Upper-bound limit analysis based on the natural element method

Transcription

1 Acta Mechanica Sinica (2012) 28(5): DOI /s RESEARCH PAPER Upper-ound limit analysis ased on the natural element method Shu-Tao Zhou Ying-Hua Liu Received: 14 Novemer 2011 / Accepted: 20 July 2012 The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag Berlin Heidelerg 2012 Astract The natural element method (NEM) is a newlydeveloped numerical method ased on Voronoi diagram and Delaunay triangulation of scattered points, which adopts natural neighour interpolation to construct trial functions in the framework of Galerkin method. Owing to its distinctive advantages, the NEM is used widely in many prolems of computational mechanics. Utilizing the NEM, this paper deals with numerical limit analysis of structures made up of perfectly rigid-plastic material. According to kinematic theorem of plastic limit analysis, a mathematical programming natural element formulation is estalished for determining the upper ound multiplier of plane prolems, and a direct iteration algorithm is proposed accordingly to solve it. In this algorithm, the plastic incompressiility condition is handled y two different treatments, and the nonlinearity and nonsmoothness of the goal function are overcome y distinguishing the rigid zones from the plastic zones at each iteration. The procedure implementation of iterative process is quite simple and effective ecause each iteration is equivalent to solving an associated elastic prolem. The otained limit load multiplier is proved to monotonically converge to the upper ound of true solution. Several enchmark examples are investigated to validate the significant performance of the NEM in the application field of limit analysis. Keywords Upper ound limit analysis Meshless method Natural neighour interpolation Natural element method Mathematical programming Iteration algorithm The project was supported y the National Foundation for Excellent Doctoral Thesis of China (200025), the Program for New Century Excellent Talents in University (NCET ) and the National Natural Science Foundation of China ( ). S.-T. Zhou Y.-H. Liu Department of Engineering Mechanics, AML, Tsinghua University, Beijing, China yhliu@mail.tsinghua.edu.cn 1 Introduction Limit analysis is a very important sudiscipline of plasticity, which can provide a powerful tool for engineering design and safety assessment of structures [1, 2]. Compared with the traditional elastic analysis, plastic limit analysis can give more economical design results, and the load-carrying capacity of structures can e rought into full play. Until now, considerale efforts have een focused on developing efficient and reliale computational methods of limit analysis, and most of these traditional and popular numerical approaches are mesh-ased, such as the finite element method (FEM) [2 7] and the oundary element method (BEM) [8]. In recent years, a novel numerical method called meshless method received much attention in computational mechanics fields [9 14]. Although in the aspects of strict mathematical proof, computational efficiency, oundary conditions treatment and actual engineering application etc., meshless method is not comparale to the mature finite element method, ut in some special fields, meshless method has some unique advantages [12]. Meshless method discretizes the domain y using only a set of scattered nodes, and does not require the connected relation etween the nodes, thus it can effectively avoid the difficulties caused y generation, distortion and reconfiguration of the mesh, and is thus a convenient and powerful tool for analysing the impact and explosion, crack propagation, metal machining and forming, fluid-solid coupling prolems. Furthermore, as an alternative and supplementary approach to mesh-ased methods, ecause of its good capaility of convergence and staility, flexile post-processing and highly accurate solution, this method has een applied to a more and more wide range of computational mechanics prolems. In order to extend the new numerical method to limit analysis, some scholars have successfully applied meshless methods to investigate the limit analysis prolems in recent years. Chen et al. [15, 16] used element-free Galerkin (EFG) method and local Petrov Galerkin (LPG) method respectively to solve two-dimensional (2D) lower ound limit analysis prolems, and otained a series of limit loads that agree

2 Upper-ound limit analysis ased on the natural element method 1399 well with the solutions otained in other literatures. Utilizing the comination of the stailized conforming nodal integration (SCNI) technique and second-order cone programming (SOCP) optimization algorithm, Le et al. [17, 18] applied EFG method to solve upper ound and lower ound limit analyses of plates and slas, respectively. Le et al. [19] used h-adaptive EFG method to study the plates with various oundary conditions in the framework of kinematic theorem of limit analysis. These papers are the newest investigations of limit analysis ased on meshless methods. As a gradually mature and applicale meshless method in computational mechanics fields, the natural element method (NEM), also called natural neighour Galerkin method, possesses some remarkale and distinctive advantages [20 22], such as strict interpolation property, high calculation efficiency and accurate imposition of piecewise linear essential oundary conditions. In addition, the distriution of irregular nodes does not affect its calculation performance. Owing to these advantages, natural neighour interpolation and natural element method have already een successfully and widely used to solve various current mechanics issues, including 2D and three-dimensional (3D) linear and nonlinear elasticity prolems [20 25], large deformation prolems [26 30], iomechanics prolems [30], metal machining and forming [24, 31], fluid mechanics prolems [32, 33], dynamic prolems [34, 35], etc. Motivated y the advantages of meshless methods stated aove, this paper firstly develops a reliale and feasile numerical method for upper ound limit analysis ased on the NEM. The corresponding mathematical programming formulation for upper ound analysis of 2D structures is estalished and the natural neighour interpolation is used to construct the displacement velocity field. A direct iterative algorithm is used to solve this formulation sujected to several equality constraints. Two different treatments are used to deal with the plastic incompressiility condition. In this algorithm, the rigid zones are distinguished from the plastic zones at each iteration step, the goal function and constraint conditions are gradually modified, and the difficulties caused y nonlinearity and nonsmoothness of the goal function are overcome. Rigorous mathematical proof of convergence indicates that the iterative process can ensure the upper ound of limit load multiplier to e otained. The moving least squares (MLS) approximation technique is used to recover and fit the nodal strain, and the smooth plastic dissipation work in the whole region is calculated and visually displayed. 2 Kinematic theorem of plastic limit analysis In plastic limit analysis, when the applied external load is increased to a certain limit value, the structure will turn into a geometrically deformale mechanism, its deformation will unlimitedly grow and therey, its load-carrying capacity will lose. This state is called as the plastic limit state of the structure [1, 2]. The upper ound limit load of a perfectly rigidplastic structure can e otained through kinematic theorem of plastic limit analysis. The theorem can e descried as follows: If a structure satisfies geometric constraint conditions and can turn into a geometrically deformale mechanism under the action of external load, and the power rate of external load is not less than the internal plastic dissipation power rate, the displacement velocity field is called as the kinematically admissile velocity field under this condition. This external load corresponding to the kinematically admissile velocity field is the upper ound of limit load, and the minimum of these upper ounds is the true limit load. The mathematical programming for upper ound limit analysis can e formulated as [5, 6] ν = min : D( ε ij )dω, (1) Ω s.t. u i t i dγ = 1, on Γ t, (2) Γ t u i,i = 0, in Ω, (3) u i = 0, on Γ u, (4) where ν is the limit load multiplier, u i denotes the kinematically admissile displacement velocity field, and D( ε ij )represents the plastic dissipation power rate in terms of the kinematically admissile strain rate ε ij. The plastic dissipation power rate is defined as D( ε ij ) = σ ij ε ij and can e written in the following form y adopting the von Mises yield criterion 2 D( ε ij ) = σ ij ε ij = εij ε ij, (5) herein, σ ij denotes the stress related to the strain rate ε ij and σ s is the yield stress of material. In Eqs. (1) (4), the following classical assumptions are usually adopted: perfectly rigid-plastic material, proportional loading, small deformation and linear geometrical relationship. Equation (2), called the normalization condition, means that the ody forces are neglected and only the external loads of surface tractions t i areappliedonthestress oundary Γ t. Equation (3) indicates that the plastic incompressiility condition should e oeyed within the whole structure. For convenience, in the following discussion, the displacement velocity u i, strain rate ε ij and plastic dissipation power rate D( ε ij ) are respectively named as displacement u i,strainε ij and plastic dissipation power D(ε ij )for short. 3 Natural element method In recent twenty years, meshless method has ecome significant research favorite and is widely used in computational mechanics fields as a novel numerical method. Up to now, more than a dozen meshless methods have een proposed, and their essential distinctions are that these methods are ased on different weighted residual methods and different trial functions [12]. The NEM [20-22] is a newly-developed meshless method which utilizes natural neighour interpolation to approximate the trial and test functions and adopts

3 1400 S.-T. Zhou, Y.-H. Liu Galerkin method to discretize and generate the stale gloal system equation. Based on Voronoi diagram and Delaunay triangulation of scattered points, the calculation of natural neighour interpolation is relatively simple, and needs neither complicated matrix operations nor any artificial parameter. The otained shape function satisfies the property of Kronecker delta, and this characteristic leads to the result that the essential oundary conditions can e easily imposed as in FEM. Meanwhile, the prolems with discontinuous field functions as well as their discontinuous derivatives can e conveniently handled. Like other meshless methods, the NEM needs the nodal information only, and can eliminate the drawacks associated with the meshing. Therefore, it can e seen that the NEM is an excellent numerical method which has comined the advantage of meshless method with the merit of finite element method. Currently, the Sison interpolant [21] and the Laplace interpolant (non-sison interpolant) [22] are two most widely used interpolation schemes which are ased upon the concept of natural neighourhood and adopted in the NEM. Before further discussions, this paper riefly reviews these two interpolation schemes. 3.1 Voronoi diagram and Delaunay triangulation Voronoi diagram and its dual structure called Delaunay triangulation are the rather attractive and useful structures descried on scattered nodes. For simplicity, supposing that a set of scattered nodes N = {n 1, n 2,, n M } are distriuted in a 2D ounded domain Ω, and this ounded domain is sudivided into a series of regions T I according to the following mathematical formula T I = { x R 2 : d(x, x I ) < d(x, x J ), J I }, (6) where d(x, x J ) is the Euclidean distance etween x I and x J. After such decomposition, the ounded domain Ω is sudivided into a series of closed and convex polygon regions, any point in regions T I is closer to node n I than to any other node n J (J I), and each region T I is associated with node n I. This unique kind of irregular and convex polygon is named Voronoi diagram, also called as first-order Voronoi diagram or Dirichlet tessellation. Adopting the analogical extension, the definition ofsecond-ordervoronoidiagramcane represented as T IJ = {x R 2 : d(x, x I ) < d(x, x J ) < d(x, x K ), J I K}. (7) Equation (7) indicates that T IJ is the locus of all points which have node x I as the nearest neighour and have node x J as the second nearest neighour. The geometrical dual structure of Voronoi diagram is the Delaunay triangulation. The circumcircle of Delaunay triangle is named as the natural neighour circumcircle, and its centre is a vertex of Voronoi structure. As shown in Fig. 1, h is the circumcenter of Delaunay triangle DT = (7, 1, 6). Each edge of Voronoi structure is the perpendicular isector of the common edge of two corresponding Delaunay triangles. Empty circumcircle criterion is a very important property of Delaunay triangulation and is usually used to determine the nearest neighour nodes, which considers that if any point lies within the circumcircle of Delaunay triangle DT = (n I, n J, n K ), n I, n J, n K are its natural neighour nodes. 1 h g 2 7 i 6 5 Fig. 1 Voronoi diagram and Delaunay triangulation 3.2 Sison interpolant x The Sison interpolant was originally introduced y Sison [36], which is extensively adopted in the NEM. Sison used the concept of second-order Voronoi cells to quantify the neighours for an inserted point x. The natural neighour shape function of this inserted point with respect to one of its natural neighour points I is calculated as the ratio of the area of A I (x) toa(x), where A I (x) is the area of the secondorder Voronoi cell and A(x) stands for the total area of the first-order Voronoi cell, that is Φ I (x) = A I(x) A(x), A(x) = f k j n A J (x). (8) J AsshowninFig.2,n = 5 is for the inserted point x. According to the definition of second-order Voronoi diagram introduced aove, the shape function of Node 1 can e written as Φ 1 (x) = A dehg. (9) A deac According to Eq. (8), the derivatives of shape function can e further deduced as Φ I, j (x) = A I, j(x) Φ I (x)a, j (x). (10) A(x) The detailed calculation formulae for areas and their derivatives of first-order and second-order Voronoi cells were stated y Sukumar et al. [21]. 3 4

4 Upper-ound limit analysis ased on the natural element method d g 1 x 7 e h a f c 3 4 By differentiating Eq. (11), the partial derivatives of Φ(x) andα(x) can e deduced as α I, j (x) Φ I (x) n α J, j (x) J=1 Φ I, j (x) = n α J (x), (12) J=1 α I, j (x) = s I, j(x) α I (x)h I, j (x). (13) h I (x) 6 Fig. 2 Computation of Sison interpolant shape functions 3.3 Laplace interpolant Different from the Sison interpolant, the Laplace interpolant, also known as non-sison interpolant, is another interpolant technique ased on the conception of the natural neighour and its calculational magnitude is one order less than the space dimension. This approximation technique was independently recommended y Belikov et al. [37] likewise y Hiyoshi and Sugihara [38, 39]. Because this approximation technique has relation to the Laplace equation, it is named as the Laplace interpolant y Hiyoshi and Sugihara. In the 2D domain, if one inserted ponit x has n natural neighour nodes, the Laplace shape function for one of its natural neighour nodes is defined as follows Φ I (x) = α I(x), α n J (x) = s J(x) h α J (x) J (x), (11) J=1 where α J (x) is the Laplace weight function and is symmetric, s I (x) is the Voronoi edge length associated with inserted ponit x and node I, andh I (x) is the Euclidean distance etween node I and inserted point x. The detailed position relationship of these variales is shown in Fig. 3. d 2 1 s 1 h 1 x s 7 h 7 e a s 2 h 2 7 c 6 5 Fig. 3 Computation of Laplace interpolant shape functions As for the procedure implementation to compute Laplace interpolant shape function and its partial derivatives, readers can refer to Sukumar et al. [22] and the references listed therein. 3.4 Properties The properties of Sison interpolant and Laplace interpolant shape functions are riefly presented in this section [20 22]. From the aforementioned definition, ecause these two interpolation schemes have the same compact support domain, the test and trial functions can e approximately expressed as the following uniform expression u h (x) = n Φ I (x)u I, (14) I=1 where u I is the vector of nodal displacements associated with n natural neighours of point x,andφ I (x) is the Sison shape function term defined in Eq. (8) or the Laplace shape function term defined in Eq. (11). The Sison and the Laplace shape functions displayed in Fig. 4 share many properties such as non-negative property, Kronecker delta property, partition of unity property and linear consistency property. Different properties etween these two interpolants are exhiited in the following aspects: (1) The Sison shape function is C everywhere except at nodal locations where it is C 0, and the Laplace shape function is C 0 everywhere in its whole support domain. (2) The Sison and the Laplace interpolants are respectively ased on the proportion of areas and lengths in two-dimensions, so the Laplace interpolant is superior to the Sison interpolant in the aspect of computational efficiency. This advantage ehaves more oviously in three-dimensions. (3) The Sison interpolant can only exactly impose the essentical oundary conditions on the convex domains, however the Laplace interpolant can also precisely impose the essentical oundary conditions on the non-convex domains. In this paper, using oth the Sison and the Laplace interpolants, the NEM is utilized to carry out the upper ound limit analysis.

5 1402 S.-T. Zhou, Y.-H. Liu a y 0.5 where K = B T DB, (16) x ε = [ε x ε y 2ε xy ] T, (17) [ ] T ŨU = u 1 v 1 u i v i u n v n, (18) Φ 1,x 0 Φ i,x 0 Φ n,x B = 0 Φ 1,y 0 Φ i,y 0 Φ n,y Φ 1,y Φ 1,x Φ i,y Φ i,x Φ n,y Φ n,x, (19) Z Z: Z Y 0.25 X c Z: Y 0.25 X Fig. 4 The shape functions of the NEM. a Nodal grid; Sion interpolant; c Laplace interpolant 4 Upper ound limit analysis ased on the NEM 4.1 The kinematic formulation using the NEM Owing to the advantages of the NEM stated aove, the NEM is used to discretize the nonlinear mathematical programming formulation expressed in Eqs. (1) (4). In the NEM, a series of scattered nodes are adopted to discretize the prolem domain, natural neighour interpolation is used to construct the trial function in the framework of Galerkin method, and the conventional three point quadrature rule is generally utilized to carry out the numerical integration. If an integral point has n natural neighouring nodes and the corresponding displacement vector of these nodes is ŨU, the strain of this integral point can e written as ε = BŨU where B stands for the strain-displacement relation matrix. The expression of ε ij ε ij in Eq. (5) can e rewritten in the following matrix form for 2D prolems ε ij ε ij = ε T Dε = ŨU T B T DBŨU = ŨU T KŨU, (15) K is a matrix associated with the natural neighouring nodes of an integral point and D is a constant matrix which has different values for different typical prolems. Assemling ŨU and K into the gloal nodal displacement vector U and the gloal matrix K, respectively, and using the Gaussian integration technique, we can deduce the discrete form of goal function as 2 3 σ s εij ε ij dω = Ω ŨU T KŨUd Ω Ω = 3 σ s ρ i J i U T K i U, (20) where Ω denotes the integral domain which is got y discretizing domain Ω. IG is the set of total integral points for a discretized structure, and ρ i and J i are respectively the integration weight and the Jacoian determinant for integral point i. Sustituting the discrete displacement form Eq. (14) into Eq. (2), we have F T U = 1, (21) F = Φ T tdγ, Γ t (22) where F denotes the equivalent nodal load vector corresponding to the prescried tractions and Φ denotes the shape function matrix. After summarizing the equations stated aove, the discrete mathematical programming formulation ased on the NEM for the upper ound limit analysis can e constructed as follows ν = min : 3 σ s ρ i J i U T K i U, (23) s.t. F T U = 1. (24) In the next section, two different measures will e introduced to handle the plastic incompressiility condition u i,i = Treatment of plastic incompressiility condition Different typical prolems have different computational formulations for dealing with the incompressiility condition

6 Upper-ound limit analysis ased on the natural element method 1403 in the plastic limit analysis. Taking plane stress and plane strain prolems for examples, we will present here two different treatments of the incompressiility condition. Regarding the plane stress prolem, it follows σ z = σ xz = σ yz = 0, ε xz = ε yz = 0andε z 0. The plastic incompressiility condition in Eq. (3) can e expressed as ε z = (ε x + ε y ), so the expression of ε ij ε ij can e written as ε ij ε ij = ε 2 x + ε 2 y + ε 2 z + 2ε 2 xy + 2ε 2 yz + 2ε 2 zx = 2(ε 2 x + ε 2 y + ε x ε y + ε 2 xy) = ε T Dε = ŨU T KŨU, (25) where K is the same as Eq. (16) and D is written as 21 0 D = (26) In this simple and skillful way, the incompressiility condition of plane stress prolem can e satisfied naturally in the goal function term y introducing a revised matrix D. It is worth pointing out that this simple way can also e used to deal with the incompressiility condition of plate and shell prolems. Considering the plane strain prolem, ε z = 0andε xz = ε yz = 0 should e satisfied, so the expression of ε ij ε ij can e written as ε ij ε ij = ε 2 x + ε 2 y + ε 2 z + 2ε 2 xy + 2ε 2 yz + 2ε 2 zx = ε 2 x + ε 2 y + 2ε 2 xy = ε T Dε = ŨU T KŨU, (27) where 10 0 D = (28) The incompressiility constraint condition of plane strain prolem can e rewritten as u i,i = ε x + ε y + ε z = ε x + ε y = B v ŨU = 0, (29) where [ ] B v = Φ 1,x Φ 1,y Φ i,x Φ i,y Φ n,x Φ n,y. (30) The revised constraint form of incompressiility condition in Eq. (29) is usually introduced into the goal function y means of the penalty function method, i.e., α v Ω (ε x + ε y ) 2 dω / 2(α v is a penalty factor). Using the Gaussian integration technique, the penalty function term can e discretized as 1 2 α ( ) 2dΩ 1 v εx + ε y = Ω 2 α v ŨU T K v ŨUd Ω Ω = 1 2 α v ρ i J i U T (K v ) i U, (31) where K v = B T v B v and K v is extended to the gloal matrix (K v ) i which has the same dimension as the gloal matrix K. Under this condition, the incompressiility condition of plane strain prolems in Eq. (31) is constructed as the quadratic form of displacement which has a format similar to the amended goal function that will e presented underneath. By the way, the penalty function method can also e adopted to treat the plastic incompressiility condition in the upper ound limit analysis of axisymmetric and 3D prolems. Up to now, we have respectively introduced two different methods to deal with the plastic incompressiility conditions for plane stress and plane strain prolems. By doing so, the unified and discrete mathematical programming formulation of upper ound limit analysis ased on the NEM for plane stress and plane strain prolems can e written as ν = min : 3 σ s ρ i J i U T K i U, (32) s.t. F T U = 1, (33) U T (K v ) i U = 0, i IG. (34) Equations (32) (34) is a nonlinear mathematical programming prolem with only equality constraints. The goal function in Eq. (32) is nonlinear and nondifferentiale, which ecomes one of the main difficulties in the upper ound limit analysis. Another difficulty is the treatment of the plastic incompressiility condition in Eq. (34). How to estalish a simple and effective solution procedure to overcome these two difficulties is very important for upper ound limit analysis. In the following sections, we will focus our attention on the construction and implementation of iterative solution algorithm. 4.3 The construction of iterative solution algorithm In this section, a direct iterative algorithm originally proposed y Zhang et al. [5] is constructed to solve the mathematical programming formulation in Eqs. (32) (34). If the strain at every integral point is non-zero, that means U T K i U 0, i IG. (35) Using the Lagrange multiplier λ to introduce the normalization condition into the goal function, the minimization prolem in Eqs. (32) (34) is equivalent to the following from min : 3 σ s ρ i J i U T K i U λ(f T U 1), (36)

7 1404 S.-T. Zhou, Y.-H. Liu s.t. U T (K v ) i U = 0, i IG. (37) According to the necessary minimization condition of Eq. (36) and (37), we have 3 σ K i U s ρ i J i = λf, (38) U T K i U F T U = 1, (39) U T (K v ) i U = 0, i IG. (40) Multiplying oth sides of Eq. (38) y U T, we arrive at 3 σ U T K i U s ρ i J i U T K i U = 3 σ s ρ i J i U T K i U = λu T F = λf T U = λ. (41) Comparing Eq. (41) with Eq. (32), we can find that Lagrange multiplier λ is a discrete solution of the limit load multiplier ν. Equations (38) (40) are nonlinear equations, and are difficult to solve directly. Considering this characteristic of Eqs. (38) (40), we design the following iterative technique to linearize them. 3 σ K i U k s ρ i J i = λ k F, (42) U Tk 1 K i U k 1 F T U k = 1, (43) U T k (K v ) i U k = 0, i IG, (44) where U k stands for the nodal displacement vector and λ k denotes the Lagrange multiplier at iterative step k, andu k 1 is the nodal displacement vector at iterative step (k 1) and is known efore the k-th iteration. However, Eq. (42) is meaningless when the strains at some integral points are zero, and in this case the iteration can not e ensured to proceed smoothly. Therefore, the value of U T K i U at each integral point should e examined firstly efore every iteration. Prior to the k-th iteration, the set IG of total integral points is respectively divided into the rigid region suset R k and the plastic region suset P k, i.e., IG = P k R k, P k = { i IG, U T k 1 K iu k 1 0 }, R k = { i IG, U T k 1 K iu k 1 = 0 }. (45) This partition of rigid region and plastic region is very important and crucial, which can remove the rigid integral points from total integral points of the goal function and ensure Eq. (42) to e meaningful at every iterative step. Under this circumstance, a constraint condition should e imposed on the rigid points as follows U T K i U = 0, i R k. (46) It is apparent that the rigid points in suset R k can automatically satisfy the plastic incompressiility condition specified y Eq. (44). For the plastic points in suset P k,theincompressiility condition should e satisfied as follows U T (K v ) i U = 0, i P k, (47) After summarizing the equations stated aove, the final iterative formula of Eqs. (32) (34) can e written as follows ρ i J i K i U k U Tk 1 K i U k 1 = λ k F, (48) F T U k = 1, (49) U T k K i U k = 0, i R k, (50) U T k (K v ) i U k = 0, i P k. (51) The minimization formulation corresponding to Eqs. (48) (51) can e reasonaly written as: ν = min : ρ i J i UT K i U U Tk 1 K i U k 1, (52) s.t. F T U = 1, (53) U T K i U = 0, i R k, (54) U T (K v ) i U = 0, i P k. (55) In Eqs. (52) (55), U denotes the unknown optimization variale at the k-th iteration. By distinguishing the rigid region from the plastic region for every iterative step, the goal function and constraint conditions are gradually modified, and the difficulties causedythe nonlinear andnondifferentiale characteristic of goal function and the treatment of incompressiility condition are overcome. In order to solve the minimization prolem sujected to several equality constraints, the following measures will e adopted: Equations (54) and (55) can e dealt with y the penalty function method, and Eq. (53) can e removed y the Lagrange multiplier method. By doing so, Equations (52) (55) are equivalent to minimizing the following prolem L k (U,λ) = ρ i J i UT K i U U T k 1 K i U k α R ρ i J i U T K i U i R k α v ρ i J i U T (K v ) i U 2λ(F T U 1), (56) where α R and α v are the penalty factors and are usually assigned from 10 5 σ s to 10 8 σ s,andλis the Lagrange multiplier. Because the plastic incompressiility condition of plane stress prolem can e satisfied y introducing a revised

8 Upper-ound limit analysis ased on the natural element method 1405 matrix D into the matrix K i, assign α v = 0 for plane stress prolem. Assemling G k = 3 σ ρ i J i K i s U T k 1 K i U k α R ρ i J i K i i R k α v ρ i J i (K v ) i and introducing the displacement oundary condition specficied y Eq. (4), we can rewrite Eq. (56) as L k (U,λ) = U T G k U 2λ(F T U 1). (57) Let L k (U,λ)/ U = 0and L k (U,λ)/ λ = 0, we have G k U = λf, (58) F T U = 1. (59) Defining U = λδ, (60) sustituting Eq. (60) into Eq. (58), we have δ = G 1 k F.By sustituting Eq. (60) into Eq. (59), λ can e otained as λ = (F T δ) 1. Sustituting δ and λ into Eq. (60), we can otain the nodal displacement vector U. Finally, the limit load multiplier can e otained as ν = 3 σ s ρ i J i U T K i U. (61) 4.4 The implementation of iterative solution algorithm Based on the construction of iterative solution algorithm presented aove, the following iterative procedure can e implemented. Step 0: In the initial state, assume the entire structure is completely plastic. Solve the initial minimization prolem as follows ν = min : ρ i J i U T K i U, (62) s.t. F T U = 1, (63) U T (K v ) i U = 0, i IG. (64) Based on the minimization conditions, the aove solution process is equivalent to solving a set of linear algeraic equations such as Eqs. (58) and (59). So, we can otain the initial nodal displacement vector U 0, and the limit load multiplier at iterative step 0 can e written as ν 0 = ρ i J i U T 0 K i U 0. (65) Step k: According to Eq. (45), firstly examine the strain value of every integration point to determine the rigid region suset R k and the plastic region suset P k. Then we solve the following minimization prolem ν = min : ρ i J i UT K i U U Tk 1 K i U k 1, (66) s.t. F T U = 1, (67) U T K i U = 0, i R k, (68) U T (K v ) i U = 0, i P k. (69) The aove solution process is equivalent to solving a set of linear algeraic Eqs. (56) (59). Thus, we can get the nodal displacement vector U k, and the limit load multiplier at iterative step k can e otained as ν k = ρ i J i U T k K i U k. (70) When the aove iterative process satisfies the following two convergence criteria, the iteration will e terminated U k U k 1 ν k ν k 1 < vol1, < vol2, (71) U k ν k where vol1 andvol2 are the error tolerances and are determined y the desired accuracy of the calculation. 5 The proof for the convergence of algorithm In this section, we will prove the convergence of the iteration process presented aove. Multiplying oth sides of Eq. (48) y U T k, and adopting the similar treatment as specified in Eq. (41), we can derive λ k = 3 σ ρ i J i UT k K i U k s. (72) U Tk 1 K i U k 1 In view of Eqs. (45), (50) and (70), we can get the following equation at iterative Step (k 1) IG = P k 1 R k 1, (73) U T k 1 K i U k 1 = 0, i R k 1, (74) ν k 1 = ρ i J i U T k 1 K i U k 1. (75) Furthermore, Eq. (75) can e modified as ν k 1 = ρ i J i U T k 1 K i U k 1 1

9 1406 S.-T. Zhou, Y.-H. Liu = 3 σ ρ i J i UT k 1 K i U k 1 s. (76) 1 U Tk 1 K i U k 1 Hence, we can see that two sequences {ν k } and {λ k } are produced in the aove iterative process. Let suset ΔR k denote the increase of rigid region suset R k (i.e., the decrease of plastic region suset P k 1 ) from the (k 1)-th iteration to the k-th iteration, we have P k ΔR k = P k 1, R k 1 ΔR k = R k. (77) Basedonthedefinition of the rigid region suset R k presented in Eq. (45), the rigid region suset at iterative Step (k + 1) can e written as R k+1 = { i IG, U T k K i U k = 0 }. (78) Because R k R k+1,wehave U T k K i U k = 0, i ΔR k. (79) So Eqs. (72) and (75) can e written as λ k = 3 σ ρ i J i UT k K i U k s U T k 1 K i U k 1 = 3 σ ρ i J i UT k K i U k s 1 U T k 1 K i U k 1 3 σ ρ i J i UT k K i U k s i ΔR k U T k 1 K i U k 1 = 3 σ ρ i J i UT k K i U k s. (80) U Tk 1 K i U k 1 1 Because U k is the minimum solution of the goal function at iterative Step k, it can e otained as λ k = = 1 ρ i J i UT k K i U k U T k 1 K i U k 1 ρ i J i UT k K i U k U T k 1 K i U k 1 ν k 1 = 3 σ ρ i J i UT k 1 K i U k 1 s. (81) 1 U Tk 1 K i U k 1 According to the Cauchy Schwarz inequality, we know that, if a i, i R, and i = 1, 2,, n, the following inequality can e estalished n n n 2 a 2 i 2 i a i i. (82) i=1 i=1 i=1 If and only if a 1 = = a i = = a n = 0or i = ka i (k is a constant), the two sides of the inequality are equal. From this theorem, it can easily e verified that λ k ν k 1 = 3 σ ρ i J i UT k K i U k s U T k 1 K i U k 1 ρ i J i U T k 1 K i U k σ ρ i J i UT k K i U k s U T k 1 K i U k 1 ρ i J i U T k 1 K i U k 1 that is 2 3 σ ρ i J i U T k K i U k s U T k 1 K i U k 1 2 ρ i J i U T k 1 K i U k 1 2 = ρ i J i U T k K i U k = ν 2 k, (83) λ k ν k 1 ν 2 k. (84) Considering Eqs. (81) and (84), we have ν k 1 ν k 1 λ k ν k 1 ν 2 k. (85) Equation (85) denotes ν k 1 ν k, which means that the sequence {ν k } decreases monotonically. Therefore, when k there must exist a constant ν which makes lim ν k = ν. (86) k Furthermore, according to Eq. (85), we can otain the following inequality ν k 1 λ k ν2 k ν k 1. (87) In view of lim ν k 1 = ν and lim (ν 2 k k k /ν k 1) = lim ν k k lim (ν k/ν k 1 ) = ν, we can easily get the following equality k y using pinching theorem lim k λ k = ν. (88) Comparing Eq. (86) with Eq. (88), we can know that the sequences {ν k } and {λ k } monotonically converge to the same value ν. 2

10 Upper-ound limit analysis ased on the natural element method 1407 Up to now, the convergence of the sequences {ν k } and {λ k } have een proved. Now we will prove the convergence of the displacement vector, U k.let A k = 1 ρ i J i K i U Tk 1 K i U k 1, Eqs. (80) and (76) can e rewritten as λ k = = 1 ρ i J i UT k K i U k U T k 1 K i U k 1 ρ i J i UT k K i U k U T k 1 K i U k 1 = U T k A k U k, (89) ν k 1 = 3 σ ρ i J i UT k 1 K i U k 1 s 1 U T k 1 K i U k 1 = U T k 1 A k U k 1. (90) Multiplying oth sides of Eq. (48) y U T k 1, we otain 3 σ ρ i J i UT k 1 K i U k s = λ k U U T Tk 1 K k 1 F = λ k. (91) i U k 1 Owing to P k 1 P k,wehave U T k 1 A k U k = 3 σ ρ i J i UT k 1 K i U k s 1 U T k 1 K i U k 1 3 σ ρ i J i UT k 1 K i U k s. (92) U Tk 1 K i U k 1 Because U k is the minimum solution of the goal function at the k-th iteration, we get ν k 1 = 3 σ ρ i J i UT k 1 K i U k 1 s 1 U T k 1 K i U k 1 3 σ ρ i J i UT k 1 K i U k s. (93) 1 U Tk 1 K i U k 1 Considering Eqs. (91) (93), the following inequality can e got ν k 1 U T k 1 A k U k λ k. (94) According to Eqs. (86) and (88), we can otain the following formula y using the pinching theorem lim k UT k 1 A k U k = ν. (95) Furthermore, we can get Δ k = (U k U k 1 ) T A k (U k U k 1 ) = U T k A k U k 2U T k 1 A k U k + U T k 1 A k U k 1. (96) Considering Eqs. (89), (90) and (95), (96) can e written as lim Δ k = lim (U k U k 1 ) T A k (U k U k 1 ) k k = v 2v + v = 0. (97) Equation (97) denotes that lim (U k U k 1 ) T K i (U k U k 1 ) = 0, i P k 1. (98) k According to the definition of rigid region suset R k 1,another equality can e otained naturally as follows (U k U k 1 ) T K i (U k U k 1 ) = 0, i R k 1. (99) Taking Eqs. (98) and (99) into account, we can otain lim (U k U k 1 ) T K i (U k U k 1 ) = 0, i IG. (100) k Because the displacement vectors U k and U k 1 in Eq. (100) are the optimal solutions of minimization mathematical programming prolem for the upper ound limit analysis, and the displacement vector U k U k 1 tends to the rigid displacement mode, the following equality must e fulfilled lim (U k U k 1 ) = 0. (101) k Equation (101) manifests that the displacement vector is also convergent. Therefore, the convergence performance of iterative solution algorithm has successfully een demonstrated. 6 Numerical examples In this section, several representative and classical numerical examples of plastic limit analysis are presented for plane prolems, which are used to verify the accuracy and reliaility in implementing the NEM-ased solution procedure outlined aove. The Sison and the Laplace interpolants are respectively used to construct the trial functions of the NEM in the framework of Galerkin method. Three point quadrature rule is employed for numerical integration of each Delaunay triangles in the NEM. Unless otherwise specified, the following structures are made up of perfectly rigid-plastic material and the material parameters are chosen as: the yield stress σ s = 200 MPa, the Young s modulus E = 210 GPa, the Poisson ratio v = 0.3. The error tolerances are assigned as vol1 = vol2 = In FEM, the same node may simultaneously elong to different elements, and the strains or stresses of the same node in different elements are usually unequal. In this case, the nodal strain or stress can not e calculated directly and

11 1408 S.-T. Zhou, Y.-H. Liu the acquisition of smooth strain or stress field in the whole region of structure needs particular treatment. Wang [40] introduced many approaches to construct the smooth stress field in the FEM. Different from those in the FEM, the shape functions of the Sison and the Laplace interpolants adopted in the NEM are nondifferentiale at the nodal locations. Hence, the nodal strains and stresses in the NEM are also unale to e calculated directly. Taara et al. [41] used the moving least squares (MLS) to interpolate the nodal displacement and recovered the nodal strains y taking appropriate derivatives of MLS interpolant, and his numerical results indicate that the MLS technique is very simple and accurate for the recovery of nodal strain. Inspired y the good idea advised y Taara, this paper firstly utilizes the NEM to gain the nodal displacements and the strains at the Gaussian points, then the MLS approximation technique is used to fit the nodal strains y using the strains at the Gaussian points. In this way, the smooth strain field in the whole region is constructed, and the smooth plastic dissipation work can e easily calculated from Eq. (5). With the help of drawing software, the distriutions of plastic dissipation work at the limit states can e visually displayed. 6.1 Four thin square plates with different defects under unidirectional tensions Four thin square plates with different defects in the state of plane stress, under the unidirectional uniform tension loads, are shown in Figs. 5a 5d with P = N and L = 2m. Due to the symmetry of structure and load, only the upper right quadrant of these thin plates is taken into account. The nodal arrangements of these plates are displayed in Fig. 6. For plates A D, their nodal numers are 541, 575, 561 and 580 respectively. Hayes and Marcal [3] utilized the displacement fields used in finite element representations to calculate the upper ound limit loads of plane stress prolems. Belytschko and Hodge [4] used the FEM to construct the piecewise quadratic equilirium stress fields and adopted the sequential unconstrained minimization technique to determine the est lower ounds on the yield-point load of plane stress prolems. By making use of the reduced-asis technique and the complex method, Zhang et al. [8] and Chen et al. [15] respectively used the symmetric galerkin oundary element method (SGBEM) and the EFG method to develop non-linear programming solution procedure, and then adopted them to otain the maximal load amplifiers for the lower ound limit analysis of 2D structures. Tale 1 summarizes the upper ound solutions otained y the numerical method developed in this paper and those availale solutions. Specially, Tale 1 also lists the elasto-plastic incremental solutions solved y the finite element analysis software AN- SYS. From the comparison of these computational results, it can e clearly seen that the present solutions are smaller than the upper ound solutions otained y Hayes and Marcal [3], and slightly larger than the lower ound solutions otained y Belytschko and Hodge [4], Zhang et al. [8], and Chen et al. [15]. The small differences of computational accuracy among these numerical results are mainly ecause different numerical methods are used to investigate their examples and different numers and distriution forms of nodes are adopted to discretize their plates. Figure 7 displays the plastic dissipation work distriutions of four square plates at the limit states. a P P c P d P L L L L Crack L 2 L 5 L 4 L 4 P P P P Fig. 5 Four thin square plates with different defects sujected to unidirectional tensions. a Plate A; Plate B; c Plate C; d Plate D a c d Fig. 6 Node discretizations for four thin square plates with different defects. a Plate A; Plate B; c Plate C; d Plate D

12 Upper-ound limit analysis ased on the natural element method 1409 Tale 1 The comparison of limit loads using different methods (P/σ s ) Plates Lower ound [4] Lower ound [8] Lower ound [15] ANSYS Upper ound [3] Present (upper ound) Sison Laplace A B C D a can e seen from Tale 1 that the results otained y these two interpolant schemes are almost the same. From Fig. 7, we can see that the plastic dissipation work distriutions using the Sison interpolant are smoother than those using the Laplace interpolant. c e g d f h Fig. 7 Plastic dissipation work distriutions of four plates at the limit states (10 6 J). a Plate A (Sison); Plate A (Laplace); c Plate B (Sison); d Plate B (Laplace); e Plate C (Sison); f Plate C (Laplace); g Plate D (Sison); h Plate D (Laplace) Here, we adopt respectively the Sison and the Laplace interpolants to carry out the limit analysis computations using the proposed solution procedure. These two interpolant schemes possess their own characteristics and advantages. It 6.2 Square plate with a central circular hole under iaxial uniform loads As another classical plane stress prolem in numerical limit analysis, square plate with a central circular hole is chosen to test the accuracy of the proposed solution procedure. The geometric parameter and load arrangements are shown in Fig. 8a, where the iaxial uniform tension loads are respectively P 1 and P 2, and the diameter of the central circular hole is one-fifth of the oundary length L. Owing to the symmetry of structure, only the upper right quarter of plate is modeled and the essential oundary conditions are imposed on the left and ottom oundaries, as depicted in Fig.8. The nodal distriution is the same as displayed in Fig. 6, and let P 1 = P 2 = N and L = 2 m in the following computation. Many scholars also investigated this representative prolem, and the comparison of the present limit load domain with their results is displayed in Fig. 9. It can e seen that the present numerical solutions of the upper-ound is slightly smaller than the upper ound result of Chen et al. [7], and slightly larger than the lower-ound results of Gross Weege [42] and Chen et al. [15, 16]. Tale 2 lists the specific comparison of different numerical solutions under three loading cases with previous numerical solutions, which demonstrates that the solutions otained y the present NEM-ased solution approach are very close to those y traditional mesh-ased numerical methods. The comparison illustrates the validity of the proposed solution procedure using the NEM. Figure 10 shows the plastic dissipation work distriutions of this square plate at the limit states when P 1 = P 2 and P 2 = P 1 /2. For the purpose of comparing the computational performance of the NEM with other meshless methods, this paper develops the computer solution codes to investigate this example y using the EFG method, and uses respectively the traditional MLS technique [10] and the revised MLS technique [43] to construct the displacement velocity field. In these two MLS interpolants, the quadratic asis function and quartic spline weight function are selected, the radius of support domain for the calculated point is set

13 1410 S.-T. Zhou, Y.-H. Liu a P L P 2 σ s Ref. [7] upper ound Sison upper ound Laplace upper ound Ref. [15] lower ound Ref. [16] lower ound Ref. [42] lower ound P 1 P L P 1 σ s Fig. 9 The comparison of the limit load domains P 2 P 2 a P 1 c d Fig. 8 Square plate with a central circular hole. a Geometric parameter and load arrangements; The upper right quarter Fig. 10 The plastic dissipation work distriutions of square plate at the limit states (10 6 J). a P 2 = P 1 (Sison); P 2 = P 1 (Laplace); c P 2 = P 1 /2 (Sison); d P 2 = P 1 /2 (Laplace) Tale 2 The comparison of different numerical solutions (P 2 /σ s ) Authors Methods Loading cases P 2 = 0 P 2 = P 1 P 2 = P 1 /2 Gross-Weege [42] Lower ound Chen et al. [15] Lower ound Chen et al. [16] Lower ound Zhang et al. [44] Lower ound Tin-Loi and Ngo [45] Lower ound ANSYS solution Tran et al. [46] Upper ound Present (NEM, Sison) Upper ound Present (NEM, Laplace) Upper ound Present (EFG, traditional MLS) Upper ound Present (EFG, revised MLS) Upper ound da Silva and Antao [47] Upper ound as 4.0 times the distance etween the calculated point and its ninth nearest node, the penalty function method is adopted to impose the essential oundary conditions of structural discretized nodes located on the left oundary, and 4 4Gaussian integration rule are used to carry out the numerical integration. These variale selections and treated methods are also used in the following investigations, and the limit loads under three loading cases for this plate are also listed in Tale 2. Figure 11 exhiits the convergence of iterative processes oth in the NEM and in the EFG for limit analysis under three loading cases. It is clearly seen that: (1) The numerical method developed in this paper has fast conver-

14 Upper-ound limit analysis ased on the natural element method 1411 gence performance. All of the limit loads decrease monotonically with the increase of iterative steps, and tend to steady minimum values after iterations. (2) The convergent curves otained using the Sison interpolant are almost coincident with those using the Laplace interpolant. (3) When the otained limit loads converge to the same values with desired computational accuracy, the iterative steps needed in the EFG are less than those needed in the NEM. Limit load P 2 σ s P2 = 0, NEM, Sison P2 = 0, NEM, Laplace P2 = P1, NEM, Sison P2 = P1, NEM, Laplace P2 = P1/2, NEM, Sison P2 = P1/2, NEM, Laplace P2 = 0, EFG, traditional MLS P2 = 0, EFG, revised MLS P2 = P1, EFG, traditional MLS P2 = P1, EFG, revised MLS P2 = P1/2, EFG, traditional MLS P2 = P1/2, EFG, revised MLS Iterative step k Fig. 11 The convergence processes of the limit loads for square plate 6.3 A tapered cantilever eam with end-loaded tangential force A tapered cantilever eam with end-loaded tangential force is chosen to study herein in the plane stress state. The detailed model size and nodal distriution are depicted in Fig. 12. This prolem was also investigated through the kinematic limit analysis y Ciria et al. [48] and Le et al. [49]. Ciria et al. otained respectively two numerical solutions as and y using uniform meshing and adaptive meshing, and Le et al. gave a numerical solution of y utilizing a cell-ased smoothing FEM. These numerical results were otained when the yield stress was assigned as σ s = 3. By nondimensionalizing the limit loads (i.e., q/σ s ), Tale 3 gives the comparison of non-dimensional limit loads using different numerical methods. In order to compare the present computational efficiency with other meshless methods, this paper develops specific computer solution codes to investigate this example y using the EFG method with traditional MLS technique. The numerical result and the computational time needed y using the EFG are also listed in Tale 3, and the computational time is counted on a same computer. It can e seen that: (1) The numerical results otained y the NEM and the EFG are in good agreement with the availale solutions. (2) The NEM ased on Sison and Laplace interpolants costs nearly half of computational time spent on the EFG. The traditional MLS technique involves a set of complicated calculations of matrix multiplications and inversions. However, the calculations for the shape functions of Sison interpolant and Laplace interpolant are relatively simple in the NEM. Thus, the computational time spent on the NEM with Sison interpolant and Laplace interpolant is less than the cost needed in the EFG with traditional MLS technique, and the NEM with Sison interpolant and Laplace interpolant has higher computational efficiency than the EFG with traditional MLS technique. a 4.4 m q =1 000 N 4.8 m 1.6 m 4.4 m Fig. 12 Model sketch and nodal distriution of a tapered cantilever eam. a Geometric size; Nodal distriution Tale 3 The comparison of limit loads using different numerical mthods Method Numerical results (q/σ s ) Computational time/s Ciria et al. [43] (uniform / adaptive meshing) / Le et al. [48] ANSYS Present (NEM, Sison) Present (NEM, Laplace) Present (EFG, traditional MLS)

15 1412 S.-T. Zhou, Y.-H. Liu Furthermore, Fig. 13 displays the plastic dissipation work distriutions of this tapered cantilever eam at the limit states. Figure 14 shows the convergence processes of the limit loads for the Sison and the Laplace interpolants. We can see that, with the increase of iteration, the limit loads decrease monotonically and tend to steady minimum values. a and the analytical solutions. The EFG is also adopted here to study this example and to compare the computational precision with the NEM, and detailed comparisons etween numerical solutions otained respectively y the NEM and the EFG, and analytical solutions are listed in Tale 4 for different ratios /a. It is ovious that: (1) No matter it is the NEM or the EFG, the otained numerical solutions can match very well with the analytical solutions. (2) The computational errors in the NEM are smaller than those in the EFG, which reveal that the NEM-ased solution method has the advantage of higher computational precision than the EFG-ased one in the limit analysis of thick-walled cylinder sujected to uniform internal pressure a Fig. 13 The plastic dissipation work distriutions of tapered cantilever eam at the limit states (10 6 J). a Using Sison interpolant; Using Laplace interpolant P Limit load q σ s Sison Laplace Fig. 15 A thick-walled cylinder under uniform internal pressure a Iterative step k Fig. 14 The convergence processes of the limit loads for tapered cantilever eam 6.4 Thick-walled cylinder sujected to uniform internal pressure As a well known example of plane strain state in the limit analysis, a thick-walled cylinder under the action of uniform internal pressure P is studied here. Considering the symmetry of structure, only the upper right quadrant of the thickwalled cylinder is modeled, as displayed in Fig. 15. The analytical solution of limit load for this prolem is P L = 2σ s 3 ln a, (102) where a and are respectively the inner and the outer radii. The limit loads of thick-walled cylinders are computed for seventeen different ratios /a (outer radius/inner radius). Different regular node distriutions are arranged for different ratios /a of cylinders. Figure 16a shows the regular nodal arrangement with 475 nodes in the case of /a = 3.0. Figure 17 indicates the comparison etween the numerical solutions Fig. 16 The nodal arrangement for /a = 3. a Regular arrangement; Irregular arrangement P σ s Analytical solution Numerical solution (Sison) Numerical solution (Laplace) a Fig. 17 The comparison of limit loads etween the numerical and analytical solutions