Contribution To The Readjustment Of Mesh On 2D Structures

Contribution To The Readjustment Of Mesh On 2D Structures

1A. Sayaou, 1Ntamack G.E, 1C.A. Moubeke, 1E. Kelmamo Sallaboui, 2S. Charif DOuazzane

1 Groupe de Mécanique et des Matériaux, GMM, Département de Physique, Faculté des Sciences, Université de Ngaoundéré, B.P : 454 Ngaoundéré, Cameroun,

2Laboratoire de Mécanique, Thermique et Matériaux, LMTM, Ecole Nationale de lIndustrie Minérale, ENIM, B.P. 753 Rabat, Maroc,

Corresponding author: MOUBEKE Claude Armand

Abstract: Before any achievement of a piece, we

technology

called

r-adaptation,

which

start a design on a computer. At this stage, we

allows changes in the position of knots of

numerically solved the mechanics milieu equations

the mesh. While maintaining the same

using the most common resolution technique

number of degrees of freedom [4, 5, 6].

known as finite element method. This technique is

an approach method set up with some

This strategy of mesh adaptation driven by

approximation errors which are important to

an estimator of error of type ZZ will be

analyze, for the best feet of the solution. This

presented

and applied to

two

different

contribution is a presentation of one of the analysis

structures in order to choose for each

techniques of these errors including the r-

structure the optimal mesh where the errors

adaptation and its effectiveness through its

application to two structures in two dimensions. Key words: structures, calculation finite element method mesh, errors analysis.

of discretization in areas of interest will be decreased.

II MATERIALS AND METHODS

To solve

a problem in

mechanics of

I.INTRODUCTION

continuous milieu we

have to determine

The r-adaptation of mesh (gearing) aim is

the unknown

moving field

u which

to improve

the

accuracy of

approximate

components are ui , and the unknown field

solutions obtained by the finite element

method [1] In principle; this method helps

of stress

with its components ij .Under

assessing the gap between the results of the

the effect of applied strength F d (area

simulations and the correct answer.

strength) and f

(volume strength).

Therefore, the control of the discrepancies

These sizes

verified the equilibrium

between the results of simulations and

equations [7]

reality remains a crucial issue in all

branches of engineering sciences. The

ij , j + f i =0 in

(1)

hypothesis, the error decreases if the step

of mesh (gearing) decreases, the adaptation

of mesh is to refine or not meshing locally

n = F d on

F

F

(2)

in order to meet the desired criteria. Two methods are commonly used: The increase in the number of nodes (knots) in the mesh,

with n the normal external vector F .

The comportment relations:

the

h-adaptation

and

the

increase of

the

ij = Cijkl kl

(3)

degree of

the

polynomial form functions,

p-adaptation [2, 3]. In this contribution, we propose to study a third adaptation of mesh

The discretized problem by finite element

The stress are determined by less

method consist of finding uk , k

square method using the formulation:

solutions of

u

= N u e

(4)

1 ~ k 2

k k

Where:

J =

2

d

(9)

uk are the approached shifts;

k

k

u e are the known shifts in levels of mesh

knots

Hence:

J = 1 T

– T b + c

(10)

N are interpolation

functions

which 2 A

verified the relation:

K U = F

(5)

The matrix

A and

the

vector b

are

[K], being the stiffness matrix;

numerically

U , the stiffs vector (of displacements);

Calculated by the finite element method

F , the external strength vector.

Strategy, using elements of reference, points

The gap between the approached solution

of Gauss and assembling matrix technical.

and the exact solution is determined by the

The minimization of the latter expression

method based on consistency flaw called

returns to solve the following system of

smoothing method [8, 9].

II.1.Evaluation of errors

In general, the exact solution is not known.

equations:

A = b

(11)

However, an estimate of the error .can be calculated Several estimators exist but we

The component of the smooth stress xx is

therefore:

xx

xx

used

an estimator simple

to implement

called ZZ1.The relative

global error

~xx

= N T

(12)

equal to [5]:

2

2

2

2

u ex u k

ex

ex k k

An estimation of

relative error

on these

2

2

2 =

= a u , u a u , u

stresses called is evaluated by the

(6)

with

u ex

au ex , u ex

relation:

~T C1 ~d T C d

au ex , u ex = T u ex C u ex d

(7)

2 =

~T C1 ~d

The exact stress

ex

is unknown but the

idea is to substitute

the

exact stress by

(13)

smoothing stress ~ .This new

approximation of stress ~u k is based on

This relation can also be written as:

n elts

shape functions [8, 9 10]. If Is a vector

2 = 2

(14)

which contains stress at mesh knots, then

~ is equal to:

e

e1

The quantity e

represent the contribution

~= N T

(8)

of the element e to the relative global .error The calculation is repeated by adapting the mesh (gearing) until the error becomes less

than

the

limit

fixed

error

by wished

precision. In our study we have used the

strategy

called

r-adaptation of

mesh

[9,

11].

Here

the

adaptation is

done by

reducing

the

waist

of elements of

the

interested zone. The fineness of the mesh

element is

obtained

by the

stiff

of mesh

knots.

The

degree

of freedom of

the

interpolation functions remains the same,

Figure 1-c: Final mesh

but the position of knots is optimized. [9.]

1. RESULTS AND ANALYSIS

   Figure 1: Numerically results of one full

   We applied this technic on two structures:

   rectangular plate meshed by 16

   one rectangular full plate and one

   triangular elements

   rectangular plate

   drilled a hole in the

   Since it is

   possible

   to bring the

   discret

   middle. By these two applications, we

   continuous gap to the level fixed in

   presented the r-adaptation strategy to be

   advance, provided they refine enough the

   used to solve a problem of concentration of

   mesh, we moved the knots of the areas of

   relative discretization error on the

   interest so as to reduce the surface of

   interestedzone element. The mesh is type Delaunay [12].The results obtained are the following:

   elements in this area [4, 5]. The relatives errors by element such as the last figure 1 shown are equidistributed in the whole of

   the

   mesh.

   The

   overall

   relative

   error

   changes

   from

   9.649%

   before

   the

   readjustment of mesh to 8.773%.after. The maximum elementary error contributed to 10%.decrease.

   The

   second

   application is

   a rectangular

   plate with a circular hole. The plate is in a state of plan stress. The radius of the hole

   is supposed

   to be

   small

   before

   the

   Figure1-a: Initial mesh

   dimension of the plate. The problem being symmetrical about x=0 and y=0 plans, we

   just have to model a quarter of the piece as

   Figure1-b: Intermediary mesh

   presented in different mesh sizes in figures 2a.-, 2b.- and 2c.

2. CONCLUSION

   In this

   contribution,

   the

   aim

   was to

   improve

   the

   accuracy of

   approximate

   solutions obtained by adjustment of mesh

   on two

   planar

   structures,

   using

   the r-

   adaptation of mesh which is to change the

   position of

   the

   initial

   mesh

   knots

   while

   Figure2-a: Initial mesh

   retaining

   the same

   number of degrees of

   freedom.

   This

   technic coupled

   with

   the

   estimator of error of type ZZ on the finite

   element

   method

   solutions

   has

   allowed to

   equi-distribute

   the

   discretization

   error on

   areas where these errors are high.

   For the full plate, the overall error changes

   from

   9.649%

   before

   the

   readjustment of

   mesh

   to 8.773%.For

   the

   perforated plate,

   the

   maximum

   error of

   an element

   that

   contributes most to the overall error goes from 6% to 4%.

   Figure 2-b: Intermediary mesh

   These

   results

   suggest

   that

   for

   both

   structures,

   the

   r-adaptation

   allow

   improvement of the accuracy of solutions by reducing the overall error.

3. REFERENCES

[1] Eric Florentin, Lionel Gendre et Julien Waeytens: «La maîtrise de l'erreur due à la discrétisation par éléments finis», LMT-Cachan.

Figure 2-c: Final mesh

[2]

A.Couët:

«Adaptation dé

maillage

anisotrope

Figure

2: Numerically

results

on one

méthode pleinement optimale basée sur un estimateur derreur hiérarchique en dimension 3 ». Mémoire 2011.

rectangular plate with a circular hole

1. Carnet, J., P. Ladevèze and D. Leguillon: « An

   meshed by 16 triangular elements.

   The overall relative errors of the elements where the r-adaptation was made decreased

   optimal mesh procedure in the finite element method. », Proc. of the Third Conf. on Math. of F.E. and Appl, 1981.

2. BabuSka, T. Strouboulis, C.S. Upadhyay et S.K. Gangaraj: «A posteriori estimation and adaptative control

   from 7.63% to 7.56%.The effect of

   of the pollution error in the h-version of the finite element

   pollution error cause increase in errors in some elements of the mesh. After several

   method», International Journal for Numerical Methods in Engineering, 38: 4207-4235, 1995.

   iterations, the final or optimal mesh is

3. BabuSka, T. Strouboulis, C.S. Upadhyay, S.K.

   retained. The results show that this mesh gives a satisfactory equi-distribution of the

   Gangaraj et K. Copps: «Validation of a posteriori error estimators by numerical approach», International Journal for Numerical Methods in Engineering, 37: 1073-1123,

   error

   by element.

   Indeed,

   we also

   noted

   1994.

   that the maximum error of an element that

4. X. Desroches: «Estimateur d'erreur de Zhu-

   contributes most to the overall error

   Zienkiewicz en élasticité 2D.», Clé: R4.10.01, Révision:

   decreased from 6 to 4%.

   3548, Date: 03/05/2010.

5. Dompierre, J., Vallet, M.-G., Bourgault, Y., Fortin,

   M. et

   Habashi,W.G.:

   «Anisotropic

   mesh

   adaptation:

   Towards user-independent, mesh-independent and solver-

   independent

   CFD.

   Part

   III:

   Unstructuredmeshes».

   International Journal for Numerical Methods in Fluids, 39, 675702.2002.

6. A.

   Alla, M.

   Fortin, F.

   Hecht,

   Z. Mghazli :

   «R-

   adaptation de

   maillage

   par

   lestimateur

   derreur

   hiérarchique.» Revue ARIMA, numéro TAM TAM'05 2006.

7. Zienkiewicz O.C., Zhu J.Z: «The super convergent patch recovery and a posteriori error estimates Part 1: the recovery technique», Int. Journal for Num. Met. in Eng., vol 33, 1331-1364, 1992.

8. Bernardi, C., Metivet, B. et Verfurth, R. (2001) :

   « Analyse

   numérique

   dindicateurs

   derreur ».

   P.-L.

   Georges, éditeur, Maillage et adaptation, Hermés, Paris, Mécanique et Ingénierie des Matériaux, chapitre 10. 251 278.

9. Zienkiewicz O.C., Zhu J.Z.: «The super convergent patch recovery and a posteriori error estimates – Part 2: error estimates and adaptivity». – Int. Journal for Num. Met. in Eng., vol 33, 1365-1382, 1992.

10. P.-L. George et H. Borouchaki : «Triangulation de Delaunay et maillage -applications aux éléments finis», Paris : Hermès, 1997.