.
composite method "IM_STRUCTURED_COMPOSITE(IM_TRIANGLE(2),
3)"
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. This
could be more stable to use 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 coulb 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).
