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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng (2011)Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.3317Computation of limit and shakedown loads using a node-basedsmoothed nite element methodH. Nguyen-Xuan1,2,*,, T. Rabczuk3, T. Nguyen-Thoi1,2, T. N. Tran4andN. Nguyen-Thanh31Department of Mechanics, Faculty of Mathematics & Computer Science, University of Science HCM,227 Nguyen Van Cu, Dist. 5, Ho Chi Minh City, Vietnam2Division of Computational Mechanics, Ton Duc Thang University, 98 Ngo Tat To St., War 19, Binh Thanh Dist.,Ho Chi Minh City, Vietnam3Institute of Structural Mechanics (ISM), Bauhaus-University Weimar, Germany4Labor fr Biomechanik, Fachhochschule Aachen, Campus Jlich Ginsterweg 1, D-52428 Jlich, GermanySUMMARYThis paper presents a novel numerical procedure for computing limit and shakedown loads of structuresusing a node-based smoothed FEM in combination with a primaldual algorithm. An associated primaldual form based on the von Mises yield criterion is adopted. The primal-dual algorithm together with aNewton-like iteration are then used to solve this associated primaldual form to determine simultaneouslyboth approximate upper and quasi-lower bounds of the plastic collapse limit and the shakedown limit. Thepresent formulation uses only linear approximations and its implementation into nite element programs isquite simple. Several numerical examples are given to show the reliability, accuracy, and generality of thepresent formulation compared with other available methods. Copyright 2011 John Wiley & Sons, Ltd.Received 10 March 2011; Revised 23 May 2011; Accepted 26 August 2011KEY WORDS: nite element method; limit analysis; node-based smoothed nite element method(NS-FEM); primaldual algorithm; shakedown analysis; strain smoothing1. INTRODUCTIONIn practical applications using the fully compatible FEM, the three-node linear triangular element(T3) and four-node linear tetrahedral element (T4) are preferred by many engineers because of theirsimplicity, robustness, less demand on the smoothness of the solution, and efciency of adaptivemesh renements for solutions of desired accuracy. However, the fully compatible FEM modelsusing T3 and T4 elements still possess certain inherent drawbacks: (i) they overestimate exces-sively the system stiffness matrix, which leads to poor accuracy in both displacement and stresssolutions, and (ii) they are subjected to locking in the problems with bending domination andincompressible materials.In the effort to develop nite element technology, Liu et al. have combined the strain smoothingtechnique [1] used in meshfree methods into the FEM to formulate a cell/element-based smoothedFEM (SFEM or CS-FEM) [27]. Applying this strain smoothing technique on smoothing domainswill help to soften the over-stiffness of the lower-order FEM model, and hence can improve signif-icantly the accuracy of solutions in both displacement and stress. In the CS-FEM, the smoothingdomains are based on the quadrilateral elements, and each element can be further subdivided into*Correspondence to: Nguyen-Xuan Hung, Department of Mechanics, Faculty of Mathematics & Computer Science,University of Science HCM, 227 Nguyen Van Cu, Dist. 5, Ho Chi Minh City, Vietnam.E-mail: nxhung@hcmus.edu.vnCopyright 2011 John Wiley & Sons, Ltd.H. NGUYEN-XUAN ET AL.one or some quadrilateral smoothing domains, as shown in Figure 1. The CS-FEM has been studiedtheoretically in [35], and further extended to the general n-sided polygonal elements (nSFEM ornCS-FEM) [6], dynamic analyses [7], incompressible materials using selective integration [8, 9],plate and shell analyses [1012], fracture mechanics problems [13, 14], and limit analysis [15].In the attempts to improve the performance of T3 and T4 elements, Liu et al. then extended thecell/element-based idea of smoothing domains in the CS-FEM to node-based, edge-based, face-based, and partly node-based ones with different applications to give, respectively, a node-basedsmoothed FEM (NS-FEM) [1622, 30], an edge-based smoothed FEM (ES-FEM) [2327, 30], anda face-based smoothed FEM (FS-FEM) [2830]. Similar to the standard FEM, these smoothed FEM(S-FEM) models also use a mesh of elements. However, the S-FEM models evaluate the weak formbased on smoothing domains created from the entities of the element mesh such as nodes(Figure 2),edges (Figure 3), or faces (Figure 4). These smoothing domains hence cover parts of adjacent ele-ments. They are linearly independent and ensure stability and convergence of the S-FEM models.Because of the use of different smoothing domains, the softening effect of strain smoothing tech-nique on the over-stiffness of the standard FEM model will be different. Therefore, each of theS-FEM models has different properties, advantages, and disadvantages [30].: field nodesx158y(c)1x749(a)y4: added nodes to form the smoothing domains 2 1 26324(d)134(b)23536Figure 1. Division of quadrilateral element into the smoothing domains (SDs) in the CS-FEM-Q4 by con-necting the mid-segment points of opposite segments of smoothing domains: (a) 1 SD; (b) 2 SDs; (c) 4 SDs;and (d) 8 SDs.: field node : centroid of triangle : mid-edge pointk(k)(k)Figure 2. Triangular elements and the smoothing domains C.k/(shaded areas) associated with nodes in theNS-FEM-T3.Copyright 2011 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2011)DOI: 10.1002/nmeNS-FEM FOR LIMIT AND SHAKEDOWN ANALYSIS OF STRUCTURESsmoothing domainsassociated withinner edgesmoothing domain associated with boundary edge: centroid of triangles : field nodeFigure 3. Triangular elements and the smoothing domains (shaded areas) associated with edges in theES-FEM-T3.: central point of elements (H, I)interface k (BCD): field nodeT4 element 1 (ABCD)ABDHICT4 element 2 (BCDE)(BCDIH)associated with interface ksmoothed domain EFigure 4. Tetrahedral elements and the smoothing domains (shaded areas) associated with face k in theFS-FEM-T4.It was proved that the S-FEM models are variationally consistent based on the modied two-eld HellingerReissner principle. However, only unknowns of the eld displacement (master eld)appear in the discretized algebraic system of equations. Therefore, it is, in general, very much dif-ferent from the so-called mixed FEM formulation [31, 32], where stresses (or strains) may also beunknowns (or also master elds). More details for a general and rigorous theoretical frameworkrelated to properties, accuracy, and convergence rates of the S-FEM models have been recentlystudied in [33].Among these S-FEM models, the NS-FEM [1621] shows some interesting properties that arevery effective for solving nonlinear problems in solid mechanics: (i) volumetric locking is signi-cantly alleviated; (ii) it possesses super-accurate and super-convergent properties of stress solutions;and (iii) the stress at nodes can be computed directly from the displacement solution without usingany post-process. The third property is similar to using just one Gauss point to compute the stressat nodes. The NS-FEM is hence a very convenient method for conducting the nonlinear computa-tional algorithm using the stress at nodes and is more computationally efcient and simpler thanthe standard FEM using the stresses at Gauss points located inside the element. On the basis ofthese crucial properties, the NS-FEM was then extended to analyze the visco-elastoplasticity prob-lems in two-dimensional (2D) and three-dimensional (3D) solids [19], fracture mechanics [20],Copyright 2011 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2011)DOI: 10.1002/nmeH. NGUYEN-XUAN ET AL.and plates [21, 22]. In this paper, the NS-FEM is further formulated for the limit and shake-down analysis of solid mechanics problems made of elasticperfectly plastic material. Recall thatthe extension of S-FEM approaches to limit and the shakedown analysis of 2D structures hasalready been investigated in previous contributions [14, 22]. In [14], a formula to compute theplastic collapse limit factor was relied on the CS-FEM in which the smoothing domains were cre-ated based on elements, and each element was then subdivided into one or several quadrilateralsmoothing domains. In [22], we used the ES-FEM in which the smoothing domains were cre-ated based on the edges of the elements to obtain the plastic collapse limit factor. Therefore, thepresent approach in which the smoothing domains are obtained based on the nodes of the elementsis basically different from the CS-FEM and ES-FEM models. The NS-FEM is particularly moregeneral than the CS-FEM and ES-FEM because the NS-FEM can solve for both 2D and 3D prob-lems of the limit and shakedown analyses, while the CS-FEM and ES-FEM are only available for2D problems.Limit and shakedown analysis has been a well-known tool for assessing the safety load factorof engineering structures. These load factors can be derived from upper bound and lower boundapproaches. The upper bound shakedown analysis is based on Koiters kinematic theorem [34](a kinematically admissible displacement eld) to determine the minimum load factor, and the lowerbound shakedown analysis is based on Melans static theorem [35] (a statically admissible stresseld) to determine the maximum load factor. However, the analytical methods to solve these twoapproaches are not available for the general problems in engineering practice [36, 37]. Therefore,various numerical methods that involve continuous, semi-continuous [38] or truly discontinuous[39, 40] approximations o