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COMPUTATIONAL MECHANICS New Trends and Applications S. Idelsohn, E. O?ate and E. Dvorkin (Eds.) ?CIMNE, Barcelona, Spain 1998

ADAPTIVE ANALYSIS FOR THE DIFFUSE ELEMENT METHOD

T. Laouar and P. Villon

Université de Technologie de Compiègne (U.T.C) Centre de Recherches de Royallieu Laboratoire LG2MS, URA 1505 du CNRS Département de Génie des Systèmes Mécaniques Division M.N.M BP 20649, 60206 Compiègne, France e-mail: laouar@utc.fr, web page: http://www.utc.fr/tlaouar e-mail: villon@utc.fr, web page: http://www.utc.fr

Key words: Diffuse Approximation, Diffuse Element, Numerical Integration, Error Estimator, Adaptive Analysis. Abstract. The Finite Element Method (FEM) has been largely required as a tool of spatial discretization of physics equations and engineering sciences. Nevertheless, it presents some limitations and disadvantages. For example, the regularity of approximate functions, and mesh generation requirements. Nayroles et al1 have proposed a method of discretization that preserves the majority of the advantages of the FEM. However it uses only a set of points of discretization easy to modify and provides continuous solutions and continuous successive derivatives. This method is named Diffuse Approximation Method (DAM)1. The DAM is based on a local technique of Moving Least Square Method (MLSM). At least four main forms of this method have been reported : 1) Belytschko et al2,3, 2) Liu et al4, 3) On?te et al5, and 4) Gravete et al6. In this paper, we present a technique of resolution using the Diffuse Element Method (DEM) with adaptive set of nodes. Nodes are generated by a "Quadtree" type decomposition of the area and the adjustment is made with the help of a posteriori knowledge of error estimate.

1

T. Laouar and P. Villon.

1

INTRODUCTION

The local Moving Last Squares Method (MLSM) technique has been studied by a number of investigators such as mathematicians, notably in the framework of the representation, approximation and modelization of surfaces from arbitrary data points. This method allows to construct regular surfaces of class at least C1 , and to adapt to random distributions of points. For the purpose to bring the global approximation problem to a local approximation problem, the weighting has been made by a rapid decrease compared to the distance from evaluation point to nodes. The MLSM is the most recommended method for the approximation of a field at the corners of a rectangular mesh7,8. The aim of this research work is to develop a method or a technique that allows to solve ordinary differential equations and partial derivatives equations (ODE and PDE) without having to use a meshing process. Several works using this technique have been developed. Belytschko et al2,3 developed the method named Element-Free Galerkin Method (EFGM). It’s essential feature is the use of an auxiliary grid, composed only of square elements, that cover the domain of the problem completely. This background grid is used in the EFGM to support numerical quadrature calculations. Liu et al4 have recently proposed a different kind of "griddles" multiple scale methods based on reproducing kernel and wavelet analysis. On?te et al5 focused on the application of the Diffuse Interpolation to fluid flow problems via a standard point collocation technique. Finally, Gravete et al6 have used a posteriori error indicator, in order to minimize the error, and the sensitivity analysis of various involved parameters. All these methods can be considered as Finite Point Methods (FPM)5 . In this paper, we first present some positive aspects of the DAM. Then we introduce the error estimation and the adaptive set of points in DEM. The numerical results indicate the improved rate of convergence and power of the method. A comparison of the results and those obtained by Zienckiewicz and Zhu9 in the context of FEM is made. 2 DIFFUSE APPROXIMATION The basic idea behind the DAM is to construct an approximation Uap(x) for all point x of a function Uex only known at points xi called nodes. The function is then constructed in each point of evaluation x by minimizing the criterion :

J x ( Uap ) = 1

2

i ∈I( x )

∑ Wi (x )( U i ? Uap( x i ))2

(1)

The weighting function Wi(x) represents the contribution of node i in the minimization criterion. The weighting functions allow to render the approximation both local and continuous. The polynomial diffuse approximation Uap(x) of the function Uex(x) is defined

U ap ( x ) =

∑ pj ( x) a j( x ) ≡ pT ( x ) a(x)

j

m

(2)

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T. Laouar and P. Villon.

Inserting (2) in (1), we obtain J x ( a ) = ∑ Wi ( x ) [ p T ( x i ) a(x) - Ui ]2

i n

(3)

The stationarity of J, in equation (3), with respect to a(x) is given by the following linear system A (x) a (x) = B (x) u where A (x) and B (x) are matrices defined as

A( x ) =

n i

(4)

∑ Wi ( x ) p( x i ) pT (x i )

(5)

and

B( x ) = [W1 ( x ) p( x 1 ),..., Wn ( x ) p( x n )]

(6)

The Shape Diffuse Functions Ni (x) ; i = 1, ........n, are given by N i (x ) =

∑ p j (x ) ( A ?1(x ) B(x) ) j i

j

m

(7)

3

DIFFUSE ELEMENT MODEL

In the variational formulation, we use the calculation of the diffuse gradient as well as a numerical integration strategy that we specify next. 3.1 Diffuse Gradient The diffuse gradient is an estimation of the derivative of a function Uap at a point x . It is called pseudo - derivative10 .

δx k δk U ap( x ) = δx k δk N( x ) {U i }

(8)

This operation is equivalent to the differentiation of the polynomial part of Uap only. It does not concern a derivative in the algebraic form because we do not have D(u.v)=D(u).v+u.D(v), this justifies the terminology of "pseudo - derivative" or diffuse gradient. 3.2 Numerical Integration We use approximate integrals with Gauss quadrature on a rectangular mesh independent of nodes. The integration has been made on the quadtrees. The diffuse approximation will be used in each Gauss points of the mesh. The method, represented in Figures 1 and 2, is named diffuse element (Four neighbor nodes are

3

T. Laouar and P. Villon.

represented). The diffuse elements model is a model in which the notion of mesh does not exist. In fact, just a set of points of discretisation and a numerical integration support (independent of this set of points) define this model. The diffuse approximation will be used in each point of integration, with neighbor nodes. We have a decoupling between the approximation and the numerical integration in the discretisation of a variational formulation (see Figures 1 and 2).

Domain of integration sphere of influence Discretization nodes

+ +

+

Integration points

+ +

Domain of integration sphere of influence

Figure 1: Numerical Integration

This model has been recently used by T. Belytschko and his collaborators2,3, under the name " element - free Galerkin method ". The definition of the model does not present meshing difficulties, but, we have other similar problems with the set of points (for example the neighbor configurations). We notes also, that it is not evident to prove the convergence and the stability of this model. In fact, the diffuse approximation being not polynomial by pieces, then the definition of the discrete space (or projection space) is more complicated. The error committed during an approximate numerical integration cannot be evaluated as in finite elements. We can show10 that the linear diffuse approximation is equivalent to a piecewise cubic approximation. Consequently, we can say that the linear diffuse approximation is richer than the linear interpolation. Nevertheless, we will see that it is difficult to show the convergence of the diffuse elements model. Indeed, we have, sometimes, incompatible spurious modes. Note that in finite elements the discontinuities of stress appear at interfaces of elements. On the other hand, in diffuse elements we use interactions between different neighbor nodes of the diffuse element to suppress these discontinuities.

4

T. Laouar and P. Villon.

+

+

Subdomain of numerical integration

+

+

Integration points

Figure 2: Integration points

4

ADAPTIVE ANALYSIS PROCEDURES WITH DIFFUSE ELEMENTS

We have been interested in already existing procedures of error estimates such as those of Zienckiewicz and Zhu9 that we have adapted11 . There exists some differences concerning the shape functions, strategy of refinement and so on...The definition of refinement strategy depends on the diffuse integration and the diffuse derivatives : the adaptive refinement being based on generation points14 . We clarify our strategy in the case of elasticity problems. 4.1 Error norm and error measure The "energy norm " written in the specific case of elasticity is

eσ ? ?2 T -1 ? $ $ = ( σ ? σ ) D ( σ ? σ) d? ? ? ? ?? ?

1

∫

(9)

A more direct measure is the so called L2 norm, which can be associated with the errors in any quantity. Thus for the displacement u, the L2 norm of the error e is

? ?2 T $ ) (u ? u $ ) d? ? e = ? (u ? u L2 ? ? ?? ?

1

∫

(10)

and for the stresses

eσ L 2 ? ?2 T $ ) (σ ? σ $ ) d? ? = ? (σ ? σ ? ? ?? ?

1

∫

(11)

the latter expression is different from the energy norm by the weighting D only. Although all the norms written above are defined on the whole domain, we note that the square of

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T. Laouar and P. Villon.

each can be obtained by summing elements contributions. Thus,

e

2

=

k =1

∑ eσ 2 k

N

(12)

where k represents the evaluation point of the subdomain ? k which union is the domain ? and N is the total number of evaluation points. Indeed, for an "optimal" set of points we shall, generally, try to make these contributions to this square of the norm equal for all domains ? k. Although the absolute value of the energy (or L2 norm ) has little physical meaning, the relative percentage error is

η= e u × 100 %

(13)

The percentage error η can be determined for the whole domain ? or for the subdomain ? k. Similarly, we can define ηL , with respect to the L2 norm in equation (13). 4.2 Error estimate To obtain acceptable results for stresses, resort is generally applied to some techniques in which it ~ is interpolated by the same function as the displacement, is assumed that the stress σ

~ = N~ σ σ?

(14)

~ is in fact a better ~? It is intuitively "obvious" that σ see section 4.3 for the determination of σ . $ and we should use it to estimate the error by approximation than σ

? ?2 ~?σ $ )T ( ~ $ ) d?? eσ = ? (σ σ?σ ? k ? ? ? ? ? k ?

1

∫

(15)

where,

~ : is a continuous stress field obtained by auto-equilibrium 10, 14. σ $ : is a Diffuse Element stress field. σ σ : is a Exact stress field.

4.3 Method based in Diffuse Approximation with equilibrium We consider the boundary value problem in the domain ? which is governed by the straindisplacement equations, the elastic stress-strain law an the equations of equilibrium, which are, respectively

6

T. Laouar and P. Villon.

?CT σ + b = 0 sur ? ? ? sur ? ?σ =Dε ?ε=Cu sur ? ? ?

(16)

In two dimensions, the above are

ε = εx

[

εy

2ε xy ? 0? ? ?? ?y ? ?? ? ?x ? ?

]T

=

strain in column matrix form

(17)

?? ? ? ?x C= ? 0 ? ?? ? ? ?y ?

= strain-displacement operator

(18)

u = ux σ = σx

[

uy

]T = displacements

(19) (20) (21) (22)

[

σy

σ xy

]T = stress in column matrix form

by

b = bx

[

]T

= body forces

D = stress-strain matrix

the conditions on the boundary

u=~ u sur Γ

(23)

In the local derivative extraction technique9 the extracted stress function is expressed in terms of continuous shape function, N(x), and nodal value, s* :

σ * = N(x) s*

(24)

The nodal values, s*, are determined from a moving least squares. The domain ? is subdivided ~ into overlapping domains ? j , j=1,...n (where n is the number of nodes ). The local stress field in

~ ~ ( x ) is interpolated by polynomials in the spatial variables ?j , σ j ~ ( x ) = P( x ) a σ j j

(25)

In one dimension, aj is a vector of coefficient of order p and P is a 1*p matrix, e.g.

7

T. Laouar and P. Villon.

P(x ) = 1, x, x2 ..., x p?1

[

]

2

(26)

The performance of this local projection is here enhanced by adding the square of the residual of the equilibrium equations. The parameters aj are then determined by minimizing

S=

∑ Wi ( P( x i ) a j

i =1

m

- σ h ( x i ))T ( P( x i ) a j - σ h ( x i )) + Wi CT ~ σ j( x ) + b( x ) ~ ?

j

(27)

where m is the number of points xi. The first term in (27) is the diffuse approximation10. The second term in (27) is the norm of equilibrium residual20 and is defined

CT ~ σ j( x ) + b(x ) ~ = ?

j

2

~ ?j

∫ (C

T

P( x ) a j + b( x )

) (C P( x) a + b( x)) d?

T T j

(28)

The minimization of S then gives

?S =0 ?a ? ( M + E) a = f1 + f2

(29)

where

M=

∑ Wi PT (x i ) P(x i )

i =1 T T T j

m

(30)

E=

~ ?j

∫ (C P( x )) (C P( x )) d? ∑ Wi P T (x i ) σ h (x i )

i =1 ~ ?j m

(31)

f1 =

(32)

f2 = ?

∫ (C P( x ))

T

T

b( x ) d? j

(33)

4.4 Refinement strategy The refinement strategy will, of course, depend on the nature of the criteria on accuracy which we aim to satisfy.

8

T. Laouar and P. Villon.

The following condition

η≤η

(34)

is to be satisfied for the whole domain. η is the permissible error. We can defined a refinement indicator9 as

ξi = ei× N

1

2 $ 2 + e 2? η? u ? ? ? ?

(35)

A refinement is necessary if ξ i > k + ε ; k is a numerical parameter. choice of k : we have done a parametric study on the factor k. First, we have used different values of k. In others terms, refine for all point whose indicator is superior to k + ε . We have made tests for k = 2, 3,.........., 10. We noticed that the best results are obtained for k = 4. This can be explained, partly, by the fact that each quadtree possesses 4 points of evaluation, and that the error is calculated in these points, see Figure 3. For weights of the numerical integration, we simply take all the quarter of the surface of the quadtree containing the point of evaluation to be integrated.

4 evaluation point with a refinement indicator ξ i > 4 3 we refine the corresponding quadtree along by adding nodes and evaluation points for each new quadtree

+

1

2

+ +

+ +

Figure 3: Refinement strategy

5 DIFFUSE INTERPOLATION AND ASYMMETRIC VARIATIONAL FORM FOR THE NUMERICAL RESOLUTION OF MECHANICAL PROBLEMS We have The DAM presents two main limitations : - Given that the DAM is not interpolant, we can not impose the exact values of a function on the boundary nodes. This is very important, for example, when we seek to solve a problem with a singularity with diffuse shape functions 10,12. - Sometimes, the undefined positivness of the rigidity matrix induces spurious modes and a bad solution. We have studied several examples of convergence and observed that a divergence occurs in several of them 12 .

9

T. Laouar and P. Villon.

To overcome these problems, we first propose a method that is a variant of the DAM which satisfies the interpolation properties called Diffuse Interpolation Method (DIM). Using the DIM, we can construct interpolant shape functions associated with the set of nodes, and consequently solve partial derivatives equations by variational or not variational methods using the DEM. First, the DIM different from the DAM by the fact that the weighting functions are singular at nodes. Second, in order to clean the spurious modes we use a method without integration by parts. This method also called weighted residual method 13 , allows us to move from a system of partial derivatives equations to an integral formulation that is not symmetrical 14,15. All details can be found in 10 and 21. 6 NUMERICAL RESULTS

The performance of the error estimators in adaptive analysis In this section, we present the performance of the error estimator in the context of adaptive analysis. The automatic adaptive refinement process used in the analysis follows the strategies. Details can be found in 10 and 14. 5.1 Example 1. Laplacian examples

The governing equation of the problem with the boundary condition is

? ? ?u = f sur ? ? ? ? ? u = g sur ?? where ? is a unit square domain ? = (0,1) × (0,1) . Example 1.1. f and g functions are chosen so that the exact solution is of the form

(36)

u( x, y) = 1 ?

cosh? ?

?

1

?Z

2 2? x +y ? ?

? ? ? ? ? Z?

(37)

cosh? ?

1?

The contour lines and the exact solution for u(x,y) and ? u / ? x are plotted in Figures 4 and 5. We have used a centered initial set of points i.e., the nodes are placed in the center of each quadtree. The problem is analyzed by the adaptive analysis procedure using the Quadratic Diffuse Element with 12 neighbor nodes (QDE12). The coefficients of the weighting functions are defined from the function of attenuation Gauss with ε = 0.001 and p = 2. The numerical integration had been made at Gauss points that are points of evaluation. The tolerance η is equal to 3% (for z = 0.03 ).

10

T. Laouar and P. Villon.

Figure 4: for z = 0.03 the Plots of

?u ?x

and

?u ?y

Figure 5: Plot u(x,y) for z = 0.3

The automatically generated meshes, by the adaptive procedure are presented in Figure 6. We note that the specified error is reached for a set of nodes containing 1360 nodes. We also note that the convergence is rapid enough since we started with a coarse set of nodes ( 20 nodes and η = 44.5 % ).

11

T. Laouar and P. Villon.

*

*

*

*

*

* * * * * * *

*

*

*

*

*

*

*

*

initial set of nodes (20 nodes η = 44.556 % ) final set of nodes (1360 nodes η = 5.0097 % ) Figure 6: Adaptive set of nodes of example 1.1 : 5 per cent aimed at in each refinement

45

40

35

30

25

20

15

10

5

0

0

500

10 0 0

1 5 00

2000

2 5 00

number of nodes

Figure 7: Convergence rate of the relative error 3 %

12

T. Laouar and P. Villon.

number of nodes

2000 1800 1600 1400 1200 1000 800 600 400 200 0 5 11 16 47 85 159 313 469 552 631 651 661 675

CPU time in secondes

Figure 8: CPU time versus number of nodes

This convergence is clearly shown in Figure 7 where we have represented the evolution of the global relative error with respect the number of discretisation nodes for η is equal 3 %. At the end Figure 8 shown the rate of CPU time. Example 1.2. f and g functions are now chosen so that the exact solution is of the form

? ? ? ? ? ? ? ? ? ? πx ? ? sin? ? ? ? 2 ? ? πy ? ? cosh? ? ? ? 2 ? ? ? ? ? ? ? ? ? ?

u( x, y) =

2

π

Arc cos

sin

πx ? 2? ? ?

? ?

2

? ?

+ sinh

πy ? 2? ? ?

? ?

(38)

2

? ?

The used element is the QDE12 ( Quadratic diffuse element with 12 neighbor nodes). The weighting coefficients are defined from the Gauss attenuation function with ε = 0.001 and p =0 2. The numerical integration has been made at four Gauss points that are points of evaluation. We choose two permissible error η equal to 3% and 5 % . We represents in Figure 9 the correspond set of nodes with the both permissible error.

13

T. Laouar and P. Villon.

(a) : 122 nodes η = 5.095 %

(b) : 1322 nodes η = 3.025 % Figure 9: Adaptive set of nodes of example 1.2 : (a) 5 per cent aimed at in each refinement ; (b) 3 per cent aimed at in each refinement

14

T. Laouar and P. Villon.

30

25

20

15

10

5

0

100

200

300

400

500

600

700

800

900

1000

Number of degrees of freedom

Figure 10: Convergence rate of the relative error

The convergence is clearly shown in the Figure 10 where we have carried out the evolution of the global relative error in function of the number of discretization nodes . The corresponding cloud of nodes is shown in Figure 9. 5.2 Example 2. Short Cantilever beam The geometric definition of the problem and the loading condition are shown in Figure 11. Plane strain conditions and Poisson's ratio of 0.25 are assumed. This problem has been used by Szabo to test the convergence rate of the h and p versions of the finite element method, as the singularities are particularly strong. It has also been used by Zienckiewicz and Zhu16 in the study of adaptive mesh refinement in finite elements for problems with singularities. The problem is analyzed using quadratic diffuse elements with 9 neighbor nodes (DEQ9). In the case of finite elements, we use the calculation obtained by Zienckiewicz et al 16, quadratic 6 node triangular elements (QTFE6). Two cases of refinement are studied : Adaptive and Uniform refinements. The obtained refinement as shown in Figure 12 with DEM is correct and acceptable compared to the mesh refinement obtained by FEM in Figure 13. The rate of convergence is shown in Figure 14 for FEM and DEM for both refinement and adaptive refinements. It is seen that the theoretically predicted convergence rate is realized in the case of both FEM with FEQT6 and DEM with DEQ9 when the refinement is uniform. The convergence of the adaptive refinement is very strong, although

we started with a very coarse set of nodes. The desired 5 % accuracy is obtained in four steps.

15

T. Laouar and P. Villon.

F=1

1.0

Figure 11: The geometric structure

initial set of nodes (12 DOF η = 16.0673 % )

final set of nodes (258 DOF η = 4.716 % )

Figure 12: Adaptive set of nodes for a short cantilever beam in DEM : quadratic diffuse elements with 9 neighbor nodes

16

T. Laouar and P. Villon.

initial mesh (40 DOF η = 27.0 % )

final mesh (286 DOF η = 4.0 % )

Figure 13: Adaptive mesh for a short cantilever beam in FEM : quadratic triangular mesh ( 5 % aimed at in each refinement ) 15

30 27 24 21 18 15 12 9 6 3 0 0 500 1000 1500 2000 2500 3000 3500 4000 Number of degrees of freedom (a) FEM DEM

17

T. Laouar and P. Villon.

30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 0 0

FEM DEM

20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 Number of degrees of freedom (b) Figure 14: Experimental rates of convergence for a cantilever beam. Adaptive refinement aiming at 5 per cent : FEM-DEM (a) Uniform refinement (b) Adaptive refinement

4 The Figure 14. shows a good convergence of the relative error. We observe that the diffuse element model converges like the finite element model. 7 DISCUSSION AND CONCLUSION

The In this paper, we have studied with success some aspects of the DIM. The Adaptive Analysis in DEM gives good results to the resolution level of PDE of the Laplacian as well as to that of thermal or mechanical problem. The adopted refinement strategy is reliable. High rates of convergence and saving in CPU time were observed. Furthermore the method appears very effective using a posteriori error estimator. Also, the addition of the equilibrium enhancement improves the robustness of the local projection schema. Without an equilibrium residual, the standard local projection can often be plagued by illconditioned equations for certain cloud of set of nodes configurations, which detracts significantly from its performance. The equilibrium enhancement eliminates this drawback completely and enables the procedure to work effectively for all cloud of set of nodes with any order interpolant. The DIM presents some positive aspects compared with DAM. The obtained results are very satisfying so to the level resolution of PDE of the Laplacian that as well as the level of thermal problem study or mechanics10 occurred problems to the level of the rigidity matrix are lessened by

18

T. Laouar and P. Villon.

using this method without integration by parts. This technique gives a satisfaisant results. Despite the loss of the symmetry of the rigidity matrix, we obtain from a very good results. Several others studies are during realization so as to palliate this integration problem. We have also developed a new linear beam diffuse element which is compared with the bilinear finite element 17,18 with success19. This element is tested on some standard beam test problems and the a stamping problem. Our results were compared with those obtained by the experimental results. REFERENCES [1] B. Nayroles, G. Touzot, P. Villon and A, Ricard, Generalizing the Finite Element Method : diffuse approximation and diffuse elements, Computational Mechanics, 298 10(5), 1-12, 1992. [2] T. Belytschko, Y.Y. Lu and L. Gu, Element-free Galerkin Methods, Int. J. Num. Meth. Eng., 37, 229-256, 1994. [3] T. Belytschko, Y.Y. Lu and L. Gu, A New Implementation of the Element-free Galerkin Method, Comput. Methods Appl. Mech. Eng., 113, 397-414, 1994. [4] W.K. Liu., S. Jun., S. Li., J. Adee and T. Belytschko, Reproducing Kernel Particle Methods for Structural Dynamics, Int. J. Num. Meth. Eng., 38, 1655-1679, 1995. [5] E. Onate, S. Idelsohn, O.C Zienckiewicz and R.L Taylor, A Finite Point Method in Computational Mechanics. Application to Conective Transport and Fluid Flow, Int. J. Num. Meth. Eng (submitted to), December 1996. [6] L. Gravete, S. Falcon and A. Ruiz, Some Results on the Diffuse Approximation using Galerkin Method, Numerical Methods in Engineering '96, 499-505, 1996. [7] P.Lancaster, Moving weighted least-squares methods, in Polynomial and Spline Approximation (B. N. Sahney, Ed.), NATO advanced Study Institue Series C, Reidel, Dordrecht, 1979, pp. 103-120. [8] P.Lancaster and K. Salkauskas, Surfaces generated by moving least-squares methods, Math. Comput. , 37, 141-158, 1981. [9] O.C. Zienckiewicz and J.Z. Zhu, The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique, Int. J. Num. Meth. Eng., 33, 1331-1364, 1992. [10] T. Laouar, Contribution àl'é tude de l'approximation diffuse : auto-adaptativité en

éé l ments diffus, Thè se de Doctorat de L'Université de Technologie de Compiè gne, (Ph.D Thesis) june 1996.

[11] T. Laouar and P. Villon, Error Estimator in Diffuse Element, 16th Canadian Congress of Applied Mechanics, CANCAM 97, June 1-5, 463-464, Québec, CANADA. [12] T. Laouar, P. Villon and F. Guyon, Interpolation Diffuse et formulation non symé trique pour me è la ré solution numé rique des problè mes mé canique, 3 Colloque National en Calcul des Structures, 20-23 mai 1997, pp 855-860, Giens, FRANCE. [13] G. Touzot and G. Dhatt, Une pré sentation de la mé thode des é lé ments finis, MALOINE SA.

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T. Laouar and P. Villon.

EDITEUR PARIS, (2ème édition), 1984. me [14] T. Laouar and P. Villon, Mé thodes des Elé ments Diffus Adaptative, 13è Congrés Fran?ais en Mécanique, 1-5 septembre 1997, Poitiers, FRANCE. [15]T. Laouar and P. Villon, On Adaptive Diffuse Element Method, 3rd EUROMECH, Solid Mechanics Conference, August 18-22, 1997, Stockholm, SWEDEN. [16]O.C. Zienckiewicz, J.Z. Zhu and N.G. Gong, effective and practical h-p version adaptive analysis procedures for the finite element method, Int. J. Num. Meth. Eng., 28, 879-891, 1989. [17] J.L Batoz and G. Dhatt, Modé lisation des structures par é lé ments finis, Tome 1. Ed. Hermès, Paris, 1990. [18]A. Ibrahimbegovic and F. Frey, Finite element analysis of linear and non-linear planar deformations of elastic initially curved beams, Int. J. Num. Meth. in Eng. vol 36 pp 32393258 (1993). [19]T. Laouar, A. Benzegaou and P. Villon, Approximation Diffuse : Elé ment diffus àquatres noeuds, Revue Européenne des éléments finis (submitted to) July 1997. [20]T. Blacker and T. Belytschko, Superconvergent patch recovery with equilibrium and conjoint interpolant enhancements, Int. J. Num. Meth. Eng., 37, 517-536, 1994. [21]T. Laouar and P. Villon, Diffuse Approximation : Adaptive procedure for the Diffuse Element Method, in press.

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Some researchers considered

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mixed finite

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