Benchmark solution
=====================

Calculation method
-----------------

The pipe support elements are SEG3.

• The position of the integration points along SEG3 is given in [:ref:`R3.01.01 <R3.01.01>`] "Shape functions and finite element integration points".

*# Coordinates of the Gauss Points along the* *SEG3* *

LongBeam = 2.0 * **pow** (3.0,0.5)

xpg = **pow** (3.0/5.0,0.5)

Points= {1: LongBeam*(1.0-xpg)* 0.5,

2: long beam* 0.5,

3: LongBeam*(1.0+xpg)* 0.5,}

• The coordinates of the integration sub-points depend on the number of layers, the number of sectors, and of course the diameter and thickness of the pipe.

*# Sub-Point Coordinates "Nbfibre = (2*Ncou+1) *(2*Nsect+1)" *

**def** CoordFibre (ifib):

Rint = Rex-Ep

*# coordinate index*

NumAng = ((ifib-1)% (2*Nsect+1))/(2.0*Nsect)

NumNeck = ((ifib-1)/(2 *Insect+1))/(2.0*Nneck)

y = (Rint+EP*NumCoU)*np.cos (2.0*np.PI*nummang)

z = - (Rint+EP*NumCoU)*np.sin (2.0*np.pi*nummang)

**return** y, z

where:

◦ Rext and Ep are the outer radius and the thickness of the pipe, respectively, specified in command AFFE_CARA_ELEM.

◦ Nsect and Ncu are the number of sectors and layers, respectively, specified in the AFFE_CARA_ELEM command.

CARA **= AFFE_CARA_ELEM (**

**MODELE =** MODELE,

**POUTRE =_F (GROUP_MA =( 'POUTRE '**, **' POUTRE0')**, **

**SECTION =' CERCLE '**, **CARA =( 'R'**, **'EP')**, **VALE =(** 10.0,1.0 **)**) **,

**TUYAU_NSEC =** 4, **TUYAU_NCOU =** 2, **)**,

**)**

• The transition matrix from pipe :math:`P0P1` to pipe :math:`P0P2` makes it possible to calculate the coordinates of the sub-points of pipe :math:`P0P2` from the previous formulas.

*# Axe_X to Trisectrice transition matrix*

un3 = 1.0/ **pow** (3.0,0.5); a2 = 1.0/ **pow** (2.0,0.5); un6 = 1.0/ **pow** (3.0,0.5); un6 = 1.0/ **pow** (3.0,0.5)

matPass = np.Array ([:ref:`[ un3, -un2, -un6 <[ un3, -un2, -un6>`],

[:ref:` un3, un2, -un6 < un3, un2, -un6>`],

[:ref:` un3, 0.0, 2.0 *un6 < un3, 0.0, 2.0*un6>`]])

For pipe :math:`P0P1` the coordinates :math:`(x,y,z)` of the fiber 'ifib' for the point 'ipt' are given by:

x = Points [ipt]

y, z = CoordFibre (ifib)

For pipe :math:`P0P2` the coordinates :math:`({x}_{t},{y}_{t},{z}_{t})` of the fiber 'ifib' for the point 'ipt' are given by:

*# According to the Trisectrice*

xx = Points [ipt]

yy, zz = CoordFibre (ifib)

xxt, yyt, zzt = np.dot (matPass, np.array ([xx, yy, zz]))


Reference quantities and results
-----------------------------------

The coordinates of all sub-points are tested.

To obtain the coordinates a table is generated from a constraint field constructed at the sub-points.

CSIEF **= CREA_CHAMP (TYPE_CHAM =' ELGA_SIEF_R '**, **OPERATION =" AFFE "**,

**MODELE =** MODELE,

**AFFE_SP =_F (CARA_ELEM =** CARA, **)**) **,** PROL_ZERO =' OUI '**,

**AFFE =_F (TOUT =" OUI "**, **** NOM_CMP =( 'SIXX '**,**) **,** VALE =**** **(** 0.0, **)**) **,**) **,**) **)**, **)**, **)**,

**)**

RESU **= CREA_RESU (TYPE_RESU =** **"" EVOL_NOLI "**, **" OPERATION =** **" AFFE "**",

**NOM_CHAM =' SIEF_ELGA '**,

**AFFE =** **_F (CHAM_GD =** CSIEF, **,** MODELE =**MODELE,** CARA_ELEM =**CARA,** INST =**0.0,**) **)**,

**)**

The coordinates are then tested using TEST_TABLE.

For example for the 'X' coordinate of pipe :math:`P0P1`:

**TEST_TABLE (REFERENCE =' ANALYTIQUE '**,

**VALE_CALC =xx**, **VALE_REFE =xx**, **=xx**, **=xx**, **TABLE =** TBEPS0, NOM_PARA COOR_X

**FILTRE =( _F (NOM_PARA =' POINT '**, **VALE_I =** **ipt**, **)**) **,**) **,

**_F (NOM_PARA =' SOUS_POINT '**, **VALE_I =** **ifib**, **)**, **)**) **)**, **)**, **

**)**


Uncertainty about the solution
----------------------------

None, exact solution.