Surface elements ==== Triangles: ELREFE TR3, TR6, TR7 ---- .. image:: images/Object_5.svg :width: 229 :height: 241 .. _RefImage_Object_5.svg: Node coordinates: .. csv-table:: "", ":math:`\xi` "," :math:`\eta`" "N1", "0.0", "0.0" "N2", "1.0", "0.0" "N3", "0.0", "1.0" "N4", "0.5", "0.0" "N5", "0.5", "0.5" "N6", "0.0", "0.5" "N7", "1/3", "1/3" +----------------------------+---------+-----------+------------+-----------+ |**Family** |**Point**|:math:`\xi`|:math:`\eta`|**Weight** | +----------------------------+---------+-----------+------------+-----------+ | | | | | | +----------------------------+---------+-----------+------------+-----------+ |FPG1 |1111 |1/3 |1/3 |1/2 | +----------------------------+---------+-----------+------------+-----------+ | | | | | | +----------------------------+---------+-----------+------------+-----------+ |FPG3 |1111 |1/6 |1/6 |1/6 | +----------------------------+---------+-----------+------------+-----------+ | |2222 |2/3 |1/6 |1/6 | +----------------------------+---------+-----------+------------+-----------+ | |3333 |1/6 |2/3 |1/6 | +----------------------------+---------+-----------+------------+-----------+ | | | | | | +----------------------------+---------+-----------+------------+-----------+ | .. code:: |1111 |1/5 |1/5 |25/ (24*4) | + + + + + + | FPG4 | | | | | +----------------------------+---------+-----------+------------+-----------+ | |2222 |3/5 |1/5 |25/ (24*4) | +----------------------------+---------+-----------+------------+-----------+ | |3333 |1/5 |3/5 |25/ (24*4) | +----------------------------+---------+-----------+------------+-----------+ | |4444 |1/3 |1/3 |-27/ (24*4)| +----------------------------+---------+-----------+------------+-----------+ | | | | | | +----------------------------+---------+-----------+------------+-----------+ |FPG6 |1111 |bbbb |bbbb |P2 | +----------------------------+---------+-----------+------------+-----------+ | |2222 |1 — 2b |bbbb |P2 | +----------------------------+---------+-----------+------------+-----------+ | |3333 |bbbb |1 — 2b |P2 | +----------------------------+---------+-----------+------------+-----------+ | |4444 |aaaa |1 — 2a |P1 | +----------------------------+---------+-----------+------------+-----------+ | |5555 |aaaa |aaaa |P1 | +----------------------------+---------+-----------+------------+-----------+ | |6666 |1 — 2a |aaaa |P1 | +----------------------------+---------+-----------+------------+-----------+ | | | | | | +----------------------------+---------+-----------+------------+-----------+ | .. code:: |1111 |1/2 |1/2 |1/6 | + + + + + + | COT3 | | | | | +----------------------------+---------+-----------+------------+-----------+ | |2222 |0000 |1/2 |1/6 | +----------------------------+---------+-----------+------------+-----------+ | |3333 |1/2 |0000 |1/6 | +----------------------------+---------+-----------+------------+-----------+ | | | | | | +----------------------------+---------+-----------+------------+-----------+ .. csv-table:: "With", "P1 = 0.11169079483905,", "P2 = 0.0549758718227661," "", "A = 0.445948490915965,", "b = 0.091576213509771" .. csv-table:: "**Family**", "**Point**", ":math:`\xi` "," "," :math:`\eta` ", "**Weight**" "", "", "", "", "" "FPG7 ", "1", "1/3", "1/3", "9/80" "", "2", "A", "A", "P1" "", "3", "1-2A", "A", "P1" "", "4", "A", "1-2A", "P1" "", "5", "B", "B", "P2" "", "6", "1-2B", "B", "P2" "", "7", "B", "1-2B", "P2" .. csv-table:: "With", "A = 0.470142064105115", "" "", "B = 0.101286507323456", "" "", "P1=0.066197076394253", "" "", "P2=0.062969590272413", "" .. csv-table:: "**Family**", "**Point**", ":math:`\xi` "," "," :math:`\eta` ", "**Weight**" "", "", "", "", "" "FPG12 ", "1", "A", "A", "P1" "", "2", "1-2A", "A", "P1" "", "3", "A", "1-2A", "P1" "", "4", "B", "B", "P2" "", "5", "1-2B", "B", "P2" "", "6", "B", "1-2B", "P2" "", "7", "C", "D", "P3" "", "8", "D", "C", "P3" "", "9", "1-C-D", "C", "P3" "", "10", "1-C-D", "D", "P3" "", "11", "C", "1-C-D", "P3" "", "12", "D", "1-C-D", "P3" .. csv-table:: "With", "A = 0.063089014491502", "" "", "B = 0.249286745170910", "" "", "C = 0.310352451033785", "" "", "D = 0.053145049844816", "" "", "P1=0.025422453185103", "" "", "P2" = 0.058393137863189", "" "", "P3 = 0.041425537809187", "" *TR3*: triangle with 3 knots .. csv-table:: "number of nodes", ": 3" "number of vertex nodes", ": 3" shape functions and first derivatives of the triangle with 3 nodes: .. csv-table:: ":math:`\left\{N\right\}` "," :math:`\left\{\partial N/\partial \xi \right\}` "," :math:`\left\{\partial N/\partial \eta \right\}`" ":math:`1-\xi -\eta` "," :math:`-1` "," :math:`-1`" ":math:`\xi` "," :math:`1` "," :math:`0`" ":math:`\eta` "," :math:`0` "," :math:`1`" *TR6*: triangle with 6 knots .. csv-table:: "number of nodes", ": 6" "number of vertex nodes", ": 3" shape functions, first derivatives of the 6-node triangle: .. csv-table:: ":math:`\left\{N\right\}` "," :math:`\left\{\mathrm{\partial }N\mathrm{/}\mathrm{\partial }\xi \right\}` "," :math:`\left\{\partial N/\partial \eta \right\}`" ":math:`\mathrm{-}(1\mathrm{-}\xi \mathrm{-}\eta )(1\mathrm{-}2(1\mathrm{-}\xi \mathrm{-}\eta ))` "," :math:`1\mathrm{-}4(1\mathrm{-}\xi \mathrm{-}\eta )` "," :math:`1-4(1-\xi -\eta )`" ":math:`-\xi (1-2\xi )` "," :math:`-1+4\xi` "," :math:`0`" ":math:`-\eta (1-2\eta )` "," :math:`0` "," :math:`-1+4\eta`" ":math:`4\xi (1-\xi -\eta )` "," :math:`4(1\mathrm{-}2\xi \mathrm{-}\eta )` "," :math:`-4\xi`" ":math:`4\xi \eta` "," :math:`\mathrm{4\eta }` "," :math:`4\xi`" ":math:`4\eta (1\mathrm{-}\xi \mathrm{-}\eta )` "," :math:`\mathrm{-}4\eta` "," :math:`4(1\mathrm{-}\xi \mathrm{-}2\eta )`" second derivatives of the 6-node triangle: .. csv-table:: ":math:`\left\{{\mathrm{\partial }}^{2}N\mathrm{/}\mathrm{\partial }{\xi }^{2}\right\}` "," :math:`\left\{{\partial }^{2}N/\partial \xi \partial \eta \right\}` "," :math:`\left\{{\mathrm{\partial }}^{2}N\mathrm{/}\mathrm{\partial }{\eta }^{2}\right\}`" "4", "4", "4" "4", "0", "0" "0", "0", "4" "-8", "-4", "0" "0", "4", "0" "0", "-4", "-8" *TR7*: triangle with 7 knots .. csv-table:: "number of nodes", ": 7" "number of vertex nodes", ": 3" 7-node triangle shape functions: .. csv-table:: ":math:`\left\{N\right\}`" ":math:`1\mathrm{-}3(\xi +\eta )+2({\xi }^{2}+{\eta }^{2})+7\xi \eta \mathrm{-}3\xi \eta (\xi +\eta )`" ":math:`\xi (\mathrm{-}1+2\xi +3\eta \mathrm{-}3\eta (\xi +\eta ))`" ":math:`\eta (\mathrm{-}1+2\xi +3\eta \mathrm{-}3\xi (\xi +\eta ))`" ":math:`4\xi (1\mathrm{-}\xi \mathrm{-}4\eta +3\eta (\xi +\eta ))`" ":math:`4\xi \eta (\mathrm{-}2+3(\xi +\eta ))`" ":math:`4\eta (1\mathrm{-}4\xi \mathrm{-}\eta +3\xi (\xi +\eta ))`" ":math:`27\xi \eta (1\mathrm{-}\xi \mathrm{-}\eta )`" first derivatives of the 7-node triangle: .. csv-table:: ":math:`\left\{\mathrm{\partial }N\mathrm{/}\mathrm{\partial }\xi \right\}` "," :math:`\left\{\mathrm{\partial }N\mathrm{/}\mathrm{\partial }\eta \right\}`" ":math:`\mathrm{-}3+4\xi +7\eta \mathrm{-}6\xi \eta \mathrm{-}3{\eta }^{2}` "," :math:`\mathrm{-}3+7\xi +4\eta \mathrm{-}6\xi \eta \mathrm{-}3{\xi }^{2}`" ":math:`\mathrm{-}1+4\xi +3\eta \mathrm{-}6\xi \eta \mathrm{-}3{\eta }^{2}` "," :math:`3\xi (1\mathrm{-}\xi \mathrm{-}2\eta )`" ":math:`3\xi (1\mathrm{-}2\eta \mathrm{-}\xi )` "," :math:`\mathrm{-}1+3\xi +4\eta \mathrm{-}6\xi \eta \mathrm{-}3{\xi }^{2}`" ":math:`4(1\mathrm{-}2\xi \mathrm{-}4\eta +6\xi \eta +3{\eta }^{2})` "," :math:`4\xi (\mathrm{-}4+3\xi +6\eta )`" ":math:`4\eta (\mathrm{-}2+6\xi +3\eta )` "," :math:`4\xi (\mathrm{-}2+3\xi +6\eta )`" ":math:`4\eta (\mathrm{-}4+6\xi +3\eta )` "," :math:`4(\mathrm{-}1\mathrm{-}4\xi \mathrm{-}2\eta +6\xi \eta +3{\xi }^{2})`" ":math:`27\eta (1\mathrm{-}2\xi \mathrm{-}\eta )` "," :math:`27\xi (1\mathrm{-}\xi \mathrm{-}2\eta )`" second derivatives of the 7-node triangle: .. csv-table:: ":math:`\left\{{\mathrm{\partial }}^{2}N\mathrm{/}\mathrm{\partial }{\xi }^{2}\right\}` "," :math:`\left\{{\mathrm{\partial }}^{2}N\mathrm{/}\mathrm{\partial }\xi \mathrm{\partial }\eta \right\}` "," :math:`\left\{{\mathrm{\partial }}^{2}N\mathrm{/}\mathrm{\partial }{\eta }^{2}\right\}`" ":math:`4\mathrm{-}6\eta` "," :math:`7\mathrm{-}6\xi \mathrm{-}6\eta` "," :math:`4\mathrm{-}6\xi`" ":math:`4\mathrm{-}6\eta` "," :math:`3\mathrm{-}6\xi \mathrm{-}6\eta` "," :math:`\mathrm{-}6\xi`" ":math:`\mathrm{-}6\eta` "," :math:`3\mathrm{-}6\xi \mathrm{-}6\eta` "," :math:`4\mathrm{-}6\xi`" ":math:`4(\mathrm{-}2+6\eta )` "," :math:`4(\mathrm{-}4+6\xi +6\eta )` "," :math:`24\xi`" ":math:`\text{24}\eta` "," :math:`4(\mathrm{-}2+6\xi +6\eta )` "," :math:`24\xi`" ":math:`24\eta` "," :math:`4(\mathrm{-}4+6\xi +6\eta )` "," :math:`4(\mathrm{-}2+6\xi )`" ":math:`\mathrm{-}54\eta` "," :math:`27(1\mathrm{-}2\xi \mathrm{-}2\eta )` "," :math:`\mathrm{-}54\xi`" Quadrangles: ELREFE QU4, QU8, QU9 ---- .. image:: images/Object_98.svg :width: 229 :height: 241 .. _RefImage_Object_98.svg: Node coordinates: .. csv-table:: "", ":math:`\xi` "," :math:`\eta`" ":math:`\mathrm{N1}` ", "-1.0", "-1.0" ":math:`\mathrm{N2}` ", "1.0", "-1.0" ":math:`\mathrm{N3}` ", "1.0", "1.0" ":math:`\mathrm{N4}` ", "-1.0", "1.0" ":math:`\mathrm{N5}` ", "0.0", "-1.0" ":math:`\mathrm{N6}` ", "1.0", "0.0" ":math:`\mathrm{N7}` ", "0.0", "1.0" ":math:`\mathrm{N8}` ", "-1.0", "0.0" ":math:`\mathrm{N9}` ", "0.0", "0.0" .. csv-table:: "**Family**", "**Point**", ":math:`\xi` "," "," :math:`\eta` ", "**Weight**" "FPG1 ", "1", "0", "0", "4" "FPG4 ", "1"," :math:`\mathrm{-}a` "," "," :math:`\mathrm{-}a` ", "1.0" "", "2"," :math:`a` "," :math:`\mathrm{-}a` ", "1.0" "", "3"," :math:`a` "," :math:`a` ", "1.0" "", "4"," :math:`\mathrm{-}a` "," :math:`a` ", "1.0" "", "", ":math:`a\mathrm{=}1\mathrm{/}\sqrt{3}` ", ", "" "FPG9 ", "1"," :math:`\mathrm{-}a` "," "," :math:`\mathrm{-}a` ", "25/81" "", "2"," :math:`a` "," :math:`\mathrm{-}a` ", "25/81" "", "3"," :math:`a` "," :math:`a` ", "25/81" "", "4"," :math:`\mathrm{-}a` "," :math:`a` ", "25/81" "", "5", "0.0"," :math:`\mathrm{-}a` ", "40/81" "", "6"," :math:`a` ", "0.0", "40/81" "", "7", "0.0", "a", "40/81" "", "8"," :math:`\mathrm{-}a` ", "0.0", "40/81" "", "9", "0.0", "0.0", "0.0", "64/81" "", "", ":math:`a\mathrm{=}0.774596669241483` ", ", "" *QU4*: quadrangle with 4 knots .. csv-table:: "number of knots", ": 4" "number of vertex nodes", ": 4" shape functions, first and second derivatives of the 4-node quadrangle: .. csv-table:: ":math:`\left\{N\right\}` "," :math:`\left\{\mathrm{\partial }N\mathrm{/}\mathrm{\partial }\xi \right\}` "," :math:`\left\{\mathrm{\partial }N\mathrm{/}\mathrm{\partial }\eta \right\}`" ":math:`(1\mathrm{-}\xi )(1\mathrm{-}\eta )\mathrm{/}4` "," :math:`\mathrm{-}(1\mathrm{-}\eta )\mathrm{/}4` "," :math:`\mathrm{-}(1\mathrm{-}\xi )\mathrm{/}4`" ":math:`(1+\xi )(1\mathrm{-}\eta )\mathrm{/}4` "," :math:`(1\mathrm{-}\eta )\mathrm{/}4` "," :math:`\mathrm{-}(1+\xi )\mathrm{/}4`" ":math:`(1+\xi )(1+\eta )\mathrm{/}4` "," :math:`(1+\eta )\mathrm{/}4` "," :math:`(1+\xi )\mathrm{/}4`" ":math:`(1\mathrm{-}\xi )(1+\eta )\mathrm{/}4` "," :math:`\mathrm{-}(1+\eta )\mathrm{/}4` "," :math:`(1-\xi )/4`" .. csv-table:: ":math:`\left\{{\partial }^{2}N/\partial {\xi }^{2}\right\}` "," :math:`\left\{{\mathrm{\partial }}^{2}N\mathrm{/}\mathrm{\partial }\xi \mathrm{\partial }\eta \right\}` "," :math:`\left\{{\mathrm{\partial }}^{2}N\mathrm{/}\mathrm{\partial }{\eta }^{2}\right\}`" "0", "1/4", "0" "0", "-1/4", "0" "0", "1/4", "0" "0", "-1/4", "0" *QU8*: quadrangle with 8 knots .. csv-table:: "number of knots", ": 8" "number of vertex nodes", ": 4" shape functions and first derivatives of the 8-node quadrangle: .. csv-table:: ":math:`\left\{N\right\}` "," :math:`\left\{\mathrm{\partial }N\mathrm{/}\mathrm{\partial }\xi \right\}` "," :math:`\left\{\mathrm{\partial }N\mathrm{/}\mathrm{\partial }\eta \right\}`" ":math:`(1\mathrm{-}\xi )(1\mathrm{-}\eta )(\mathrm{-}1\mathrm{-}\xi \mathrm{-}\eta )\mathrm{/}4` "," :math:`(1\mathrm{-}\eta )(2\xi +\eta )\mathrm{/}4` "," :math:`(1\mathrm{-}\xi )(\xi +2\eta )\mathrm{/}4`" ":math:`(1+\xi )(1\mathrm{-}\eta )(\mathrm{-}1+\xi \mathrm{-}\eta )\mathrm{/}4` "," :math:`(1\mathrm{-}\eta )(2\xi \mathrm{-}\eta )\mathrm{/}4` "," :math:`\mathrm{-}(1+\xi )(\xi \mathrm{-}2\eta )\mathrm{/}4`" ":math:`(1+\xi )(1+\eta )(\mathrm{-}1+\xi +\eta )\mathrm{/}4` "," :math:`(1+\eta )(2\xi +\eta )\mathrm{/}4` "," :math:`(1+\xi )(\xi +2\eta )\mathrm{/}4`" ":math:`(1\mathrm{-}\xi )(1+\eta )(\mathrm{-}1\mathrm{-}\xi +\eta )\mathrm{/}4` "," :math:`\mathrm{-}(1+\eta )(\mathrm{-}2\xi +\eta )\mathrm{/}4` "," :math:`(1\mathrm{-}\xi )(\mathrm{-}\xi +2\eta )\mathrm{/}4`" ":math:`(1\mathrm{-}{\xi }^{2})(1\mathrm{-}\eta )\mathrm{/}2` "," :math:`\mathrm{-}\xi (1\mathrm{-}\eta )` "," :math:`\mathrm{-}(1\mathrm{-}{\xi }^{2})\mathrm{/}2`" ":math:`(1+\xi )(1\mathrm{-}{\eta }^{2})\mathrm{/}2` "," :math:`(1\mathrm{-}{\eta }^{2})\mathrm{/}2` "," :math:`\mathrm{-}\eta (1+\xi )`" ":math:`(1\mathrm{-}{\xi }^{2})(1+\eta )\mathrm{/}2` "," :math:`\mathrm{-}\xi (1+\eta )` "," :math:`(1\mathrm{-}{\xi }^{2})\mathrm{/}2`" ":math:`(1\mathrm{-}\xi )(1\mathrm{-}{\eta }^{2})\mathrm{/}2` "," :math:`\mathrm{-}(1\mathrm{-}{\eta }^{2})\mathrm{/}2` "," :math:`\mathrm{-}\eta (1\mathrm{-}\xi )`" second derivatives of the 8-node quadrangle: .. csv-table:: ":math:`\left\{{\mathrm{\partial }}^{2}N\mathrm{/}\mathrm{\partial }{\xi }^{2}\right\}` "," :math:`\left\{{\mathrm{\partial }}^{2}N\mathrm{/}\mathrm{\partial }\xi \mathrm{\partial }\eta \right\}` "," :math:`\left\{{\mathrm{\partial }}^{2}N\mathrm{/}\mathrm{\partial }{\eta }^{2}\right\}`" ":math:`(1-\eta )/2` "," :math:`(1-2\xi -2\eta )/4` "," :math:`(1\mathrm{-}\xi )\mathrm{/}2`" ":math:`(1-\eta )/2` "," :math:`-(1+2\xi -2\eta )/4` "," :math:`(1+\xi )\mathrm{/}2`" ":math:`(1+\eta )/2` "," :math:`(1+2\xi +2\eta )/4` "," :math:`(1+\xi )\mathrm{/}2`" ":math:`(1+\eta )/2` "," :math:`-(1-2\xi +2\eta )/4` "," :math:`(1\mathrm{-}\xi )\mathrm{/}2`" ":math:`-1+\eta` "," :math:`\xi` "," :math:`0`" ":math:`0` "," :math:`-\eta` "," :math:`\mathrm{-}1\mathrm{-}\xi`" ":math:`-1-\eta` "," :math:`-\xi` "," :math:`0`" ":math:`0` "," :math:`\eta` "," :math:`\mathrm{-}1+\xi`" *QU9*: quadrangle with 9 knots .. csv-table:: "number of nodes", ": 9" "number of vertex nodes", ": 4" shape functions and first derivatives of the 9-node quadrangle: .. csv-table:: ":math:`\left\{N\right\}` "," :math:`\left\{\partial N/\partial \xi \right\}` "," :math:`\left\{\mathrm{\partial }N\mathrm{/}\mathrm{\partial }\eta \right\}`" ":math:`\xi \eta (\xi \mathrm{-}1)(\eta \mathrm{-}1)\mathrm{/}4` "," :math:`(2\xi \mathrm{-}1)\eta (\eta \mathrm{-}1)\mathrm{/}4` "," :math:`\xi (\xi \mathrm{-}1)(2\eta \mathrm{-}1)\mathrm{/}4`" ":math:`\xi \eta (\xi +1)(\eta \mathrm{-}1)\mathrm{/}4` "," :math:`(2\xi +1)\eta (\eta \mathrm{-}1)\mathrm{/}4` "," :math:`\xi (\xi +1)(2\eta \mathrm{-}1)\mathrm{/}4`" ":math:`\xi \eta (\xi +1)(\eta +1)\mathrm{/}4` "," :math:`(2\xi +1)\eta (\eta +1)\mathrm{/}4` "," :math:`\xi (\xi +1)(2\eta +1)\mathrm{/}4`" ":math:`\xi \eta (\xi \mathrm{-}1)(\eta +1)\mathrm{/}4` "," :math:`(2\xi \mathrm{-}1)\eta (\eta +1)\mathrm{/}4` "," :math:`\xi (\xi \mathrm{-}1)(2\eta +1)\mathrm{/}4`" ":math:`(1\mathrm{-}{\xi }^{2})\eta (\eta \mathrm{-}1)\mathrm{/}2` "," :math:`\mathrm{-}\xi \eta (\eta \mathrm{-}1)` "," :math:`(1\mathrm{-}{\xi }^{2})(2\eta \mathrm{-}1)\mathrm{/}2`" ":math:`\xi (\xi +1)(1\mathrm{-}{\eta }^{2})\mathrm{/}2` "," :math:`(2\xi +1)(1\mathrm{-}{\eta }^{2})\mathrm{/}2` "," :math:`\mathrm{-}\xi \eta (\xi +1)`" ":math:`(1\mathrm{-}{\xi }^{2})\eta (\eta +1)\mathrm{/}2` "," :math:`\mathrm{-}\xi \eta (\eta +1)` "," :math:`(1\mathrm{-}{\xi }^{2})(2\eta +1)\mathrm{/}2`" ":math:`\xi (\xi \mathrm{-}1)(1\mathrm{-}{\eta }^{2})\mathrm{/}2` "," :math:`(2\xi \mathrm{-}1)(1\mathrm{-}{\eta }^{2})\mathrm{/}2` "," :math:`\mathrm{-}\xi \eta (\xi \mathrm{-}1)`" ":math:`(1\mathrm{-}{\xi }^{2})(1\mathrm{-}{\eta }^{2})` "," :math:`\mathrm{-}2\xi (1\mathrm{-}{\eta }^{2})` "," :math:`\mathrm{-}2\eta (1\mathrm{-}{\xi }^{2})`" second derivatives of the 9-node quadrangle: .. csv-table:: ":math:`\left\{{\mathrm{\partial }}^{2}N\mathrm{/}\mathrm{\partial }{\xi }^{2}\right\}` "," :math:`\left\{{\mathrm{\partial }}^{2}N\mathrm{/}\mathrm{\partial }\xi \mathrm{\partial }\eta \right\}` "," :math:`\left\{{\mathrm{\partial }}^{2}N\mathrm{/}\mathrm{\partial }{\eta }^{2}\right\}`" ":math:`\eta (\eta \mathrm{-}1)\mathrm{/}2` "," :math:`(\xi \mathrm{-}1\mathrm{/}2)(\eta \mathrm{-}1\mathrm{/}2)` "," :math:`\xi (\xi \mathrm{-}1)\mathrm{/}2`" ":math:`\eta (\eta \mathrm{-}1)\mathrm{/}2` "," :math:`(\xi +1\mathrm{/}2)(\eta \mathrm{-}1\mathrm{/}2)` "," :math:`\xi (\xi +1)\mathrm{/}2`" ":math:`\eta (\eta +1)\mathrm{/}2` "," :math:`(\xi +1\mathrm{/}2)(\eta +1\mathrm{/}2)` "," :math:`\xi (\xi +1)\mathrm{/}2`" ":math:`\eta (\eta +1)\mathrm{/}2` "," :math:`(\xi \mathrm{-}1\mathrm{/}2)(\eta +1\mathrm{/}2)` "," :math:`\xi (\xi \mathrm{-}1)\mathrm{/}2`" ":math:`\mathrm{-}\eta (\eta \mathrm{-}1)` "," :math:`\mathrm{-}\xi (2\eta \mathrm{-}1)` "," :math:`1\mathrm{-}{\xi }^{2}`" ":math:`1\mathrm{-}{\eta }^{2}` "," :math:`\mathrm{-}\eta (2\xi +1)` "," :math:`\mathrm{-}\xi (\xi +1)`" ":math:`\mathrm{-}\eta (\eta +1)` "," :math:`\mathrm{-}\xi (\mathrm{2\eta }+1)` "," :math:`1\mathrm{-}{\xi }^{2}`" ":math:`1\mathrm{-}{\eta }^{2}` "," :math:`\mathrm{-}\eta (2\xi \mathrm{-}1)` "," :math:`\mathrm{-}\xi (\xi \mathrm{-}1)`" ":math:`\mathrm{-}2(1\mathrm{-}{\eta }^{2})` "," :math:`4\xi \eta` "," :math:`\mathrm{-}2(1\mathrm{-}{\xi }^{2})`"