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h de2437.D,)24]eYO24](@ A *-'I��&C.DH KM&(,)K-@ A K-*M3S.+*-,)'L/1'}&(37.D*M,+3Q/1]![^2O&(37.D' w �24,+2O3B24&A *-'+HV*-'0y?/O3BE�&(37.D*M,)N¤/OKM*tËSÊÍÌI2ON�ÉBÊn/O3BE ÉSÌr�u��HB*t,+&Cc4H7.?[B&(KU.D@B,)*t')HB2r��'.DHV*?')*-c A *-37.+'�ÑBÊ9&(3 5 24]CE;� ³I³?³?³I³?³I³?³�³I³?³?³I³?³I³?³?³I³?³�³I³?³I³?³?³ h ¶

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A *-'+H$](*UY4*M]('�N_24,!.DHV*�.+HB,+*M*�[B,+2 5 ](* A '�&(3$�{TV/ A [B](* � �ij?³?³I³?³�³I³?³I³?³?³ �4� »� º ��*~/OKU.D&(2O38��EB&Z�q@V'+&(2O3L[B,)2 5 ]C* A ��&C.+HLÇ�È j ³ j � ��.+HB*�*M,+,+2O, w ]C*MNÁ.0y�/13BE�.DHB*

A *-'+H w ,)&(c4H7.0ye2 5 .0/1&(3B*MEiN_2O,�](*MYO*-] ¶ ��&C.+H h · h µ�c4,)&(Ei[^2O&(37.D'M�I³?³�³I³?³I³?³?³ � h1j�l¶ ��*~/OKU.D&(2O38��EB&Z�q@V'+&(2O3�[B,+2 5 ](* A ��&Z.DH�Ç�È j ³ j4j � �n.+HB*F*-,),+24, w ](*UNÁ.oyR/O3VE

.DHV* A *M'+H w ,+&(cOHS.0y�2 5 .0/O&C3B*-ELN_24,e]C*MY4*M] ¶ ��&Z.DH h � » ¶ cO,+&(EL[^2O&(37.D' 5S6 .DHB*,+*M'+&CEB@Q/O];. 6 [�*?*M,+,)24,�*-').+& A /1.D2O, w j ³¥�}.D24]C*-,+/O3BK-*~yU�n³I³?³I³?³?³I³?³�³I³?³I³?³?³ � h �

� ¹ ��*~/OKU.D&(2O38��EB&Z�q@V'+&(2O3�[B,)2 5 ]C* A ��&C.+H�Ç�È j ³ j4j � � A *-'+HV*-'�2 5 .D/O&(3V*-E 576.DHV*�*-,+,)24,�*-'#.D& A /1.+24, 5 /O'+*ME�2O3�]C28K-/O]a��&(,+&CK0HB](*U.�[B,)2 5 ]C* A ' w ]C*MNÁ.oy���&C.+HhOj � �Ic4,)&(Eu[�24&C3S.+'!/O3VE 576 .+HB*�:q&C*-3B|S&(*U��&(KMvU�#:qHS@t. 6 [�*�*M,+,+2O,n*M').D& A /�.D24,w ,)&(c4H7.0yn��&C.DH hOj � � c4,+&CEF[^24&C37.D'�/1.>](*UY4*M] ¶ ��³I³?³?³I³?³I³?³?³I³?³�³I³?³I³?³?³ � hOh

�l· de2437Y4*MKM.+&(2438�EB&C�^@B')&(243F[B,)2 5 ]C* A N_,+2 A �{TV/ A [V](*��R��&Z.DHuÇ�È � j Û�Ú �V.DHB*A *-'+H w ](*UNÁ.oyx��&C.+H � hOj µ?c4,+&CEL[�24&C3S.+'!/O3VE�'+24][email protected]&C243 w ,+&Cc4H7.oy�2 5 .D/O&(3B*ME�/1.](*UY4*M]�ºB��³I³?³I³?³?³I³�³?³I³?³I³?³?³I³?³I³?³�³I³?³?³I³?³I³?³?³I³?³�³I³?³I³?³?³ � h �

� » de2437Y4*MKM.+&(2438�EB&C�^@B')&(243F[B,)2 5 ]C* A N_,+2 A �{TV/ A [V](*��R��&Z.DHuÇ�È � j Û�ü-�V.DHB*A *-'+H w ](*UNÁ.oy;��&C.+H �l· ¹ � c4,+&CER[�24&(37.+'x/O3BEI.DHB*n'+24][email protected]&C243 w ,+&Cc4H7.oy92 5 .0/O&C3B*-E/1.�]C*MY4*M]�ºB��³I³?³?³I³�³?³I³?³I³?³?³I³?³I³?³�³I³?³?³I³?³I³?³?³I³?³�³I³?³I³?³?³ � h º

� µ �{TV/ A [V](*�º�m�.DHV* A *-')H w ](*UNÁ.oy>��&Z.DH h ¶ ¹4h c4,+&CE�[^24&C37.D'�/O3BE�.+HB*�](*UY4*-]K-@V,)Y4*M' w ,+&(cOHS.0yU� ³I³�³?³I³?³I³?³?³I³?³I³?³�³I³?³?³I³?³I³?³?³I³?³�³I³?³I³?³?³ � h ¶

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h � �{TV/ A [V](* ¹ m �924&(')'+243g*-bS@Q/1.+&(243$&(3$/G'+](&Z.�EB2 A /O&(3���&Z.DH$'+24][email protected]&C243iÔiÈý �¢Ù��')&(3 woþ� yU��.DHV* A *-')H w ]C*MNÁ.0y}��&Z.DH hOj4¹ º�c4,)&(E�[^2O&(37.D't/O3BE£.DHB*$*-,),+24,w ,)&(c4H7.0ynN_24,�](*UY4*-]�µ8�^³�³?³I³?³I³?³?³I³?³I³?³�³I³?³?³I³?³I³?³?³I³?³�³I³?³I³?³?³ � h ·

h4h de2437Y4*MKM.+&(2438�EB&C�^@B')&(243�[V,+2 5 ]C* A &(3��{��'+HB/O[^*ME�EB2 A /O&C3;�{.DHV* A *M'+H�2O3](*UY4*M] ¶tw ]C*MNÁ.0ya��&C.DH ¶ h � h 3B28EV*-'n/O3BE � j7h º · .+,+&\/13Bc4](*M'-�4.+HB*�](*MYO*-]�K-@V,)Y4*M'w ,)&(c4H7.0ye243F.DHB*�'D/ A *�](*UY4*M]��ÿ³?³?³I³?³I³?³�³I³?³?³I³?³I³?³?³I³?³�³I³?³I³?³?³ � h1»

h � de2437Y4*MKM.+&(2438�EB&C�^@B')&(243�[V,+2 5 ]C* A &(3��{��'+HB/O[^*ME�EB2 A /O&C3;�{.DHV* A *M'+H�2O3](*UY4*M]�º w ](*MNÁ.0y ��&C.+H$�Oº ¶ j 3B2�EB*-'>/O3VE ¹ · µ » .D,+&(/O3Bc4]C*-'M�8.DHV*?]C*MYO*-]�K-@V,)Y4*M'w ,)&(c4H7.0ye243F.DHB*�'D/ A *�](*UY4*M]��ÿ³?³?³I³?³I³?³�³I³?³?³I³?³I³?³?³I³?³�³I³?³I³?³?³ � h µ

h º de2437Y4*MKM.+&(2438�EB&C�^@B')&(243t[B,+2 5 ](* A &(3GEB2 A /O&C3G��&Z.DHt]\/ 6 *M,+'M��.+HB*�2 5 .0/O&C3B*-E](2�K~/1](] 6 ,+*U`Q3B*-E A *M'+H�/�NÁ.D*-, ¶ ]C*MYO*-]('?2ON�,)*M`Q3V* A *-37. w ](*MNÁ.0y���&C.DH ¹4j � ¶3B2�EB*-'>/O3VE �4� » µ h .D,)&\/O3Vc4](*M'-�V.+HB*�](*MYO*-]�KM@B,)YO*-' w ,+&Cc4H7.oyn243i.DHV*?'+/ A *�](*MYO*-]¢� � � j

h ¶ ��2 A 24cO*-3B*M24@B'�,+*M'+*M,)Y42O&(,-�8.DHB*������ A *M'+Hi243F,)*M`Q3B* A *M37.�]C*MYO*-] ¹ w .D24[�y��&C.+H ¶ º � h µ}3B2�EB*M'-�Q.DHB*?K-2O3BK-*M3S.+,D/1.+&(243$](*UY4*M]�K-@B,#Y4*-' w 5 2O.+.+2 A yn&(3i.DHB*[B](/O3B*�� � È j N_2O,!.DHB*�K~/1'+*�2ONa](2r� 5 &(2�EB*-c4,+/OEQ/1.+&(243F,+/1.D*1�ª³I³?³�³I³?³I³?³?³ � � h

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��HB* A *M'+H�K~/O3 5 * A /O&C3S.D/O&(3V*-E�c4]C2 5 /O]C] 6 K-2438N_24, A &(3BcF2O,?]C*MNÁ.�.+2 5 *G3B2438�K-2438N_24, A &(3Bc5 *M.��n*-*-3s.+HB*?'+@ 5 EB2 A /O&(3V'-�!��HB*I]\/1.).D*M,>[�24')'+& 5 &(](&Z. 6 &C'>N¤/1').D*M,�/13BE�K-/O3 5 *I@V'+*-EgN_24, A 24,#.0/O,. 6 [^*t/O[V[B,+2~T8& A /1.D&C243B'M�}��3 5 2O.+H�K-/O'+*M'-�a*MY4*M, 6 [B,)28KM*-')'+24,?|O*M*-[B'I/O3�/O,+,+/ 6 2ONnN¤/1K-*-'�243�.DHV*&(37.+*-,)N¤/1K-*-' 5 *U.��e*M*-3G.DHB*�')@ 5 EV2 A /1&(3B'M�a��3�.DHV*!3B2438�K-2438N_24, A &(3Bc�K~/O')*!.DHB&C'x/O]C](2r��'{@V'{.D2�KM2 A �[B@V.+*�N_24, *UTV/ A [B]C*�&(37.+*-c4,+/O]('�](&C|O*o È �QÊÊÉ;Ì ¯ ������HV*-,+* ¹ &C' N¤/OKM*�243t.DHB*�'+@ 5 EB2 A /O&C3u&(37.+*-,)N¤/1K-*O���Ê�/O3BEkÉ;ÌI/O,+* 5 /O'+&C'�N_@B3BKM.+&(243V'�/O'+')28KM&\/1.+*-Es��&Z.DHs.DHV*�EB&C�^*-,)*-37.�'+&CEB*-'�2ON�.+HB*�&(37.D*M,)N¤/OKM*O�?��3.DHV*�K-243VN_2O, A &(3Bc�K~/O')*�.+HB*!&C3S.+*-,#N¤/OK-*!N¤/OKM*-'�/O,)*!24,)EB*-,)*-EG&(3}.+HB*�'D/ A *e�e/ 6 N_,+2 Aï5 2O.DHG')&(EB*M'x2ON.DHV*�&(37.D*M,)N¤/OKM*-'-�{��HB&('�c4&ZY4*M'x.DHB*!N¤/OKM*�K-24,),+*M'+[�243BEB*M3BK-*�N_,+2 A .DHV*!.��e2?'+&CEB*-'�2ON�.+HB*�&(37.D*M,)N¤/OKM*-'M��.>&('�@B')*-EiN_24,�*UTV/ A [B](*�.+2 A /O|1*�.DHB* A *-'+HiKM243VN_24, A &C3Bc 5 *U.��e*M*-3�.+HB*�'+@ 5 EB2 A /O&C3B'-�

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H yQÝ~z­�L���0Þ�Ð N�Ó'Ô û ëÝ�Ùüéhä3Ø-ä3á�á�Þ"áeä3á ã×LØ�Ý�à_ëè Ø`ÞQP�ßFÞ�Ù�ËFòÞ"Ø`ÞQP�ßFÞ�Ù�Ü�Þ"Ø"à5ä3Ý�ߣèüä3Ø�á$Þ�òÆÞ"á�Þ"è�Þ"ß@à�Ùä3ß~ò�à_ë@Þ"ßÎä$ò$ò3Ý�à�Ý5×Lß~ä3á�á�â�Ø`ÞQP�ßFÞ�ÙrÝ�ßZ×LØ-òÞ"Ørà�×ré~Ø�×�ò3Ú@Ü�Þ+ä]è�Þ�Ù�ë�à_ëhä3àMÝ�ÙkÜ�×Lß æ ×LØ"è+Ý�ßã[ñ"Þ"à ô Þ�Þ"ßà�ë@Þ>Ù"Ú;ñ�ò×Lèüä3Ý�ßhÙ î+û ë@Þ æ ×Lá�á�× ô Ý�ßãüÝ�ÙcØ�Ú ß�Ý�ßüéhä3Ø�ä3á�á�Þ"álñ�â[Þ�T3Þ"Ø�â>é~Ø�×*Ü�Þ�Ù�Ù*×LغRB Ø`ÞQP�ßFÞ�ËFòÞ"Ø`ÞQP�ßFÞ>à�ë@Þcá�×*Ü�ä3á�á�â©èüä3Ø�á$Þ�ò�Þ"á�Þ"è�Þ"ß@à_Ù�àB ä$ò$ò3Ý�à'Ý5×Lß~ä3á�á�â©Ø�ÞQP�ßFÞ>Ú ß@à'Ý�álá�×*Ü�ä3á�Ü�×Lß æ ×LØ�è+Ý�à�âüÝ�Ù.Ø`Þ�äÜ�ë@Þ�òeàB Ù*Þ"ß~ò�éhäÜ�á3ä*ãÞ�Ù]à�×Îà�ë@Þ�ßFÞ"Ý ãLëhñ"×LØ�Ý�ßã�Ù"Úhñ�ò×Lèüä3Ý�ßhÙ æ ×LØ�Þ�T3Þ"Ø�âÎÝ�ß@à�Þ"Ø æ äÜ�Þ å ã3ÝUT*Ý�ßãnë@× ôèüä3ß@â©à'Ý�è�Þ�Ù+Þ�äÜ�ë�× æ à_ë@Þ æ äÜ�Þ�Ù>Ý�ß�à_ëhä3à#Ý�ß@à�Þ"Ø æ äÜ�Þ>ëhäLÙcñ"Þ�Þ"ß�ñ�Ý�Ù*Þ�Ü"à�Þ�òeàB Ø`Þ�Ü�Þ"ÝUT3Þ�Ù"Ý�è+Ý�á ä3Ø>Ý�ß æ ×LØ�èüä3à'Ý5×Lß�ä3ß~ò[Ü�ë@Þ�Ü�á©Ý æ Ø`ÞQP�ßFÞ"è�Þ"ß@à�Ý�Ù.ßFÞ�Þ�òÞ�òeàB Ý æ â$Þ�Ù å à_ë@Þ"ßÖØ`ÞQP�ßFÞ]à�ë@Þ�Þ"á�Þ"è�Þ"ß@à�Ù�à_ëhä3àrßFÞ�Þ�òÎØ�ÞQP�ßFÞ"è�Þ"ß@àÊä3ß~òuØ`ÞQP�ßFÞ�ä$ò$ò3Ý�à�Ý5×Lß~ä3á�á�â åÚß@à�Ý�álá�×*Ü�ä3á1Ü�×Lß æ ×LØ"è+Ý�à'âüÝ�Ù.Ø`Þ�äÜ�ë@Þ�òeàB Ø`Þ�é@Þ�ä3à#à_ë@Þrä$ñ"×eT3Þ7é~Ø�×*Ü�Þ�Ù�Ù>Ú ß@à'Ý�á¤ßF×�Ù"Ú;ñ�ò×Lèüä3Ý�ß]ëhäLÙOé@Þ"Ø æ ×LØ"è�Þ�òüØ�ÞQP�ßFÞ"è�Þ"ß@à î

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» ·

��HB*�EQ/1.D/>'#.D,)@BKM.+@B,+*M'9.+HQ/1.��n*n3B*M*-EB*MER&C3�24,+EB*M,�.D2�& A [B](* A *M37.���]Cc424,+&Z.DH A R� � /O,+*�EV&('+KM@B'+')*-E&(3g�8*MKM.+&(243 ¶ ��ºB�

��NÁ.+*-,�K-2O3VN_24, A &C. 6 &('�,)*~/OK0HV*-Eg�n*�@B'+*I g*M��&Á��.D2GK0HB*MK0|i.+HB*�](27/1E 5 /O]\/13BK-*1����HB*�&(3B[B@8.&('I.DHB*Fc4,+/O[BH 2ON�.+HB*F').D/O,).+&(3Bc A *-')H���&Z.DH 3B2�EB*M' 5 *M&(3Bcs.+HB*F*-]C* A *-37.D'M���xY4*M, 6 3B2�EB*FHQ/1'�n*-&(cOHS.-�;��HB&(K0H�&('�.DHB*G3S@ A}5 *-,�2ONnK0HB&C](EB,)*-3s.DHB/1.�.DHB*�*M](* A *M3S.�HB/O'-�}�Ne/O3�*-]C* A *-37.�HQ/O'�.+2A &(c4,+/1.D*t&(3�24,+EB*M,R.D2 A /O&C37.0/O&C3�.+HB*�](24/OE 5 /O](/O3BKM*L.DHV*�*-]C* A *-37. A &(c4,+/1.D*M'R/1](243Bc$��&C.+H�&C.+'K0HB&C](EB,)*-3;��� A 24,+*�EB*U.0/O&C](*MEFEB*-')K-,+&C[V.D&C243i&('�c4&ZY4*M3g&C3���*-KM.+&(243 ¶ ��ºB�

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¶ � � � h �½��](* A *-37.�,)*M`Q3B* A *M37.�/O]Cc424,)&C.DH A '

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� *�')24,).i.+HB*�*-EBc4*M'i243¬.+HB*�').D/O,).+&(3Bc A *-'+H 576 ](*M3BcO.+H©/13BE¬.+HB*-3¬'+*M](*MKM.iN_24,F*MYO*-, 6*-]C* A *-37.�.DHB*G]C243Bc4*M').�*-EVc4*t/O'?/�®),)*M`Q3V* A *-37.?*-EBc4*M°B�t�>*U`Q3B* A *M3S.I*-EVc4*G&('�.DHB*G*MEBc4*�.DHQ/�.��&(]C] 5 * 5 &(')*-KU.D*-E�&ZN�3V*-*-EV*-E;� �.G&C' A /1&(37.0/O&C3B*-E�N_24,}*MYO*-, 6 *-](* A *M37.GEB@B,+&C3Bcs.DHV*i/OEQ/O[V.+&CYO*[B,)28KM*-'+'Mm w�� y?&ZN!�e*FEV2 5 &(')*-KU.D&(2O3;�{.DHV*-3�.DHV*�,+*U`Q3B* A *-37.�*MEBc4*M'�&(3�.+HB*�3B*U�¿*M](* A *M3S.+'�/O,)*'+*M](*MKM.D*MEg.+2 5 *�24[B[�24'+&Z.D*�.D2G.DHB*�3B*U�«Y4*M,).+*UT^�q/13BE w h ye&CN9�e*IEB2t@B3B&CN_2O, A ,)*M`Q3V* A *-37.~�Q.DHV*-3.DHV*t,)*M`Q3V* A *-37.?*-EBc4*M'�&(3�.DHB*G3B*U�Ã*M](* A *-37.+'�/O,+*G&(3VHB*-,)&C.D*MEsN_,+2 A .DHB*G[Q/1,+*-37. w ')*-* � &Cc4@B,+*¹ yo����HB*I,+*U`Q3B* A *-37.�*MEBc4*I&('�3V2O.�*oT8[B](&CK-&C.+] 6 '#.D24,)*-E;���.�&C' A /O&(37.D/O&(3B*ME 5 *M.��n*-*M3�*-](* A *M37.~��'Y4*M,).+&(K-*M' j /O3BE � ����HB&(' A *-/O3B'�.DHQ/�.!/1NÁ.+*-, ,+*U`Q3B* A *-37.-�8.DHV*�3B*U��*-]C* A *-37.D'nHQ/~Y4*�YO*-,#.D&(KM*-'�/1''+HV2l��3g243 � &(cO@B,+* ¹ �

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»4»

refinement edge

K

uniform refinement

bisection

0

1 2 0

1

2

0 1

2

0 1

2

0 1

2

01

2

refinement edgerefinement edge

K1.node[0]:=v1, K1.node[1]:=v2, K1.node[2]:=vm

Note that : K0.node[0]:=v2, K0.node[1]:=v0, K0.node[2]:=vm

v0:=K0.node[1], v1:=K1.node[0], v2:=K0.node[0]

Also, to derefine K0 and K1 into K we have :

V0 V1

V2

vm

K0 K1

K0 K1

K2

K3

� &(cB� ¹ ����*M`Q3B* A *M37.~�OEV*U��,)*M`Q3V* A *-37.~�7/13BE A /O&(37.+*-3Q/O3VK-*�2ONV.+HB* ,+*M`B3B* A *-37.x*MEBc4* &(3 h ���}�-��HB*,+*U`Q3B* A *M3S.>*MEBc4*�&('e|O*-[8.>/O]C�e/ 6 ' 5 *M.��n*-*M3�]C28K-/O]�YO*-,#.D&(KM*-' j /O3BE � � � HB*-3 5 &(')*-KU.D&(3Vc.DHV*e3B*U�¯,)*M`Q3V* A *-37.x*MEBc4*-'x/O,+*n24[B[�24'+&Z.D*�.DHB*n3B*M� YO*-,).+*UTRà7á}/O3VERN_24,9@B3B&CN_2O, A ,)*M`Q3B*o�A *-37.�.DHV* 6 /1,+*R|1*-[V.?/1'�&(3�.+HB*�[Q/O,)*-37.~����HV*�EB*U�,+*M`B3B* A *-37.�2ON�*M](* A *-37.+'�2 5 .D/O&(3B*ME576 @V3B&CN_24, A ] 6 ,)*M`Q3B* A *M37.�&('?'+& A &C]\/O,�.+2g.DHV*uEB*o��,+*U`Q3B* A *M3S.}2ON�*-]C* A *-37.D'I2 5 .D/O&(3B*ME576i5 &(')*-KU.D&(2O3;m{àVâ�ã�ä4åVâ�æ¢ç�è1é8ê�ë�â7ì��qàqí�ã�ä4å;í�æ¢ç�è1éVê�ëoí~ì��;/O3BEiàVî�ã�ä4åVî�æ¢ç�è1é8ê�ë�î7ì��

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» µ

��HB*>,+*M[^*-/1.D*ME�/O[B[V](&(K-/1.D&C243�2ON^.DHB*�/O](cO24,+&Z.DH A EB2�*-' 3B2O. ](*-/OEt.+2I*M](* A *M3S. ')HQ/O[�*�EB*Mc4*-3��*-,+/1.D&C243;� � 24,!/}c4&CYO*-3F*-]C* A *-37. w '+*-* � &(c4@B,)* · y�243B*�K~/13$cO*M.�/1. A 24'#.eN_24@V,e. 6 [^*M'>21Na')HQ/O[�*-'M���HB&CK0H�/O,)*t.+HB*u24,)&(c4&C3Q/O]a243B*O�aEB*-3V2O.D*ME 576 ÄI�{/13BE�/O]C'+2gñu�xòG�a/O3BE�ðg�{/O'I'+HB2r��3�2O3�.DHV*`Qc4@V,+*O�

A AA

A

B CD D

A A

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µ j

case 5 :

ME = {3, 4}

Right sub-element, K1 :

K1.type = (K.type + 1)%3

switch (ME for face V1 V3 V2)

case 1 :

case 4 :

K1.node = {V2, V1, V3, VM}

if (K.type == 2) ME = {1, 3}

K1.node = {V1, V3, V2, VM}

if (K.type == 2) ME = {2, 4}

K1.node = {V2, V3, VM, V1}

V0V1

V2

V3

K0 K1

edge 0

edge 1

edge 2

edge 3 edge 4

edge 5

K

VMV0V1

V2

V3

Left sub-element, K0 :

K0.type = (K.type + 1)%3

case 2 :

case 3 :

switch (ME for face V0 V2 V3)

K0.node = {V0, V2, V3, VM}

if (K.type == 2) ME = {2, 4}

K0.node = {V3, V0, V2, VM}

if (K.type == 2) ME = {1, 3}

case 5 : K0.node = {V3, V2, VM, V0}

ME = {3, 4}

else ME = {2, 1}

else ME = {1, 2}

else ME = {1, 2}

else ME = {2, 1}

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