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Quadrangular plane element under integrated stabilized


Date:
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Organization (S): EDF-R & D/AMA
Handbook of Référence
R3.06 booklet: Machine elements and thermal for the continuous mediums
Document: R3.06.10
Quadrangular element at a point of integration,
stabilized by the method “Assumed Strain)

Summary:

Despite everything the potential of the linear isoparametric quadrilateral element for calculation by finite elements,
the application of a traditional bilinear formulation to define its field of displacement led to
poor results. If under integration of the element allows to improve its performances, it makes
however to appear parasitic modes which make calculations unstable. This document shows them
principal stages of a method of stabilization of calculations named “assumed strain method” and explains
the way in which it was established in code Code_Aster the computer. Many results resting on
the stabilized element are compared and commented on in order to conclude on the performances from the method.

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1
Setting at fault of the element 2D QUAD4:

In order to propose the need for a more powerful plane element, we carried out calculations
being based on two cases tests, which make it possible to highlight blockings of the element
isoparametric quadrilateral with 4 nodes.

1.1
Case test n°1

The first series of calculations results from the modeling of a beam fixed and subjected to an effort
of shearing fsy at its loose lead [Figure 1.1-a]. This beam has the characteristics
material following: E = 100, = 0.4999. This case test takes as a starting point that written in [bib4] (assumption of
plane deformations).


y

Boundary conditions:



Displacements on AD:


D
C

Ux (A) = Uy (A) = 0

Ux (D) = 0

fsx = 8 * L * y/B ²

fsx

X
O

fsy
B
Efforts on BC:



Fy = fsy = 1 - 4 * y ²/B ²

Fx = 0


With

B


Geometry:


L

L = 100

B = 50
Appear 1.1-a

We carried out this calculation seven times by multiplying the number of meshs on each edge by two
each time:

Etc

1
2 X 2
4 X 4

(A Number of meshs on an edge)

With the exit of each calculation, we brought the arrow out of C with that closer to the solution
“exact” of the theory of the beams of Timoshenko.

[Figure 1.1-b] shows us that the convergence of calculation towards the theoretical solution is largely
insufficient taking into account the number of meshs used for modeling.
However, by carrying out calculations in plane constraints, we note that the results of
calculation converge towards the theoretical solution satisfactorily ([Figure 1.1-c], convergence
quadratic).
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Convergence of the Q4 element
(Deformation planes)
Plane deformations
Exact solution (Tim bone henko)
4,5
4
3,5
3
He Vc
2,5
E
C
Fl
2
1,5
1
0,5
0
0
10
20
30
40
50
60
70
NR

Appear 1.1-b: convergence of plane element QUAD4 in deformation

CONVERGENCE OF THE Q4 ELEMENT
(Plane Constraints)
Exact solution (Timoshenko)
5
4,5
4
3,5
ARROW Vc
3
2,5
0
2
4
6
8
10
NR

Appear 1.1-c: convergence of element QUAD4 in plane constraints
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This stress is said to dominant inflection. Through the first series of calculations, we put
in obviousness impossibility for the QUAD4 of representing the modes of deformation in inflection
[Figure 1.1-d] in plane deformation and for a coefficient close to 0,5. This results in one
excessive rigidity of the element due under the terms of shearing of the operator discretized gradient.

QUAD4
QUAD8

(important shearing)

Appear 1.1-d

1.2
Case test n°2

The second series of calculation is based on the modeling of a notched sample [Figure 1.2-a]
solicited by an imposed displacement.
Imposed displacement: Dy = 1

Characteristics material:



= 0.4999
5

E = 200 Gpa



y = 0,1

AND = 10


Criterion of plasticity:
5


Von Mises, plasticity with
linear isotropic work hardening.



1

E

T

y

E

Appear 1.2-a

For the display of the results, each QUAD4 was cut out in 4 zones containing each one a point
of Gauss. It is the value of the constraint in this point which we display.
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Reference solution: QUAD8_INCO
(element treating the incompressibility perfectly).

yy

Solution QUAD4:





yy

The results show that the QUAD4 converges only at the price of strong oscillations of constraints with
center of each element. If these oscillations make it possible the element to put in agreement its
nodal displacements and its plastic deformation with constant volume, they return the results
unrealistic persons.
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2
Under integration of the QUAD4

Despite everything the potential of the isoparametric quadrilateral element for calculation by elements
stop, the application of a traditional bilinear formulation to define its field of
displacement, leads to poor results. This is explained by various reasons:

· it has an excessive rigidity (“lock”), at the time of a stress of which the cross-bending part
is important.
· the traditional bilinear formulation of the field of displacement is very sensitive to the distortion
grid and presents severe “
locking
” when one applies it to a material
incompressible.

A solution with these problems of numerical blocking consists in calculating the matrix of rigidity by
the intermediary of a reduced integration. The principle of this method is to consider at the time of
numerical diagram of integration less points of integrations than one usually should not any for
to evaluate the matrices of exact rigidity of the element. On the basis of isoparametric element QUAD4
[R3.03.02], one modifies the number of points of integration as well as the weight and the co-ordinates of these
the last, to create the under-integrated element which we will name in this document: QUAS4



QUAD4 QUAS4



A number of points of integration: 4

A number of points of integration: 1



Weight of each point: 1

Weight of the point: 4






1
1

Co-ordinates: (X, y) = (0,0)
Co-ordinates: (X, y) = (+/-
, +/-
)
3
3

2.1 Formulation

In the center, the operator discretized gradient used to calculate the matrix of rigidity is form
following:
T
B
0
X

B = Bc =
T
0
B y







éq 2.1-1
T
T
B y B X
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With
T

bx =
NR
NR, X (0) =
= = 0
X
éq
2.1-2
T

by =
NR
NR, y (0) =
= = 0
y

and NR represents (N1, N2, N3, N4), the vector of the functions of form. Let us recall that the vectors
B represent the simplest shape of the operator discretized gradient under-integrated introduced by
Hallquist [bib1] and which is based on the evaluation of derived from the isoparametric functions of form with
the origin of the reference frame (,)

That is to say:

btx = 1 ([y2 - y4), (y3 - 1
y), (y4 - y2), (1
y - y3)] = Constant on


element
2A

éq
2.1-3
bty = 1 ([x4 - x2), (1
X - x3), (x2 - x4), (x3 - 1
X)] = Constant on


element
2A

With = (1/2). (x2 - x4) (.y3 - y)
1 + (x3 - X)
1 * (y4 - y2): Surface of the element
The matrix of rigidity is written:

T
K E = Kc = A. Bc. C. Bc éq
2.1-4
C being:

· that is to say the elastic matrix of behavior for calculations in elasticity
· that is to say the tangent matrix for the plastic designs. Let us note that during such calculations, it is
the integration of the law of behavior at the point of Gauss (in the center in our case) who
determine the value of the coefficients of C.

Finally, the internal forces are written:

T
F
= Ke.
int
U = Bc. C éq
2.1-5
U: Vector of nodal displacements.
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2.2
Failure of calculation in Aster: parasitic modes

The calculations launched on the case test n°1 with such an element fail. The stage of the calculation which consists with
to reverse K E to determine displacement nodal cannot be crossed. Indeed, in the center,
stamp rigidity is singular. By displaying the core of K E, we note that his dimension
is not any more three but five. Two vectors were added to the core and returned the inversion of
stamp impossible rigidity. Appearance of these two additional vectors in the core of
K E is directly related to the fact that we choose the center like only point of integration.
In other words, there are two fields of nodal displacements others that the fields of displacement
corresponding to the rigid movement of solid cancelling the internal forces. These modes represent
modes out of sand glass of the QUAD4 [Figure 2.2-a]. Thereafter, one of the stages of stabilization will consist
to enrich the operator gradient discretized in order to return K E invertible.

Modes of rigid solid
Modes out of sand glass

(hourglass modes)

Dim de Ker [K

E] = 3 + 2

Appear 2.2-a

2.3
Graphic interpretation of the problem involved in under integration

The problem of the under-integration of the QUAD4 is related to its modes of nodal displacements out of sand glass
(stress in inflection). [Figure 2.3-a] shows us that, on such modes, the co-ordinates of
center remain unchanged. If this is in agreement with the theory of the beams (i.e in pure inflection, xx =
0 on neutral fiber), the classical theory of the finite elements do not make it possible to differentiate deformed state
and not deformed of an element in such a case. Therefore these modes are also called
modes with null energy, which on the level of Code_Aster, make calculations unrealizable.








Appear 2.3-a

This problem originates in a value of the nonsignificant deformation in the center of
deformation of the QUAD4. The next chapter will describe the step which will allow us
to calculate the deformation of the QUAD4 whatever the modes of displacement of these nodes.
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3
Stabilization of the QUAD4 at a point of Gauss

The stabilization of the QUAS4 takes place in two stages:

1) To enrich the operator gradient discretized Bc and to thus allow to calculate the energy of
deformation related to the modes of displacements out of sand glass (hourglass modes);
2) To interpolate a field of deformation/constraints allowing to account for
deformation/of the constraints on the whole of the element while integrating the QUAD4 into
center (assumed strain stabilization).

3.1
Variational principle of the problem

That Ci is drawn from the weak form of the variational principle of Hu-Washizu:

(
T
, &,) =
&.
D +
T

.

T
ext.
& D
D
&
F
éq
3.1-1



(S -) -
.
= 0



With
& (X, T) = B (X) .d & (T)

Stabilization “Assumed strain” is based on the fact that the postulated constraint is selected
orthogonal with the difference between the symmetrical part of the gradient speed and the rate of deformation.
Consequently, that enables us to write:

T
D
&.
T
B.
D -
T
D
&. ext.
F
= 0


E

And thus that:
F int =
T
B. (&)
D

E

3.2
Enrichment of the operator discretized gradient

To enrich the operator gradient discretized Bc consists in making a new operator B of it while adding to him
a third component which, like bx and by, is a vector of IR4. However, like
the initial operator Bc correctly calculates the gradient of the linear fields of displacement,
new component must be orthogonal with the latter. Stages of the calculation of this operator
nouveau riche are detailed in the paragraph [§1.2] of the document joint entitled: “Report
bibliographical “.

Note: In this report, the new operator B connects the tensor rate of deformation & and it
vector nodal speeds d&i. This formulation allows us in the continuation of the report of
to reason in term of increments of displacement, and consequently to deal with problems
incrémentaux, carried out on several steps of time. For the elastic designs (solutions obtained in
a step of time) we formulate the problem starting from nodal displacements: the operator B connects
then tensor of the nodal deformations and displacements ui.
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The new operator B is based on the expression of the field of displacement of the QUAD4 written by
Belytschko and Bachrach [bib2]:

ui = (T
T
T
T
+ X B
. X + y B
. y + H) ui
éq
3.2-1
And is written:
T
T
B
H
0

X +

, X

B = Bc + Bn =
T
0
B y +
T
H, y
éq
3.2-2
T
T
T
T

B y + H, y
B X + H, X

With:
= 1/[
4 T - (tT .x) bx - (tT .y) by], = 1/[
4 H - (T
h. X) bx - (T
h. y) by]
1
btx =
([y2 - y4), (y3 - y1), (y4 - y2), (y1 - y3)]
2A

1
bty =
([x4 - x2), (x1 - x3), (x2 - x4), (x3 - x1)]
2A
H =


, are the co-ordinates of reference. ui is the vectors of nodal displacements and H are them
values taken by the function H with the four nodes.
T
D = [ux, uy]
Let us note what can be expressed directly according to the nodal co-ordinates:

X (
2 y3 - y)
4 + X (
3 y4 - y)
2 + X (
4 y2 - y)
3


1 X (
3
1
y - y)
4 + X (
4 y3 - y)
1 + X (
1 y4 - y)
3
=
éq
3.2-3
4 X (
4
1
y - y)
2 + X (
1 y2 - y)
4 + X (
2 y4 - y)

1
X (
1 y3 - y)
2 + X (
2
1
y - y)
3 + X (
3 y2 - y)
1

The shape [éq 3.2-2] of the operator B is equivalent to that of the QUAD4. However this writing
particular of the operator allows to differentiate the terms to be integrated into the center and the terms of
stabilization. It is only while intervening on the value of these terms of stabilization that us
let us can improve the performances of the element.
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3.3
Interpolation of the field of deformation:

The general shape of a field “assumed strain” within a QUAD4 is given by Belytschko and
Bindeman [bib3]. It is following form:


Q E H
Q E H


+


X (center)
X 1, X + y 2, y
+


X (center)
X (stab)

assumed strain

=
+ Q E H
Q E H


y (center)
X 2, X +

y 1, y
=
+
éq 3.3-1


y (center)
y (stab)

2
+ Q E H
Q E H
2
2
xy (center)
X 3, y + y 3, X

+
xy (center)
xy (stab)




With:
qx =. ux
qy =. uy

and 1
E, e2, 3
E which vary according to the consideration physics of each author [Tableau 3.3-1].
Each triplet of values characterizes an element and gives place to a particular interpolation of
deformation:

Element
e1 E 2e3
QUAD4
1 0 1
ASMD
1/2 - 1/2 1
ASBQI
1 - 0
ASOI
1 - 1 0
ASOI (1/2)
1/2 - 1/2 0
Table 3.3-1
Thus we can deduce the expression from the operator discretized gradient rising from the field from
deformation supposed ((stab)) element. We note this new Bn operator.


T
H X
0
,

QUAD4:
Bn = 0
T
,
H y

T
T
,
H y
,
H X

T
H, X
-
T
H

, y

ASBQI:
Bn = -
T
T
H, X
H, y
éq
3.3-2




0
0


1
T
1
T

,
H X
-, hy
2
2

ASOI (1/2):
Bn = - 1
T
1
T
H
X
H y
2,
2,


0
0





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This writing enables us to test the various elements easily.
The matrix of rigidity is written then in the following way:

stab
K E = Kc + K

éq
3.3-3

With:
T
K C = B C B

D

C
C

=
A.B

D.C. Bc éq
3.3-4
E
And

K stab = T
Bc

C Bn D + T
Bn
B

C C D +
D T
Bn
B

C N D


E
E
E

éq
3.3-5
4
=
(
JAC I) (T
.
B. C. B (I)
C
N
+ T
B (I). C.
N
Bc + T
B (I). C. B (I)
N
N
)
i=1
Finally we calculate the forces intern in the following way:

int
F
= B. (C. ((center) + (stab)))
4
éq
3.3-6
=
(
JAC I). ((
Bc + Bn (I)

)
(

.

C (
. C + stab (I))))
i=1

Notes:
Although the calculation of K stab requires a sum on the four points of Gauss, the integration of
law of behavior which determines the value of the terms of C, is carried out in the center. In addition, them
equations [éq 3.3-4] and [éq 3.3-6] shows us that calculations remain relatively bulky. One
solution with this problem (not yet established in Code_Aster) consists in carrying out calculations in
placing itself in a reference mark turning with the element (cf [§3.4] of the bibliographical report). This has
for advantage:

· to remove the calculation of the cross terms: Bc. C. Bn and Bn. C. Bc;
· better a processing of blocking in transverse shearing;
· a writing of the law of behavior adapted better to the problems including of
geometrical non-linearities.

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4
Integration of the element in Code_Aster

Element QUAS4, has two families of points of Gauss. The first family consists of one
not being in the center and whose weight is worth four. Elementary calculations are done naturally
with the first family. The calculated options are identical to that calculated by the QUAD4. One
thus calculate in the center of the element the internal constraints and variables, as well as the behavior
tangent. Within the framework of a post processing, each element presents constant fields.

The second family is identical to that of the traditional QUAD4 and comprises 4 points of Gauss. She is
used to calculate the matrix of stabilization. To store the forces of stabilization, one adds to
stress field at the point of Gauss 6 components.

This element is activated by choosing modelings “C_PLAN_SI” or “D_PLAN_SI” in
AFFE_MODELE for meshs QUAD4. Only calculations in small deformations are possible. It
remain an important limitation: in version 7.2, two types of stabilization are programmed:
ASBQI and ASOI (1/2), but are not accessible by a key word in the command file. It
is necessary to activate them to modify parameter PROJ in routine NMAS 2D.

5
Description of the contribution of element QUAS4

In order to evaluate the contribution of element QUAS4, we used it to carry out calculations of the cases tests
number one and two (cf [§3]).

5.1
Case test n°1 (SSLP106)

Convergence of calculation towards the theoretical solution
(calculation in elastic strain with = 0,4999)
ASOI (1/2)
Timoshneko solution
QUAD4 and QUAS4
4,5
4
3,5
Vc
3
2,5
2
0
1
2
3
4
5
6
7
8
9
10
NR

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Among the elements appearing in [Tableau 3.3-1], only element ASOI (1/2) could be established with
success. Moreover calculations converge only for the case test of the right beam, modelled by one
regular grid. Same the calculations carried out on a divergent voluntarily deformed grid
[Figure 5.1-a].








Regular grid
Irregular grid

Convergence of calculation
Divergence of calculation
Appear 5.1-a

Concerning the calculations carried out on a regular grid, we note a clear improvement of
results. The theoretical solution is reached with a network of sixty four elements. This us
reasonably allows to say that the blocking of element QUAD4 disappeared. This test corresponds to
test SSLP106 of the base of tests of Code_Aster.

5.2
Case test n°2 (SSNP123)

Recall of the problem: With a mesh of the type QUAD4, calculation reveals the important ones
oscillations of constraints on the grid. We will compare the exits results of the grid
QUAD4 with those of grid QUAS4:

Calculation in elasticity in the plan: isovaleurs of
yy












· QUAD4
· QUAS4
· = 0,4999
· = 0,4999


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Plastic designs in the plan: isovaleurs of
yy

· QUAD4
· QUAS4

· = 0,3
· = 0,3

Each computing time indicated is an average carried out on three launching.
For the visualization of the results, each QUAD4 was divided into four zones containing each one one
not Gauss. The QUAS4 as for them are not redivisés.
The tests carried out are voluntarily severe (near to 0,5). Indeed the goal here is to reach them
limits of element QUAD4 in order to note the performances of the new element.

5.3 Comments

We note through this series of calculations that the results obtained with elements QUAD4
comprise strong oscillations of constraints. Results resulting from calculations resting on
QUAS4 do not have almost any more oscillations. When they appear, they are very localized.
That it is for the QUAD4 or the QUAS4, displacement calculated with node A are identical.
Indeed, the operator gradient discretized of the QUAS4 is deduced from the field of deformation of the QUAD4
(cf [éq 3.2-2]). Consequently, that it is with a grid with one or the other of these elements, the structure
have same rigidity.

Taking into account the severity of the test and in order to account to us for the quality of the provided results
by the QUAS4, we carried out calculations being pressed on quadratic meshs such as
QUAD8, QUAS8 (QUAD8 under integrated with 4 points of Gauss) or the QUAD8_INCO,
element treating the incompressibility perfectly. The comparison of elements with interpolation
quadratic with elements with linear interpolation little direction has. Such tests were carried out
in order to have a reference solution and they allowed us to purely make an assessment
qualitative on element QUAS4.
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Results of reference: calculations in elasticity in the plan with quadratic elements

· QUAD8
· QUAS8

· = 0,4999
· = 0,4999

· Dy (A) = 6,0E-2
· Dy (A) = 6,01E-2

· Duration of calculation: 5,6 S
· Duration of calculation: 3,9 S

· QUAD8_INCO

· = 0,4999

· Dy (A) = 6,01E-2

· Duration of calculation: 5,5 S

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Plastic designs with quadratic elements

· QUAD8
· QUAS8
· = 0,3
· = 0,3
· Dy (A) = 1,01E-1
· Dy (A) = 1,01E-1
· Duration of calculation: 17 S
· Duration of calculation: 13,1 S
· QUAD8_INCO
· = 0,4999
· Dy (A) = 1,01E-1
· Duration of calculation: 34,4 S
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A refinement of the grid for each calculation at summer carried out. This refinement consists in doubling it
a many meshs present on each stop notch. We summarize the results in
following tables:

Elastic design Sol ref.: Depl.
deformation plan
6,01E-02




Node A
= 0,4999







Numbers
504
2016
8064
Meshs

Time calculating Depl Noeud A Time calculating Depl Noeud A Time calculating Depl Noeud A
QUAD8_INCO
5,5 6,01E-02
18 6,01E-02


QUAD8
5,6 6,00E-02
15,5 6,01E-02


QUAD8 IF
3,9 6,01E-02
10,2 6,01E-02


QUAD4
3,3 2,70E-02
8,4 3,60E-02


QUAS4
2,72 2,70E-02
5,77 3,60E-02 17,3
4,77E-02

Table 5.3-1: Calculations in elasticity = 0,4999

Calculation
deformation
Ground ref.: Depl.
plane plastic
Node A
1,01E-01




= 0,3







Numbers
504
2016
8064
Meshs
Depl Node
Depl Node

Time calculating Depl Noeud A Time calculating
Calculating time
With
With
QUAD8
17 1,01E-01 75,1 1,01E-01

QUAD8 IF
13,1 1,01E-01 62,8 1,01E-01

QUAD4
6,6 9,32E-02 22 1,00E-01

QUAS4
6,5 9,18E-02 17,33 1,00E-01 133,2 1,01E-01
Table 5.3-2: Plastic designs = 0,3

Calculation
deformation
Ground ref.: Depl.
plane plastic
Node A
1,01E-01




= 0,4999







Numbers
504
2016
8064
Meshs
Depl Node

Time calculating Depl Noeud A Time calculating
Calculating time Depl Noeud A
With
QUAD8_INCO
34,4 1,01E-01 240,8 1,01E-01

QUAD8
21,7 8,84E-02 88,17 1,01E-01

QUAD8 IF
20,6 1,01E-01 87,83 1,01E-01

QUAD4
5,1 2,06E-02 13,07 2,25E-02

QUAS4
4,28 2,06E-02 9,87 2,25E-02 43,75 2,79E-02

Table 5.3-3: Plastic designs = 0,4999
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In addition, we traced the value of the constraint yy along a segment AB crossing the notch
for a basic grid with 504 meshs.

B
With

Value of the constraint along AB

Plastic designs (= 0,3)

350

300

250

] 200

Pa

[
1M 150

[

yy

100

50

0

0
1
2
3
4
5
6

- 50

X-coordinate

QUAS4
QUAD4
QUAD8
QUAS8

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Value of the constraint along AB

Plastic designs (= 0,4999)

1500

1000

500

yy [Mpa]
0


0
1
2
3
4
5
6

- 500

- 1000

X-coordinate

QUAS4
QUAD4
QUAD8_INCO
QUAS8
QUAD8

Comments on the results

Qualitative aspects:

· In spite of the richness of the field of displacement of mesh QUAD8, the majority of the results
oscillations present.
· In spite of under integration of the QUAS8, the oscillations appear on meshs QUAS8
for certain plastic designs.
· Whatever the element used (except for the QUAD8_INCO), under integration remains
essential, particularly for the materials of which approaches 0,5.
· These calculations lead us to establish the following report: the QUAS4 remains stable with respect to
oscillations whatever the parameters of calculation used.

Quantitative aspect:

We always note that the value of the displacement of node A resulting from a grid QUAS4 remains
between 3 and 5 times weaker than the reference solution (slow convergence). This value remains
nevertheless identical to that calculated with a grid QUAD4.

The profiles traced along the notch enable us to affirm that the values of constraints
calculated with a grid QUAS4 are of much better quality than those calculated with one
grid QUAD4. By taking the computing times of the QUAD4, mesh QUAS4 allows a reduction
substantial of the duration of calculation:


Saving of time (time of ref. = time QUAD4)
Grid

504
2016
= 0,3
26% 35%
Elasticity
= 0,4999
18% 31%
= 0,3
16% 21%
Plasticity
= 0,4999
18% 24%

Let us notice in the passing that the savings of time seem to increase with the number of meshs. In
plasticity in particular, this saving of time depends as a large majority on the iteration count of Newton
necessary to the convergence of calculation within each step of time.
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6 Conclusions

In elasticity as in plasticity, that it is for materials of which is worth 0,3 or 0,5, the new one
element QUAS4 remains stable and the always realistic results, without any oscillations of constraints.
This, we noted it, is far from being the case for the QUAD4. The stability of this new element
vis-a-vis cases tests as severe as those presented in this report/ratio is comparable with that of
quadratic element QUAD8 under integrated.

On the other hand, this element with the convergence of a linear element in terms of a number of DDL. It is necessary
thus to net with a sufficient smoothness to collect the gradients of constraints of the solution
sought. This refinement necessary must be put out of balance with the saving of time induced by
under-integration.

On the treated examples, the QUAS4 allowed a saving of time of significant calculation of about 20%
on average for laws of elastic and elastoplastic behaviors. Let us note that these laws are
relatively inexpensive to integrate. Savings of time much more important are awaited
for laws more difficult to integrate.

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7 Bibliography

[1]
OJ HALLQUIST: Theoretical manual for DYNA 3D. UC1D-19401 Lawrence Livermore
National Lab., University off California, 1983.
[2]
T. BELYTSCHKO and W.E. BACHRACH: Efficient implementation off quadrilaterals with high
coarse-mesh accuracy, Comput. Methods Appl. Mech. Engrg. 43 (1986) 279-301.
[3]
T. BELYTSCHKO and L.P. BINDEMAN: Assumed strain stabilization off the 4-node
quadrilateral with 1-point quadrature for nonlinear problems. Comput. Methods Appl. Mech.
Eng., 88:311 - 340, 1991
[4]
J.L. BATOZ and G. DHATT: Modeling of the structures by finite elements volume 2, beams
and plates. HERMES 1990.

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