GMLS approximations in the EFG method - CiteSeerX

International Workshop on MeshFree Methods 2003

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GMLS approximations in the EFG method: applications to C1 structural problems. Carlos Tiago(1) and Vitor Leit˜ao(2) Abstract: Continuity of the generalized displacement and stress fields are preserved by meshless methods, such as the EFG (Element Free Galerkin) , as long as an appropriate basis together with an appropriate weight function are used. Nevertheless, the possibility of using approximation functions built not only on information from the unknown functions at the nodal points but also of its derivatives, may provide more efficient procedures in the numerical solution of boundary value problems governed by fourth order differential equations. The GMLS (Generalized Moving Least Squares Method) is, as its name suggests, a generalization of the moving least squares concept (used in the EFG method) which takes into account, to build the approximation, both the approximation values and the corresponding derivatives. In this work, application of the GMLS concept to thin beams and plates is carried out. Comparisons with the standard MLS (Moving Least Squares) are presented. Keywords: meshless, EFG, GMLS, C1 .

1

Introduction

Problems governed by a differential equation of order higher than two are usual in engineering, e.g., beams, thin plates and shells. The Finite Element Method (FEM) approach to this kind of problems relies in the use of elements with several degrees of freedom (dof) per node. These generalized displacements can be displacements, rotations, curvatures or may even not posses a explicit physical meaning. The continuity requirements, even for the generalized displacements, is rarely achieved. As the continuity of the generalized stress field is imposed in a weak form, these are discontinuous. Generating continuous shape functions is not an easy task and several approaches have been proposed. The diversity of the proposals is such that not less than 88 different elements for plate bending are listed in a 1984 paper by Hrabok and Hrudey [5]. 1 Instituto 2 Instituto

Superior T´ecnico, Av. Rovisco Pais, 1049-001 Lisboa ([email protected]). Superior T´ecnico, Av. Rovisco Pais, 1049-001 Lisboa ([email protected]).

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Carlos Tiago, Vitor Leit˜ao

On the contrary, the systematic approach of the EFG [3] allows to deal, using exactly the same procedure, with all these problems, regardless of the order of the governing differential equations. The ease of use and generality of the EFG is due to the possibility of generating displacement fields approximations to a given continuity degree. Additionally, it is also possible to ensure continuity of the generalized stress fields. C1 problems were first analyzed with EFG by Krysl and Belytschko to static analysis of thin plates [6] and thin shells [7] and by the authors to vibration analysis of thin beams and plates [9], among others. The results presented in these articles demonstrate that the EFG is a efficient tool to solve this type of problems. Taking into account the need of imposing boundary conditions directly associated with the first derivatives of the displacement field, Atluri et al [1] presented a generalization of the MLS (Moving Least Squares) approximation which uses information not only of the function itself but also its derivatives: the GMLS (Generalised Moving Least Squares). This approximation was used in the context of the MLPG (Meshless Local Petrov-Galerking) [2] method and was applied to static analysis of thin beams. The objectives of the present work are: 1. to establish the relative performance of the GMLS versus MLS. 2. to extend the use of the GMLS to 2D problems. 3. to use the GMLS approximation in the EFG method context. Section 2 presents a brief revision of the GMLS approximation. Although the MLS approximation is also used here for the sake of comparison, it will not be described here. This can be found, among others, in the works of Belytschko et al [3], Atluri et al [1] and the authors [9]. In section 3 the weak form of the equilibrium equations is used to obtain the governing system of equations. The essential boundary conditions are also imposed in the weak sense through a suitable ortogonal transformation technique. Numerical examples are presented in section 4 followed by conclusions and discussion in section 5.

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The generalized moving least squares approximation

The notation used here is similar to the one used by Atluri et al in the original work on GMLS approximation [1]. Consider a two dimensional domain, Ω, containing a given set of scattered nodes xi (1 ≤ i ≤ n). Over this set a continuous function, u, assumes the values uˆi , θˆ ix and θˆ iy , i. e., u(xi ), ∂u(xi ) ∂x

i) and ∂u(x ∂y . The GMLS approximation of u over Ω, u
(x), can be written as:

u
(x) = ΨTu (x)uˆ + ΨTθx (x)ˆtx + ΨTθy (x)ˆty

(1)

where ΨTu (x) = pT A−1 (x)PT w(0,0) (x),

uˆ =

ΨTθx (x) = pT A−1 (x)PTx w(1,0) (x),

ˆtx =

ΨTθy (x) = pT A−1 (x)PTy w(0,1) (x),

ˆty =

  

uˆ1 uˆ2 . . . uˆn θˆ 1x θˆ 2x . . . θˆ nx θˆ 1y θˆ 2y . . . θˆ ny

T T T

.

International Workshop on MeshFree Methods 2003

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p(x) is a linearly independent basis of m functions,   pT (x) = p1 (x) p2 (x) . . . pm (x) and A(x) = PT w(0,0) P + PTx w(1,0) Px + PTy w(0,1) Py   B(x) = PT w(0,0) PTx w(1,0) PTy w(0,1) . The matrices in these expressions are given by T  P = p(x1 ) p(x2 ) . . . p(xn )  T ∂p(x1 ) ∂p(x2 ) ∂p(xn ) Px = ... ∂x ∂x ∂x T  ∂p(x1 ) ∂p(x2 ) ∂p(xn ) Py = ... ∂y ∂y ∂y and w(0,0) , w(1,0) and w(0,1) are diagonal n × n matrices that collect the weight functions, wi (x).

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Governing equations

The GMLS approximation (1) for the displacements field resembles the usual interpolation used in the FEM. Following the standard procedure based on a weak form of the equilibrium equations, a relationship KU = F

(2)

is obtained which involves all degrees of freedom of the approximation. For the imposition of the essential boundary conditions a weak form of the kinematic restraints can be written, resulting from here a relationship of the form GT U = q,

(3)

where G is not a distribution matrix, as in the FEM. This is a consequence of the non interpolatory character of the approximation (does not possess the Kronecker delta property: ΦI (xJ ) = δIJ ). Several methods are possible to overcome this drawback. In this work an ortogonal transformation of coordinates was used [4, 8].

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Numerical tests

In the tests carried out, the nodes for the approximation are always equally spaced. The background cells for the numerical integrations are placed between the nodes. Gaussquadrature is used. The weighting function used in this work is [1]   s 1 − x − xi 2 /dm2 i , if x − xi  ≤ dm i (4) w(x − xi ) = 0, if x − xi  > dm i where dm i is the size of the domain of influence the ith node.

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Carlos Tiago, Vitor Leit˜ao

p(x) = p0 sin

 πx  l

x A

B

E, I, l

z Figure 1: Properties of beam. 1 number of dof 0, 001

0, 010

0, 100

1, 000 1 × 100 1 × 10−1

L2 (MLS) H 1 (MLS) H 2 (MLS) L2 (GMLS) H 1 (GMLS) H 2 (GMLS)

1 × 10−2 1 × 10−3 1 × 10−4 1 × 10−5 1 × 10−6 1 × 10−7 1 × 10−8

Figure 2: Results for sinusoidal load for a quadratic basis.

4.1 Euler-Bernoulli beam Consider the clamped-simply supported beam represented in Figure 1. Ten point integration Gauss-Legendre quadrature was used. The following norms were used to measure the errors: √ 2 Ω |unum −uexact | dΩ Relative L2 error norm: √ 2 √ Relative H 1 error norm: √ Relative

H2

error norm:

Ω |uexact |

dΩ

 2 +|u 2 num −uexact | dΩ  2 2 Ω |uexact | +|uexact | dΩ

Ω |unum −uexact |

√

  2 +|u 2  2 num −uexact | +|unum −uexact | dΩ   2 2 2 Ω |uexact | +|uexact | +|uexact | dΩ

Ω |unum −uexact |

√

In Figure 2 the MLS

and GMLS  results for the h refinement are given for a quadratic basis, i. e., (pT = 1 x x2 ). The ratio between the size of the domain of influence and the nodal spacing was equal to 3,0. In Figure 3 the results (error norm L2 ) using a GMLS approximation for a quadratic, cubic and quartic basis, are shown.

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International Workshop on MeshFree Methods 2003

1 number of dof 0, 001

0, 010

0, 100

1, 000 1 × 100 1 × 10−1 1 × 10−2 1 × 10−3 1 × 10−4

quadratic basis cubic basis quartic basis

1 × 10−5 1 × 10−6 1 × 10−7 1 × 10−8

Figure 3: Results of L2 error norm with p refinement in beam using GMLS.

4.2 Kirchhoff square plate A simply supported square thin plate with length a, thickness h, material properties E and ν, uniform load p0 , was analyzed using MLS and GMLS. The energy norm, χ, and the relative error of the energy norm, re , are defined as 1 2 1 χ − χexact  T χ = χ Dχ dΩ re = num . (5) 2 Ω χexact  where χ is the vector of curvatures and D represents the generalized constitutive relationship. The results obtained for re are presented in Figure 4 for both approximations with the following basis:

 quadratic pT = 1 x y x2 xy y2 ,  cubic pT = 1 x y x2 xy y2 x3 x2 y y2 x y3 ,  quartic pT = 1 x y x2 xy y2 x3 x2 y y2 x y3 x4 x3 y x2 y2 xy3 y4 . The ratio between the size of the domain of influence and the nodal spacing was equal to 2,0, 3,0 and 4,0 for the quadratic, cubic and quartic basis, respectively.

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Conclusions

In this work an implementation of the EFG method by means of the generalization of the moving least squares concept, GMLS, was applied to 1D and 2D structural analysis problems where C1 continuity is required. The 1D examples show clearly that higher rates of convergence can be achieved by the GMLS when compared to the MLS. For the 2D examples the GMLS leads, in general, to better results than the conventional MLS but convergence is worse. This is, probably, due to numerical difficulties (ill conditioning, machine precision, etc). Further work on these issues is currently under development. Acknowledgements: This work has been partially supported by FCT through projects ”Financiamento Plurianual” and POCTI/33066/ECM/2000.

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Carlos Tiago, Vitor Leit˜ao

0, 001

0, 010

1 number of dof

0, 100

1, 000 1 × 100 1 × 10−1

MLS (quadratic)

1 × 10−2

MLS (cubic)

1 × 10−3

MLS (quartic)

1 × 10−4

GMLS (quadratic)

1 × 10−5

GMLS (cubic) GMLS (quartic)

1 × 10−6 1 × 10−7

Figure 4: Results for the relative error on the energy norm, re , using MLS and GMLS.

References [1] S. N. Atluri, J. Y. Cho, and H.-G. Kim. Analysis of thin beams, using the meshless local Petrov-Galerkin method, with generalized moving least squares interpolations. Comput. Mech., 24:334–347, 1999. [2] S. N. Atluri and T.-L. Zhu. The meshless local Petrov-Galerkin (MLPG) approach for solving problems in elasto-statics. Comput. Mech., 25:169–179, 2000. [3] T. Belytschko, Y. Lu, and L. Gu. Element-Free Galerkin Methods. Int. J. Numer. Meth. Engng., 37:229–256, 1994. [4] F. C. G¨unter and W. K. Liu. Implementation of boundary conditions for meshless methods. Comput. Methods Appl. Mech. Engrg., 163:205–230, 1998. [5] M. M. Hrabok and T. M. Hrudey. A review and catalogue of plate bending finite elements. Computers & Structures, 19(3):479–495, 1984. [6] P. Krysl and T. Belytschko. Analysis of thin plates by the element-free Galerking method. Comput. Mech., 17:26–35, 1995. [7] P. Krysl and T. Belytschko. Analysis of thin shells by the element-free Galerking method. Int. J. Solids Structures, 33(20–22):3057–3080, 1996. [8] A. El Ouatouti and D. A. Johnson. A new approach for numerical modal analysis using the element-free method. Int. J. Numer. Meth. Engng., 46:1–27, 1999. [9] C. Tiago and V.M.A. Leit˜ao. Analysis of free vibration problems with the elementfree Galerkin method. In IXth International Conference on Numerical Methods in Continuum Mechanics (to be presented at), 2003.