Volumic elements ==== Tetrahedra: ELREFE TE4, T10 ---- .. image:: images/100012360000182F0000170C3B52F571AC5ECFD5.svg :width: 312 :height: 297 .. _RefImage_100012360000182F0000170C3B52F571AC5ECFD5.svg: Node coordinates: .. csv-table:: "", ":math:`x` "," :math:`y` "," :math:`z`" ":math:`\mathrm{N1}` ", "0. ", "1. ", "0." ":math:`\mathrm{N2}` ", "0. ", "0. ", "1." ":math:`\mathrm{N3}` ", "0. ", "0. ", "0." ":math:`\mathrm{N4}` ", "1. ", "0. ", "0." ":math:`\mathrm{N5}` ", "0. ", "0.5", "0.5" ":math:`\mathrm{N6}` ", "0. ", "0. ", "0.5" ":math:`\mathrm{N7}` ", "0. ", "0.5", "0." ":math:`\mathrm{N8}` ", "0.5", "0.5", "0.5", "0." ":math:`\mathrm{N9}` ", "0.5", "0. ", "0.5" ":math:`\mathrm{N10}` ", "0.5", "0. ", "0." **Shape functions:** 4-knot formula :math:`\{\begin{array}{}{w}_{1}(x,y,z)=y\\ {w}_{2}(x,y,z)=z\\ {w}_{3}(x,y,z)=1-x-y-z\\ {w}_{4}(x,y,z)=x\end{array}` 10-knot formula .. math:: :label: eq-None \begin{array}{}{w}_{1}=y(2y-1)\\ {w}_{2}=z(2z-1)\\ {w}_{3}=(1-x-y-z)(1-2x-2y-2z)\\ {w}_{4}=x(2x-1)\\ {w}_{5}=4yz\end{array} **Numerical integration formula:** Formula with 1 point, order 1 in :math:`x,y,z`: (FPG1) .. csv-table:: "**Point**", ":math:`x` "," :math:`y` "," "," :math:`z` ", "**Weight**" "1"," :math:`1/4` "," :math:`1/4` "," "," :math:`1/4` "," :math:`1/6`" Formula with 4 points, order 2 in :math:`x,y,z`: (FPG4) .. csv-table:: "**Point**", ":math:`x` "," :math:`y` "," "," :math:`z` ", "**Weight**" "1"," :math:`a` "," :math:`a` "," "," :math:`a` "," :math:`1/24`" "2"," :math:`a` "," :math:`a` "," "," :math:`b` "," :math:`1/24`" "3"," :math:`a` "," :math:`b` "," "," :math:`a` "," :math:`1/24`" "4"," :math:`b` "," :math:`a` "," "," :math:`a` "," :math:`1/24`" with: :math:`a=\frac{5-\sqrt{5}}{\text{20}}`, :math:`b=\frac{5+3\sqrt{5}}{\text{20}}` Formula with 5 points, order 3 in :math:`x,y,z`: (FPG5) .. csv-table:: "**Point**", ":math:`x` "," :math:`y` "," "," :math:`z` ", "**Weight**" "1"," :math:`a` "," :math:`a` "," "," :math:`a` "," :math:`-2/15`" "2"," :math:`b` "," :math:`b` "," "," :math:`b` "," :math:`3/40`" "3"," :math:`b` "," :math:`b` "," "," :math:`c` "," :math:`3/40`" "4"," :math:`b` "," :math:`c` "," "," :math:`b` "," :math:`3/40`" "5"," :math:`c` "," :math:`b` "," "," :math:`b` "," :math:`3/40`" With: :math:`a=0\text{.}\text{25}`, :math:`b=\frac{1}{6}`, :math:`c=0\text{.}5` Formula with 15 points, order 5 in :math:`x,y,z`: (FPG15) .. csv-table:: "**Point**", ":math:`x` "," :math:`y` "," "," :math:`z` ", "**Weight**" "1"," :math:`a` "," :math:`a` "," "," :math:`a` "," :math:`8/405`" "2 3 4 5", ":math:`{b}_{1}` :math:`{b}_{1}` :math:`{b}_{1}` :math:`{c}_{1}` "," :math:`{b}_{1}` :math:`{b}_{1}` :math:`{c}_{1}` :math:`{b}_{1}` "," :math:`{b}_{1}` :math:`{c}_{1}` :math:`{b}_{1}` :math:`{b}_{1}` "," :math:`\frac{2\text{665}-\text{14}\sqrt{\text{15}}}{\text{226}\text{800}}`" "6 7 8 9", ":math:`{b}_{2}` :math:`{b}_{2}` :math:`{b}_{2}` :math:`{c}_{2}` "," :math:`{b}_{2}` :math:`{b}_{2}` :math:`{c}_{2}` :math:`{b}_{2}` "," :math:`{b}_{2}` :math:`{c}_{2}` :math:`{b}_{2}` :math:`{b}_{2}` "," :math:`\frac{2\text{665}+\text{14}\sqrt{\text{15}}}{\text{226}\text{800}}`" "10 11 12 13 14 15", ":math:`d` :math:`d` :math:`e` :math:`d` :math:`e` :math:`e` "," :math:`d` :math:`e` :math:`d` :math:`e` :math:`d` :math:`e` "," :math:`e` :math:`d` :math:`d` :math:`e` :math:`e` :math:`d` "," :math:`\frac{5}{\text{567}}`" with: .. csv-table:: ":math:`a=0\text{.}\text{25}` "," :math:`\begin{array}{}{b}_{1}=\frac{7+\sqrt{\text{15}}}{\text{34}}\\ {b}_{2}=\frac{7-\sqrt{\text{15}}}{\text{34}}\end{array}` "," :math:`\begin{array}{}{c}_{1}=\frac{\text{13}-3\sqrt{\text{15}}}{\text{34}}\\ {c}_{2}=\frac{\text{13}+3\sqrt{\text{15}}}{\text{34}}\end{array}` "," :math:`\begin{array}{}d=\frac{5-\sqrt{\text{15}}}{\text{20}}\\ e=\frac{5+\sqrt{\text{15}}}{\text{20}}\end{array}`" Pentahedra: ELREFE PE6, P15, P15, P18, P21 ---- N19 ROAD N20 ROAD N21 ROAD .. image:: images/Shape1.gif .. _RefSchema_Shape1.gif: Node coordinates: .. csv-table:: "", ":math:`x` "," :math:`y` "," :math:`z`" ":math:`\mathrm{N1}` ", "-1. ", "1. ", "0." ":math:`\mathrm{N2}` ", "-1. ", "0. ", "1." ":math:`\mathrm{N3}` ", "-1. ", "0. ", "0." ":math:`\mathrm{N4}` ", "1. ", "1. ", "0." ":math:`\mathrm{N5}` ", "1. ", "0. ", "1." ":math:`\mathrm{N6}` ", "1. ", "0. ", "0." ":math:`\mathrm{N7}` ", "-1. ", "0.5", "0.5." ":math:`\mathrm{N8}` ", "-1. ", "0. ", "0.5." ":math:`\mathrm{N9}` ", "-1. ", "0.5", "0." ":math:`\mathrm{N10}` ", "0. ", "1. ", "0." ":math:`\mathrm{N11}` ", "0. ", "0. ", "1." ":math:`\mathrm{N12}` ", "0. ", "0. ", "0." ":math:`\mathrm{N13}` ", "1. ", "0.5", "0.5" ":math:`\mathrm{N14}` ", "1. ", "0. ", "0.5" ":math:`\mathrm{N15}` ", "1. ", "0.5", "0." ":math:`\mathrm{N16}` ", "0. ", "0.5", "0.5" ":math:`N17` ", "0. ", "0. ", "0.5" ":math:`\mathrm{N18}` ", "0. ", "0.5", "0." ":math:`N19` ", "-1. ", "1/3", "1/3" ":math:`N20` ", "1. ", "1/3", "1/3" ":math:`N21` ", "0. ", "1/3", "1/3" **Shape functions:** 6-knot formula .. math:: :label: eq-None \begin{array}{}{w}_{1}=\frac{1}{2}y(1-x)\\ {w}_{2}=\frac{1}{2}z(1-x)\\ {w}_{3}=\frac{1}{2}(1-y-z)(1-x)\end{array} 15-knot formula .. math:: :label: eq-None \begin{array}{}{w}_{1}=y(1-x)(2y-2-x)/2\\ {w}_{2}=z(1-x)(2z-2-x)/2\\ {w}_{3}=(x-1)(1-y-z)(x+2y+2z)/2\\ {w}_{4}=y(1+x)(2y-2+x)/2\\ {w}_{5}=z(1+x)(2z-2+x)/2\\ {w}_{6}=(-x-1)(1-y-z)(-x+2y+2z)/2\\ {w}_{7}=2yz(1-x)\\ {w}_{8}=2z(1-y-z)(1-x)\end{array} 18 knot formula .. math:: :label: eq-None \begin{array}{}{w}_{1}=xy(x-1)(2y-1)/2\\ {w}_{2}=xz(x-1)(2z-1)/2\\ {w}_{3}=x(x-1)(z+y-1)(2z+2y-1)/2\\ {w}_{4}=xy(x+1)(2y-1)/2\\ {w}_{5}=xz(x+1)(2z-1)/2\\ {w}_{6}=x(x+1)(z+y-1)(2z+2y-1)/2\\ {w}_{7}=2xyz(x-1)\\ {w}_{8}=-2xz(x-1)(z+y-1)\\ {w}_{9}=-2xy(x-1)(z+y-1)\end{array} **6-point numerical integration formulas** (order 3 in :math:`x`, order 2 in :math:`y` and :math:`z`) (FPG6) .. csv-table:: "**Point**", ":math:`x` "," :math:`y` "," "," :math:`z` ", "**Weight**" "1"," :math:`-1/\sqrt{3}` ", "0.5", "0.5"," :math:`1/6`" "2"," :math:`-1/\sqrt{3}` ", "0. ", "0.5"," :math:`1/6`" "3"," :math:`-1/\sqrt{3}` ", "0.5", "0. "," :math:`1/6`" "4"," :math:`1/\sqrt{3}` ", "0.5", "0.5"," :math:`1/6`" "5"," :math:`1/\sqrt{3}` ", "0. ", "0.5"," :math:`1/6`" "6"," :math:`1/\sqrt{3}` ", "0.5", "0. "," :math:`1/6`" **8-point numerical integration formula:** **(FPG8)** 2 Gauss points in :math:`x` (order 3). 4 Hammer points in :math:`y` and :math:`z` (3rd order). .. csv-table:: "**Point**", ":math:`x` "," :math:`y` "," "," :math:`z` ", "**Weight**" "1"," :math:`-a` "," :math:`1/3` "," "," :math:`1/3` "," :math:`-27/96`" "2"," :math:`-a` ", "0.6", "0.2"," :math:`25/96`" "3"," :math:`-a` ", "0.2", "0.6"," :math:`25/96`" "4"," :math:`-a` ", "0.2", "0.2"," :math:`25/96`" "5"," :math:`+a` "," :math:`1/3` "," "," :math:`1/3` "," :math:`-27/96`" "6"," :math:`+a` ", "0.6", "0.2"," :math:`25/96`" "7"," :math:`+a` ", "0.2", "0.6"," :math:`25/96`" "8"," :math:`+a` ", "0.2", "0.2"," :math:`25/96`" With :math:`a=0.577350269189626` **21-point numerical integration formula:** **(FPG21)** 3 Gauss points in :math:`x` (order 5). 7 Hammer points in :math:`y` and :math:`z` (order 5 in :math:`y` and :math:`z`). .. csv-table:: "**Point**", ":math:`x` "," :math:`y` "," "," :math:`z` ", "**Weight**" "1"," :math:`-\alpha` "," :math:`1/3` "," "," :math:`1/3` "," :math:`{c}_{1}\frac{9}{80}`" "2 3 4", ":math:`-\alpha` :math:`-\alpha` :math:`-\alpha` "," :math:`a` :math:`1-\mathrm{2a}` :math:`a` "," :math:`a` :math:`a` :math:`1-\mathrm{2a}` "," :math:`{c}_{1}(\frac{155+\sqrt{15}}{2400})`" "5 6 7", ":math:`-\alpha` :math:`-\alpha` :math:`-\alpha` "," :math:`b` :math:`1-\mathrm{2b}` :math:`b` "," :math:`b` :math:`b` :math:`1-\mathrm{2b}` "," :math:`{c}_{1}(\frac{155-\sqrt{15}}{2400})`" "8", "0. "," :math:`1/3` "," :math:`1/3` "," :math:`{c}_{2}\frac{9}{80}`" "9 10 11", "0. 0. 0."," :math:`a` :math:`1-\mathrm{2a}` :math:`a` "," :math:`a` :math:`a` :math:`1-\mathrm{2a}` "," :math:`{c}_{2}(\frac{155+\sqrt{15}}{2400})`" "12 13 14", "0. 0. 0."," :math:`b` :math:`1-\mathrm{2b}` :math:`b` "," :math:`b` :math:`b` :math:`1-\mathrm{2b}` "," :math:`{c}_{2}(\frac{155-\sqrt{15}}{2400})`" "15"," :math:`\alpha` "," :math:`1/3` "," "," :math:`1/3` "," :math:`{c}_{1}\frac{9}{80}`" "16 17 18", ":math:`\alpha` :math:`\alpha` :math:`\alpha` "," :math:`b` :math:`1-\mathrm{2a}` :math:`a` "," :math:`a` :math:`a` :math:`1-\mathrm{2a}` "," :math:`{c}_{1}(\frac{155+\sqrt{15}}{2400})`" "19 20 21", ":math:`\alpha` :math:`\alpha` :math:`\alpha` "," :math:`b` :math:`1-\mathrm{2b}` :math:`b` "," :math:`b` :math:`b` :math:`1-\mathrm{2b}` "," :math:`{c}_{1}(\frac{155-\sqrt{15}}{2400})`" with: .. csv-table:: ":math:`\alpha =\sqrt{\frac{3}{5}}` "," :math:`{c}_{1}=\frac{5}{9}` "," :math:`{c}_{2}=\frac{8}{9}` "," :math:`a=\frac{6+\sqrt{15}}{21}` "," :math:`b=\frac{6-\sqrt{15}}{21}`" 27-point numerical integration formula (FPG27): see [bib3]_. .. csv-table:: "**Point**", ":math:`x` "," :math:`y` "," "," :math:`z` ", "**Weight**" "**1**", "0.0", "0.895512822481133", "0.052243588759434", "0.027191062410231" "**2**", "0.0", "0.052243588759434", "0.052243588759434", "0.895512822481133", "0.027191062410231" "**3**", "0.0", "0.052243588759434", "0.052243588759434", "0.027191062410231" "**4**", "0.0", "0.198304865473555", "0.270635256143164", "0.040636041641641220" "**5**", "0.0", "0.198304865473555", "0.531059878383280", "0.040636041641641220" "**6**", "0.0", "0.270635256143164", "0.531059878383280", "0.040636041641641220" "**7**", "0.0", "0.531059878383280", "0.270635256143164", "0.040636041641641220" "**8**", "0.0", "0.531059878383280", "0.198304865473555", "0.040636041641641220" "**9**", "0.0", "0.270635256143164", "0.198304865473555", "0.040636041641641220" "**10**", "0.936241512371697", "0.3333333333333333333", "0.33333333333333333", "0.05027512371697", "0.050275140937507" "**11**", "0.948681147283254", "0.841699897299232", "0.079150051350384", "0.011774414962347" "**12**", "0.948681147283254", "0.079150051350384", "0.841699897299232", "0.011774414962347" "**13**", "0.948681147283254", "0.079150051350384", "0.079150051350384", "0.011774414962347" "**14**", "0.600638052820557", "0.054831294873304", "0.308513201856883", "0.041951149272741" "**15**", "0.600638052820557", "0.054831294873304", "0.636655503269814", "0.041951149272741" "**16**", "0.600638052820557", "0.308513201856883", "0.636655503269814", "0.041951149272741" "**17**", "0.600638052820557", "0.636655503269814", "0.308513201856883", "0.041951149272741" "**18**", "0.600638052820557", "0.636655503269814", "0.054831294873304", "0.041951149272741" "**19**", "0.600638052820557", "0.308513201856883", "0.054831294873304", "0.041951149272741" "**20**", "-0.93624151212371697", "0.3333333333333333333", "0.33333333333333333", "0.05027512371697", "0.050275140937507" "**21**", "-0.948681147283254", "0.841699897299232", "0.079150051350384", "0.011774414962347" "**22**", "-0.948681147283254", "0.079150051350384", "0.841699897299232", "0.011774414962347" "**23**", "-0.948681147283254", "0.079150051350384", "0.079150051350384", "0.011774414962347" "**24**", "-0.600638052820557", "0.054831294873304", "0.308513201856883", "0.041951149272741" "**25**", "-0.600638052820557", "0.054831294873304", "0.636655503269814", "0.0419511492820557", "0.041951149272741" "**26**", "-0.600638052820557", "0.308513201856883", "0.636655503269814", "0.0419511492820557", "0.041951149272741" "**27**", "-0.600638052820557", "0.636655503269814", "0.308513201856883", "0.041951149272741" "**28**", "-0.600638052820557", "0.636655503269814", "0.054831294873304", "0.041951149272741" "**29**", "-0.600638052820557", "0.308513201856883", "0.054831294873304", "0.041951149272741" Hexahedra: ELREFE HE8, H20, H27 ---- .. image:: images/Shape2.gif .. _RefSchema_Shape2.gif: .. image:: images/Shape3.gif .. _RefSchema_Shape3.gif: Node coordinates: .. csv-table:: "", ":math:`x` "," :math:`y` "," :math:`z`" "N1", "-1. ", "-1. ", "-1." "N2", "1. ", "-1. ", "-1." "N3", "1. ", "1. ", "-1." "N4", "-1. ", "1. ", "-1." "N5", "-1. ", "-1. ", "1." "N6", "1. ", "-1. ", "1." "N7", "1. ", "1. ", "1." "N8", "-1. ", "1. ", "1." "N9", "0. ", "-1. ", "-1." "N10", "1. ", "0. ", "-1." "N11", "0. ", "1. ", "-1." "N12", "-1. ", "0. ", "-1." "N13", "-1. ", "-1. ", "0." "N14", "1. ", "-1. ", "0." "N15", "1. ", "1. ", "0." "N16", "-1. ", "1. ", "0." "N17", "0. ", "-1. ", "1." "N18", "1. ", "0. ", "1." "N19", "0. ", "1. ", "1." "N20", "-1. ", "0. ", "1." "N21", "0. ", "0. ", "-1." "N22", "0. ", "-1. ", "0." "N23", "1. ", "0. ", "0." "N24", "0. ", "1. ", "0." "N25", "-1. ", "0. ", "0." "N26", "0. ", "0. ", "1." "N27", "0. ", "0. ", "0." **Shape functions:** 8-knot formula .. math:: :label: eq-None \begin{array}{}{w}_{1}=\frac{1}{8}(1-x)(1-y)(1-z)\\ {w}_{2}=\frac{1}{8}(1+x)(1-y)(1-z)\\ {w}_{3}=\frac{1}{8}(1+x)(1+y)(1-z)\\ {w}_{4}=\frac{1}{8}(1-x)(1+y)(1-z)\end{array} 20-knot formula .. math:: :label: eq-None \begin{array}{}{w}_{1}=\frac{1}{8}(1-x)(1-y)(1-z)(-2-x-y-z)\\ {w}_{2}=\frac{1}{8}(1+x)(1-y)(1-z)(-2+x-y-z)\\ {w}_{3}=\frac{1}{8}(1+x)(1+y)(1-z)(-2+x+y-z)\\ {w}_{4}=\frac{1}{8}(1-x)(1+y)(1-z)(-2-x+y-z)\\ {w}_{5}=\frac{1}{8}(1-x)(1-y)(1+z)(-2-x-y+z)\\ {w}_{6}=\frac{1}{8}(1+x)(1-y)(1+z)(-2+x-y+z)\\ {w}_{7}=\frac{1}{8}(1+x)(1+y)(1+z)(-2+x+y+z)\\ {w}_{8}=\frac{1}{8}(1-x)(1+y)(1+z)(-2-x+y+z)\\ {w}_{9}=\frac{1}{4}(1-{x}^{2})(1-y)(1-z)\\ {w}_{10}=\frac{1}{4}(1-{y}^{2})(1+x)(1-z)\end{array} 27-knot formula .. math:: :label: eq-None \begin{array}{}{w}_{1}=\frac{1}{8}x(x-1)y(y-1)z(z-1)\\ {w}_{2}=\frac{1}{8}x(x+1)y(y-1)z(z-1)\\ {w}_{3}=\frac{1}{8}x(x+1)y(y+1)z(z-1)\\ {w}_{4}=\frac{1}{8}x(x-1)y(y+1)z(z-1)\\ {w}_{5}=\frac{1}{8}x(x-1)y(y-1)z(z+1)\\ {w}_{6}=\frac{1}{8}x(x+1)y(y-1)z(z+1)\\ {w}_{7}=\frac{1}{8}x(x+1)y(y+1)z(z+1)\\ {w}_{8}=\frac{1}{8}x(x-1)y(y+1)z(z+1)\\ {w}_{9}=\frac{1}{4}(1-{x}^{2})y(y-1)z(z-1)\\ {w}_{\text{10}}=\frac{1}{4}x(x+1)(1-{y}^{2})z(z-1)\\ {w}_{\text{11}}=\frac{1}{4}(1-{x}^{2})y(y+1)z(z-1)\\ {w}_{\text{12}}=\frac{1}{4}x(x-1)(1-{y}^{2})z(z-1)\\ {w}_{\text{13}}=\frac{1}{4}x(x-1)y(y-1)(1-{z}^{2})\\ {w}_{\text{14}}=\frac{1}{4}x(x+1)y(y-1)(1-{z}^{2})\end{array} Gauss quadrature formula with 2 points in each direction (order 3) (FPG8) .. csv-table:: "**Point**", ":math:`x` "," :math:`y` "," "," :math:`z` ", "**Weight**" "1"," :math:`-1/\sqrt{3}` "," :math:`-1/\sqrt{3}` "," "," :math:`-1/\sqrt{3}` ", "1." "2"," :math:`-1/\sqrt{3}` "," :math:`-1/\sqrt{3}` "," "," :math:`1/\sqrt{3}` ", "1." "3"," :math:`-1/\sqrt{3}` "," :math:`1/\sqrt{3}` "," "," :math:`-1/\sqrt{3}` ", "1." "4"," :math:`-1/\sqrt{3}` "," :math:`1/\sqrt{3}` "," "," :math:`1/\sqrt{3}` ", "1." "5"," :math:`1/\sqrt{3}` "," :math:`-1/\sqrt{3}` "," "," :math:`-1/\sqrt{3}` ", "1." "6"," :math:`1/\sqrt{3}` "," :math:`-1/\sqrt{3}` "," "," :math:`1/\sqrt{3}` ", "1." "7"," :math:`1/\sqrt{3}` "," :math:`1/\sqrt{3}` "," "," :math:`-1/\sqrt{3}` ", "1." "8"," :math:`1/\sqrt{3}` "," :math:`1/\sqrt{3}` "," "," :math:`1/\sqrt{3}` ", "1." **Gauss quadrature formula with 3 points in each direction (order 5): (FPG27)** .. csv-table:: "**Point**", ":math:`x` "," :math:`y` "," "," :math:`z` ", "**Weight**" "1"," :math:`-\alpha` "," :math:`-\alpha` "," "," :math:`-\alpha` "," :math:`{c}_{1}^{3}`" "2"," :math:`-\alpha` "," :math:`-\alpha` ", "0. "," :math:`{c}_{1}^{2}{c}_{2}`" "3"," :math:`-\alpha` "," :math:`-\alpha` "," "," :math:`\alpha` "," :math:`{c}_{1}^{3}`" "4"," :math:`-\alpha` ", "0. "," :math:`-\alpha` "," :math:`{c}_{1}^{2}{c}_{2}`" "5"," :math:`-\alpha` ", "0. ", "0. "," :math:`{c}_{1}{c}_{2}^{2}`" "6"," :math:`-\alpha` ", "0. "," :math:`\alpha` "," :math:`{c}_{1}^{2}{c}_{2}`" "7"," :math:`-\alpha` "," :math:`\alpha` "," "," :math:`-\alpha` "," :math:`{c}_{1}^{3}`" "8"," :math:`-\alpha` "," :math:`\alpha` ", "0. "," :math:`{c}_{1}^{2}{c}_{2}`" "9"," :math:`-\alpha` "," :math:`\alpha` "," "," :math:`\alpha` "," :math:`{c}_{1}^{3}`" "10", "0. "," :math:`-\alpha` "," :math:`-\alpha` "," :math:`{c}_{1}^{2}{c}_{2}`" "11", "0. "," :math:`-\alpha` ", "0. "," :math:`{c}_{1}{c}_{2}^{2}`" "12", "0. "," :math:`-\alpha` "," :math:`\alpha` "," :math:`{c}_{1}^{2}{c}_{2}`" "13", "0. ", "0. "," :math:`-\alpha` "," :math:`{c}_{1}{c}_{2}^{2}`" "14", "0. ", "0. ", "0. "," :math:`{c}_{2}^{3}`" "15", "0. ", "0. "," :math:`\alpha` "," :math:`{c}_{1}{c}_{2}^{2}`" "16", "0. "," :math:`\alpha` "," :math:`-\alpha` "," :math:`{c}_{1}^{2}{c}_{2}`" "17", "0. "," :math:`\alpha` ", "0. "," :math:`{c}_{1}{c}_{2}^{2}`" "18", "0. "," :math:`\alpha` "," :math:`\alpha` "," :math:`{c}_{1}^{2}{c}_{2}`" "19"," :math:`\alpha` "," :math:`-\alpha` "," "," :math:`-\alpha` "," :math:`{c}_{1}^{3}`" "20"," :math:`\alpha` "," :math:`-\alpha` ", "0. "," :math:`{c}_{1}^{2}{c}_{2}`" "21"," :math:`\alpha` "," :math:`-\alpha` "," "," :math:`\alpha` "," :math:`{c}_{1}^{3}`" "22"," :math:`\alpha` ", "0. "," :math:`-\alpha` "," :math:`{c}_{1}^{2}{c}_{2}`" "23"," :math:`\alpha` ", "0. ", "0. "," :math:`{c}_{1}{c}_{2}^{2}`" "24"," :math:`\alpha` ", "0. "," :math:`\alpha` "," :math:`{c}_{1}^{2}{c}_{2}`" "25"," :math:`\alpha` "," :math:`\alpha` "," "," :math:`-\alpha` "," :math:`{c}_{1}^{3}`" "26"," :math:`\alpha` "," :math:`\alpha` ", "0. "," :math:`{c}_{1}^{2}{c}_{2}`" "27"," :math:`\alpha` "," :math:`\alpha` "," "," :math:`\alpha` "," :math:`{c}_{1}^{3}`" with: .. csv-table:: ":math:`\alpha =\sqrt{\frac{3}{5}}` "," :math:`{c}_{1}=\frac{5}{9}` "," :math:`{c}_{2}=\frac{8}{9}`" Pyramids: ELREFE PY5, P13, P19 ---- .. image:: images/10000201000003E8000003215D64B9F61CD54C22.png :width: 373 :height: 298 .. _RefImage_10000201000003E8000003215D64B9F61CD54C22.png: The blue nodes are in the middle of the faces, the red one is in the middle of the cell. The square base is made up of the quadrangle :math:`{N}_{1}{N}_{2}{N}_{3}{N}_{4}` and :math:`{N}_{5}` is the top of the pyramid. .. csv-table:: "", ":math:`x` "," :math:`y` "," :math:`z`" ":math:`{N}_{1}` ", "1. ", "0. ", "0." ":math:`{N}_{2}` ", "0. ", "1. ", "0." ":math:`{N}_{3}` ", "—1. ", "0. ", "0." ":math:`{N}_{4}` ", "0. ", "—1. ", "0." ":math:`{N}_{5}` ", "0. ", "0. ", "1." ":math:`{N}_{6}` ", "0.5", "0.5", "0.5", "0." ":math:`{N}_{7}` ", "—0.5", "0.5", "0." ":math:`{N}_{8}` ", "—0.5", "—0.5", "0." ":math:`{N}_{9}` ", "0.5", "—0.5", "0." ":math:`{N}_{10}` ", "0.5", "0. ", "0.5" ":math:`{N}_{11}` ", "0. ", "0.5", "0.5" ":math:`{N}_{12}` ", "—0.5", "0. ", "0.5" ":math:`{N}_{13}` ", "0. ", "—0.5", "0.5" ":math:`{N}_{14}` ", "0. ", "0. ", "0" ":math:`{N}_{15}` ", "1/3", "1/3", "1/3" ":math:`{N}_{16}` ", "-1/3", "1/3", "1/3" ":math:`{N}_{17}` ", "-1/3", "-1/3", "1/3" ":math:`{N}_{18}` ", "1/3", "-1/3", "1/3" ":math:`{N}_{19}` ", "0", "0", "0", "0.2" **Shape functions:** 5-knot formula :math:`\begin{array}{}{w}_{1}=\frac{(-x+y+z-1)(-x-y+z-1)}{4(1-z)}\\ {w}_{2}=\frac{(-x-y+z-1)(x-y+z-1)}{4(1-z)}\\ {w}_{3}=\frac{(x+y+z-1)(x-y+z-1)}{4(1-z)}\\ {w}_{4}=\frac{(x+y+z-1)(-x+y+z-1)}{4(1-z)}\\ {w}_{5}=z\end{array}` 13-knot formula :math:`\begin{array}{}{w}_{1}=\frac{(-x+y+z-1)(-x-y+z-1)(x-0\text{.}5)}{2(1-z)}\\ {w}_{2}=\frac{(-x-y+z-1)(x-y+z-1)(y-0\text{.}5)}{2(1-z)}\\ {w}_{3}=\frac{(x-y+z-1)(x+y+z-1)(-x-0\text{.}5)}{2(1-z)}\\ {w}_{4}=\frac{(x+y+z-1)(-x+y+z-1)(-y-0\text{.}5)}{2(1-z)}\\ {w}_{5}=\mathrm{2z}(z-0\text{.}5)\\ {w}_{6}=-\frac{(-x+y+z-1)(-x-y+z-1)(x-y+z-1)}{2(1-z)}\\ {w}_{7}=-\frac{(-x-y+z-1)(x-y+z-1)(x+y+z-1)}{2(1-z)}\end{array}` :math:`\begin{array}{}{w}_{8}=-\frac{(x-y+z-1)(x+y+z-1)(-x+y+z-1)}{2(1-z)}\\ {w}_{9}=-\frac{(x+y+z-1)(-x+y+z-1)(-x-y+z-1)}{2(1-z)}\\ {w}_{\text{10}}=\frac{z(-x+y+z-1)(-x-y+z-1)}{1-z}\\ {w}_{\text{11}}=\frac{z(-x-y+z-1)(x-y+z-1)}{1-z}\\ {w}_{\text{12}}=\frac{z(x-y+z-1)(x+y+z-1)}{1-z}\\ {w}_{\text{13}}=\frac{z(x+y+z-1)(-x+y+z-1)}{1-z}\end{array}` Numerical integration formula with 5 points of order 2 (FPG5): .. csv-table:: "**Point**", ":math:`x` "," :math:`y` "," "," :math:`z` ", "**Weight**" "1"," :math:`\mathrm{0,5}` "," :math:`0` "," "," :math:`{h}_{1}` "," :math:`{p}_{1}`" "2"," :math:`0` "," :math:`\mathrm{0,5}` "," "," :math:`{h}_{1}` "," :math:`{p}_{1}`" "3"," :math:`–\mathrm{0,5}` "," :math:`0` "," "," :math:`{h}_{1}` "," :math:`{p}_{1}`" "4"," :math:`0` "," :math:`–\mathrm{0,5}` "," "," :math:`{h}_{1}` "," :math:`{p}_{1}`" "5"," :math:`0` "," :math:`0` "," "," :math:`{h}_{1}` "," :math:`{p}_{1}`" with: :math:`{h}_{1}=0.1531754163448146` :math:`{h}_{2}=0.6372983346207416` :math:`{p}_{1}=\frac{2}{15}` Numerical integration formula with 6 order 3 points (FPG6): .. csv-table:: "**Point**", ":math:`x` "," :math:`y` "," "," :math:`z` ", "**Weight**" "1"," :math:`0` "," :math:`0` "," "," :math:`{h}_{1}` "," :math:`{p}_{1}`" "2"," :math:`0` "," :math:`0` "," "," :math:`{h}_{2}` "," :math:`{p}_{2}`" "3"," :math:`–a` "," :math:`0` "," "," :math:`{h}_{3}` "," :math:`{p}_{3}`" "4"," :math:`0` "," :math:`–a` "," "," :math:`{h}_{3}` "," :math:`{p}_{3}`" "5"," :math:`0` "," :math:`a` "," "," :math:`{h}_{3}` "," :math:`{p}_{3}`" "6"," :math:`a` "," :math:`0` "," "," :math:`{h}_{3}` "," :math:`{p}_{3}`" With: :math:`a=0.5610836110587396` :math:`{p}_{1}=0.1681372559485071` :math:`{p}_{2}=0.07500000404404333` :math:`{p}_{3}=0.1058823516685291` :math:`{h}_{1}=0.1681372559485071` :math:`{h}_{2}=0.00000000567585` :math:`{h}_{3}=0.1058823516685291` 10-point numerical integration formula (FPG10) of order 4, see [:ref:`2 `]: .. csv-table:: "**Point**", ":math:`x` "," :math:`y` "," "," :math:`z` ", "**Weight**" "1"," :math:`0` "," :math:`0` "," "," :math:`{h}_{1}` "," :math:`{w}_{1}`" "2"," :math:`0` "," :math:`0` "," "," :math:`{h}_{2}` "," :math:`{w}_{2}`" "3"," :math:`-a` "," :math:`-a` "," "," :math:`{h}_{3}` "," :math:`{w}_{3}`" "4"," :math:`-a` "," :math:`a` "," "," :math:`{h}_{3}` "," :math:`{w}_{3}`" "5"," :math:`a` "," :math:`a` "," "," :math:`{h}_{3}` "," :math:`{w}_{3}`" "6"," :math:`a` "," :math:`-a` "," "," :math:`{h}_{3}` "," :math:`{w}_{3}`" "7"," :math:`-b` "," :math:`0` "," "," :math:`{h}_{4}` "," :math:`{w}_{4}`" "8"," :math:`0` "," :math:`-b` "," "," :math:`{h}_{4}` "," :math:`{w}_{4}`" "9"," :math:`0` "," :math:`b` "," "," :math:`{h}_{4}` "," :math:`{w}_{4}`" "10"," :math:`b` "," :math:`0` "," "," :math:`{h}_{4}` "," :math:`{w}_{4}`" With: :math:`a=0.3252907781991163` :math:`b=0.65796699712169` :math:`{h}_{1}=0.6772327888861374` :math:`{h}_{2}=0.1251369531087465` :math:`{h}_{3}=0.3223841495782137` :math:`{h}_{4}=0.0392482838988154` :math:`{w}_{1}=0.07582792211376127` :math:`{w}_{2}=0.1379222683930349` :math:`{w}_{3}=0.07088305859288367` :math:`{w}_{4}=0.04234606044708394` Numerical integration formula with 15 Gauss points (FPG15) of order 5: .. csv-table:: "**Point**", ":math:`x` "," :math:`y` "," "," :math:`z` ", "**Weight**" "**1**", "0.0", "0.0", "0.0", "0.7298578807825067", "0.0456235799393942674" "**2**", "0.0", "0.0", "0.0", "0.300401020813769", "0.112931409661816" "**3**", "0.0", "0.0", "0.0", "0.0000000064917722", "0.03913635721904967" "**4**", "-0.3532630157731623", "-0.3532630157731623", "0.125", "0.050960862086209874681" "**5**", "-0.3532630157731623", "0.3532630157731623", "0.125", "0.050960862086209874681" "**6**", "0.3532630157731623", "0.3532630157731623", "0.125", "0.05096086209874681" "**7**", "0.3532630157731623", "-0.3532630157731623", "0.125", "0.050960862086209874681" "**8**", "-0.705117122727788277", "0.531059878383280", "0.061111907062023", "0.0264472678827788277", "0.02644726771976367" "**9**", "0.0", "-0.705117121227788277", "0.061111907062023", "0.02644726771976367" "**10**", "0.0", "0.705117121227788277", "0.061111907062023", "0.02644726771976367" "**11**", "0.705117122727788277", "0.0", "0.061111907062023", "0.02644726771976367" "**12**", "-0.432882864103541", "0.0", "0.4236013371197248", "0.011774414962347" "**13**", "0.0", "-0.432882864103541", "0.4236013371197248", "0.011774414962347" "**14**", "0.0", "0.432882864103541", "0.4236013371197248", "0.041951149272741" "**15**", "0.432882864103541", "0.0", "0.4236013371197248", "0.041951149272741" Numerical integration formula with 24 Gauss points (FPG24) of order 6: .. csv-table:: "**Point**", ":math:`x` "," :math:`y` "," "," :math:`z` ", "**Weight**" "**1**", "0.0", "0.0", "0.0", "0.8076457976939595", "0.01697526244176133" "**2**", "0.0", "0.0", "0.0", "0.0017638088528196", "0.0107023421167942" "**3**", "0.0", "0.0", "0.0", "0.1382628064637306", "0.0797197029683492" "**4**", "0.0", "0.0", "0.0", "0.4214239119356371", "0.0687071134661012" "**5**", "-0.4172976755573542", "-0.4172976755573542", "0.097447341025462", "0.024633755573542", "0.02463375557353542", "0.024633755573542" "**6**", "-0.4172976755573542", "0.4172976755573542", "0.097447341025462", "0.024633755573542", "0.02463375557353542", "0.4172976755573542", "0.097447341025462", "0.024633755573542" "**7**", "0.4172976755573542", "0.4172976755573542", "0.097447341025462", "0.024633755573542", "0.02463375557353542", "0.4172976755573542", "0.097447341025462", "0.024633755573542" "**8**", "0.4172976755573542", "-0.4172976755573542", "0.097447341025462", "0.024633755573542", "0.02463375557353542", "0.024633725573542" "**9**", "-0.2169627046883496", "-0.2169627046883496", "0.5660745906233009", "0.02105846883496", "0.02105838883496", "0.0210583863632544886" "**10**", "-0.2169627046883496", "0.2169627046883496", "0.5660745906233009", "0.0210583846883496", "0.02105838883496", "0.0210583863632544886" "**11**", "0.2169627046883496", "0.2169627046883496", "0.5660745906233009", "0.02105838883496", "0.0210583868632544886" "**12**", "0.2169627046883496", "-0.2169627046883496", "0.5660745906233009", "0.02105838883496", "0.0210583868632544886" "**13**", "-0.565680854444256755", "0.0", "0.0294777308457207", "0.0248000862596322" "**14**", "0.0", "-0.565680858544256755", "0.0294777308457207", "0.0248000862596322" "**15**", "0.0", "0.565680858544256755", "0.0294777308457207", "0.0248000862596322" "**16**", "0.565680854444256755", "0.0", "0.0294777308457207", "0.0248000862596322" "**17**", "-0.498079091780705", "0.0", "0.2649158632121295", "0.049254923117951295125", "0.04925492311795127" "**18**", "0.0", "-0.498079079091780705", "0.2649158632121295", "0.04925492311795127" "**19**", "0.0", "0.498079099091780705", "0.2649158632121295", "0.04925492311795127" "**20**", "0.498079091780705", "0.0", "0.2649158632121295", "0.04925492311795125", "0.04925492311795127" "**21**", "-0.9508994872144825", "0.0", "0.0", "0.048249070631936", "0.0028934404244966" "**22**", "0.0", "-0.950899484872144825", "0.048249070631936", "0.0028934404244966" "**23**", "0.0", "0.950899484872144825", "0.048249070631936", "0.0028934404244966" "**24**", "0.9508994872144825", "0.0", "0.048249070631936", "0.0028934404244966"