Functions | |
| void | gauss_line (const int aOrder, Vector< real > &aWeights, Matrix< real > &aPoints) |
| void | gauss_quad (const int aOrder, Vector< real > &aWeights, Matrix< real > &aPoints) |
| void | gauss_hex (const int aOrder, Vector< real > &aWeights, Matrix< real > &aPoints) |
| void | gauss_hex14 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| void | gauss_hex34 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| void | gauss_hex6 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| void | gauss_penta1 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| void | gauss_penta11 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| void | gauss_penta16 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| void | gauss_penta29 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| void | gauss_penta40 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| void | gauss_penta5 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| void | gauss_penta50 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| void | gauss_penta71 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| void | gauss_penta8 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| void | gauss_pyra1 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| void | gauss_pyra125 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| Conical-product Gauss rule on the reference pyramid ( base [-1,1]^2 at zeta = 0, apex ( 0, 0, 1 ), as in cl_IF_PYRA5.hpp ): 5-point Gauss-Legendre in xi and eta, scaled by ( 1 - zeta ), times 5-point Gauss-Jacobi(2,0) in zeta, which absorbs the ( 1 - zeta )^2 volume factor. | |
| void | gauss_pyra27 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| Conical-product Gauss rule on the reference pyramid ( base [-1,1]^2 at zeta = 0, apex ( 0, 0, 1 ), as in cl_IF_PYRA5.hpp ): 3-point Gauss-Legendre in xi and eta, scaled by ( 1 - zeta ), times 3-point Gauss-Jacobi(2,0) in zeta, which absorbs the ( 1 - zeta )^2 volume factor. | |
| void | gauss_pyra64 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| Conical-product Gauss rule on the reference pyramid ( base [-1,1]^2 at zeta = 0, apex ( 0, 0, 1 ), as in cl_IF_PYRA5.hpp ): 4-point Gauss-Legendre in xi and eta, scaled by ( 1 - zeta ), times 4-point Gauss-Jacobi(2,0) in zeta, which absorbs the ( 1 - zeta )^2 volume factor. | |
| void | gauss_pyra8 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| Conical-product Gauss rule on the reference pyramid ( base [-1,1]^2 at zeta = 0, apex ( 0, 0, 1 ), as in cl_IF_PYRA5.hpp ): 2-point Gauss-Legendre in xi and eta, scaled by ( 1 - zeta ), times 2-point Gauss-Jacobi(2,0) in zeta, which absorbs the ( 1 - zeta )^2 volume factor. | |
| void | gauss_quad12 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| void | gauss_quad20 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| void | gauss_quad28 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| void | gauss_quad37 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| void | gauss_quad8 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| void | gauss_tet1 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| void | gauss_tet10 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| void | gauss_tet11 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| void | gauss_tet15 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| void | gauss_tet165 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| 11th order interpolation | |
| void | gauss_tet20 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| void | gauss_tet220 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| 13th order interpolation | |
| void | gauss_tet236 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| 14th order interpolation | |
| void | gauss_tet24 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| void | gauss_tet31 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| void | gauss_tet35 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| void | gauss_tet4 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| void | gauss_tet46 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| 8th order interpolation | |
| void | gauss_tet5 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| void | gauss_tet56 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| void | gauss_tet81 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| 10th order interpolation | |
| void | gauss_tri1 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| void | gauss_tri12 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| 6th order interpolation | |
| void | gauss_tri15 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| 7th order interpolation | |
| void | gauss_tri16 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| 8th order interpolation | |
| void | gauss_tri19 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| 9th order interpolation | |
| void | gauss_tri25 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| 10th order interpolation | |
| void | gauss_tri28 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| 11th order interpolation | |
| void | gauss_tri3 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| void | gauss_tri33 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| 12th order interpolation | |
| void | gauss_tri37 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| 13th order interpolation | |
| void | gauss_tri4 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| void | gauss_tri42 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| 14th order interpolation | |
| void | gauss_tri49 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| 15th order interpolation | |
| void | gauss_tri55 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| 16th order interpolation | |
| void | gauss_tri60 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| 17th order interpolation | |
| void | gauss_tri67 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| 18th order interpolation | |
| void | gauss_tri7 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| 5th order interpolation | |
| void | gauss_tri73 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| 19th order interpolation | |
| void | gauss_tri79 (Vector< real > &aWeights, Matrix< real > &aPoints) |
| 20th order interpolation | |
| void belfem::integration::gauss_hex | ( | const int | aOrder, |
| Vector< real > & | aWeights, | ||
| Matrix< real > & | aPoints ) |
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| void belfem::integration::gauss_line | ( | const int | aOrder, |
| Vector< real > & | aWeights, | ||
| Matrix< real > & | aPoints ) |
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Conical-product Gauss rule on the reference pyramid ( base [-1,1]^2 at zeta = 0, apex ( 0, 0, 1 ), as in cl_IF_PYRA5.hpp ): 5-point Gauss-Legendre in xi and eta, scaled by ( 1 - zeta ), times 5-point Gauss-Jacobi(2,0) in zeta, which absorbs the ( 1 - zeta )^2 volume factor.
Exact for polynomials up to degree 9; the weights sum to 4/3. Regenerated 2026-09-03: the stored coordinates did not integrate degree 1 ( the weights already were these ). Construction as in 10.1108/02644400410554362
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Conical-product Gauss rule on the reference pyramid ( base [-1,1]^2 at zeta = 0, apex ( 0, 0, 1 ), as in cl_IF_PYRA5.hpp ): 3-point Gauss-Legendre in xi and eta, scaled by ( 1 - zeta ), times 3-point Gauss-Jacobi(2,0) in zeta, which absorbs the ( 1 - zeta )^2 volume factor.
Exact for polynomials up to degree 5; the weights sum to 4/3. Regenerated 2026-09-03: the stored coordinates did not integrate degree 1 ( the weights already were these ). Construction as in 10.1108/02644400410554362
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Conical-product Gauss rule on the reference pyramid ( base [-1,1]^2 at zeta = 0, apex ( 0, 0, 1 ), as in cl_IF_PYRA5.hpp ): 4-point Gauss-Legendre in xi and eta, scaled by ( 1 - zeta ), times 4-point Gauss-Jacobi(2,0) in zeta, which absorbs the ( 1 - zeta )^2 volume factor.
Exact for polynomials up to degree 7; the weights sum to 4/3. Regenerated 2026-09-03: the stored coordinates did not integrate degree 1 ( the weights already were these ). Construction as in 10.1108/02644400410554362
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Conical-product Gauss rule on the reference pyramid ( base [-1,1]^2 at zeta = 0, apex ( 0, 0, 1 ), as in cl_IF_PYRA5.hpp ): 2-point Gauss-Legendre in xi and eta, scaled by ( 1 - zeta ), times 2-point Gauss-Jacobi(2,0) in zeta, which absorbs the ( 1 - zeta )^2 volume factor.
Exact for polynomials up to degree 3; the weights sum to 4/3. Regenerated 2026-09-03: the stored coordinates did not integrate degree 1 ( the weights already were these ). Construction as in 10.1108/02644400410554362
| void belfem::integration::gauss_quad | ( | const int | aOrder, |
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| Matrix< real > & | aPoints ) |
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11th order interpolation
B. Vioreanu, V. Rokhlin : Spectra of Multiplication Operators as a Numerical Tool SIAM J. Sci. Comput., vol. 36, no. 1, 2014 https://doi.org/10.1137/110860082
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13th order interpolation
B. Vioreanu, V. Rokhlin : Spectra of Multiplication Operators as a Numerical Tool SIAM J. Sci. Comput., vol. 36, no. 1, 2014 https://doi.org/10.1137/110860082
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14th order interpolation
Linbo Zhang, Tao Cui, Hui Liu : A set of symmetric quadrature rules on triangles and tetrahedra Journal of Computational Mathematics, vol. 27, no. 1, pp. 89-96, 2009 https://www.jstor.org/stable/43693493
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8th order interpolation
F.D. Witherden, P.E. Vincent : On the identification of symmetric quadrature rules for finite element methods Computers & Mathematics with Applications, vol. 69, no. 10, pp. 1232–1241, 2015 https://doi.org/10.1016/j.camwa.2015.03.017
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10th order interpolation
F.D. Witherden, P.E. Vincent : On the identification of symmetric quadrature rules for finite element methods Computers & Mathematics with Applications, vol. 69, no. 10, pp. 1232–1241, 2015 https://doi.org/10.1016/j.camwa.2015.03.017
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6th order interpolation
G.R. Cowper : Gaussian quadrature formulas for triangles Numerical Methods in Engineering, vol. 7, no. 3, pp. 405–408, 1973 https://doi.org/10.1002/nme.1620070316
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7th order interpolation
M.E. Laursen, M. Gellert : Some criteria for numerically integrated matrices and quadrature formulas for triangles International Journal for Numerical Methods in Engineering, vol. 12, no. 1, pp. 67–76, 1978 https://doi.org/10.1002/nme.1620120107
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8th order interpolation
J.N. Lyness, D. Jespersen : Moderate Degree Symmetric Quadrature Rules for the Triangle IMA Journal of Applied Mathematics, vol. 15, no. 1, pp. 19-32, 1975 https://doi.org/10.1093/imamat/15.1.19
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9th order interpolation
J.N. Lyness, D. Jespersen : Moderate Degree Symmetric Quadrature Rules for the Triangle IMA Journal of Applied Mathematics, vol. 15, no. 1, pp. 19-32, 1975 https://doi.org/10.1093/imamat/15.1.19
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10th order interpolation
F.D. Witherden, P.E. Vincent : On the identification of symmetric quadrature rules for finite element methods Computers & Mathematics with Applications, vol. 69, no. 10, pp. 1232–1241, 2015 https://doi.org/10.1016/j.camwa.2015.03.017
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11th order interpolation
Hong Xiao, Zydrunas Gimbutas : A numerical algorithm for the construction of efficient quadrature rules in two and higher dimensions Computers & Mathematics with Applications, vol. 59, no. 2, pp. 663–676, 2010 https://doi.org/10.1016/j.camwa.2009.10.027
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12th order interpolation
Hong Xiao, Zydrunas Gimbutas : A numerical algorithm for the construction of efficient quadrature rules in two and higher dimensions Computers & Mathematics with Applications, vol. 59, no. 2, pp. 663–676, 2010 https://doi.org/10.1016/j.camwa.2009.10.027
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13th order interpolation
Hong Xiao, Zydrunas Gimbutas : A numerical algorithm for the construction of efficient quadrature rules in two and higher dimensions Computers & Mathematics with Applications, vol. 59, no. 2, pp. 663–676, 2010 https://doi.org/10.1016/j.camwa.2009.10.027
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14th order interpolation
Hong Xiao, Zydrunas Gimbutas : A numerical algorithm for the construction of efficient quadrature rules in two and higher dimensions Computers & Mathematics with Applications, vol. 59, no. 2, pp. 663–676, 2010 https://doi.org/10.1016/j.camwa.2009.10.027
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15th order interpolation
F.D. Witherden, P.E. Vincent : On the identification of symmetric quadrature rules for finite element methods Computers & Mathematics with Applications, vol. 69, no. 10, pp. 1232–1241, 2015 https://doi.org/10.1016/j.camwa.2015.03.017
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16th order interpolation
Hong Xiao, Zydrunas Gimbutas : A numerical algorithm for the construction of efficient quadrature rules in two and higher dimensions Computers & Mathematics with Applications, vol. 59, no. 2, pp. 663–676, 2010 https://doi.org/10.1016/j.camwa.2009.10.027
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17th order interpolation
Hong Xiao, Zydrunas Gimbutas : A numerical algorithm for the construction of efficient quadrature rules in two and higher dimensions Computers & Mathematics with Applications, vol. 59, no. 2, pp. 663–676, 2010 https://doi.org/10.1016/j.camwa.2009.10.027
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18th order interpolation
F.D. Witherden, P.E. Vincent : On the identification of symmetric quadrature rules for finite element methods Computers & Mathematics with Applications, vol. 69, no. 10, pp. 1232–1241, 2015 https://doi.org/10.1016/j.camwa.2015.03.017
5th order interpolation
P.C. Hammer, O.J. Marlowe, A.H. Stroud : Numerical Integration Over Simplexes and Cones Mathematical Tables and Other Aids to Computation, vol. 10, no. 55, pp. 130-137, 1956 https://doi.org/10.1090/S0025-5718-1956-0086389-6
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19th order interpolation
D.A. Dunavant : High Degree Efficient Symmetrical Gaussian Quadrature Rules for the Triangle Article in International Journal for Numerical Methods in Engineering, vol. 21, no. 6, pp. 1129-1148, 1985 https://doi.org/10.1002/nme.1620210612
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20th order interpolation
Hong Xiao, Zydrunas Gimbutas : A numerical algorithm for the construction of efficient quadrature rules in two and higher dimensions Computers & Mathematics with Applications, vol. 59, no. 2, pp. 663–676, 2010 https://doi.org/10.1016/j.camwa.2009.10.027