SSNV173… Bar fissured with X-FEM

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SSNV173 Barreau fissured with X-FEM

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S. GENIAUT, P. MASSIN

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Manual of Validation

V6.04 booklet: Linear statics of the voluminal structures

HT-66/05/005/A

Organization (S):

EDF-R & D/AMA

Manual of Validation
V6.04 booklet: Linear statics of the voluminal structures
Document: V6.04.173

SSNV173 Barreau fissured with X-FEM

Summary

The purpose of this test is to validate two aspects of elementary calculation within the framework of X-FEM [R7.02.12]:

· the integration of a discontinuous size thanks to a under-cutting of the element,
· the enrichment of the functions of form by the Heaviside function.

This test brings into play a parallelepipedic bar fissured on all its section, subjected to a displacement
imposed, which has as a consequence the total opening of the fissure and the separation of the two parts of
structure.

The influence of the mesh and the boundary conditions is also studied.

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SSNV173 Barreau fissured with X-FEM

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Problem of reference

1.1 Geometry

The structure is a right with low square and healthy parallelepiped. Dimensions of the bar
(see [Figure 1.1-a]) are: LX = 5, LY = 5 and LZ = 25. It does not comprise any fissure.

The fissure will be introduced by functions of levels (level sets) directly into the file
order using operator DEFI_FISS_XFEM [U4.82.08]. The fissure is present in the middle of

the structure by the means of its representation by 2 level sets LSN and LST (see [Figure 1.1-b]) of which them
equations are as follows:

fissure)

plan

(for

LSN

éq

1.1-1

fissure)

melts

(for

LST

éq

1.1-2

Thus, the bottom of fissure is located to Z = LZ/2 and at X = 10. It is thus located apart from the structure, it
who allows to have the structure completely cut into two.

Geometry of the bar and positioning of the fissure

1.2

Properties of material

Young modulus: E= 205000 MPa

Poisson's ratio:

= 0.3

1.3

Boundary conditions and loadings

The nodes of the lower face of the bar are embedded and a displacement is imposed on those of
the higher face. One wishes to show here the possibility of separating a finite element into two with
X-FEM.

Appear 1.1-a

Appear 1.1-b

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SSNV173 Barreau fissured with X-FEM

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Modeling a: only one finite element

2.1

Characteristics of the mesh

The structure is modelized by only one finite element of type HEXA8. The fissure is thus present at
center of this element by the means of the level sets.

2.2

Boundary conditions

Considering that the nodes close to the fissure, i.e. here the 8 nodes of the mesh are enriched by
ddls additional, the boundary conditions are written a little differently. That relates to
the enrichment of the conventional functions of forms [R7.02.12] by the function Heaviside H (X).

To impose a null displacement on the nodes of the lower face amounts writing a linear relation
between the ddls. For each node, one imposes DCX H1X = 0 (idem according to y and Z) where DCX is the ddl
conventional and H1X the enriched ddl.

For the nodes of the higher face, one imposes a displacement according to Z being worth 10

- 6

and no one following

two other directions, i.e. DCX + H1X = 0, DCY + H1Y = 0 and DCZ + H1Z = 10

- 6

2.3 Resolution

analytical

The solution of such a problem is of course obvious. It is seen well that mechanically speaking, both
left the structure will be detached: the lower part will have a null displacement and the part
higher an overall movement equal to imposed displacement will have (see [Figure 2.3-b]).

In our case, it does not occur anything according to X and y, and according to symmetries of the problem, one can
to study only on one vertical edge (for example that joining the noted nodes N1 and N2).
There are four conditions to satisfy:

· displacement of null the N1 node
· displacement of the point medium lower A

no one

· displacement of the node N2 equal to imposed displacement, noted uz
· displacement of the point medium higher A

equal to uz

According to the formulation, these four conditions are written respectively:

(

)

(

)

dcz

The analytical solution is then the following one: all the ddls according to X and are null there and all the ddls
according to Z uz/2 is worth.

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SSNV173 Barreau fissured with X-FEM

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Results of modeling A

3.1 Values

tested

One tests the values of displacement after convergence of the iterations of the operator
STAT_NON_LINE. It is checked that one finds the values well determined with [§2.3].

Identification Reference

Aster %

difference

DCX for all the nodes

0.00

DCY for all the nodes

0.00

DCZ for all the nodes

0.5E-6

0.00

H1X for all the nodes

0.00

H1Y for all the nodes

0.00

H1Z for all the nodes

0.5E-6

0.00

To test all the nodes in only once, one test the MIN and the MAX of the column.

3.2 Comments

It is noticed that the values of the ddls Heaviside in Z are not null because there is discontinuity of
field of displacement following this direction to the interface.

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SSNV173 Barreau fissured with X-FEM

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Modeling b: Several finite elements

4.1

Characteristics of the mesh

One discretizes the structure in 5 finite elements HEXA8

The nodes on both sides of the fissure are nodes nouveau riches, therefore them
three central meshs having such nodes are they also enriched.
Only the two extreme meshs are conventional meshs having only
conventional nodes.

One will be able to thus impose boundary conditions on the extreme meshs
in the usual manner.

4.2

Boundary conditions

The boundary conditions applied represents the same phenomenon
physics that for modeling A. One embeds the nodes of the face
lower and one imposes a displacement of the nodes of the face
higher:

Lower face (Nodes N1, N6, N11, N16): DX = 0, DY = 0 and DZ = 0
Higher face (Nodes N21, N22, N23, N24):DX = 0, DY = 0 and DZ = uz.
This constitutes the 1st case of loading.

In fact, one takes freedom to move the higher part of the structure according to the three directions,
one will thus choose like 2

ème

case of loading:

Lower face (Nodes N1, N6, N11, N16): DX = 0, DY = 0 and DZ = 0
Higher face: DX = ux, DY = uy and DZ = uz

ux = 10

- 6

uy = 2. 10

- 6

uz = 3. 10

- 6

4.3 Resolution

analytical

The solution of such a problem is of course still obvious. In a way similar to [§2.3], one can
to show that the solution is as follows:

Nodes of the lower face (stage n°0):

DX = 0, DY = 0 and DZ = 0.

Nodes of the stage n°1:

DX = 0, DY = 0 and DZ = 0.

Nodes of the stages n°2 and 3:

DCX = ux/2, DCY = uy/2, DCZ = uz/2,

H1X = ux/2, H1Y = uy/2 and H1Z = uz/2.

Nodes of the stage n°4:

DX = ux, DY = uy and DZ = uz.

Nodes of the higher face (stage n°5):

DX = ux, DY = uy and DZ = uz.

4.4 Functionalities

tested

Controls

DEFI_FISS_XFEM

MODI_MODELE_XFEM

AFFE_CHAR_MECA LIAISON_XFEM

Appear 4.1-a: Mesh

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SSNV173 Barreau fissured with X-FEM

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Results of modeling B

5.1 Values

tested

One tests the values of displacement after convergence of the iterations of the operator
STAT_NON_LINE. It is checked that one finds the values well determined with [§4.3] for the 2 cases of

loadings. One obtains the following table for the 2

ème

case of loading.

Identification Reference

Aster %

difference

DX in N1 (stage n°0)

0.00

DY in N1 (stage n°0)

0.00

DZ in N1 (stage n°0)

0.00

N2 DX (stage n°1)

0.00

N2 DY (stage n°1)

0.00

N2 DZ (stage n°1)

0.00

DCX in N3 (stage n°2)

0.5E-6

0.00

DCY in N3 (stage n°2)

1.0E-6

0.00

DCZ in N3 (stage n°2)

1.5E-6

0.00

H1X in N3 (stage n°2)

0.5E-6

0.00

H1Y in N3 (stage n°2)

1.0E-6

0.00

H1Z in N3 (stage n°2)

1.5E-6

0.00

DCX in N4 (stage n°3)

0.5E-6

0.00

DCY in N4 (stage n°3)

1.0E-6

0.00

DCZ in N4 (stage n°3)

1.5E-6

0.00

H1X in N4 (stage n°3)

0.5E-6

0.00

H1Y in N4 (stage n°3)

1.0E-6

0.00

H1Z in N4 (stage n°3)

1.5E-6

0.00

DX in N5 (stage n°4)

1.0E-6

0.00

DY in N5 (stage n°4)

2.0E-6

0.00

DZ in N5 (stage n°4)

3.0E-6

0.00

DX in N21 (stage n°5)

1.0E-6

0.00

DY in N21 (stage n°5)

2.0E-6

0.00

DZ in N21 (stage n°5)

3.0E-6

0.00

One does not test with each time only one node of each stage.

5.2 Comments

It is noticed that the values of the ddls Heaviside in X, y and Z are not null because there is discontinuity
field of displacement following these three directions to the interface.

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SSNV173 Barreau fissured with X-FEM

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Modeling C: Several elements and interfaces leaning

6.1

Characteristics of the mesh and the interface

One considers a structure of dimensions LX = 5 m, LY = 5 m and LZ = 25 Mr. This structure are
discretized with 5 finite elements HEXA8. One is interested in a plane interface of normal

1
1

passing by point A of co-ordinates (5, 5

, 5). [Figure 6.1-a] a zoom of the 2 shows

ème

element

where the trace of the interface is represented in red.

Appear 6.1-a: Mesh C and zoom

The interface is characterized by the level set normal having for Cartesian equation:

lsn

Note:

The parameter

a strong influence on the problem has. If 0 or 1 are worth, then point A coincides

with a node, and the interface passes by this node. If

is nonnull, but small in front of 1, the interface

will separate the element into 2 part, very different volumes. In this situation, enrichment

N9 node by the Heaviside function becomes almost useless, and led to very small pivots

during the factorization of the matrix of rigidity. That results in a significant loss of
a number of decimals and with a false result. For

= a 0.1 (either point A accounting for 10% of

the edge), one loses already 8 decimals (default value causing a fatal error) and for

0.05, one lose 10 decimals. The idea thus consists in not enriching the N9 node by the function

Heaviside when such cases arise. An algorithm of detection was set up, based

on the volumetric ratio for an element crossed into two. This problem makes it possible to test the good
operation of the algorithm, when the parameter

becomes small.

In the continuation, one will take

= 0.02. This value leads to the loss of 13 decimals at the time of

factorization (before the installation of a special processing).

The algorithm will be validated if calculation proceeds normally, without loss of decimals in

factorization. One will check also solution displacement in a selected node.

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SSNV173 Barreau fissured with X-FEM

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6.2

Boundary conditions

The boundary conditions are the same one as those of modeling B. One embeds the nodes of
the lower face and one impose a displacement of traction on the nodes of the higher face:

Lower face (Nodes N21, N22, N23, N24): DX = 0, DY = 0 and DZ = 0
Higher face (Nodes N13, N14, N15, N16): DX = 0, DY = 0 and DZ = 10

- 6

6.3 Functionalities

tested

The determination of the nodes where Heaviside enrichment is superabundant is carried out in
order MODI_MODELE_XFEM [U4.44.11] and the cancellation of these ddls is carried out by the control
AFFE_CHAR_MECA [U4.44.01] with key word LIAISON_XFEM=' OUI'.

Controls

MODI_MODELE_XFEM

AFFE_CHAR_MECA LIAISON_XFEM

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SSNV173 Barreau fissured with X-FEM

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Results of modeling C

7.1 Values

tested

The good course of calculation makes it possible a priori to validate the case. Moreover, one tests the values of
displacement with the N1 node of co-ordinates (0, 5, 0) as well as contact pressures (LAGS_C). One
make sure that the displacement of the N1 node is correct and that contact pressures all are
null (loading of traction).

Identification Reference

Aster %

difference

MIN (LAGS_C) 0.00

0.00

MAX (LAGS_C) 0.00

0.00

DCZ in N1

5.E-7

0.00

H1Z in N1

5.E-7

0.00

Recall:

DCZ is the component according to Z of the conventional ddl

H1Z is the component according to Z of the enriched ddl

Summaries of the results

The objectives of this test are achieved:

It is a question of validating the taking into account of enrichment by the Heaviside function of
conventional functions of form.

Moreover, modeling B allowed the validation of the operand LIAISON_XFEM which allows

to remove the ddls nouveau riches in excess.