Comput Mech (2014) 54:1111–1128DOI 10.1007/s00466-014-1043-z

ORIGINAL PAPER

Upper bound shakedown analysis with the nodal natural elementmethod

Shutao Zhou · Yinghua Liu · Dongdong Wang ·Kai Wang · Suyuan Yu

Received: 21 January 2014 / Accepted: 10 May 2014 / Published online: 8 June 2014© Springer-Verlag Berlin Heidelberg 2014

Abstract In this paper, a novel numerical solution proce-dure is developed for the upper bound shakedown analy-sis of elastic-perfectly plastic structures. The nodal naturalelement method (nodal-NEM) combines the advantages ofthe NEM and the stabilized conforming nodal integrationscheme, and is used to discretize the established mathemat-ical programming formulation of upper bound shakedownanalysis based on Koiter’s theorem. In this formulation, thedisplacement field is approximated by using the Sibson inter-polation and the difficulty caused by the time integrationis solved by König’s technique. Meanwhile, the nonlinearand non-differentiable characteristic of objective function is

S. ZhouInstitute of Nuclear and New Energy Technology, The KeyLaboratory of Advanced Reactor Engineering and Safety,Ministry of Education, Tsinghua University,Beijing 100084, Chinae-mail: zhoust1983@163.com

Y. Liu (B)Department of Engineering Mechanics, AML,Tsinghua University, Beijing 100084, Chinae-mail: yhliu@mail.tsinghua.edu.cn

D. WangDepartment of Civil Engineering, Xiamen University,Xiamen 361005, Fujian, Chinae-mail: ddwang@xmu.edu.cn

K. WangDepartment of Mechanical and Electronic Engineering,Jinan Engineering Vocational Technical College,Jinan 250200, Shandong, Chinae-mail: wwwkai@163.com

S. Yu (B)Department of Thermal Engineering, Center for Combustion Energy,Tsinghua University, Beijing 100084, Chinae-mail: suyuan@tsinghua.edu.cn

overcome by distinguishing non-plastic areas from plasticareas and modifying associated constraint conditions andgoal function at each iteration step. Finally, the objectivefunction subjected to several equality constraints is linearizedand the upper bound shakedown load multiplier is obtained.This direct iterative process can ensure the shakedown load tomonotonically converge to the upper bound of true solution.Several typical numerical examples confirm the efficiencyand accuracy of the proposed method.

Keywords Shakedown analysis · Kinematic theorem ·Sibson interpolation · Stabilized conforming nodalintegration · Nodal natural element method

1 Introduction

Shakedown analysis provides a direct method to investigatethe plastic collapse behavior of engineering structure withvariable repeated loading and an effective measure to calcu-late the structural load-carrying capacity. Consequently, it iswidely used for the optimization design and the safety evalua-tion of engineering structure [1,2]. As an extension of plasticlimit analysis, shakedown analysis is built on two fundamen-tal and critical shakedown theorems, which are respectivelyproved by Melan [3] and Koiter [4], and are well knownas the lower bound or static theorem and the upper boundor kinematic theorem. In the lower bound shakedown the-orem, the statically admissible residual stress field is opti-mized to calculate the maximum shakedown load. Whilein the upper bound shakedown theorem, the kinematicallyadmissible strain field is assumed to obtain the minimumshakedown load. Additionally, the novel and attractive dual-ity approach connected these two theorems has been proved,and can simultaneously be used to determine the upper and

123

1112 Comput Mech (2014) 54:1111–1128

the lower bound loads for the shakedown analysis. For adetailed description of the duality approach of shakedownanalysis, we refer to the literature of Kamenjarzh and Mer-zljakov [5,6], Khoi et al. [7,8], Tran et al. [9] and Tran [10],etc. As the rapid development in the last several decades,many scholars have devoted great efforts to the field of shake-down analysis, and their investigations have far exceededthe classical shakedown theorem category. The influencesof strain or working hardening, nonlinear geometry, ther-mal, dynamic and moving loads, uncertainties have beenwidely considered, the yield criterions included von Mises’scriterion, Tresca criterion, unified strength theory, Mohr–Coulomb criteria, Drucker–Prager criteria and Hill’s criteriaare usually employed, the applications in the aspects of dam-aged structures, composite and frictional materials, Shapememory alloys (SMAs) structures, pressurized pipelines andpavements have been thoroughly investigated. Simultane-ously, various remarkable and significant research fruits havebeen published, and the relevant summaries and overviewsfor the developments and applications of shakedown analysiscan be found from Stein et al. [11], Zhang [12], Hachemi andWeichert [13], Yan and Hung [14], Maier et al. [15], Borinoand Polizzotto [16], Tran et al. [17], Carvelli [18], Li and Yu[19], Li and Yu [20], Feng and Liu [21], Feng and Sun [22],Carvelli et al. [23], Chen et al. [24], Gorss-Weege [25], Shiau[26], Konig and Maier [27], and the references listed therein.

The numerical method is a major tool in the field ofshakedown analysis. With the rapid development of differentcomputational techniques, a lot of efficient and crediblenumerical approaches have been developed in the imple-mentation of shakedown analysis. The traditional numericalcalculation tool is mesh-based, such as the finite elementmethod (FEM) [6–14,16–20,23–25] or boundary elementmethod (BEM) [28–30]. In the past two decades, meshlessmethods among others have achieved significant develop-ments, such as Belytschko et al. [31] (element free Galerkinmethod), Liu et al. [32] (reproducing kernel particle method),Liszka [33] (hp cloud method), Atluri and Zhu [34] (localPetrov-Galerkin method), Sukumar et al. [35] (natural ele-ment method), Zhang et al. [36] (least-squares colloca-tion method), Liu and Wang [37] (radial point interpolationmethod), Gu [38] (moving kriging interpolation method),Kim et al. [39] (meshfree point collocation method), Bessaet al. [40] (reproducing kernel peridynamics method), Yoonand Song [41,42] (extended particle difference method), andso on. Different from those mesh-based numerical meth-ods, the meshless methods only need the discretized infor-mation of nodes in the problem domain regardless of theconnecting relationship among those nodes, and can over-come the drawbacks with regard to the meshing. As an alter-native and supplementary computational tool to the mesh-based methods, applying meshless method to investigate theshakedown analysis problem is an interesting and significant

work. Recently, Chen et al. [43] and Chen et al. [44] respec-tively applied element-free Galerkin (EFG) method and localPetrov–Galerkin (MLPG) method to investigate static shake-down analysis problems of the unbounded and bounded kine-matic hardening structures, and obtained a series of lowerbound shakedown load multipliers which show excellentagreement with the results in the available literatures. Inmost of current meshless methods, the numerical integra-tion usually carried out over the background meshes or cellstructures. As an effective way, the nodal integration tech-nique was proposed to eliminate the background meshesand made the numerical integration more efficiency. Beis-sel and Belytschko [45] introduced the nodal integrationinto the EFG method (EFG), and dealt with spatial insta-bility by adding the stabilization item. However, the numer-ical results shown in their paper are less accuracy than thatobtained by using the EFG. Chen et al. [46,47] proposed astabilized conforming nodal integration (SCNI) with strainsmoothing [48] stabilization to eliminate the spatial instabil-ity for the nodal integration of Galerkin mesh-free methods.Their numerical results indicate that the proposed methodowns some unique characteristics, such as good accuracy,high efficiency, robust for the irregular nodal discretization,no demand of the additional control parameter. Wang andChen [49,50] further developed locking-free SCNI formula-tions for Mindlin-Reissner plates and curved beams. LaterWang and Chen [51] and Wang and Lin [52] proposed a sub-domain stabilized conforming integration (SSCI) with Her-mite reproducing kernel (HRK) approximation for the staticand dynamic analyses of thin plates. Yoo et al. [53] intro-duced the stabilized conforming nodal integration schemeinto the natural element method (NEM), named this numer-ical method as the nodal-NEM and used it to investigate thetwo-dimensional elastic problems involved small and largedeformations, material incompressibility and discontinuity.Puso et al. [54] proposed a modified SCNI (MSCNI), andcompare it with SCNI adopted Laplace interpolant and afinite element Q1P0 in the eigenanalysis of a block and thelarge deformation evaluation of a statically compressed rub-ber billet. The nodal-NEM is a fascinating numerical method,which combines the advantages of the SCNI and the NEM.The natural neighbour interpolation used in the NEM andthe nodal-NEM does not concern the complex matrix inver-sion, without any uncertain user-defined parameters, and thegained shape function satisfies the property of delta func-tion so that the essential boundary condition can be imposedeasily and precisely. The strain smoothing technique and thestabilized conforming nodal integration scheme used in thenodal-NEM are respectively beneficial to improve the com-putational efficiency as well as the accuracy.

The attractive advantages of the nodal-NEM are adoptedin this work to establish a stabilized and effective numeri-cal solution algorithm for the kinematic shakedown analysis

123

Comput Mech (2014) 54:1111–1128 1113

problems. According to the Koiter’s theorem and the nodal-NEM, the nonlinear mathematical programming formulationfor the kinematic shakedown analysis of elastoplastic struc-tures is constructed, and the Sibson interpolation is utilized toapproximate the variable cumulative displacement fields inthe framework of Galerkin method. This nonlinear minimiza-tion problem is subjected to several equality constraints andits goal function is nonlinear and non-differentiable, the twomain difficulties of which are how to construct the kinemat-ically admissible plastic strain field and how to disposal thetime integral. A direct iterative algorithm is adopted hereinto solve this minimization problem. As similar to the upperbound limit analysis of plane structure, this algorithm intro-duces the plastic incompressibility condition into the mini-mized goal function by two different ways and overcomes thenonlinear and non-differentiable characteristic of the objec-tive function by distinguishing non-plastic areas from plasticareas and revising the associated constraint conditions andgoal function at every iterative step. The numerical resultsillustrate that the developed numerical algorithm can effec-tively ensure the monotonously convergence of shakedownloads to the upper bounds of true solutions, and verify thehigh efficiency and the good accuracy of this algorithm.

The organization of the rest of this paper is as follows.The kinematic theorem of shakedown analysis is outlined inSect. 2. A summary of the nodal natural element method isgiven in Sect. 3. Section 4 presents the upper bound shake-down analysis based on the nodal-NEM. The effusiveness ofthe proposed method is verified in Sect. 5 through numericalexamples. Finally the conclusion is drawn in Sect. 6.

2 Kinematic theorem of shakedown analysis

The practical engineering structure usually subjects to theeffect of repeated or cycling external load which varies in thespecific range. The following three structural responses willtake place successively for elastoplastic structure with theincreasing of applied loads: (1) when the external repeatedload is less than the elastic limit, the structural deformationis purely elastic; (2) when the load is larger than the elas-tic limit and lower than a critical limit at the same time,the plastic deformations occurs in some local parts of thestructure at the beginning of several load cycles. Afterwards,the further development of plastic deformation terminatesand the gradual increase of the residual stress ceases, thestructure just generates elastic deformation, and the varia-tion between stress and strain completely follows the purelyelastic response law in the surplus load cycles. This state isknown as shakedown and the relevant critical limit is namedas “shakedown limit”; (3) when the load is greater than shake-down limit, the structure enters into the non-shakedown state.At this time, the plastic flow develops unlimitedly and the

structure damages owe to the alternating plasticity (low-cyclefatigue) or incremental collapse (ratcheting). Whether struc-ture is shakedown or not relates to the load varying range andthe structural material properties.

As proved by Koiter [4] in 1956, the kinematic shakedownanalysis theorem can be stated as [2,4]: if exists a kinemat-ically admissible plastic strain rate cycle, which makes thestructural work rate produced by the applied external loadsless than the rate of plastic dissipation power, then structuralshakedown will take place. Suppose an elastoplastic struc-ture with domain Ω is applied on a set of quasi-static loadsP̃ which independently vary within the following specifiedscopes:

P̃ = [P̃1, P̃2, . . . , P̃v

]T

= s∗ [μ1 P1, μ2 P2, . . . , μv Pv

]T(1)

μ−i ≤ μi ≤ μ+

i , (i = 1, 2, . . . , v) (2)

where s∗ is the load multiplier, v denotes the total numberof varying loads P̃ , μi P i (i = 1, 2, . . . , k) stands for thebasic load, μi represents the variation factor with the lowerbound μ−

i and the upper bound μ+i . Accordingly, the struc-

tural upper bound shakedown load multiplier can be deter-mined according to above-mentioned kinematic shakedowntheorem, and the associative mathematical programming for-mulation for the upper bound shakedown analysis can bewritten as:

s = minε̇

pi j ,�ui

T∫

0

dt∫

Ω

D(ε̇

pi j

)dΩ (3)

s.t.

T∫

0

dt∫

Ω

σ ei j ε̇

pi j dΩ = 1 (4)

�εpi j =

T∫

0

ε̇pi j dt = 1

2

(�ui, j + �u j,i

)in Ω (5)

�ui =T∫

0

u̇i dt in Ω (6)

ε̇pi i = 0 in Ω (7)

�ui = 0 on Γu (8)

where s is the shakedown load multiplier, u̇i and ε̇pi j respec-

tively denote the displacement velocity field and plastic strainrate, �ui and �ε

pi j respectively represent the cumulative dis-

placement field and cumulative plastic strain after one load-ing cycle during the time interval [0, T ], σ e

i j is the vir-tual pure elastic stress field caused by the applied externaltraction, Γu represents the structural displacement boundary,

D(ε̇

pi j

)stands for the plastic dissipation work rate and can

123

1114 Comput Mech (2014) 54:1111–1128

be formulated as D(ε̇

pi j

)= σi j ε̇

pi j . Herein, σi j represents

the stress field associated with the plastic strain rate ε̇pi j . By

adopting the von Mises yield criterion and the associativeflow rule, the plastic dissipation work rate can be expressedas follows:

D(ε̇

pi j

)= σi j ε̇

pi j =

√2

3σs

√ε̇

pi j ε̇

pi j (9)

where σs stands for the yield stress of material.In Eqs. (3)–(8), the following classical assumptions are

adopted: (1) the structural material is ideal plastic, and thehardening and softening effects are not considered; (2) thestructural deformation is small and the geometric effect isignored; (3) the structural material satisfies the Drucker pos-tulate, which means the yield surface is convex, and the plas-tic strain rate coincides with the direction of outward normal;(4) the elastic modulus is constant and the properties of mate-rial are not affected by temperature; (5) the loading process isquasi-static, there is no dynamic effect; (6) the factors relatedto the time are ignored. Eqs. (3)–(8) indicate that the kine-matic shakedown analysis can be summarized to solve thenonlinear minimization mathematical programming prob-lem subjected to several equality constraints. Equation (4)is the normalization condition, Eq. (5) stands for the geomet-rical compatibility equation, Eq. (6) represents the cumula-tive displacement over a time cycle, Eq. (7) denotes the plas-tic incompressibility condition, Eq. (8) is the displacementboundary condition.

3 Nodal natural element method with Sibsoninterpolation

The nodal natural element method (nodal-NEM) [53] is arecently developed variant of the NEM, which uses the weakform of global Galerkin method to generate its system gov-erned equation and adopts the natural neighbour interpolationtechnique to approximate its trail function. Since the naturalneighbour interpolation has been introduced by Sibson [55],Braun and Sambridge [56] applied it for the solution of par-tial differential equations and Sukumar et al. [35] employedthis approximation for the solid mechanics problem in theGalerkin framework and named this numerical method as thenatural element method (NEM). Sukumar et al. [57] furtherinvestigated the two-dimensional solid mechanic problemsby using non-Sibson interpolation technique in the NEM.From then on, the NEM has attracted considerable interestsand has been more and more widely used in the computa-tional mechanic field. Cueto et al. [58] overviewed the mostnotable advances of the NEM for the application of linearand non-linear problems in the solid, fluid and biomechanicsmechanics. In the traditional NEM, the domain integration

is carried out over the background mesh, which has beenconsidered to be far from the optimal selection. Under thiscircumstance, Yoo et al. [53] introduced the stabilized con-forming nodal integration scheme into the NEM based on thenon-Sibson interpolation and coined its name as the nodal-NEM. In contrast to the traditional NEM adopted the back-ground mesh integration, the application of the nodal-NEMin two-dimensional plane solid mechanic problem shows thesignificant improvement in the aspects of efficiency and accu-racy. In the nodal-NEM, the derivatives of shape function arenot needed, the strain smoothing technique is used to approx-imate the nodal strain, the stabilized conforming nodal inte-gration scheme is adopted to execute the numerical integra-tion, the nodal stresses and strains can be calculated directlywhile those ones in the NEM are needed to be recovered bythe utilization of values at integration points. Unlike the non-Sibson interpolation adopted by Yoo et al. [53], this paperuses the Sibson interpolation to construct the trail functionin the nodal-NEM. Before going further, this paper brieflyreviews the Sibson interpolation technique and the stabilizedconforming nodal integration scheme.

3.1 Sibson interpolation

At present, there exist several different interpolation tech-niques associated with the notion of natural neighbourhood,such as Thiessen interpolation [59,60], Sibson interpolation[35,55], Laplace interpolation (non-Sibson interpolation)[57,61,62], psdudo-NEM interpolation [63]. In addition,a modified version of Sibson interpolation was proposedby Cueto et al. [64], which is based on the notion of α-shapes, and achieves the linear interpolation both over convexand non-convex boundaries. These natural neighbour-basedinterpolation techniques possess their own distinctive char-acteristics and applications.

The Sibson interpolation relays on the construction ofVoronoi diagram and Delaunay triangulation. Assume a setof scattered nodes N = {n1, n2, . . . , nN P } are used to dis-cretize the bounded problem domain Ω of two-dimensionalsolid Mechanics. The Voronoi diagram, which is also calledfirst-order Voronoi diagram or Dirichlet tessellation, is aunique decomposition of problem domain into a series ofregions TI defined as:

TI ={

x ∈ R2 : d(x, x I ) < d(x, x J ),∀J �= I}

(10)

herein, d(x, x J ) represents the Euclidean distance betweenx I and x J . Each region TI is related to node nI , and any pointwithin these regions TI are nearer to node nI than to anyother nodes n J ∈ N (J �= I ). The Delaunay triangulation isthe geometrical dual structure of the Voronoi diagram, whichis usually used as the background integral cell in the tradi-

123

Comput Mech (2014) 54:1111–1128 1115

2

1

3

4

5

6

7x

a

bc

d

e

fg

h

Fig. 1 Computation of Sibson interpolant shape function

tional NEM. Empty circumcircle criterion is an importantproperty of the Delaunay triangulation and can be adoptedto search the natural neighbours. This criterion describes asthat if any point locates inside the circumcircle of Delau-nay triangle DT = (nI , n J , nK ), the three nodes nI , n J , nK

are the natural neighbours of this point. Take the insertedpoint x displayed in Fig. 1 into account, point x simultane-ously locates inside the circumcircles of Delaunay trianglesDT = (1, 5, 6), DT = (1, 6, 7) and DT = (1, 7, 2), thus thenodes 1, 5, 6, 7, 2 are the natural neighbours of point x andthe relevant number of natural neighbours is np = 5.

The notion of second-order Voronoi cells are adopted bySibson [55] to quantify the neighbours for an inserted pointx , and the second-order Voronoi diagram for each regionTI J associated with a nodal-neighbour-pair (nI , n J ) can bedefined as:

TI J = {x ∈ R2 : d(x, x I )

< d(x, x J ) < d(x, xK ),∀J �= I �= K } (11)

herein, TI J denotes the locus of points that respectively havenode x I and x J as the closest and second closest neighbournodes. The Sibson shape function of point x with respect tonode I is formulated as the ratio of area AI (x) for Tx I andA (x) for Tx , that is:

ΦI (x) = AI (x)

A (x), A (x) =

np∑

J

AJ (x) (12)

Take the depiction listed in Fig. 1 for example, the Sibsonshape function Φ1 (x) can be calculated as:

Φ1 (x) = Aabf gh

Aabcde(13)

The detailed computational implementation of Sibson shapefunction was provided by Sukumar [35], in which the inter-ested readers can refer to the computational procedure and the

relevant bibliographies therein. The obtained Sibson shapefunction possesses the properties of positivity, interpolation,partition of unity and linear consistency, and the relevantmathematical formulas of those properties can be describedas:

0 ≤ ΦI (x) ≤ 1, ΦI (x J ) = δI J ,

np∑

I=1

ΦI (x) = 1,

np∑

I=1

ΦI (x) xI = x (14)

Furthermore, Sibson shape functions are C∞ everywhereapart from the nodal locations where they are C0, and theycan ensure the linear precision on the convex boundary. Fig-ure 2a, b respectively illustrate the supported domain and theassociative image of Sibson shape function for node I .

Accordingly, the trail function of unknown displacementfield variable can be approximated as following form:

uh (x) =np∑

I=1

ΦI (x) uI (15)

where uI (I = 1, 2, . . . np) stand for the nodal displacementvectors related to np natural neighbours of point x.

3.2 Stabilized conforming nodal integration scheme

Consider a problem domain Ω that is discretized by a groupof nodes with total numbers N P . According to the defin-ition of Voronoi diagram, the problem domain Ω can be

subdivided into a series of sub-domain Ωr

(Ω = N P∪

r=1Ωr

).

According to the stabilized conforming nodal integration(SCNI) scheme proposed by Chen et al. [46], a smoothedstrain at the location of node xr , denoted by ε̃i j (xr ), can becalculated as:

ε̃i j (xr ) = 1

Ar

∫

Ωr

εi j (xr ) dΩ

= 1

2Ar

∫

Ωr

{∂ui

∂x j+ ∂u j

∂xi

}dΩ

= 1

2Ar

∫

Γr

{ui n j + u j ni

}dΓ (16)

whereΓr and Ar respectively denote the boundaries and areasof the sub-domain Ωr with respect to node xr , and ni ’s arethe unit outward normals of boundaries Γr as displayed inFig. 3.

123

1116 Comput Mech (2014) 54:1111–1128

Fig. 2 The shape function ofSibson interpolation. a nodalgrid. b Shape function Φ (x) fornode I

x

y

I

0.5

0.5

0.5−

0.5−

a b

rx

rΓ

n

1n

2n

rΩ

Fig. 3 The depiction of nodal sub-domain in two dimensions

Substituting the trail function of displacement field intoEq. (16), we can obtain:

ε̃ (xr ) =Gr∑

I=1

B̃ I (xr ) uI (17)

with the following defined matrices:

uI = [u1I u2I

]T(18)

ε̃ (xr ) = [ε̃11 (xr ) ε̃22 (xr ) 2ε̃12 (xr )

]T(19)

B̃ I (xr ) = 1

Ar

∫

Γr

⎛

⎝ΦI n1 0

0 ΦI n2

ΦI n2 ΦI n1

⎞

⎠ dΓ (20)

where Gr stands for a set of nodes in which their relatednatural neighbours contain node xr .

So far, some applications of SCNI with the natural neigh-bour interpolation techniques to the field of computationalmechanics have been published. Yoo et al. [53] used thenodal-NEM with SCNI and non-Sibson interpolant to studythe two-dimensional elastic problems, and their obtainednumerical results for the problems of small and large defor-mations, material incompressibility and discontinuity are allin good agreement with the exact solutions and appear no

obvious oscillations, at least for quasi-static analysis. Pusoet al. [54] studied MSCNI and SCNI with Laplace inter-polant, finite element Q1P0 through the eigen-analysis andlarge deformation simulations. Their MSCNI results com-pare very well with the finite element, while their SCNIresults exhibit the spurious at first eigenmode in the eigen-analysis of a block and demonstrate significant oscillationand eventually corrupted result at large strains in the largedeformation evaluation of a statically compressed rubber bil-let. According to the assumptions listed in Sect. 2 and thederivation given in Sect. 4, the minimization problem forthe upper bound shakedown analysis can be equivalent tosolving the linear equations with small deformation assump-tion. Additionally, the Sibson interpolant is another accu-rate and effective natural neighbour interpolation techniqueas the Laplace interpolant. Thus, using the nodal-NEM withSCNI and Sibson interpolant for the upper bound shakedownanalyses of two-dimensional structures is expected to obtainthe similar accurate and stable numerical results as those byYoo et al. [53]. The numerical results and the iterative con-vergence processes provided in Sect. 5 will demonstrate theaccuracy and the stability of proposed numerical method.

4 Discrete formulation of upper bound shakedownanalysis with nodal-NEM

The discretization of nonlinear mathematical programmingproblem expressed in Eqs. (3)–(8) can be performed by usingthe above-mentioned nodal-NEM. In the displacement-basednodal-NEM, the natural neighbor interpolation is adoptedto construct the trail function, and the stabilized conform-ing nodal integration scheme is introduced in the Galerkinimplementation. Refer to the above-mentioned subdivisionof sub-domain Ωr and Eq. (17), the smoothing cumulativeplastic strain field at the location of node xr can be approxi-mated as follows:

123

Comput Mech (2014) 54:1111–1128 1117

�ε̃p (xr ) =Gr∑

I=1

B̃ I (xr )�uI = B̃ (xr )�u (21)

where �u =Gr∑

I=1�uI and B̃ (xr ) =

Gr∑

I=1B̃ I (xr ) respec-

tively denote the global nodal cumulative displacement vec-tor during a loading cycle and global strain-displacementrelation matrix corresponding to those natural neighboringnodes used for the strain smoothing of node xr .

4.1 Treatment of incompressibility condition

Taking plane stress problem into account in the kinematicshakedown analysis, it follows ˙̃εp

xz (xr ) = ˙̃εpyz (xr ) = 0

and ˙̃εpz (xr ) �= 0. The plastic incompressibility condition in

Eq. (7) can be written as ˙̃εpz (xr ) = −

[ ˙̃εpx (xr ) + ˙̃εp

y (xr )],

therefore ˙̃εpi j (xr ) ˙̃εp

i j (xr ) can be formulated as follows:

˙̃εpi j (xr ) ˙̃εp

i j (xr )

=[ ˙̃εp

x (xr )]2 +

[ ˙̃εpy (xr )

]2 +[ ˙̃εp

z (xr )]2

+2[ ˙̃εp

xy (xr )]2 + 2

[ ˙̃εpyz (xr )

]2 + 2[ ˙̃εp

zx (xr )]2

= 2

{[ ˙̃εpx (xr )

]2 +[ ˙̃εp

y (xr )]2

+˙̃εpx (xr ) ˙̃εp

y (xr ) +[ ˙̃εp

xy (xr )]2}

=[ ˙̃εp

(xr )]T

D ˙̃εp(xr ) (22)

where

˙̃εp(xr ) =

[ ˙̃εpx (xr ) ˙̃εp

y (xr ) 2 ˙̃εpxy (xr )

]T(23)

D =⎡

⎣2 1 01 2 00 0 0.5

⎤

⎦ (24)

By this way, the incompressibility condition for the kine-matic shakedown analysis of plane stress problem can beautomatically satisfied in the matrix expression form of˙̃εp

i j (xr ) ˙̃εpi j (xr ).

Regarding to the plane strain problem in the kinematicshakedown analysis, ˙̃εp

xz (xr ) = ˙̃εpyz (xr ) = 0 and ˙̃εp

z (xr ) =0 should be satisfied. Thus ˙̃εp

i j (xr ) ˙̃εpi j (xr ) can be written as:

˙̃εpi j (xr ) ˙̃εp

i j (xr )

=[ ˙̃εp

x (xr )]2 +

[ ˙̃εpy (xr )

]2 +[ ˙̃εp

z (xr )]2

+2[ ˙̃εp

xy (xr )]2 + 2

[ ˙̃εpyz (xr )

]2 + 2[ ˙̃εp

zx (xr )]2

=[ ˙̃εp

x (xr )]2 +

[ ˙̃εpy (xr )

]2 + 2[ ˙̃εp

xy (xr )]2

=[ ˙̃εp

(xr )]T

D ˙̃εp(xr ) (25)

with the following defined matrix:

D =⎡

⎣1 0 00 1 00 0 0.5

⎤

⎦ (26)

The incompressibility condition in Eq. (7) for the kinematicshakedown analysis of plane strain problem can be expressedas follows:

˙̃εpi i (xr ) = ˙̃εp

x (xr ) + ˙̃εpy (xr ) + ˙̃εp

z (xr ) = ˙̃εpx (xr )

+˙̃εpy (xr ) = Bv ˙̃εp

(xr ) = 0 (27)

where

Bv = [1 1 0

](28)

Herein, the penalty function method is used to introduceEq. (27) into the goal function, and the penalty function item

is set asαv∫ T

0

∫ [ ˙̃εi ip(xr )

]2/2dΩdt , whereαv is the penalty

factor. The matrix form of penalty function item is writtenas:

1

2αv

T∫

0

∫ [ ˙̃εpi i (xr )

]2dΩdt

= 1

2αv

T∫

0

∫ [ ˙̃εp(xr )

]TDv ˙̃εp

(xr ) dΩdt (29)

where Dv = BTv Bv.

Through substituting Eq. (22) or Eq. (25) into the goalfunction Eq. (3), and writing the item σ e

i j ε̇pi j in normaliza-

tion condition Eq. (4) as the matrix form [σ̃ e (xr )]T ˙̃εp(xr ),

the previous goal function, normalization condition and thepenalty function item for the plastic incompressibility con-dition can respectively be discretized using the nodal-NEMas:

√2

3σs

T∫

0

∫ √[ ˙̃εp(xr )

]TD ˙̃εp

(xr )dΩdt

=√

2

3σs

T∫

0

N P∑

r=1

∫ √[ ˙̃εp(xr )

]TD ˙̃εp

(xr )dΩr dt

=√

2

3σs

T∫

0

N P∑

r=1

Ar

√[ ˙̃εp(xr )

]TD ˙̃εp

(xr )dt (30)

123

1118 Comput Mech (2014) 54:1111–1128

T∫

0

∫ [σ̃ e (xr )

]T ˙̃εp(xr ) dΩdt

=T∫

0

N P∑

r=1

∫ [σ̃ e (xr )

]T ˙̃εp(xr ) dΩr dt

=T∫

0

N P∑

r=1

Ar[σ̃ e (xr )

]T ˙̃εp(xr ) dt (31)

1

2αv

T∫

0

∫ [ ˙̃εp(xr )

]TDv ˙̃εp

(xr ) dΩdt

= 1

2αv

T∫

0

N P∑

r=1

∫ [ ˙̃εp(xr )

]TDv ˙̃εp

(xr ) dΩr dt

= 1

2αv

T∫

0

N P∑

r=1

Ar

[ ˙̃εp(xr )

]TDv ˙̃εp

(xr ) dt (32)

where Ar denotes the integral weight and is the area of sub-domain Ωr .

By replacing the plastic incompressibility conditionEq. (7) by using the penalty function item Eq. (32) andtemporarily omit the factor αv/2, the conclusive discretizedformulation of minimum optimized kinematic shakedownanalysis based on the nodal-NEM for plane stress and planestrain problems can ultimately be written as the follows:

s = min˙̃εp

(xr ),�u

√2

3σs

T∫

0

N P∑

r=1

Ar

√[ ˙̃εp(xr )

]TD ˙̃εp

(xr )dt

(33)

s.t.

T∫

0

N P∑

r=1

Ar[σ̃ e (xr )

]T ˙̃εp(xr ) dt = 1 (34)

�ε̃p (xr ) =T∫

0

˙̃εp(xr ) dt = B̃ (xr )�u in Ω (35)

�u =T∫

0

u̇dt in Ω (36)

T∫

0

N P∑

r=1

Ar

[ ˙̃εp(xr )

]TDv ˙̃εp

(xr ) dt = 0 (37)

�u = 0 on Γu (38)

The above mathematical programming formation containsthe time integration and satisfies a series of equality con-straints, and the plastic incompressibility condition is ful-filled by introducing a matrix D into the goal function for

the plane stress problem and importing a penalty functionitem into the goal function for the plane strain problem.

4.2 Time integration

The time integration in the nonlinear mathematical program-ming problem Eqs. (33)–(38) is a potential computational dif-ficulty. The König’s technique [2] is adopted in this paper toresolve this issue.

Under the repeated loading, the load domain�

Ω can beconsidered as a hyper polyhedron constructed by the defi-nition of a sequence of load vertices Pk (k = 1, 2, . . . , l).When v groups of independently varied basic loads areapplied to a structure, the relevant numbers of load ver-tices are l = 2v . König’s technique stated that [2]: if astructure reaches the shakedown state under the sequenceof load vertices Pk (k = 1, 2, . . . , l), then it will shakedown

under the whole load domain�

Ω . If the stress caused by loadvertices Pk (k = 1, 2, . . . , l) let the structure enter into theyield state, and the time of duration for this stress state is

τk

(∑lk=1 τk = T

), the plastic strain increment produced

by load vertices Pk (k = 1, 2, . . . , l) during the time τk canbe obtained as follows:

ε̃pk (xr ) =

∫

τk

˙̃εp(xr ) dt (k = 1, 2, . . . , l) (39)

After one loading cycle with time interval [0, T ], the struc-tural cumulative plastic strain can be calculated as:

�ε̃p (xr ) =l∑

k=1

ε̃pk (xr ) (40)

As a result, the nonlinear mathematical programming withthe elimination of time variable integration for kinematicshakedown analysis can be written as the following form:

s = minε̃

pk (xr ),�u

√2

3σs

l∑

k=1

N P∑

r=1

Ar

√[ε̃

pk (xr )

]TDε̃

pk (xr ) (41)

l∑

k=1

N P∑

r=1

Ar[σ̃ e

k (xr )]T

ε̃pk (xr ) = 1 (42)

�ε̃p (xr ) =l∑

k=1

ε̃pk (xr ) = B̃ (xr )�u (43)

l∑

k=1

N P∑

r=1

Ar[ε̃

pk (xr )

]TDvε̃

pk (xr ) = 0 (44)

�u = 0 on Γu (45)

In Eqs. (41)–(45), both the plastic incompressibility condi-tion and time integration are treated. In the following discus-sion, we establish a direct iterative algorithm to linearize the

123

Comput Mech (2014) 54:1111–1128 1119

above minimum optimization problem with several equalityconstraints and then construct the kinematically admissibleplastic strain flied and present a computational formula forthe upper bound shakedown load multiplier.

4.3 Derivation of the iterative algorithm

The nonlinear and nondifferentiable characteristic of theobjective function is similar to the kinematic limit analy-sis [65] and will pose considerable difficulty in the solu-tion process. A computational algorithm, which has success-fully and effectively adopted in the kinematic shakedownanalysis [12,18–20,23], is used to search the minimizationupper bound shakedown loads herein. The detailed strategyis: the Lagrangian multiplier method is used to insert the nor-malization condition Eq. (42) and geometrical compatibilityEq. (43) into the goal function Eq. (41), and the incompress-ibility condition Eq. (44) is dealt with the above-mentionedpenalty function method. Consequently, the mathematicalprogramming problem depicted in Eqs. (41)–(45) is equiv-alent to the following unconstrained minimization problem:

L(ε̃

pk (xr ) ,�u, λ, Lr

)

=√

2

3σs

l∑

k=1

N P∑

r=1

Ar

√[ε̃

pk (xr )

]TDε̃

pk (xr )

+λ

{

1 −l∑

k=1

N P∑

r=1

Ar[σ̃ e

k (xr )]T

ε̃pk (xr )

}

+N P∑

r=1

LTr

[l∑

k=1

ε̃pk (xr ) − B̃ (xr ) (�u)

]

+1

2αv

l∑

k=1

N P∑

r=1

Ar[ε̃

pk (xr )

]TDvε̃

pk (xr ) (46)

where λ and Lr stand for the Lagrangian multipliers. Forthe plane stress problem, because the plastic incompress-ibility condition has been directly satisfied in the nonlinear

item√[

ε̃pk (xr )

]TDε̃

pk (xr ) of Eq. (46), penalty factor is set

as αv = 0 in this case. For the plane stress problem, penaltyfactor is usually assigned as αv = (

105 ∼ 108)σs.

In terms of the Kuhn-Tucker stationary condition [66],and let ∂L

∂ ε̃pk (xr )

= 0, ∂L∂(�u)

= 0, ∂L∂λ

= 0 and ∂L∂ Lr

= 0, the

following equations can be derived:

√2

3σs

Dε̃pk (xr )√[

ε̃pk (xr )

]TDε̃

pk (xr )

+ αv Dvε̃pk (xr ) − λσ̃ e

k (xr )

+ (Ar )−1 Lr = 0 (47)

N P∑

r=1

B̃T

(xr ) Lr = 0 (48)

l∑

k=1

N P∑

r=1

Ar[σ̃ e

k (xr )]T

ε̃pk (xr ) = 1 (49)

l∑

k=1

ε̃pk (xr ) − B̃ (xr ) (�u) = 0 (50)

Obviously, Eq. (47) has the square root item and is not trivialto be directly solved. In the view of this feature, the lineariza-tion of Eqs. (47)–(50) is performed as follows:√

2

3σs

Dε̃phk (xr )

√[ε̃

p(h−1)

k (xr )]T

Dε̃p(h−1)

k (xr )

+αv Dvε̃phk (xr ) − λh σ̃ e

k (xr ) + (Ar )−1 Lh

r = 0 (51)N P∑

r=1

B̃T

(xr ) Lhr = 0 (52)

l∑

k=1

N P∑

r=1

Ar[σ̃ e

k (xr )]T

ε̃phk (xr ) = 1 (53)

l∑

k=1

ε̃phk (xr ) = B̃ (xr ) (�u)h (54)

In Eqs. (51)–(54), ε̃p(h−1)

k (xr ) denotes the plastic strainincrement at iterative step (h − 1) and its value is knownbefore the hth iteration, ε̃

phk (xr ), (�u)h , λh and Lh

r arerespectively stand for the plastic strain increment, cumulativedisplacement field, Lagrangian multipliers, and their valuesare unknown at the hth iteration. Thus, Eqs. (51)–(54) arejust the linear equations of ε̃

phk (xr ), λh , Lh

r and (�u)h .It should be pointed out that the linear iterative formula

in Eqs. (51)–(54) is unreasonable and the iteration can not beimplemented uninterruptedly and smoothly when the plasticstrain increment at some nodes are zeroes. Thus before the

h th iteration, the value of

√[ε̃

p(h−1)

k (xr )]T

Dε̃p(h−1)

k (xr )

should be checked to ensure it is non-zero. We define

X̃ hk (xr ) =

√[ε̃

p(h−1)

k (xr )]T

Dε̃p(h−1)

k (xr ), and divide the

nodes N P into two subsets by adopting the following iden-tified rules:

N P = (N P)hE ∪ (N P)h

P (55)

(N P)hE =

{r ∈ N P, X̃ h

k (xr ) = 0}

(56)

(N P)hP =

{r ∈ N P, X̃ h

k (xr ) �= 0}

(57)

where (N P)hE denotes the non-plastic zone subset where no

plastic dissipation work generates, and (N P)hP stands for the

plastic zone subset where the plastic dissipation work pro-duces. Eqs. (55)–(57) indicate that the non-plastic zones aredistinguished from the plastic zones by checking whether thevalue of X̃ h

k (xr ) equals to zero or non-zero. Due to the fact

123

1120 Comput Mech (2014) 54:1111–1128

that the numerical error exists in the practical calculation,we use the following recognition criterion when load vertexk are applied: if the value of X̃ h

k (xr ) for node xr at itera-tion step h (h ≥ 1) is greater than a small positive decimalγ , node xr is considered as in the plastic states and locatesin the plastic zones; if the value of X̃ h

k (xr ) for node xr atiteration step h equals to or is smaller than γ , node xr isthought of in the non-plastic states and locates in the non-plastic zones. Usually, we set γ = (

10−5 ∼ 10−8)

X in the

actual computation, where X = 1l×N P

l∑

k=1

N P∑

r=1X̃1

k (xr ) and

X̃1k (xr ) =

√[ε̃

p0k (xr )

]TDε̃

p0k (xr ).

For the sake of simplicity, an intermediate matrix Hhkr is

defined as follows:

H̃hk (xr ) =

√2

3

σs

X̃ hk (xr )

D + αv Dv (58)

As a result, the iterative formula in Eqs. (47)–(50) can con-cisely be written as:

H̃hk (xr ) ε̃

phk (xr ) − λh σ̃ e

k (xr ) + (Ar )−1 Lh

r = 0 (59)N P∑

r=1

B̃T

(xr ) Lhr = 0 (60)

l∑

k=1

N P∑

r=1

Ar[σ̃ e

k (xr )]T

ε̃phk (xr ) = 1 (61)

l∑

k=1

ε̃phk (xr ) = B̃ (xr ) (�u)h (62)

In order to solve Eqs. (59)–(62) with four unknown vari-ables ε̃

phk (xr ), λh , Lh

r and (�u)h , and obtain the upper boundshakedown load multiplier, we propose the following com-putational flow chart:

(a) Subtracting Eq. (59) corresponding to a load vertexremarked as m, from all the other equations to the loadvertex k, we have:

ε̃phk (xr ) =

[H̃

hk (xr )

]−1 {λh [σ̃ e

k (xr ) − σ̃ em (xr )

]

+H̃hm (xr ) ε̃ph

m (xr )}

(63)

(b) Inserting Eq. (63) into Eq. (62), we obtain:

ε̃phm (xr ) =

[H̃

hm (xr )

]−1{

l∑

k=1

[H̃

hk (xr )

]−1}−1

{

B̃ (xr ) (�u)h − λhl∑

k=1

[H̃

hk (xr )

]−1 [σ̃ e

k (xr ) − σ̃ em (xr )

]}

(64)

(c) Switch the position of subscript marked as m and k inEq. (64), (64) becomes:

ε̃phk (xr ) =

[H̃

hk (xr )

]−1{

l∑

m=1

[H̃

hm (xr )

]−1}−1

{

B̃ (xr ) (�u)h + λhl∑

m=1

[H̃

hm (xr )

]−1 [σ̃ e

k (xr ) − σ̃ em (xr )

]}

(65)

(d) Respectively substituting Eq. (65) into Eqs. (59) and (61),we get:

Lhr = Ar

{l∑

m=1

[H̃

hm (xr )

]−1}−1

{

λhl∑

m=1

[H̃

hm (xr )

]−1σ̃ e

m (xr ) − B̃ (xr ) (�u)h

}

(66)

l∑

k=1

N P∑

r=1

Ar[σ̃ e

k (xr )]T [H̃

hk (xr )

]−1{

l∑

m=1

[H̃

hm (xr )

]−1}−1

{

B̃ (xr ) (�u)h + λhl∑

m=1

[H̃

hm (xr )

]−1 [σ̃ e

k (xr ) − σ̃ em (xr )

]}

= 1 (67)

(e) Substituting Lagrangian multiplier Lhr in Eq. (66) back

into Eq. (60), we have:

λhN P∑

r=1

Ar B̃T

(xr )

{l∑

m=1

[H̃

hm (xr )

]−1}−1

l∑

m=1

[H̃

hm (xr )

]−1σ̃ e

m (xr )

=N P∑

r=1

Ar B̃T

(xr )

{l∑

m=1

[H̃

hm (xr )

]−1}−1

B̃ (xr ) (�u)h (68)

(f) Define (�u)h = λhδh , herein, δh has the same dimensionas the global cumulative displacement vector. Substitute(�u)h into Eq. (68), the linear equation of δh can be writ-ten as follows:

K hδh = Fh (69)

where

K h =N P∑

r=1

Ar B̃T

(xr )

{l∑

m=1

[H̃

hm (xr )

]−1}−1

B̃ (xr )

(70)

123

Comput Mech (2014) 54:1111–1128 1121

Fh =N P∑

r=1

Ar B̃T

(xr )

{l∑

m=1

[H̃

hm (xr )

]−1}−1

l∑

m=1

[H̃

hm (xr )

]−1σ̃ e

m (xr ) (71)

(g) Introduce the displacement boundary condition Eq. (45)and solve the linear algebraic equation Eq. (69), δh can becalculated. Substitute the obtained δh into Eq. (67), theLagrangian multiplier λh can be calculated as:

λh = 1

l∑

k=1

N P∑

r=1Ar[σ̃ e

k (xr )]T [H̃

hk (xr )

]−1{

l∑

m=1

[H̃

hm (xr )

]−1}−1 {

B̃ (xr ) δh +l∑

m=1

[H̃

hm (xr )

]−1 [σ̃ e

k (xr ) − σ̃ em (xr )

]}

(72)

(h) Substitute the obtained δh and λh into Eq. (65), the plasticstrain increment at iterative step h can be written as:

ε̃phk (xr ) = λh

[H̃

hk (xr )

]−1{

l∑

m=1

[H̃

hm (xr )

]−1}−1

{

B̃ (xr ) δh +l∑

m=1

[H̃

hm (xr )

]−1 [σ̃ e

k (xr ) − σ̃ em (xr )

]}

(73)

(i) Finally, the shakedown load multiplier of h th iterationcan be written as:

sh =√

2

3σs

l∑

k=1

N P∑

r=1

Ar

√[ε̃

phk (xr )

]TDε̃

phk (xr ) (74)

4.4 The iteration procedure

According to the derivation presented above, the followingiterative procedure can be constructed to evaluate the upperbound shakedown multiplier:Iterative Step 0: The iterative process is initialized with thehypothesis that the whole structure is in the purely plasticstate. Under this assumption, the structure is differentiableeverywhere and the plastic strain rate is nonzero at this itera-tive step. Set X̃0

k (xr ) = 1 (r = 1, 2, . . . N P; k = 1, 2, . . . l)

and H̃0k (xr ) =

√23σs D + αv Dv, and solve the Eqs. (59)–

(62) by implementing the solution process listed in Eqs. (63)–(74), we can successively obtain the Lagrangian multiplierλ0 and the plastic strain increment ε̃

p0k (xr ) of load vertex k

for node xr at step 0. And the initial upper bound shakedownmultiplier can be calculated as following:

s0 =√

2

3σs

l∑

k=1

N P∑

r=1

Ar

√[ε̃

p0k (xr )

]TDε̃

p0k (xr ) (75)

Iterative Step h (h = 1, 2, . . .): Based on the computationalresults of step h − 1, we check the value of X̃ h

k (xr ) =√[ε̃

p(h−1)

k (xr )]T

Dε̃p(h−1)

k (xr ), and determine the non-

plastic subset (N P)hE and the plastic subset (N P)h

P by uti-lizing the identified rules listed in Eqs. (55)–(57) before theiteration of step h. Then calculating the coefficient matrix

H̃hk (xr ) and solving the Eqs. (59)–(62), λh , ε̃

phk (xr ) and sh

at h th iteration can respectively be obtained by the relevantequations expressed in Eqs. (72)–(74).

The above iterative process will be terminated until thefollowing convergence criterion is fulfilled:∥∥(�u)h − (�u)h−1

∥∥∥∥(�u)h−1

∥∥ < vol1 or

∣∣sh − sh−1∣∣

sh−1 < vol2 (76)

where vol1 and vol2 represent the error tolerances deter-mined by the anticipated computational accuracy.

5 Numerical examples

In this section, four representative shakedown analysisnumerical examples are presented to demonstrate the per-formance of the present method, where the comparisonsof accuracy and efficiency between the nodal-NEM andthe NEM are given. The Sibson interpolant is utilized toapproximate the trial function of the nodal-NEM and theNEM in the Galerkin framework. For the numerical inte-gration strategy, 2 point quadrature scheme is adopted oneach boundary of the sub-domain in the nodal-NEM, and 3point quadrature rule is utilized over each Delaunay trian-gle region in the NEM. Additionally, for the further com-parisons of accuracy and efficiency of the nodal-NEM andthe NEM between the FEM, four-nodal isoparametric ele-ment is selected to do the upper bound shakedown analy-ses in the last two examples, and 2 × 2 quadrature rule isadopted over each finite element. The structures are madeup of elastic-perfectly plasticity material with the von Misesyield criterion, and the computational cost listed below iscounted from the same computer (Pentium 4, 3.0 GHz). The

123

1122 Comput Mech (2014) 54:1111–1128

Fig. 4 Square plate with acentral circular hole. aGeometry and loading. bComputational model. c Nodalarrangement (361 nodes)

1P 1P

/ 5L

2mL =

2P

2P

2P

1P

a b c

corresponding parameters are: yield stress σs = 200 MPa,Young’s modulus E = 2.1×105 MPa, Poisson ratio v = 0.3,basis load P = 1,000 N/m, error tolerances vol1 = vol2 =1.0 × 10−4.

5.1 Square plate with a central circular hole under thefluctuating biaxial uniform loads

As a benchmark numerical example in the field of limit andshakedown analysis with plane stress hypothesis, a squareplate with a central circular hole is selected to perform theupper bound shakedown analysis herein. Owing to the struc-tural symmetry of geometry and loading, the upper rightquadrant of this plate is modeled, and the detailed geom-etry parameter, the loading distribution and the computa-tional model adopted in this paper are respectively shownin Fig. 4a, b. The biaxial uniform stretch loads P1 and P2 areapplied on boundaries, both of which can change indepen-dently between 0 and certain maximum magnitudes Pmax

1and Pmax

2 . Figure 4c displays the arrangement of 361 nodesfor the structural discretization of this plate.

This well-known and representative example has beeninvestigated by many scholars by adopting different tech-niques. Belytschko [67] did the pioneer work and gained thelower bound shakedown load factors by utilizing the equi-librated triangular FEM in conjunction with nonlinear opti-mization programming to construct the residual stress fields.Corradi and Zavelani [68] obtained the lower and upperbound shakedown load factors by applying the FEM and lin-ear programming strategy on the basis of Bleich and Melan’stheorem and the relevant duality properties. Nguyen and Pal-gen [69] developed quadrilateral equilibrium finite element,over where the von Mises yield criterion are fulfilled in theaverage, to study the statical shakedown analysis problem.Genna [70] presented a numerical scheme for the evaluationof lower bound safety factor by using isoparametric 8-nodedisplacement elements and a local, posteriori linearizationof the yield surface. Stein et al. [11] incorporated the over-lay model into a finite element system, and employed it toinvestigate the static shakedown problem consisting of elas-tic, perfectly plastic materials. Gross-Weege [25] proposed a

numerical assessment of safety factor based on an extensionof Melan’s statical shakedown theorem and the finite elementdisplacement method. Zouain et al. [71] constructed an algo-rithm derived by coupling a Newton formula with a returnmapping procedure to study the shakedown analysis problemwith a nonlinear yield function, and implemented the rele-vant numerical solution procedure in the finite element mod-els. Garcea et al. [72] obtained the discrete formulation byusing the mixed triangular finite element with interpolation ofboth stresses and displacement variables, and performed theshakedown analysis of two-dimensional flat structures. Tin-Loi and Ngo [73] carried out the static shakedown analysisby applying the p-version quadrilateral finite element. Addi-tionally, many other researchers also studied this example,such as Carvelli et al. [23], Zhang and Raad [74], Liu et al.[30], Krabbenhoft et al. [75], Chen et al. [43] (EFG), Chen[76] (MLPG) and Tran et al. [9], etc.

Table 1 summarizes the published shakedown loads underthree special load combinations, and Fig. 5 compares the pre-sented shakedown load domain with those available ones. Itis clearly demonstrated that the present results lie betweenthose available lower bounds and upper bounds and matchwell with them, and the results respectively obtained by usingthe nodal-NEM and the NEM are almost equal to each other.In order to compare the computational efficiency of the nodal-NEM and the NEM, the statistics of their computational timesin three loading cases are listed in Table 2. It is noted thatthe nodal-NEM is more efficient than the NEM, and thetime spent by using the nodal-NEM is just about quarterof that cost by adopting the NEM. Figure 6 plots the itera-tive convergence processes of shakedown limit loads for thisexample, and it indicates that all of the calculated shakedownloads monotonically decrease to steady minimum values afterabout 60–70 iterative steps.

5.2 Thick-walled cylinder subjected to repeated uniforminternal pressure

As shown in Fig. 7, a thick-walled cylinder is subjected torepeated uniform internal pressure P . This problem is a clas-sical plane strain example in shakedown analysis. In view

123

Comput Mech (2014) 54:1111–1128 1123

Table 1 The shakedown limitloads compared with othernumerical methods (PSD/σs)

Authors Methods Loading cases

P2 = P1 P2 = P1/2 P2 = 0

Zouain et al. [71] 0.429 0.500 0.594

Belytschko [67] Lower bound 0.431 0.501 0.571

Nguyen and Palgen [69] Lower bound 0.431 0.514 0.557

Tran et al. [9] 0.434 0.505 0.601

Tin-Loi and Ngo [73] Lower bound 0.436 0.506 0.604

Garcea et al. [72] 0.438 0.508 0.604

Gross-Weege [25] Lower bound 0.446 0.524 0.614

Stein et al. [11] Lower bound 0.453 0.539 0.624

Chen [76] (MLPG) Lower bound 0.463 0.536 0.634

Liu et al. [30] Lower bound 0.477 0.549 0.647

Genna [70] Lower bound 0.478 0.566 0.653

Chen et al. [43] (EFG) Lower bound 0.480 0.553 0.649

Present NEM Upper bound 0.480 0.554 0.653

nodal-NEM Upper bound 0.480 0.554 0.653

Corradi and Zavelani [68] Upper bound 0.504 0.579 0.654

Carvelli et al. [23] Upper bound 0.518 0.607 0.696

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

P 2/ σ

s

P1/σs

Carvelli et al.

Nodal-NEM Chen et al.(EFG) Zhang et al.

Gross-Weege Stein et al.

upper boundNEM upper bound

upper boundlowerboundlowerbound

Chen(MLPG) lowerboundlowerboundlowerbound

Fig. 5 Comparison of shakedown load domains between differentcomputational results

Table 2 Comparison of the computational times (s)

Methods Loading cases

P2 = P1 P2 = P1/2 P2 = 0

NEM 1354 1415 1530

nodal-NEM 309 320 390

of the structural symmetric property, only the upper rightquadrant of the thick-walled cylinder is modeled as shown inFig. 7, where a and b denote the internal and external radii,respectively. When the variation range of applied pressure is

0 10 20 30 40 50 60 700.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

P SD / σ

s

Iterative step h

P2=0 NEM

P2=0 Nodal-NEM

P2=P1/2 NEM

P2=P1/2 Nodal-NEM

P2=P1 NEM

P2=P1 Nodal-NEM

Fig. 6 The iterative convergence processes of shakedown limit loadsfor the square plate with a central circular hole

between −Pmax and Pmax, the relevant analytical solution ofshakedown limit load is:

PSD = σs√3

(1.0 − a2

b2

)(77)

Several different ratios of b/a are adopted to perform theshakedown analysis, and the corresponding node distribu-tions are used to discretize the computational model. Figure 8illustrates the arrangement of 475 nodes when b/a = 2.8.Figure 9 shows the comparison of numerical results respec-tively obtained by using the nodal-NEM and the NEM with

123

1124 Comput Mech (2014) 54:1111–1128

P

b = 0.2m

a

Fig. 7 A thick-walled cylinder under uniform internal pressure

Fig. 8 Distribution of 475 nodes for thick-walled cylinder (b/a = 2.8)

analytical solutions of shakedown loads, and Table 3 givesthe detailed comparisons of accuracy and efficiency betweensome of them. It can be seen that the numerical results are ingood agreement with the analytical results, and the numericalerrors in the nodal-NEM is smaller than those in the NEM.Table 3 also obviously indicates that the computational costsfor the nodal-NEM are about one-ninth to one-fourth of thosespent by the NEM. Thus, the nodal-NEM provides a bet-ter overall performance of computational accuracy and effi-ciency than the NEM in the upper bound shakedown analysisof thick-walled cylinder. Finally, Fig. 10 displays the distri-bution of plastic dissipation work rate for the studied thick-walled cylinder when b/a = 2.8 by using the nodal-NEM,and it really reveals the failure mode of this thick-walledcylinder at shakedown limit state.

5.3 An arch beam under the effect of vertical uniformtraction

Figure 11 shows an arch beam under the action of vertical uni-form traction P is selected to investigate herein in the plane

1.6 2.0 2.4 2.8 3.2 3.6 4.0

0.4

0.5

0.6

0.7

0.8

0.9

1.0

P SD / σ

s

b /a

NEM Nodal-NEM

Analytical solution

Fig. 9 Comparison of shakedown loads between analytical and numer-ical solutions

stress state, and the applied traction varies from 0 to Pmax.The discrete model is depicted in Fig. 12 with 495 nodes.Taking advantage of the combination of reduced basis tech-nique and complex method, Liu et al. [30] got the lowerbound shakedown load PSD = 35.24 MPa = 0.1762σs

by adopting the symmetric Galerkin boundary element, andChen et al. [43] obtained the lower bound shakedown loadPSD = 33.91 MPa ≈ 0.1698σs by using element freeGalerkin method and non-linear programming. The upperbound shakedown loads respectively obtained by adoptingthe nodal-NEM, the NEM and the FEM in this paper are0.1874σs, 0.1912σs and 0.1913σs, which are slightly greaterthan the results of Liu et al. [30] and Chen et al. [43] andagree well with them in reasonable precision. Additionally,the computational costs by using the nodal-NEM, the NEMand the FEM are 936, 3,067 and 203 s, which also evidentlyindicate that the nodal-NEM is more efficient than the NEMand is less efficient than the FEM, because the computationaleffort spent by using the nodal-NEM is just one-third of thatcost by using the NEM and is about five times of that costby using the FEM. In the calculations of nodal strain andnodal plastic dissipation work rate, the nodal-NEM is muchmore convenient and direct than the NEM and the FEM. Therelevant distribution of plastic dissipation work rate for archbeam at shakedown limit state by using the nodal-NEM isdepicted in Fig. 13.

5.4 Grooved rectangular plate under the effect of varyingtension and bending moment

In this example, we study a grooved rectangular plate underthe effect of bending moment PM and uniform tensionPN which both independently vary between 0 and Pmax.The detailed information of geometrical dimension, loading

123

Comput Mech (2014) 54:1111–1128 1125

Table 3 Comparisons of accuracy and efficiency between the nodal-NEM and the NEM

b/a Analyticalsolution(PSD/σs)

NEM nodal-NEM

Numericalsolution(PSD/σs)

Error (%) Time (s) Numericalsolution(PSD/σs)

Error (%) Time (s)

1.6 0.352 0.360 2.27 2,390 0.358 1.70 672

2.2 0.458 0.481 5.02 3,082 0.471 2.84 657

2.8 0.504 0.538 6.75 4,074 0.523 3.77 572

3.4 0.527 0.563 6.83 7,931 0.552 4.74 879

4.0 0.541 0.587 8.50 8,072 0.571 5.55 1010

work rate: 0 508 1017 1525 2033 2542 3050 3558 4066 4575

Fig. 10 Distribution of plastic dissipation work rate for thick-walledcylinder at shakedown limit state when b/a = 2.8(106N m)

2.6m 9.0m 9.0m 2.6m

1.5m

4.5m

2.0m

P

9.75m13.6m

1.85m

Fig. 11 The geometric model of arch beam subjected to vertical uni-form traction

arrangement and nodal discretization are depicted in Fig. 14.By adopting a primal-dual algorithm and an edge-basedsmoothed FEM using three-node linear triangular elements(ES-FEM-T3), Tran et al. [9] investigated this example and

Fig. 12 495 node distribution of arch beam

work rate: 0.00 50.07 100.14 150.22 200.29 250.36 300.43 350.50 400.57 450.65

Fig. 13 Distribution of plastic dissipation work rate for arch beam atshakedown limit state (106N m)

got the shakedown load PSD = 0.23603σs. Vu [77] also stud-ied the same example and obtained the upper bound shake-down load PSD = 0.23494σs. The upper bound shakedownloads by the nodal-NEM, the NEM and the FEM are PSD =0.24403σs, PSD = 0.24397σs and PSD = 0.24603σs. It isobserved that the present results match well with those avail-able ones. The relevant computational times of the nodal-NEM and the NEM are 4,086,14,690 and 1306 s, respectively.

When the error tolerances vol1 and vol2 are determinedby the anticipated computational accuracy, the shakedownloads obtained in our proposed iterative solution method mayonly influence by the value of X̃0

k (xr ) (r = 1, 2, . . . N P ;k = 1, 2, . . . l). In our original manuscript, the value ofX̃0

k (xr ) is assigned as 1. In order to discuss the influence ofthe value of X̃0

k (xr ) on the obtained shakedown loads in thisiterative algorithm, we additionally set the values of X̃0

k (xr )

as 10−6, 10−3, 103, 106, and respectively adopt the NEM,the nodal-NEM and the FEM to carry out the upper bound

123

1126 Comput Mech (2014) 54:1111–1128

L

RR=0.25L

L=1m

NP

MPa b

Fig. 14 The geometry, loading and nodal discretization of groovedrectangular plate. a Sketch of geometry and loading b Discretization of661 nodes

0 10 20 30 40 50 60 70 80 90 100

0.2

0.3

0.4

0.5

0.6

0.7

PSD

/ σs

Iterative step h

X 0

k( x

r) = 1.0e-6 : NEM , nodal-NEM , FEM ;

X 0

k( x

r) = 1.0e-3 : NEM , nodal-NEM , FEM ;

X 0

k( x

r) = 1.0 : NEM , nodal-NEM , FEM ;

X 0

k( x

r) = 1.0e3 : NEM , nodal-NEM , FEM ;

X 0

k( x

r) = 1.0e6 : NEM , nodal-NEM , FEM .

Fig. 15 The iterative convergence processes of shakedown limit loadsfor grooved rectangular plate

shakedown analysis to compare the convergence processeswith that of X̃0

k (xr ) = 1 in the last example by using thenodal discretization displayed in Fig. 14. The relevant itera-tive convergence processes of upper bound shakedown loadsfor grooved rectangular plate are depicted in Fig. 15. It canbe seen that: (1) All of the iterative convergence processesare stable and monotonically convergent. (2) The iterativeconvergence processes are completely coincident with eachother when different values of X̃0

k (xr ) are assigned in thesame numerical method, which means the proposed iterativealgorithm is robust for the value of X̃0

k (xr ).

6 Conclusion

A novel numerical solution procedure to estimate the upperbound shakedown limit loads of elastic-perfectly plasticstructures was presented. This procedure relates to the com-bination of a direct iterative algorithm and the nodal-NEM,

which was compared with the NEM and the FEM for the pre-cision and efficiency. A summary of this paper is remarkedas follows:

1. In the nodal-NEM, the stabilized conforming nodal inte-gration scheme is employed, where the strain smoothingtechnique is used to compute the nodal strain, and conse-quently the derivatives of shape function are not requiredto calculate and the nodal strain can be directly obtained.While in the NEM, the Delaunay triangulation is usedas the background mesh to carry out the numerical inte-gration, and the nodal strain can not be obtained directlyand usually needs to be recovered from the strain valuesat integral points by making use of some fitting tech-niques. Thus, for the upper bound shakedown analysisrequired to construct the smoothing plastic strain field,the nodal-NEM shows greater advantage than the NEMin improving the computational efficiency.

2. In the present mathematical programming formulationfor the kinematic shakedown analysis, the plastic incom-pressibility conditions are handled in two different waysbased on the different control conditions and the formatsof plane stress and plane strain problems. The obsta-cle triggered by the time integration is resolved by theKönig’s proposed technique. By means of the stabilizedconforming nodal integration scheme, a direct iterativeprocess is developed to solve the objective function sub-jected to several equality constraints. The linearizationof objective function is achieved by distinguishing theunidentified non-plastic areas from the plastic areas andrevising the relevant constraint conditions and objectivefunction at each iterative step. The proposed iterativeprocess is very simple and effective, because the mini-mum optimization problem at each iterative step is equiv-alent to solving the associated linear equations.

3. Numerical examples show that the numerical results bythe nodal-NEM, the NEM and the FEM are all in excel-lent agreement with the reference solutions, and at thesame time demonstrate that the proposed iterative algo-rithm has very favorable accuracy and stability in theactual computations and is robust for the value of X̃0

k (xr ).The comparison of computational time states that thenodal-NEM is much more efficient than the NEM andis less efficient than the FEM under the same condition.The costs of the nodal-NEM are just about one-ninth toone-third of those by the NEM and are about three tofive times of those by the FEM in different examples.The advantage of the nodal-NEM is that it is much moreconvenient and direct than the NEM and the FEM incalculating the value of nodal strain and nodal plasticdissipation work rate. Additionally, the visualized distri-butions of structural plastic dissipation work rates at the

123

Comput Mech (2014) 54:1111–1128 1127

shakedown limit states obtained by using the nodal-NEMare displayed.

The present kinematic shakedown analysis with nodal-NEM solution procedure and direct iterative algorithm canbe straightforwardly extended to the kinematic shakedownproblems with other yield and loading conditions.

Acknowledgments S. Zhou is supported by the Chinese Postdoc-toral Science Foundation (2013M540934). Y. Liu is supported by theNational Science Foundation for Distinguished Young Scholars ofChina (11325211). D. Wang is supported by the National Natural Sci-ence Foundation of China (11222221). K. Wang is supported by theProject of Shandong Province Higher Educational Science and Tech-nology Program (J13LG51).

References

1. Xu BY, Liu XS (1985) Plastic limit analysis of structures. ChinaArchitecture and Building Press, Beijing

2. Chen G, Liu YH (2006) Numerical theories and engineering meth-ods for structural limit and Shakedown analyses. Science Press,Beijing

3. Melan E (1938) Zur Plastizitä t des räumlichen Kontinuums. Ing-Arch 9:115–126

4. Koiter WT (1956) A new general theorem on shakedown of elastic-plasitc structures. Proc Konink Ned Akad Wet B 59:24–34

5. Kamenjarzh J, Merzljakov A (1994) On kinematic method inshakedown theory. I: duality of extremum problems. Int J Plast10(4):363–380

6. Kamenjarzh J, Merzljakov A (1994) On kinematic method inshakedown theory. II: modified kinematic method. Int J Plast10(4):381–392

7. Khoi VD, Yan AM, Hung ND (2004) A dual form for discretizedkinematic formulation in shakedown analysis. Int J Solids Struct41(1):267–277

8. Khoi VD, Yan AM, Hung ND (2004) A primal-dual algorithm forshakedown analysis of structures. Comput Methods Appl MechEng 193(42–44):4663–4674

9. Tran TN, Liu GR, Nguyen-Xuan H, Nguyen-Thoi T (2010) Anedge-based smoothed finite element method for primal-dual shake-down analysis of structures. Int J Numer Meth Eng 82(7):917–938

10. Tran TN (2011) A dual algorithm for shakedown analysis of platebending. Int J Numer Meth Eng 86(7):862–875

11. Stein E, Zhang GB, Konig JA (1992) Shakedown with nonlin-ear strain-hardening including structural computation using finite-element method. Int J Plast 8(1):1–31

12. Zhang YG (1995) An iteration algorithm for kinematic shakedownanalysis. Comput Methods Appl Mech Eng 127(1–4):217–226

13. Hachemi A, Weichert D (1998) Numerical shakedown analysis ofdamaged structures. Comput Methods Appl Mech Eng 160:57–70

14. Yan AM, Hung ND (2001) Kinematical shakedown analysiswith temperature-dependent yield stress. Int J Numer Meth Eng50(5):1145–1168

15. Maier G, Pan LG, Perego U (1993) Geometric effects on shake-down and ratchetting of axisymmetric cylindrical shells subjectedto variable thermal loading. Eng Struct 15:453–465

16. Borino G, Polizzotto C (1996) Dynamic shakedown of structureswith variable appended masses and subjected to repeated excita-tions. Int J Plast 12(2):215–228

17. Tran TN, Kreissig R, Staat M (2009) Probabilistic limit and shake-down analysis of thin plates and shells. Struct Saf 31(1):1–18

18. Carvelli V (2004) Shakedown analysis of unidirectional fiber rein-forced metal matrix composites. Comput Mater Sci 31(1–2):24–32

19. Li HX, Yu HS (2006) A non-linear programming approach to kine-matic shakedown analysis of composite materials. Int J NumerMeth Eng 66(1):117–146

20. Li HX, Yu HS (2006) A nonlinear programming approach to kine-matic shakedown analysis of frictional materials. Int J Solids Struct43(21):6594–6614

21. Feng XQ, Liu XS (1997) On shakedown of three-dimensionalelastoplastic strain-hardening structures. Int J Plast 12(10):1241–1256

22. Feng XQ, Sun QP (2007) Shakedown analysis of shape memoryalloy structures. Int J Plast 23(2):183–206

23. Carvelli V, Cen ZZ, Liu YH, Maier G (1999) Shakedown analysisof defective pressure vessels by a kinematic approach. Arch ApplMech 69(9–10):751–764

24. Chen HF, Ure J, Li TB, Chen WH, Mackenzie D (2011) Shakedownand limit analysis of 90◦ pipe bends under internal pressure, cyclicin-plane bending and cyclic thermal loading. Trans ASME J PressVessel Technol 88(5–7):213–222

25. Gross-Wedge J (1997) On the numerical assessment of the safetyfactor of elastic-plastic structures under variable loading. Int JMech Sci 39(4):417–433

26. Shiau SH (2001) Numerical methods for shakedown analysis ofpavements under moving surface loads. PhD thesis, University ofNewcastle, NSW, Australia

27. Konig JA, Maier G (1981) Shakedown analysis of elastoplas-tic structures: a review of recent developments. Nucl Eng Des66(1):81–95

28. Panzeca T (1992) Shakedown and limit analysis by the boundaryintegral equation method. Eur J Mech A-Solids 11(5):685–699

29. Zhang XF, Liu YH, Cen ZZ (2004) Boundary element methods forlower bound limit and shakedown analysis. Eng Anal BoundaryElem 28(8):905–917

30. Liu YH, Zhang XF, Cen ZZ (2005) Lower bound shakedown analy-sis by the symmetric Galerkin boundary element method. Int J Plast21(1):21–42

31. Belytschko T, Lu YY, Gu L (1994) Element free Galerkin method.Int J Numer Meth Eng 37(2):229–256

32. Liu WK, Jun S, Zhang YF (1995) Reproducing kernel particlemethod. Int J Numer Meth Fluids 20(8–9):1081–1106

33. Liszka TJ, Duarte CAM, Tworzydlo WW (1996) hp-meshlesscloud method. Comput Methods Appl Mech Eng 139(1–4):263–288

34. Atluri SN, Zhu T (1998) A new meshless local Petrov-Galerkin(MLPG) approach in computational mechanics. Comput Mech22(2):117–127

35. Sukumar N, Moran B, Belytschko T (1998) The natural elementmethod in solid mechanics. Int J Numer Meth Eng 43(5):839–887

36. Zhang XG, Liu XH, Song KZ, Lu MW (2001) Least-squares col-location meshless method. Int J Numer Meth Eng 51(9):1089–1100

37. Liu GR, Wang JG (2002) A point interpolation meshlessmethod based on radial basis functions. Int J Numer Meth Eng54(11):1623–1648

38. Gu L (2003) Moving Kriging interpolation and element freeGalerkin method. Int J Numer Meth Eng 56(1):1–11

39. Kim DW, Yoon YC, Liu WK, Belytschko T (2007) Extrinsic mesh-free approximation using asymptotic expansion for interfacial dis-continuity of derivative. J Comput Phys 221(1):370–394

40. Bessa MA, Foster JT, Belytschko T, Liu WK (2013) A mesh-free unification: reproducing kernel peridynamics. Comput Mech.doi:10.1007/s00466-013-0969-x

41. Yoon YC, Song JH (2013) Extended particle difference method forweak and strong discontinuity problems: part I. Derivation of theextended particle derivative approximation for the representation

123

1128 Comput Mech (2014) 54:1111–1128

of weak and strong discontinuities. Comput Mech. doi:10.1007/s00466-013-0950-8

42. Yoon YC, Song JH (2013) Extended particle difference method forweak and strong discontinuity problems: part II. Formulations andapplications for various interfacial singularity problems. ComputMech. doi:10.1007/s00466-013-0951-7

43. Chen SS, Liu YH, Cen ZZ (2008) Lower bound shakedown analysisby using the element free Galerkin method and nonlinear program-ming. Comput Methods Appl Mech Eng 197:3911–3921

44. Chen SS, Liu YH, Li J, Cen ZZ (2010) Performance of the MLPGmethod for static shakedown analysis for bounded kinematic hard-ening structures. Eur J Mech A/Solids 30(2):183–194

45. Beissel S, Belytschko T (1996) Nodal integration of the element-free Galerkin method. Comput Methods Appl Mech Eng 139(1–4):49–74

46. Chen JS, Wu CT, Yoon S, You Y (2001) A stabilized conformingnodal integration for Galerkin mesh-free methods. Int J NumerMeth Eng 50(2):435–466

47. Chen JS, Yoon S, Wu CT (2002) Non-linear version of stabilizedconforming nodal integration for Galerkin mesh-free methods. IntJ Numer Meth Eng 53(12):2587–2615

48. Chen JS, Wu CT, Belytschko T (2000) Regularization of mater-ial instabilities by meshfree approximations with intrinsic lengthscales. Int J Numer Meth Eng 47(7):1303–1322

49. Wang D, Chen JS (2004) Locking-free stabilized conforming nodalintegration for Mindlin–Reissner plate. Comput Methods ApplMech Eng 193(12–14):1065–1083

50. Wang D, Chen JS (2006) A locking-free meshfree curved beam for-mulation with the stabilized conforming nodal integration. ComputMech 39(1):83–90

51. Wang D, Chen JS (2008) A Hermite reproducing kernel approxima-tion for thin plate analysis with sub-domain stabilized conformingintegration. Int J Numer Meth Eng 74(3):368–390

52. Wang D, Lin Z (2011) Dispersion and transient analyses of Hermitereproducing kernel Galerkin meshfree method with sub-domainstabilized conforming integration for thin beam and plate struc-tures. Comput Mech 48(1):47–63

53. Yoo JW, Moran B, Chen JS (2004) Stabilized conforming nodalintegration in the natural-element method. Int J Numer Meth Eng60:861–890

54. Puso MA, Chen JS, Zywicz E, Elmer W (2008) Meshfree and finiteelement nodal integration methods. Int J Numer Meth Eng 74:416–446

55. Sibson R (1980) Vector identity for the dirichlet tessellation. MathProc Camb Philos Soc 87:151–155

56. Braun J, Sambridge M (1995) A numerical method for solvingpartial differential equations on highly irregular evolving grids.Nature 376:655–660

57. Sukumar N, Moran B, Semenov AY, Belikov VV (2001) Naturalneighbour Galerkin methods. Int J Numer Meth Eng 50(1):1–27

58. Cueto E, Sukumar N, Calvo B, Martinez MA, Cegonino J, DoblareM (2003) Overview and recent advances in natural neighbourGalerkin methods. Arch Comput Methods Eng 10(4):307–384

59. Gonzalez D, Cueto E, Doblare M (2004) Volumetric lockingin natural neighbour Galerkin methods. Int J Numer Meth Eng61(4):611–632

60. Thiessen AH (1911) Precipitation averages for large areas. MonWeather Rep 39:1082–1084

61. Belikov VV, Ivanov VD, Kontorovich VK, Korytnik SA (1997)The non-Sibsonian interpolation: a new method of interpolation ofthe values of a function on an arbitrary set of points. Comput MathMath Phys 37(1):9–15

62. Hiyoshi H, Sugihara K (1999) Two generalizations of an interpolantbased on Voronoi diagrams. Int J Shape Model 5(2):219–231

63. Alfaro I, Yvonnet J, Chinesta F, Cueto E (2007) A study on theperformance of natural neighbour-based Galerkin methods. Int JNumer Meth Eng 71(12):1436–1465

64. Cueto E, Doblare M, Gracia L (2000) Imposing essential boundaryconditions in the natural element method by means of density-scaled a-shapes. Int J Numer Meth Eng 49:519–546

65. Zhou ST, Liu YH (2012) Upper-bound limit analysis based on thenatural element method. Acta Mech Sin 28(5):1398–1415

66. Himmelblau DM (1972) Appl Nonlinear Progr. McGraw-Hill BookCompany, New York

67. Belytschko T (1972) Plane stress shakedown analysis by finite ele-ments. Int J Mech Sci 14(9):619–625

68. Corradi L, Zavelani A (1974) A linear programming approach toshakedown analysis of structures. Comput Methods Appl MechEng 3(1):37–53

69. Nguyen DH, Palgen L (1980) Shakedown analysis by displacementmethod and equilibrium finite elements. Trans Can Soc Mech Eng61:34–40

70. Genna F (1988) A nonlinear inequality, finite element approachto the direct computation of shakedown load safety factors. Int JMech Sci 30(10):769–789

71. Zouain N, Borges L, Silveira JL (2002) An algorithm for shake-down analysis with nonlinear yield functions. Comput MethodsAppl Mech Eng 191(23–24):2463–2481

72. Garcea G, Armentano G, Petrolo S, Casciaro R (2005) Finiteelement shakedown analysis of two-dimensional structures. Int JNumer Meth Eng 63(8):1174–1202

73. Tin-Loi F, Ngo NS (2007) Performance of a p-adaptive finite ele-ment method for shakedown analysis. Int J Mech Sci 49(10):1168–1178

74. Zhang TG, Raad L (2002) An eigen-mode method in kinematicshakedown analysis. Int J Plast 18(1):71–90

75. Krabbenhoft K, Lyamin AV, Sloan SW (2007) Bounds to shake-down loads for a class of deviatoric plasticity models. ComputMech 39(6):879–888

76. Chen SS (2009) Lower-bound limit and Shakedown analysis basedon meshless methods. PhD thesis, Tsinghua University, Beijing,People’s Republic of China

77. Vu DK (2001) Dual limit and shakedown analysis of structures,dissertation. Universite de Liege, Belgium

123