Code_Aster ®
Version
5.0
Titrate:
SSLV109 - Full Cylindre in pressure nonuniform mode 1

Date:
23/09/02
Author (S):
X. DESROCHES Key
:
V3.04.109-C Page:
1/16

Organization (S): EDF/AMA

Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
Document: V3.04.109

SSLV109 - Full Cylindre in nonuniform pressure
mode 1

Summary:

This test validates all the elements of Fourier (triangles and quadrangles of degrees 1 and 2) in elasticity.
functionalities are as follows:

· variable pressure in space,
· imposed displacements,
· matrices of rigidity Fourier mode 1,
· constraints with the nodes Fourier mode 1,
· recombination of Fourier on displacements and constraints (modeling A),
· transverse isotropic material (modeling F).

The test has a quadratic analytical solution in displacements.

The interest of the test lies in:

· the comparison between solution calculated and analytical solution on the various finite elements,
· the comparison between the results and Code PERMAS on elements TRIA6 (modeling A).

Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A

Code_Aster ®
Version
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Titrate:
SSLV109 - Full Cylindre in pressure nonuniform mode 1

Date:
23/09/02
Author (S):
X. DESROCHES Key
:
V3.04.109-C Page:
2/16

1
Problem of reference

1.1 Geometry

L = 12
With
C
p = R
Z
R = 1
R
B
D


The modelled field is ACDB (plane = 0).

1.2
Material properties

E = 72 NR/M2

= 0.3

1.3
Boundary conditions and loadings

ur ()
With
= uz ()
With
= uq ()
With
= 0
uz (A)
B
= 0

R
p = p
cos
R

with p = 1. and R = 1 applied in Z = 12

1.4 Conditions
initial

Without object for the static analysis.

Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A

Code_Aster ®
Version
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Titrate:
SSLV109 - Full Cylindre in pressure nonuniform mode 1

Date:
23/09/02
Author (S):
X. DESROCHES Key
:
V3.04.109-C Page:
3/16

2
Reference solution

2.1
Method of calculation used for the reference solution

M
p
2

U
,
=
,
cos with
,
=
+
2
R (R Z
)
(ur Z)
(ur Z)
Z
R
2EI
2ER
p
U
,
=
,
cos with
,
= -

Z (R Z
)
v (R Z)
v (R Z)
rz
2EI
M
p
2

U
2
(R, Z,)
=
(wr, Z) (- sin) with (wr, Z) =
Z -
R
2EI
2ER

p
All the constraints are null except zz (R, Z) = - R.
R

The data were selected in such way that (
u.a.) = (
U 0, L) = 1.

Displacements are thus written here:

(
z2
R 2
R Z
z2
R 2
U R, Z) =
+
v (R, Z) = -
(
W R, Z) =
-

144
480
72
144
480

and:

zz (R, Z) = - R

2.2
Results of reference

U, v, W, zz
in
R = 0., 0.5, 1.
Z = 0., 6., 12.

U, U, U
R
Z

in R = 0., Z = 6., 12., = 45°

2.3
Uncertainty on the solution

Analytical solution.

2.4 References
bibliographical

[1]
PERMAS-HS. Axisymmetric Continua with arbitrary loads. Stuttgart 1985. INTES publication
n°224 p 42 - 49.

Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A

Code_Aster ®
Version
5.0
Titrate:
SSLV109 - Full Cylindre in pressure nonuniform mode 1

Date:
23/09/02
Author (S):
X. DESROCHES Key
:
V3.04.109-C Page:
4/16

3 Modeling
With

3.1
Characteristics of modeling

Number of the nodes:
With = N1
B = N3
C = N13


D = N15
E = N7
F = N8
G = N9

blocked node
With
E
C
face
blocked
F
y (Z)
in Z
B
G
D
X (R)


Limiting conditions:

DDL_IMPO:
(NOEUD: WITH DX = 0.
DY = 0.
DZ = 0.)
face AB
(GROUP_NO: AB DY = 0.)

Pressure on the face CD: PRES_REP (GROUP_MA: Boils NEAR: p)

p being defined by AFFE_CHAR_MECA_F by p (X) = ­ X

3.2
Characteristics of the grid

A number of nodes: 15

A number of meshs and types: 4 TRIA6, 1SEG3 on segment CD

3.3 Functionalities
tested

Commands




AFFE_MODELE
MECANIQUE
“AXIS_FOURIER”
TOUT

DEFI_FONCTION
NOM_PARA
“X”

AFFE_CHAR_MECA_F
DDL_IMPO
NOEUD

PRES_REP
GROUP_MA

CALC_MATR_ELEM
“RIGI_MECA”
MODE_FOURIER

CALC_CHAM_ELEM
“SIGM_ELNO_DEPL”
MODE_FOURIER

COMB_CHAM_NO
COMB_FOURIER

COMB_CHAM_ELEM
COMB_FOURIER


3.4 Remarks

The number of the mode of Fourier not affecting the loading, key word MODE_FOURIER is not
necessary in command CALC_VECT_ELEM (by defect, it is regarded as being worth 0).
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A

Code_Aster ®
Version
5.0
Titrate:
SSLV109 - Full Cylindre in pressure nonuniform mode 1

Date:
23/09/02
Author (S):
X. DESROCHES Key
:
V3.04.109-C Page:
5/16

4
Results of modeling A

4.1 Values
tested

Node Size
Reference
Aster %
difference

B
U
2.0833 10­3 2.0835
10­3 9.10­3
v
0.
DDL_IMPO

W
­ 2.0833 10­3
­ 2.0834 10­3 3.10­3

zz
­ 1. ­ 1.000001
6.10­4

E
U 0.25
0.250001
5.10­4
v
0.
­ 4.10­6

W
0.25
0.250000
2.10­4

zz
0.
­ 1.10­4


F
U 0.250521
0.250522
4.10­4
v
­ 0.04166 ­ 0.041668 3.10­3
W
0.0249479
0.0249479
­ 6.10-3

zz
­ 0.5 ­ 0.50005 1.10­2

G
U 0.252083
0.252084
3.10­4
v
­ 0.083333
­ 0.083333 ­ 7.10­3
W
0.247917 0.247916
1.10­4

zz
­ 1. ­ 1.
9.10­3

C
U 1. 1.0006
6.10­3
v
0.
­ 5.10­5

W
1.
1.00006
6.10­3

zz
0.
­ 1.1 10­3


D
U 1.00208
1.00215
7.10­3
v
­ 0.16666 ­ 0.166691 1.5 10­2
W
0.99791 0.997981
1.9 10­2

zz
­ 1. ­ 0.999 0.1

4.2 Remarks

The analytical solution is found with a precision < 0.02% for displacements and < 0.1%
for the constraints.

With a numerical formula of integration at 6 points of GAUSS (instead of 3) to calculate the stiffness,
one would find the relation except for 10­10 (like PERMAS).
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A

Code_Aster ®
Version
5.0
Titrate:
SSLV109 - Full Cylindre in pressure nonuniform mode 1

Date:
23/09/02
Author (S):
X. DESROCHES Key
:
V3.04.109-C Page:
6/16

5 Modeling
B

5.1
Characteristics of modeling

Number of the nodes:
With = N1
B = N3
C = N13


D = N15
E = N7
F = N8
G = N9

blocked node
C
With
E
y (Z)
face
blocked
F
in Z
B
G
D
X (R)


Limiting conditions:

DDL_IMPO:
(NOEUD: WITH DX = 0.
DY = 0.
DZ = 0.)
face AB
(GROUP_NO: AB DY = 0.)

Pressure on the face CD: PRES_REP (GROUP_MA: Boils NEAR: p)

p being defined by AFFE_CHAR_MECA_F by p (X) = ­ X

5.2
Characteristics of the grid

A number of nodes: 15

A number of meshs and types: 2 QUAD8, 1 SEG3 on segment CD

5.3 Functionalities
tested

Commands




AFFE_MODELE
MECANIQUE
“AXIS_FOURIER”
TOUT

DEFI_FONCTION
NOM_PARA
“X”

AFFE_CHAR_MECA_F
DDL_IMPO
NOEUD

PRES_REP
GROUP_MA

CALC_MATR_ELEM
“RIGI_MECA”
MODE_FOURIER

CALC_CHAM_ELEM
“SIGM_ELNO_DEPL”
MODE_FOURIER


5.4 Remarks

The number of the mode of Fourier not affecting the loading, key word MODE_FOURIER is not
necessary in command CALC_VECT_ELEM (by defect, it is regarded as being worth 0).
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A

Code_Aster ®
Version
5.0
Titrate:
SSLV109 - Full Cylindre in pressure nonuniform mode 1

Date:
23/09/02
Author (S):
X. DESROCHES Key
:
V3.04.109-C Page:
7/16

6
Results of modeling B

6.1 Values
tested

Node Size Reference
Aster %
difference

B
U
2.0833 10­3 2.0833
10­3 10.10­11
v
0.
DDL_IMPO

W
­ 2.0833 10­3
­ 2.0833 10­3
10.10­11

zz
­ 1. ­ 1 10.10­11

E
U 0.25
0.25
10.10­11
v
0.
­ 3.9 10­14

W
0.25
0.25
10.10­11

zz
0.
­ 4.10­12


F
U 0.250521
0.250521
10.10­11
v
­ 0.04166 ­ 0.04166 10.10­11
W
0.0249479
0.0249479
10.10­11

zz
­ 0.5 ­ 0.5 10.10­11

G
U 0.252083
0.252083
10.10­11
v
­ 0.08333 ­ 0.08333 10.10­11
W
0.247917 0.247917
10.10­11

zz
­ 1. ­ 1. 10.10­10

C
U 1. 1. 10.10­11
v
0.
­ 4.3.10­14

W
1.
1
10.10­11

zz
0.
6.10­12


D
U 1.00208
1.00208
10.10­11
v
­ 0.16666 ­ 0.16666 10.10­11
W
0.99791 0.99791
10.10­11

zz
­ 1. ­ 1. 10.10­11

6.2 Remarks

The analytical solution is found with 10 or 11 significant digits.

Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A

Code_Aster ®
Version
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Titrate:
SSLV109 - Full Cylindre in pressure nonuniform mode 1

Date:
23/09/02
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X. DESROCHES Key
:
V3.04.109-C Page:
8/16

7 Modeling
C

7.1
Characteristics of modeling

Number of the nodes:
With = N1
B = N3
C = N13


D = N15
E = N7
F = N8
G = N9

blocked node
With
E
C
face
y (Z)
blocked
F
in Z
B
G
D
X (R)


Limiting conditions:

DDL_IMPO:
(NOEUD: WITH DX = 0.
DY = 0.
DZ = 0.)
face AB
(GROUP_NO: AB DY = 0.)

Pressure on the face CD: PRES_REP (GROUP_MA: Boils NEAR: p)

p being defined by AFFE_CHAR_MECA_F by p (X) = ­ X

7.2
Characteristics of the grid

A number of nodes:15

A number of meshs and types: 2 QUAD9, 1 SEG3 on segment CD

7.3 Functionalities
tested

Commands




AFFE_MODELE
MECANIQUE
“AXIS_FOURIER”
TOUT

DEFI_FONCTION
NOM_PARA
“X”

AFFE_CHAR_MECA_F
DDL_IMPO
NOEUD

PRES_REP
GROUP_MA

CALC_MATR_ELEM
“RIGI_MECA”
MODE_FOURIER

CALC_CHAM_ELEM
“SIGM_ELNO_DEPL”
MODE_FOURIER


7.4 Remarks

The number of the mode of Fourier not affecting the loading, key word MODE_FOURIER is not
necessary in command CALC_VECT_ELEM (by defect, it is regarded as being worth 0).

Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A

Code_Aster ®
Version
5.0
Titrate:
SSLV109 - Full Cylindre in pressure nonuniform mode 1

Date:
23/09/02
Author (S):
X. DESROCHES Key
:
V3.04.109-C Page:
9/16

8
Results of modeling C

8.1 Values
tested

Node Size
Reference
Aster %
difference

B
U
2.0833 10­3 2.0833
10­3 10.10­11
v
0.
DDL_IMPO

W
­ 2.0833 10­3
­ 2.0833 10­3
10.10­11

zz
­ 1. ­ 1 10.10­10

E
U 0.25 0.25 10.10­11
v
0.
­ 3.9 10­14

W
0.25
0.25
10.10­11

zz
0.
­ 4.10­12


F
U 0.250521
0.250521
10.10­11
v
­ 0.04166
­ 0.04166
10.10­11
W
0.0249479
0.0249479
10.10­11

zz
­ 0.5 ­ 0.5 10.10­11

G
U 0.252083
0.252083
10.10­11
v
­ 0.08333
­ 0.08333
10.10­11
W
0.247917
0.247917
10.10­11

zz
­ 1. ­ 1. 10.10­11

C
U 1.
1.
10.10­11
v
0.
­ 4.3.10­14

W
1.
1.
10.10­11

zz
0.
6.10­12


D
U 1.00208
1.00208
10.10­11
v
­ 0.16666
­ 0.16666
10.10­11
W
0.99791
0.99791
10.10­11

zz
­ 1. ­ 1. 10.10­11

8.2 Remarks

The analytical solution is found with 10 or 11 significant digits.

Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A

Code_Aster ®
Version
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Titrate:
SSLV109 - Full Cylindre in pressure nonuniform mode 1

Date:
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X. DESROCHES Key
:
V3.04.109-C Page:
10/16

9 Modeling
D

9.1
Characteristics of modeling

Number of the nodes:
With = N1
B = N1129
C = N1369


D = N2169
E = N141
F = N705
G = N1269

blocked node
With
E
C
face
y (Z)
blocked
F
in Z
B
G
D
X (R)


Limiting conditions:

DDL_IMPO:
(NOEUD: WITH DX = 0.
DY = 0.
DZ = 0.)
face AB
(GROUP_NO: AB DY = 0.)

Pressure on the face CD: PRES_REP (GROUP_MA: Boils NEAR: p)

p being defined by AFFE_CHAR_MECA_F by p (X) = ­ X

9.2
Characteristics of the grid

A number of nodes: 2169

A number of meshs and types: 1920 QUAD4, 8 SEG2 on segment CD

9.3 Functionalities
tested

Commands




AFFE_MODELE
MECANIQUE
“AXIS_FOURIER”
TOUT

DEFI_FONCTION
NOM_PARA
“X”

AFFE_CHAR_MECA_F
DDL_IMPO
NOEUD

PRES_REP
GROUP_MA

CALC_MATR_ELEM
“RIGI_MECA”
MODE_FOURIER

CALC_CHAM_ELEM
“SIGM_ELNO_DEPL”
MODE_FOURIER


9.4 Remarks

The number of the mode of Fourier not affecting the loading, key word MODE_FOURIER is not
necessary in command CALC_VECT_ELEM (by defect, it is regarded as being worth 0).
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A

Code_Aster ®
Version
5.0
Titrate:
SSLV109 - Full Cylindre in pressure nonuniform mode 1

Date:
23/09/02
Author (S):
X. DESROCHES Key
:
V3.04.109-C Page:
11/16

10 Results of modeling D

10.1 Values
tested

Node Size
Reference
Aster %
difference

B
U
2.0833 10­3 2.0919
10­3
0.41
v
0.
DDL_IMPO

W
­ 2.0833 10­3
­ 2.0674 10­3
0.76

zz
­ 1. ­ 1.00974
0.97

E
U 0.25 0.2498 0.07
v
0.
­ 2.7.10­10

W
0.25
0.2498 0.07

zz
0. 0.0090
0.90

F
U 0.250521
0.250347
0.07
v
­ 0.04166
­ 0.04164
0.07
W
0.0249479
0.0249317 0.06

zz
­ 0.5 ­ 0.51005
2.01

G
U 0.252083
0.251911
0.07
v
­ 0.083333
­ 0.083273
0.07
W
0.247917
0.247752 0.07

zz
­ 1. ­ 1.0103
1.03

C
U 1.
0.99927
0.07
v
0.
­ 2.3.10­7

W
1.
0.99928 0.07

zz
0. 0.0088


D
U 1.00208
1.001357
0.07
v
­ 0.16666
­ 0.16653
0.08
W
0.99791
0.997208 0.06

zz
­ 1. ­ 1.0027
0.27

10.2 Remarks

To obtain a precision of about 1% on the constraints, it is necessary to model
structure very finely (8 elements radially and 240 axially).

Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A

Code_Aster ®
Version
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Titrate:
SSLV109 - Full Cylindre in pressure nonuniform mode 1

Date:
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X. DESROCHES Key
:
V3.04.109-C Page:
12/16

11 Modeling
E

11.1 Characteristics of modeling

Number of the nodes:
With = N1
B = N2421


E = N121
F = N1331
G = N2541

blocked node
With
E
y (Z)
face
blocked
F
in Z
B
G
X (R)


Limiting conditions:

DDL_IMPO:
(NOEUD: WITH DX = 0.
DY = 0.
DZ = 0.)
face AB
(GROUP_NO: AB DY = 0.)

Pressure on face EG: PRES_REP (GROUP_MA: Boils NEAR: p)

p being defined by AFFE_CHAR_MECA_F by p (X) = ­ X

11.2 Characteristics of the grid

A number of nodes: 2541

A number of meshs and types: 4800 TRIA3, 20 SEG2 on segment EG

11.3 Functionalities
tested

Commands




AFFE_MODELE
MECANIQUE
“AXIS_FOURIER”
TOUT

DEFI_FONCTION
NOM_PARA
“X”

AFFE_CHAR_MECA_F
DDL_IMPO
NOEUD

PRES_REP
GROUP_MA

CALC_MATR_ELEM
“RIGI_MECA”
MODE_FOURIER

CALC_CHAM_ELEM
“SIGM_ELNO_DEPL”
MODE_FOURIER


11.4 Remarks

To decrease the number of nodes, one modelled the structure for y 6.
The precision on the results is nevertheless less than for elements QUAD4.

Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A

Code_Aster ®
Version
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Titrate:
SSLV109 - Full Cylindre in pressure nonuniform mode 1

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:
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13/16

12 Results of modeling E

12.1 Values
tested

Node Size
Reference
Aster %
difference

B
U
2.0833 10­3 2.0228
10­3
2.90
v
0.
DDL_IMPO

W
­ 2.0833 10­3
­ 2.1143 10­3
1.49

zz
­ 1. ­ 1.0078
0.78

E
U 0.25 0.24949 0.20
v
0.
­ 2.4 10­5

W
0.25
0.24950 0.20

zz
0.
­ 5.3 10­2


F
U 0.250521
0.25003 0.19
v
­ 0.04166 ­ 0.04161
0.13
W
0.249479 0.24900 0.19

zz
­ 0.5 ­ 0.49738
0.52

G
U 0.252083
0.251605
0.19
v
­ 0.083333 ­ 0.083157
0.21
W
0.247917 0.247449 0.19

zz
­ 1. ­ 0.9814
1.86

12.2 Remarks

The precision on displacements is lower than 3%, that on the constraints lower than 2%.

On this example, the TRIA3 converge definitely less quickly than the QUAD4 towards the solution
exact.

Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A

Code_Aster ®
Version
5.0
Titrate:
SSLV109 - Full Cylindre in pressure nonuniform mode 1

Date:
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:
V3.04.109-C Page:
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13 Modeling
F

13.1 Characteristics of modeling

Number of the nodes:
With = N1
B = N3
C = N13


D = N15
E = N7
G = N9



blocked node
With
N4
E
N12
C

y (Z)
face
blocked N2
N14
in Z
B
G
N12
D
X (R)


Limiting conditions:

DDL_IMPO:
(NOEUD: WITH DX = 0.
DY = 0.
DZ = 0.)
face AB
(GROUP_NO: AB DY = 0.)

Pressure on the face CD: PRES_REP (GROUP_MA: Boils NEAR: p)

p being defined by AFFE_CHAR_MECA_F by p (X) = ­ X

13.2 Characteristics of the grid

A number of nodes: 15

A number of meshs and types: 2 QUAD8, 1SEG3 on segment CD

13.3 Functionalities
tested

Commands




AFFE_MODELE
MECANIQUE
“AXIS_FOURIER”
TOUT

DEFI_MATERIAU
ELAS_ORTH

DEFI_FONCTION
NOM_PARA
“X”

AFFE_CHAR_MECA_F
DDL_IMPO
NOEUD

PRES_REP
GROUP_MA

CALC_MATR_ELEM
“RIGI_MECA”
MODE_FOURIER

CALC_CHAM_ELEM
“SIGM_ELNO_DEPL”
MODE_FOURIER


13.4 Remarks

The number of the mode of Fourier not affecting the loading, key word MODE_FOURIER is not
necessary in command CALC_VECT_ELEM (by defect, it is regarded as being worth 0).
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A

Code_Aster ®
Version
5.0
Titrate:
SSLV109 - Full Cylindre in pressure nonuniform mode 1

Date:
23/09/02
Author (S):
X. DESROCHES Key
:
V3.04.109-C Page:
15/16

14 Results of modeling F

14.1 Values
tested

Node Size
Reference
Aster %
difference

N2
U 2.6041666
2.6041666
1.10­9
W
­ 2.6041666 ­ 2.6041666
1.10­9
With
zz
0.
­ 1.10­13

B
zz
­ 1. ­ 1. 3.10­10

N4
U 0.0625 0.0625 4.10­10
W
0.0625
0.0625
4.10­10
E U
0.25
0.25
3.10­10
W
0.25
0.25
3.10­10

zz
0.
­ 1. 10­14

G v
­ 0.083333 ­ 0.083333
2.10­10

zz
­ 1. ­ 1. 2.10­11
N10 U
0.5625 0.5625 2.10­10
W
0.5625 0.5625 2.10­10
N12 v
­ 0.125 ­ 0.125 1.10­10
C U
1. 1. 1.10­10
W
1. 1. 1.10­10

zz
0.
7.10­14

N14 v
­ 0.083333 ­ 0.083333 9.10­11
D v
­ 0.166666 ­ 0.166666 9.10­11

zz
­ 1. ­ 1. 1.10­11

Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A

Code_Aster ®
Version
5.0
Titrate:
SSLV109 - Full Cylindre in pressure nonuniform mode 1

Date:
23/09/02
Author (S):
X. DESROCHES Key
:
V3.04.109-C Page:
16/16

15 Summary of the results

The elements of command 2 give the analytical solution.

The elements of command 1 converge slowly towards the solution and require very fine grids.
Times calculations remain however reasonable.
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A

Outline document