Flores et al 2011a - Revision history

Cinmemj: Cinmemj moved page Draft Samper 793606474 to Flores et al 2011a

2018-04-12T11:46:57Z

Cinmemj moved page Draft Samper 793606474 to Flores et al 2011a

Cinmemj at 11:45, 12 April 2018

2018-04-12T11:45:13Z

Cinmemj at 11:44, 12 April 2018

2018-04-12T11:44:23Z

Cinmemj at 11:43, 12 April 2018

2018-04-12T11:43:59Z

Cinmemj at 11:42, 12 April 2018

2018-04-12T11:42:42Z

Cinmemj at 11:07, 12 April 2018

2018-04-12T11:07:01Z

Cinmemj at 11:05, 12 April 2018

2018-04-12T11:05:40Z

Cinmemj at 11:03, 12 April 2018

2018-04-12T11:03:34Z

Cinmemj at 10:57, 12 April 2018

2018-04-12T10:57:06Z

Cinmemj at 10:54, 12 April 2018

2018-04-12T10:54:41Z

@@ Line 2,149: / Line 2,149: @@
 <span id='_Toc284943829'></span>
-==9 References==
+==References==
-<span id='_Ref284604132'></span>:[[#cite-_Ref284604132|[1]]] E. Ortega, R. Flores and E. Oñate. ''Innovative numerical tools for the simulation of parachutes''. ''CIMNE publication,'' PI341, 2010.
+<span id='_Ref284604132'></span>[[#cite-_Ref284604132|[1]]] E. Ortega, R. Flores and E. Oñate. ''Innovative numerical tools for the simulation of parachutes''. ''CIMNE publication,'' PI341, 2010.
-<span id='_Ref284604146'></span>:[[#cite-_Ref284604146|[2]]] E. Ortega, R. Flores and E. Oñate. ''A 3D low-order panel method for unsteady aerodynamic problems''. ''CIMNE publication,'' PI343, 2010.
+<span id='_Ref284604146'></span>[[#cite-_Ref284604146|[2]]] E. Ortega, R. Flores and E. Oñate. ''A 3D low-order panel method for unsteady aerodynamic problems''. ''CIMNE publication,'' PI343, 2010.
-<span id='_Ref284616324'></span>:[[#cite-_Ref284616324|[3]]] R. Rossi. ''Light weight structures: Structural analysis and coupling issues''. PhD thesis, University of Bologna, 2005.
+<span id='_Ref284616324'></span>[[#cite-_Ref284616324|[3]]] R. Rossi. ''Light weight structures: Structural analysis and coupling issues''. PhD thesis, University of Bologna, 2005.
-<span id='_Ref284616569'></span><span id='_Ref284688748'></span>:[[#cite-_Ref284688748|[4]]] J. Drukker and. D.G. Roddeman. ''The wrinkling of thin membranes: Part 1 -theory''. Journal of Applied Mechanics, 54:884–887, 1987.
+<span id='_Ref284616569'></span><span id='_Ref284688748'></span>[[#cite-_Ref284688748|[4]]] J. Drukker and. D.G. Roddeman. ''The wrinkling of thin membranes: Part 1 -theory''. Journal of Applied Mechanics, 54:884–887, 1987.
-<span id='_Ref284688750'></span>:[[#cite-_Ref284688750|[5]]] J. Drukker and. D.G. Roddeman. ''The wrinkling of thin membranes: Part 2 -'' ''numerical analysis''. Journal of Applied Mechanics, 54:888–892, 1987.
+<span id='_Ref284688750'></span>[[#cite-_Ref284688750|[5]]] J. Drukker and. D.G. Roddeman. ''The wrinkling of thin membranes: Part 2 -'' ''numerical analysis''. Journal of Applied Mechanics, 54:888–892, 1987.
-<span id='_Ref284688881'></span>:[[#cite-_Ref284688881|[6]]] R. Rossi, R. Vitaliani. ''Numerical coupled analysis of flexible structures subjected to the fluid action''. In 5<sup>th</sup> PhD ''symposium'', Delft, 2004.
+<span id='_Ref284688881'></span>[[#cite-_Ref284688881|[6]]] R. Rossi, R. Vitaliani. ''Numerical coupled analysis of flexible structures subjected to the fluid action''. In 5<sup>th</sup> PhD ''symposium'', Delft, 2004.
-<span id='_Ref284617109'></span>:[[#cite-_Ref284617109|[7]]] T. Belytschko, W.K. Liu, and B. Moran. ''Nonlinear finite elements for continua and structures''. John Wiley & Sons, 2000.
+<span id='_Ref284617109'></span>[[#cite-_Ref284617109|[7]]] T. Belytschko, W.K. Liu, and B. Moran. ''Nonlinear finite elements for continua and structures''. John Wiley & Sons, 2000.
-<span id='_Ref284617545'></span>:[[#cite-_Ref284617545|[8]]] O.C. Zienkiewicz and R.L Taylor. ''The finite element method, Volume I''. Butterworth-Heinemann, 2000.
+<span id='_Ref284617545'></span>[[#cite-_Ref284617545|[8]]] O.C. Zienkiewicz and R.L Taylor. ''The finite element method, Volume I''. Butterworth-Heinemann, 2000.
-<span id='_Ref284690146'></span>:[[#cite-_Ref284690146|[9]]] R.L. Taylor. ''Finite element analysis of membrane structures''. Technical report, CIMNE, 2001
+<span id='_Ref284690146'></span>[[#cite-_Ref284690146|[9]]] R.L. Taylor. ''Finite element analysis of membrane structures''. Technical report, CIMNE, 2001
-<span id='_Ref284690302'></span>:[[#cite-_Ref284690302|[10]]] R. Rossi. ''A finite element formulation for 3D membrane structures including a wrinkling modified material model''. Technical Report 226, CIMNE, 2003.
+<span id='_Ref284690302'></span>[[#cite-_Ref284690302|[10]]] R. Rossi. ''A finite element formulation for 3D membrane structures including a wrinkling modified material model''. Technical Report 226, CIMNE, 2003.
-<span id='_Ref284690882'></span>:[[#cite-_Ref284690882|[11]]] R.M. Pauletti, D.M. Guirardi and T.E. Deifeld. ''Argyri’s natural membrane finite element revisited''. Proceedings of the International Conference on Textile composites and Inflatable Structures, Barcelona, 2005.
+<span id='_Ref284690882'></span>[[#cite-_Ref284690882|[11]]] R.M. Pauletti, D.M. Guirardi and T.E. Deifeld. ''Argyri’s natural membrane finite element revisited''. Proceedings of the International Conference on Textile composites and Inflatable Structures, Barcelona, 2005.
-<span id='_Ref284691523'></span>:[[#cite-_Ref284691523|[12]]] P. Contri and B.A. Schrefler. ''A geometrically nonlinear finite element analysis of wrinkled membrane surfaces by a no-compression material model''. Communications in applied numerical methods, 4:5–15, 1988.
+<span id='_Ref284691523'></span>[[#cite-_Ref284691523|[12]]] P. Contri and B.A. Schrefler. ''A geometrically nonlinear finite element analysis of wrinkled membrane surfaces by a no-compression material model''. Communications in applied numerical methods, 4:5–15, 1988.
-<span id='_Ref284691525'></span>:[[#cite-_Ref284691525|[13]]] R. Ziegler, W. Wagner and K. Bletzinger. ''A multisurface concept for the finite element analysis of wrinkled membranes''. Proceedings of the 4<sup>th</sup> International colloquium on computation of shell & spatial structures, IASS – IACM, 2000.
+<span id='_Ref284691525'></span>[[#cite-_Ref284691525|[13]]] R. Ziegler, W. Wagner and K. Bletzinger. ''A multisurface concept for the finite element analysis of wrinkled membranes''. Proceedings of the 4<sup>th</sup> International colloquium on computation of shell & spatial structures, IASS – IACM, 2000.
-<span id='_Ref284691526'></span>:[[#cite-_Ref284691526|[14]]] J. Hornig. ''Analyse der Faltenbildung in Membranen aus unterschiedlichen Material''. PhD thesis, Technischen Universität Berlin, 2004.
+<span id='_Ref284691526'></span>[[#cite-_Ref284691526|[14]]] J. Hornig. ''Analyse der Faltenbildung in Membranen aus unterschiedlichen Material''. PhD thesis, Technischen Universität Berlin, 2004.
-<span id='_Ref284943732'></span>:[[#cite-_Ref284943732|[15]]] CIMSA Ingeniería y Sistemas. Web page: [http://www.cimsa.com/ http://www.cimsa.com/] (checked February 8<sup>th</sup> 2011).
+<span id='_Ref284943732'></span>[[#cite-_Ref284943732|[15]]] CIMSA Ingeniería y Sistemas. Web page: [http://www.cimsa.com/ http://www.cimsa.com/] (checked February 8<sup>th</sup> 2011).

@@ Line 2,143: / Line 2,143: @@
 <span id='_Toc284943828'></span>
-==8 Acknowledgements==
+==Acknowledgements==
 The authors wish to thank the support from CIMSA Ingeniería y Sistemas <span id='cite-_Ref284943732'></span>[[#_Ref284943732|[15]]] for providing sample parachute geometries as well as experimental measurements used during the development of the code.

@@ Line 2,125: / Line 2,125: @@
 <div id="_Ref284534609" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 <span style="text-align: center; font-size: 75%;">'''Fig. 19''' – Computational domain geometry. Structure shown in dark grey and surrounding atmosphere in light gray</span></div>
 The pressure field corresponding to the explosion of a charge of 10kg of TNT has been computed with an Euler flow solver and used as prescribed load history on the inner wall of the structure. In a regular desktop computer the code is able to advance 10 time steps approximately every 9s. Given to short time scales involved (the time it takes for the shockwave to leave the domain is on the order of 10ms) the CPU time for a complete simulation is in the neighbourhood of 10 minutes (the exact value depends on the particular conditions of each simulation, as the stability limit changes as the mesh deforms). The explicit algorithm proves itself extremely efficient when dealing with fast transient events. <span id='cite-_Ref284603098'></span>[[#_Ref284603098|Fig. 20]] shows four snapshots of the temporal evolution of the structural deformations.

@@ Line 1,992: / Line 1,992: @@
 ===6.1 Ram air parachute modelling===
   [[Image:draft_Samper_793606474-image138.png|600px]]
 <div id="_Ref284524976" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-<span style="text-align: center; font-size: 75%;">'''Fig. 15 - Fastwing parachute. Initial configuration and final deformed shape'''</span></div>
+<span style="text-align: center; font-size: 75%;">'''Fig. 15''' - Fastwing parachute. Initial configuration and final deformed shape</span></div>
   [[Image:draft_Samper_793606474-image139.png|600px]]
 <div id="_Ref284525112" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-<span style="text-align: center; font-size: 75%;">'''Fig. 16 – Fastwing parachute right turn manoeuvre. Starting from the steady state the right brake is pulled. Yaw angle increases form left to right'''</span></div>
+<span style="text-align: center; font-size: 75%;">'''Fig. 16''' – Fastwing parachute right turn manoeuvre. Starting from the steady state the right brake is pulled. Yaw angle increases form left to right</span></div>
-−
-The code has also been used to predict the dynamic response of the parachute in order to simulate, for example, manoeuvres. The choice of an explicit scheme allows for instant transition from static to dynamic analysis if an unsteady flow solver is available. It must be stressed that the time step used by the structural solver is very short while for most applications the high-order modes of the mechanical response are of little relevance. Therefore it is not necessary to update the aerodynamic loads at every step in order to obtain a satisfactory global response. This saves processing time on the part of the flow solver. <span id='cite-_Ref284525112'></span>[[#_Ref284525112|Fig. 16]] illustrates a right turn manoeuvre.
 <span id='_Toc284943824'></span>
@@ Line 2,013: / Line 2,015: @@
 This is a simple inflation test aimed at exploring the capabilities of the code to simulate parachute deployment and inflation. Direct calculation of the pressure field around the partially inflated canopy is an extremely challenging task. Therefore, semi-empirical correlations have been used for the pressure field. Deployment takes place in two steps. Initially the apex of the canopy is pulled in the direction of the incident wind. Once the lines and canopy have been stretched the inflation phase begins. The parachute is discretized with an unstructured grid of 3390 triangular elements and 2040 cable elements modelling the suspension lines and fabric reinforcements. <span id='cite-_Ref284526381'></span>[[#_Ref284526381|Fig.  17]] shows the inflation stage, including the final steady configuration.
   [[Image:draft_Samper_793606474-image140.png|600px]]
 <div id="_Ref284526381" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-<span style="text-align: center; font-size: 75%;">'''Fig.  17 – Several stages of the parachute deployment'''</span></div>
+<span style="text-align: center; font-size: 75%;">'''Fig.  17''' – Several stages of the parachute deployment</span></div>
 <span id='_Toc284943825'></span>
@@ Line 2,024: / Line 2,027: @@
 By virtue of the wrinkling model the code is able to simulate the interaction between flexible fabrics and rigid objects. In the example shown a square membrane 3m across is released on top of a rigid circular disc with a diameter of 2m. The fabric is subject to the action of gravity, contact forces and damping forces introduced to simulate the effect of the air. The definition of the rigid surface is analytic so it does not need to be discretized (i.e. it does not enter the mesh). The membrane has been meshed with an unstructured grid of 8262 elements. The complete process, 4s of real time, can be simulated in just 9s in a current mid-range desktop computer. <span id='cite-_Ref284528141'></span>[[#_Ref284528141|Fig. 18]] shows several snapshots of the process.
- [[Image:draft_Samper_793606474-image141.png|600px]]
 <div id="_Ref284528141" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-<span style="text-align: center; font-size: 75%;">'''Fig. 18 - Draping simulation. Snapshots taken every 0,4s'''</span></div>
+<span style="text-align: center; font-size: 75%;">'''Fig. 18''' - Draping simulation. Snapshots taken every 0,4s</span></div>
 <span id='_Toc284943826'></span>
@@ Line 2,114: / Line 2,118: @@
 The example presented corresponds to a conceptual design for a blast containment device. The overall dimensions of the structure are 16x11x4,5m. Due to the double symmetry only a quarter of the geometry has been modelled. The mesh contains 2376000 tetrahedral elements and 79440 triangular membrane elements. The volume elements are located inside the cells of the structure and in the surrounding atmosphere. The volume of atmosphere meshed extends 9m outside the structure (see <span id='cite-_Ref284534609'></span>[[#_Ref284534609|Fig. 19]]).
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
@@ Line 2,119: / Line 2,124: @@
 <div id="_Ref284534609" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-<span style="text-align: center; font-size: 75%;">'''Fig. 19 – Computational domain geometry. Structure shown in dark grey and surrounding atmosphere in light gray'''</span></div>
+<span style="text-align: center; font-size: 75%;">'''Fig. 19''' – Computational domain geometry. Structure shown in dark grey and surrounding atmosphere in light gray</span></div>
 The pressure field corresponding to the explosion of a charge of 10kg of TNT has been computed with an Euler flow solver and used as prescribed load history on the inner wall of the structure. In a regular desktop computer the code is able to advance 10 time steps approximately every 9s. Given to short time scales involved (the time it takes for the shockwave to leave the domain is on the order of 10ms) the CPU time for a complete simulation is in the neighbourhood of 10 minutes (the exact value depends on the particular conditions of each simulation, as the stability limit changes as the mesh deforms). The explicit algorithm proves itself extremely efficient when dealing with fast transient events. <span id='cite-_Ref284603098'></span>[[#_Ref284603098|Fig. 20]] shows four snapshots of the temporal evolution of the structural deformations.
@@ Line 2,127: / Line 2,132: @@
 <div id="_Ref284603098" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-<span style="text-align: center; font-size: 75%;">'''Fig. 20 - Inflatable structure subject to blast loading. Time between frames: 2ms'''</span></div>
+<span style="text-align: center; font-size: 75%;">'''Fig. 20''' - Inflatable structure subject to blast loading. Time between frames: 2ms</span></div>
 <span id='_Toc284943827'></span>

@@ Line 1,958: / Line 1,958: @@
-{| style="width: 84%;border-collapse: collapse;"
+{| style="width: 84%;border-collapse: collapse;font-size:85%;"
 |-
 |  style="border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: top;"|

← Older revision	Revision as of 11:46, 12 April 2018
(No difference)

@@ Line 1,828: / Line 1,828: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <span id='ZEqnNum682903'></span>  <math>y=a {\cos h\left(\frac{x}{a}\right)\mbox{ }\mbox{ }\mbox{ }\mbox{ };\mbox{ }\mbox{ }\mbox{ }\mbox{ }s= a \sin h\left(\frac{x}{a}\right)</math>
+| <span id='ZEqnNum682903'></span>  <math>y=a \cos h\left(\frac{x}{a}\right)\mbox{ }\mbox{ }\mbox{ }\mbox{ };\mbox{ }\mbox{ }\mbox{ }\mbox{ }s= a \sin h\left(\frac{x}{a}\right)</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (5.1)
@@ Line 1,888: / Line 1,888: @@
 <div id="_Ref284435822" class="center" style="width: auto; margin-left: auto; margin-right: auto;font-size:75%;">
-<span style="text-align: center; font-size: 75%;">'''Table 1''' - Computed geometry for the catenary problem</span></div>
+'''Table 1''' - Computed geometry for the catenary problem</div>

@@ Line 1,828: / Line 1,828: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <span id='ZEqnNum682903'></span>  <math>y=acosh\left(\frac{x}{a}\right)\mbox{ }\mbox{ }\mbox{ }\mbox{ };\mbox{ }\mbox{ }\mbox{ }\mbox{ }s=</math><math>asinh\left(\frac{x}{a}\right)</math>
+| <span id='ZEqnNum682903'></span>  <math>y=a {\cos h\left(\frac{x}{a}\right)\mbox{ }\mbox{ }\mbox{ }\mbox{ };\mbox{ }\mbox{ }\mbox{ }\mbox{ }s= a \sin h\left(\frac{x}{a}\right)</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (5.1)
@@ Line 1,851: / Line 1,851: @@
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-<span style="text-align: center; font-size: 75%;">'''Fig. 11 - Catenary problem definition'''</span></div>
+<span style="text-align: center; font-size: 75%;">'''Fig. 11''' - Catenary problem definition</span></div>
 The cable has been discretized with 20 linear elements. In order to check also the behaviour of the membrane formulation a strip made of triangular elements has been suspended the same way as the cable. The strip is modelled with a mesh of 20x4 triangles. The relevant properties chosen are:
@@ Line 1,869: / Line 1,870: @@
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-  [[Image:draft_Samper_793606474-image133.png|600px]] </div>
+  [[Image:draft_Samper_793606474-image133.png|400px]] </div>
 <div id="_Ref284436048" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-<span style="text-align: center; font-size: 75%;">'''Fig. 12 - Catenary problem. Deformed shape'''</span></div>
+<span style="text-align: center; font-size: 75%;">'''Fig. 12''' - Catenary problem. Deformed shape</span></div>
 {| style="width: 65%;border-collapse: collapse;"
 |-
 |  style="border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: top;"|
@@ Line 1,885: / Line 1,887: @@
 |}
 <span style="text-align: center; font-size: 75%;"><span id="fn-1"></span>([[#fnc-1|<sup>1</sup>]])  Value for the triangular elements is an average. Due to the constraints imposed by the discretisation the deformed shape is not perfectly cylindrical.</span>
@@ Line 1,911: / Line 1,912: @@
 Reference values can be found in the literature for the vertical displacement of the central point of the membrane for an applied pressure of 100KPa (Pauletti 2005). The solution was computed using an unstructured mesh containing 344 triangular elements.
 {| style="width: 89%;border-collapse: collapse;"
 |-
 |  style="border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: top;"|
@@ Line 1,923: / Line 1,925: @@
 |  style="border: 1pt solid black;text-align: center;vertical-align: top;"|34,8
 |}
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-<span style="text-align: center; font-size: 75%;">'''Table 2 – Central deflection of the membrane (mm)'''</span></div>
+<span style="text-align: center; font-size: 75%;">'''Table 2''' – Central deflection of the membrane (mm)</span></div>
 While there is not a single definite reference solution in the literature, the result obtained compares well with the two values presented. It must be stressed that slight differences must be expected because the deformations experienced by the membrane exceed the strict limits of validity of linear elasticity.
@@ Line 1,934: / Line 1,936: @@
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-<span style="text-align: center; font-size: 75%;">'''Fig. 13 – Henky’s problem. Deformed shape'''</span></div>
+<span style="text-align: center; font-size: 75%;">'''Fig. 13''' – Henky’s problem. Deformed shape</span></div>
 <span id='_Toc284943821'></span>
@@ Line 1,954: / Line 1,956: @@
 A mesh composed of 16x16 squares is used for each side of the airbag. Each square has then been divided into 4 equal triangles in order to eliminate mesh orientation effects. The total number of triangular elements is therefore 2048. The next table shows the comparison of the result from PUMI_MEM with several sources <span id='cite-_Ref284691523'></span>[[#_Ref284691523|[12]]]<span id='cite-_Ref284691525'></span>[[#_Ref284691525|[13]]]<span id='cite-_Ref284691526'></span>[[#_Ref284691526|[14]]]. The differences are negligible.
 {| style="width: 84%;border-collapse: collapse;"
@@ Line 1,969: / Line 1,972: @@
 |  style="border: 1pt solid black;text-align: center;vertical-align: top;"|216,2
 |}
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-<span style="text-align: center; font-size: 75%;">'''Table 3 – Central displacement of the airbag (mm)'''</span></div>
+<span style="text-align: center; font-size: 75%;">'''Table 3''' – Central displacement of the airbag (mm)</span></div>
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
@@ Line 1,978: / Line 1,981: @@
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-<span style="text-align: center; font-size: 75%;">'''Fig. 14 – Square airbag. Deformed shape'''</span></div>
+<span style="text-align: center; font-size: 75%;">'''Fig. 14''' – Square airbag. Deformed shape</span></div>
 <span id='_Toc284943822'></span>

@@ Line 1,310: / Line 1,310: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>(\xi ,\eta ,\zeta )=\left((\boldsymbol{x-}\boldsymbol{x}^\boldsymbol{1})\cdot \boldsymbol{e}_\boldsymbol{1},(\boldsymbol{x-\right. </math><math>\left. }\boldsymbol{x}^\boldsymbol{1})\cdot \boldsymbol{e}_\boldsymbol{2},(\boldsymbol{x-\right. </math><math>\left. }\boldsymbol{x}^\boldsymbol{1})\cdot \boldsymbol{n}\right)</math>
+| <math>(\xi ,\eta ,\zeta )=\left((\boldsymbol{x-}\boldsymbol{x}^{1})\cdot \boldsymbol{e}_{1},(\boldsymbol{x}-\boldsymbol{x}^{1})\cdot \boldsymbol{e}_{2},(\boldsymbol{x}-\boldsymbol{x}^{1})\cdot \boldsymbol{n}\right)</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (4.42)

@@ Line 897: / Line 897: @@
 \eta_3 & 0\\
 -{\xi}_3 & {\xi }_2
-\end{array}\right] \left[\begin{array}{ccc} -1 & 1 & 1\\-1 & 0 & 1 \end{array}\right]= \frac{1}{\xi_2\eta_3}\left[\begin{array}{ccc}-{\xi }_3 & {\xi }_3 & 0\\
+\end{array}\right] \left[\begin{array}{ccc} -1 & 1 & 1\\-1 & 0 & 1 \end{array}\right]= \frac{1}{\xi_2\eta_3}\left[\begin{array}{ccc}-{\eta }_3 & {\eta }_3 & 0\\{\xi }_3-{\xi }_2 & -{\xi }_3 & {\xi }_2\end{array}\right]
 {\xi }_3-{\xi }_2 & -{\xi }_3 & {\xi }_2\end{array}\right]
 </math>
 |}