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A Ck continuous generalized finite element formulation applied to

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  • Comput Mech (2009) 44:377393DOI 10.1007/s00466-009-0376-5

    ORIGINAL PAPER

    A Ck continuous generalized finite element formulation appliedto laminated Kirchhoff plate model

    Clovis Sperb de Barcellos Paulo de Tarso R. Mendona Carlos A. Duarte

    Received: 5 December 2008 / Accepted: 30 January 2009 / Published online: 10 March 2009 Springer-Verlag 2009

    Abstract A generalized finite element method based on apartition of unity (POU) with smooth approximation func-tions is investigated in this paper for modeling laminatedplates under Kirchhoff hypothesis. The shape functions arebuilt from the product of a Shepard POU and enrichmentfunctions. The Shepard functions have a smoothness degreedirectly related to the weight functions adopted for their eval-uation. The weight functions at a point are built as productsof C edge functions of the distance of such a point to eachof the cloud boundaries. Different edge functions are inves-tigated to generate Ck functions. The POU together withpolynomial global enrichment functions build the approxi-mation subspace. The formulation implemented in this paperis aimed at the general case of laminated plates composed ofanisotropic layers. A detailed convergence analysis is pre-sented and the integrability of these functions is alsodiscussed.

    Keywords Generalized finite element method Partitionof unity method Kirchhoff plate FEM Ck continuousapproximation functions

    C. S. de Barcellos (B) P. de Tarso R. Mendona (B)Mechanical Engineering Department, Federal University of SantaCatarina, 88040-900 Florianpolis, SC, Brazile-mail: [email protected]

    P. de Tarso R. Mendonae-mail: [email protected]

    C. A. Duarte2122 Newmark Civil Engineering Laboratory,University of Illinois at Urbana-Champaign, 205 North Mathews Ave.,Urbana, IL 62801-2352, USAe-mail: [email protected]

    1 Introduction

    In the last decade a number of meshless procedures havebeen proposed in the FEM community. These include, amongseveral others, the smoothed particle hydrodynamics method[27], the diffuse element method [37], wavelet Galerkinmethod [1], the element free Galerkin method (EFGM) [12],reproducing Kernel particle method (RKPM) [32], the mesh-less local Petrov-Galerkin method [2], the natural elementmethod [46], partition of unity method [3], and the hp-cloudsmethods e.g. [20,21]. The latter has the further appeal of nat-urally introducing a procedure for performing p-adaptivity,in a very flexible way, avoiding the construction of functionsby sophisticated hierarchical techniques. The advantages ofthese procedures are, however, balanced by increased com-putational cost since a mesh is still needed for integrationpurposes and, at each integration point, the partition of unity(POU) must be independently computed since the coveringof each point is arbitrary.

    The hp-clouds approximations have been proved to bemore efficient than others like the EFGM [21,22], and forthis reason they were used in [26,36]. But all these meshlessmethods present some disadvantages regarding the imposi-tion of boundary conditions and high computational costs.In order to ameliorate the cost of numerical integration andthe implementation difficulties of mesh free methods, Odenet al. [38], proposed that, instead of using circles or rectan-gles for defining the clouds around each node, it would bemore convenient to use linear finite element meshes. Here theclouds associated with a node i is built by the union of theelements connected to this node. This concept greatlyreduces the number of floating point operations, since thePOU is known beforehand and allows standard integrationroutines for integrating the nodal matrices. This new schemeled to a generalized finite element method (GFEM).

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    Independently, Babuka and coworkers proposed essentiallythe same procedure, initially named as special finite elementmethod [34], and later as the partition of unity finite elementmethod [4]. A similar philosophy is inserted in the works ofBelytschko and Black [10], and Mos et al. [35] for discontin-uous solutions and is called extended finite element method(XFEM). The p-enrichment, as in other meshfree methodslike hp-clouds, are performed nodally, which suggests anadaptive scheme to provide automatic control of approxima-tion errors. Several contributions have been proposed, amongthem are the works of Strouboulis et al. [45], Babuka et al.[4], Belytschko et al. [11], Liu et al. [32], and Barros et al.[5]. In such procedures, the solution improvement is reachedby only performing nodal enrichment without excessivelyincreasing the computational effort even in presence of stressconcentration [19], thus reducing the possible need to per-form a mesh refinement in this type of problems.

    The usual GFEM scheme leads only to C0 approxima-tion functions. On the other hand, there are several mod-els, like the Kirchhoff plate, which require solutions to beat least C1 continuous. This requirement has led to severalfinite element formulations which release such a need underthe cost of lower accuracy and/or consistency. Many otherformulations are based on mixed or hybrid variational prin-ciples for coping with such difficulties like, e.g. [7]. In manycircumstances, the Mindlins and Reissners models, whichrequires only C0 continuity, have been dominant over thelast decades, e.g. [15]. In recent years, some meshless meth-ods have been proposed for solving Kirchhoff plate and shellmodels [8,9,16,30,31,33,43]. Again, another approach toreduce the numerical integration costs and boundary con-ditions difficulties of the meshless methods was proposedby Edwards [24,25], in which a finite element mesh is usedto build arbitrarily smooth approximation functions whichhave the same support of corresponding global finite elementLagrangean shape functions on the same mesh. This schemehas an important restriction for requiring the clouds to beconvex, which is not always possible to guarantee. Aim-ing at removing this limitation, Duarte et al. [18] used theso-called boolean R-function of Shapiro [41]. Latter, Barroset al. [6] discusses this procedure for linear elasticity prob-lems. The arbitrary continuity is based on the type of selectededge functions and on the value of a parameter of a booleanfunction.

    In addition, the higher degree of regularity has the advan-tage of enhancing the definition of error measures as is alsopointed out by [11], for the case of the reproducing kernelparticle method (RKPM), and also allows strong residualevaluation.

    Presently, the authors make use of an extension of theEdwards approach, utilized by Duarte [18], for convex andnon-convex supports with the aid of the so-called R-functions [40,41], on GFEM with Ck approximating

    functions, in triangular unstructured meshes. These sets ofapproximation functions are applied for solving some Kirch-hoff plate problems. The influence of the type of integrationrules, Gaussian or triangular rules, is analyzed. Several typesof cloud edge functions are implemented and tested.

    The remainder of this paper is outlined as follows: Sect. 2summarizes the partition of unity concepts; Sect. 3 presentsthe hp-cloud partition of unity functions and their enrich-ment; Sect. 4 develops the construction of weighting func-tions based on several cloud edge functions in order to achieveC and Ck continuity on the approximation functions;Sect. 5 presents a summary of the laminated Kirchhoff platemodel and Sect. 6 presents results of the proposed formula-tion in order to test its behavior under several conditions.

    2 Partition of unity and approximation functions

    Here, the basic idea is to employ weight functions whichare zero at the clouds boundaries, together with its first (orhigher) normal derivative in order to lead to a C1 (or highercontinuity) partition of unity as it is performed in thehp-clouds method. Aiming to summarize such procedure, letus consider a conventional triangular finite element mesh,{Ke}NEe=1 (NE being the number of elements Ke) defined byN nodes, {x}N=1, in an open bounded domain R2(x).To each of these nodes, one denotes the interior of the unionof the finite elements sharing it as cloud, , = 1, . . . , N ,as is usual in the GFEM. Over each cloud, Ck appropriateweight functions are evaluated and used in the Shepardsmoving least square method [42] scheme for generating apartition of unity.

    Let an open bounded domain R2(x), here definedas the plate mid-surface, and N be an open covering of thisdomain made of the set of N clouds , associated with nodesx . In other words, the closure of the domain is containedin the union of the clouds closures :

    N

    =1 (1)

    One denotes the set of nodes as QN = {x0, x1, . . ., xN } ={x}N=0. Consider next a set of functions SN = {(x)}N=1,each of which having the corresponding cloud as its com-pact support. If each one of these functions is such that(x) Ck0 (), k 0 and

    Ni=1 (x) = 1, x ,

    and all compact subset of intersects only a finite numberof supports, the set {(x)}, = 1, . . . , N is said to be apartition of unity subordinated to the covering N . The firstrequirement indicates that the function is non-zero onlyover its respective cloud and is, at least, k times continu-ously differentiable.

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    Let () = span{Li}iI() denote the local functionsubspaces defined on , = 1, . . . , N , where I(), =1, . . . , N , are index sets and {Li}iI() is a set of enrich-ment functions for this cloud. Without loss of generality, theconsidered sets {Li}iI(), = 1, . . . , N , are the polyno-mial basis functions satisfying

    Pp() p () (2)where Pp denotes the space of polynomials of degree less orequal to p, e.g.,

    linear basis: {1, (x x), (y y)} or quadratic basis: {1, (x x), (y y), (x x)2,

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