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3rd Reading January 16, 2013 15:57 WSPC/0219-8762 196-IJCM 1340004 International Journal of Computational Methods Vol 10, No (2013) 1340004 (15 pages) c World Scientific Publishing Company DOI: 10.1142/S0219876213400045 COMPUTATION OF LIMIT LOAD USING EDGE-BASED SMOOTHED FINITE ELEMENT METHOD AND SECOND-ORDER CONE PROGRAMMING C V LE∗ Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF WARSAW on 01/23/15 For personal use only Department of Civil Engineering International University, VNU-HCM, Vietnam lvcanh@hcmiu.edu.vn H NGUYEN-XUAN Department of Mechanics Faculty of Mathematics and Computer Science University of Science, VNU-HCM, Vietnam H ASKES Department of Civil and Structural Engineering The University of Sheffield, Sheffield S1 3JD, UK T RABCZUK Institute of Structural Mechanics Bauhaus-University Weimar Marienstrasse 15, 99423 Weimar T NGUYEN-THOI Department of Mechanics Faculty of Mathematics and Computer Science University of Science, VNU-HCM, Vietnam Received 17 January 2011 Accepted 10 June 2011 Published 18 January 2013 This paper presents a novel numerical procedure for limit analysis of plane problems using edge-based smoothed finite element method (ES-FEM) in combination with secondorder cone programming In the ES-FEM, the discrete weak form is obtained based on the strain smoothing technique over smoothing domains associated with the edges of the elements Using constant smoothing functions, the incompressibility condition only needs to be enforced at one point in each smoothing domain, and only one Gaussian point is required, ensuring that the size of the resulting optimization problem is kept to a minimum The discretization problem is transformed into the form of a second-order cone ∗ Corresponding author 1340004-1 3rd Reading January 16, 2013 15:57 WSPC/0219-8762 196-IJCM 1340004 C V Le et al programming problem which can be solved using highly efficient interior-point solvers Finally, the efficacy of the procedure is demonstrated by applying it to various benchmark plane stress and strain problems Keywords: Collapse load; limit analysis; SFEM; ES-FEM; SOCP Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF WARSAW on 01/23/15 For personal use only Introduction Upper bound limit analysis has been widely used to provide upper-bound estimates of the load required to cause collapse of a body or structure One of the key conditions in the upper-bound computational limit analysis is that the flow rule (or incompressibility condition in plane strain) is required to hold everywhere in the problem domain This requirement can be easily met using constant strain finite elements However, it is well-known that the low-order displacement finite elements exhibit volumetric locking phenomena in the kinematic formulations It is, therefore, desirable to develop an efficient method that can overcome the locking problem while allowing the flow rule to be met easily Recently, Le et al [2010d] proposed a numerical kinematic formulation using the cell-based smoothed finite element method (CS-FEM) to furnish good (approximate) upper-bound solutions They have shown that when smoothed strains (using one cell version CS-FEM1) are used in the kinematic formulation the volumetric locking can be removed, the flow rule only needs to be enforced at any one point in each smoothing cell, and it is guaranteed to be satisfied everywhere in the problem domain Following this line of research, the main objective of this paper is to develop a computational limit analysis procedure which combines the edge-based smoothed finite element method (ES-FEM) with second-order cone programming (SOCP) to approximate upper-bound solutions for plane problems The strain smoothing technique, which was originally proposed by Chen et al [2001] to stabilize a direct nodal integration in mesh-free methods, has been applied to the FEM settings to formulate various smoothed finite element methods (SFEM) [Liu and Nguyen-Thoi (2010)], including CS-FEM [Liu et al (2007a); Liu et al (2007b); Nguyen-Xuan et al (2008)], node-based SFEM (NS-FEM) [Liu et al (2009)], and the ES-FEM [Liu et al (2009); Nguyen-Xuan et al (2009)] and the face-based SFEM (FS-FEM) [Nguyen-Thoi et al (2009)] Each of these smoothed FEM has different characters and properties, and has been used successfully in solid mechanics [Nguyen-Xuan et al (2008); Nguyen-Thoi et al (2010); Nguyen-Xuan et al (2010)] Theoretical study of these SFEM was also reported in Liu et al (2007b) and Nguyen-Xuan et al [2008] In general, these SFEM are able to remove locking problems while accurate solutions can be obtained with minimal computational effort Tran et al [2010] has applied the ES-FEM to limit and shakedown analysis problems using Koiter’s theorem, in which fictitious elastic stresses are assumed However, in this paper the ES-FEM is formulated associated with Markov’s kinematic theorem, and the resulting optimization problem is cast in the form of a SOCP problem so that a large-scale problem can be solved 1340004-2 3rd Reading January 16, 2013 15:57 WSPC/0219-8762 196-IJCM 1340004 Computation of Limit Load Using Edge-Based Smoothed Finite Element Method efficiently [Le et al (2009); Le et al (2010a); Le et al (2010b); Le et al (2010c) and references therein] Therefore, the present approach is different from the previous ES-FEM approach given in Tran et al [2010] The paper is organized as follows The next section briefly describes the edgebased smoothed finite element method The kinematic limit analysis formulation is then recalled and ES-FEM based discretization problem is then formulated as a SOCP in Sec Numerical examples are provided in Sec to illustrate the performance of the proposed procedure Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF WARSAW on 01/23/15 For personal use only Brief of the ES-FEM In ES-FEM, basing on the mesh of elements, we further discretize the problem domain into smoothing domains based on edges of the elements such that Ω ≈ Ned Ωk and Ωi ∩ Ωj = for i = j, in which Ned is the total number of edges of k=1 all elements in the entire problem domain Moreover, ES-FEM shape functions are identical to those in the FEM However, instead of using compatible strains, the ESFEM uses strains smoothed over local smoothing domains These local smoothing domains are constructed based on edges of elements as shown in Fig A strain smoothing formulation is now defined by the following operation ˜hk = h ∇s uh (x)φk (x)dΩ, (x)φk (x)dΩ = Ωk (1) Ωk where ∇s uh are the compatible strains of the approximate fields, uh , and φk (x) is a distribution (or smoothing) function that is positive and normalized to unity φk (x) dΩ = (2) Ωk For simplicity, the smoothing function φk is taken as φk (x) = 1/Ak , x ∈ Ωk 0, (3) otherwise, edge-based smoothing cell Ωk Γk centroid element i element j Fig (Color online) Smoothing cell Ωk connected to edge k of triangular elements 1340004-3 3rd Reading January 16, 2013 15:57 WSPC/0219-8762 196-IJCM 1340004 C V Le et al where Ak is the area of the smoothing domain Ωk , and is calculated by Ak = Ωk dΩ = Nek Aje , (4) j=1 where Nek is the number of elements around the edge k (Nek = for the boundary edges and Nek = for interior edges) and Aje is the area of the jth element around the edge k Since interior edges are formed by two neighboring elements, the smoothed strains in the smoothing domain Ωk can be determined by Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF WARSAW on 01/23/15 For personal use only ˜ k1 dk1 + B ˜ k2 dk2 = B ˜ k dk , ˜hk = B (5) where dk1 and dk2 are nodal displacement vectors of element e1 and element e2 , respectively, dk is the displacement vector of the nodes associated with edge k, and ˜ kj (j = 1, 2) are the strain-displacement matrices defined by B ˜ kj kj ˜n,x N1,x N ˜ kj = ˜ kj ˜ kj B (6) N N n,y 1,y kj kj ˜ kj N ˜ kj N ˜n,x ˜n,y N N 1,y 1,x with j ˜ kj = Ae N I,α 3Ak NI (x)nα (x)dΓ, (7) Γkj ˜ kj is the smoothed version of shape function derivative N kj , nα is the where N I,α I,α normal vector and Γkj is boundaries of element j associated with edge k It is worth noting that the ES-FEM is different from the standard FEM by two key points: (1) FEM uses the compatible strain on the element, while ES-FEM uses the smoothed strain on the smoothing domain; and (2) the assembly process of FEM is based on elements, while that of ES-FEM is based on smoothing domain Ωk Limit Analysis Based on ES-FEM 3.1 Kinematic formulation Consider a rigid-perfectly plastic body of area Ω ∈ R2 with boundary Γ, which is subjected to body forces f and to surface tractions g on the free portion Γt of Γ The constrained boundary Γu is fixed and Γu ∪ Γt = Γ, Γu ∩ Γt = Let u˙ = [ u˙ v˙ ]T be plastic velocity or flow fields that belong to a space Y of kinematically admissible velocity fields [Ciria et al (2008); Christiansen (1996)], where u˙ and v˙ are the velocity components in x- and y-direction, respectively The external work rate associated with a virtual plastic flow u˙ is expressed in the linear form as f T u˙ dΩ + ˙ = F (u) Ω gT u˙ dΓ Γt 1340004-4 (8) 3rd Reading January 16, 2013 15:57 WSPC/0219-8762 196-IJCM 1340004 Computation of Limit Load Using Edge-Based Smoothed Finite Element Method ˙ = 1}, the collapse load multiplier λ+ can be Upon defining C = {u˙ ∈ Y | F (u) determined by the following mathematical programming λ+ = D( ˙ ) dΩ, ˙ u∈C Ω (9) where strain rates ˙ are given by ˙xx ˙ = ˙yy = Lu˙ (10) Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF WARSAW on 01/23/15 For personal use only γ˙ xy with L is the differential operator ∂ ∂x L= ∂ ∂y ∂ ∂y ∂ ∂x (11) The plastic dissipation D( ˙ ) is defined by D( ˙ ) = max σ : ˙ ≡ σ : ˙ ψ(σ)≤0 (12) in which σ represents the admissible stresses contained within the convex yield surface and σ represents the stresses on the yield surface associated to any strain rates ˙ through the plasticity condition In the framework of a limit analysis problem, only plastic strains are considered and they are assumed to obey the normality rule ˙ = µ˙ ∂ψ , ∂σ (13) where the plastic multiplier µ˙ is non-negative and the yield function ψ(σ) is convex In this study, the von Mises failure criterion is used + σ2 − σ σ σxx plane stress xx yy + 3σxy − σ0 yy ψ(σ) = (14) (σ − σ )2 + σ − σ plane strain, xx yy xy where σ0 is the yield stress Then the power of dissipation can be formulated as a function of strain rates as Capsoni and Corradi [1997] D( ˙ ) = σ0 ˙T Θ ˙, 1340004-5 (15) 3rd Reading January 16, 2013 15:57 WSPC/0219-8762 196-IJCM 1340004 C V Le et al Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF WARSAW on 01/23/15 For personal use only where 1 2 Θ= −1 0 plane stress (16) −1 plane strain Note that condition (13) acts as a kinematic constraint which confines the vectors of admissible strain rates For plane strain problems, the yield surface ψ(σ) is unbounded, and the incompressibility condition χT ˙ = 0, where χ = [ 1 ]T , must be introduced to ensure that the plastic dissipation D( ˙ ) is finite [Andersen et al (1998); Christiansen and Andersen (1999)] 3.2 ES-FEM discretization and solution procedure In the ES-FEM formulation, the plastic dissipation is expressed as Ned DES−FEM = σ0 Ωk k=1 ˙h ˜˙ hT k Θ ˜ k dΩ (17) in which ˜˙ hk can be obtained from Eq (5) When three-node triangular elements are used, ˜˙ hk are constant over the smoothing domain Ωk , and hence Eq (17) can be rewritten as Ned DES−FEM = σ0 Ak k=1 ˙h ˜˙ hT k Θ ˜k (18) Consequently the optimization problem (9) associated with the ES-FEM can now be rewritten as Ned λ+ = ˙h ˜˙ hT k Θ ˜k σ0 Ak k=1 s.t u˙ h = on Γu ˙h F (u ) = (19) which will search the nodal velocities vector The above limit analysis problem is a nonlinear optimization problem with equality constraints In fact, the problem can be reduced to the problem of minimizing a sum of norms as Ned σ0 Ak ρβ , λ+ = β = 1, k=1 s.t u˙ h = ˙h F (u ) = 1340004-6 on Γu , (20) 3rd Reading January 16, 2013 15:57 WSPC/0219-8762 196-IJCM 1340004 Computation of Limit Load Using Edge-Based Smoothed Finite Element Method where ρβ are additional variables defined by ρ1 √0 ρ = ρ2 = √ ˜˙ hk 0 ρ3 ρ2 = ρ1 ρ2 = (˜˙hxxk − ˜˙hyyk ) for plane stress, for plane strain h γ˜˙ xyk (21) (22) Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF WARSAW on 01/23/15 For personal use only Introducing auxiliary variables t1 , t2 , , tNed , optimization problem (20) can be cast as a SOCP problem Ned + λ = σ0 Aj tj j=1 h u˙ = s.t F (u˙ h ) = ρβ i ≤ ti on Γu (23) i = 1, 2, , Ned , where the third constraint in problem (23) represents quadratic cones Numerical Examples In this section, the performance of the proposed solution procedure is illustrated via a number of benchmark problems in which analytical and other numerical solutions are available All examples are considered in either plane stress or plane strain state and the von Mises criterion is exploited Because distorted meshes does not necessarily result in better solutions [Le et al (2010d)], and for convergence study, regular meshes will be used in examples 4.1 Square plate with a central circular hole The first example deals with a square plate with a central circular hole which is subjected to biaxial uniform loads p1 and p2 as shown in Fig 2(a), where L = 10 m Owing to symmetry, only the upper-right quarter of the plate is modeled, see Fig 2(b) Symmetry conditions are enforced on the left and bottom edges Analytical solutions are available [Gaydon and McCrum (1954)] for the case p1 when p2 = 0, namely λ = 0.8 Therefore, to enable objective validation, the σ0 procedure was first applied to this case Numerical solutions obtained for different models with variation of N are compared with those obtained using CS-FEM, as shown in Table It can be seen that solutions obtained by using ES-FEM are more accurate than those obtained by using CS-FEM2, CS-FEM4 and FEMQ4, and are slightly higher than results obtained by using CS-FEM1 (CS-FEMk– SFEM 1340004-7 3rd Reading January 16, 2013 15:57 WSPC/0219-8762 196-IJCM 1340004 C V Le et al p2 L p1 p1 Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF WARSAW on 01/23/15 For personal use only L/5 p2 N = 10 (a) (b) Fig A square plate with a circular hole: (a) geometry and loading, (b) finite element mesh (N is the number of elements along the horizontal symmetry axis, and thus a measure for the mesh density) Table Collapse load multiplier of the plate with variation of N (p2 = 0) N ×N Formulations CS-FEM1 CS-FEM2 CS-FEM4 FEMQ4 Present method 6×6 12 × 12 24 × 24 48 × 48 Analytical solution 0.8151 0.8216 0.8226 0.8238 0.8217 0.8047 0.8078 0.8085 0.8090 0.8077 0.8017 0.8035 0.8038 0.8041 0.8030 0.8006 0.8018 0.8019 0.8021 0.8013 0.800 four-noded quadrilateral element with k smoothing cells or subcells, whereby k = 1, and 4) When 288 T3 elements (N = 12) are used, the proposed method provides a solution of 0.8077 compared with 0.7980 obtained in Tran et al [2010], where ES-FEM was used in combination with a primal-dual algorithm for shakedown analysis of structures proposed by Vu et al [2004] It is evident that the method presented by Tran et al [2010] provides a lower solution than the actual collapse multiplier, and therefore it does not guarantee an upper-bound on the actual collapse load On the contrary, the proposed method can provide strict upper bounds on the actual collapse multiplier, as shown in Table Moreover, the present solution procedure with the use of SOCP is more efficient and robust since just less than 30 secondsa were taken to solve the optimization problem with up to 32,834 variables and 21,123 constraints The convergence rate is also illustrated in Fig It is evident that all numerical solutions converge to the exact solution as the mesh size h tends to zero a The code is written using MATLAB and was run using a 2.8 GHz Pentium PC running Microsoft XP 1340004-8 3rd Reading January 16, 2013 15:57 WSPC/0219-8762 196-IJCM 1340004 Computation of Limit Load Using Edge-Based Smoothed Finite Element Method 0.6 log10(Relative error in collapse load) Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF WARSAW on 01/23/15 For personal use only 0.4 ~1.13 0.2 ~1.15 ~1.16 −0.2 ~1.36 −0.4 −0.6 −0.8 FEMQ4 CS−FEM1 CS−FEM2 CS−FEM4 ES−FEM ~1.59 −1 −1.2 −1.4 −1.2 −1 −0.8 −0.6 log10(mesh size h) −0.4 −0.2 Fig (Color online) Convergence rate with p2 = (values indicated in the figure are approximated slopes and the results of CS-FEM and FEMQ4 were obtained in Le et al [2010]) It can also be observed that convergence rate of the ES-FEM is higher than that of CS-FEM2, CS-FEM4, and FEM Although the ES-FEM converges slower than CS-FEM1, it guarantees stable solutions when Koiter’s kinematic theorem is used [see Trans et al (2010)] while the CS-FEM1 does not Moreover, the ES-FEM is based on triangular meshes that can be generated more easily than quadrilateral meshes used in the CS-FEM Next, the geometric effect of the circular hole was examined by considering various values of ratio R/L The obtained solutions were reported in Table It can be observed that in case when 288 elements are used, p1 = and p2 = 0, the present solutions are in good agreement with those obtained previously Table compares the best solutions obtained using the present method with solutions obtained previously by different limit analysis approaches (kinematic or Table Collapse multiplier: p1 = and p2 = R/L 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Heitzer [1999] Tran [2008] [Trans et al (2010)] Present method 0.8951 0.7879 0.6910 0.5720 0.4409 0.2556 0.1378 0.0565 0.0193 0.9017 0.8015 0.7022 0.5914 0.4012 0.2425 0.1254 0.0523 0.0123 0.8932 0.7967 0.6930 0.5760 0.4011 0.2429 0.1277 0.0521 0.0133 0.9054 0.8077 0.7084 0.5961 0.4120 0.2524 0.1343 0.0573 0.0155 1340004-9 3rd Reading January 16, 2013 15:57 WSPC/0219-8762 196-IJCM 1340004 C V Le et al Table Collapse load multiplier with different loading cases and N = 48 compared with previously obtained solutions Loading cases Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF WARSAW on 01/23/15 For personal use only Approach Authors p2 = p1 p2 = p1 /2 p2 = Kinematic (upper bound) da Silva and Antao [2007] Le et al [2010d] Present method 0.899 0.895 0.896 0.915 0.911 0.911 0.807 0.801 0.801 Mixed formulation Analytical solution Zouain et al [2002] Gaydon and McCrum [1954] 0.894 — 0.911 — 0.803 0.800 Static (lower bound) Chen et al [2008] Gross-Weege [1997] Belytschko [1972] Nguyen-Dang and Palgen [1979] 0.874 0.882 — 0.704 0.899 0.891 — — 0.798 0.782 0.780 0.564 Fig (Color online) Collapse mechanisms (original shape in green) for loading case p2 = static) using FEM, CS-FEM or element-free Galerkin simulations The present solutions are in excellent agreement to those obtained by Le et al (2010d) when p2 = and p2 = p1 /2 The collapse mechanism for the case when p2 = is shown in Fig 4, where the deformation is calculated by multiplying the computed collapse velocity by a suitable time scale and then adding it to the original grid The plastic dissipation distribution is also displayed in Fig 4.2 Prandtl’s punch problem This classical plane strain problem was originally investigated by Prandtl [1920], which consists of a semi-infinite rigid-plastic von Mises medium under a punch 1340004-10 3rd Reading January 16, 2013 15:57 WSPC/0219-8762 196-IJCM 1340004 Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF WARSAW on 01/23/15 For personal use only Computation of Limit Load Using Edge-Based Smoothed Finite Element Method Fig (Color online) Plastic dissipation distribution for loading case p2 = 1 H B Fig Prandtl’s punch: geometry and loading load, as shown in Fig For a load of 2τ0 , the analytical collapse multiplier is λ = + π = 5.142 Due to symmetry, only half of the foundation is considered The rectangular region of B = and H = was considered sufficiently large to ensure that rigid elements show up along the entire boundary The punch is represented by a uniform vertical load and appropriate boundary conditions were applied This problem exhibits volumetric locking when the standard T3-FEM with full integration is used However, when the ES-FEM is applied the locking problem can be eliminated similar to CS-FEM1 presented in Le et al [2010d] This can be explained by the fact that using the smoothed strain rates which are constant over a smoothing cell, the flow rule (or incompressibility condition in plane strain) only needs to be enforced at any one point in each smoothing cell, and as a result the number of constraints is kept to a minimum not to cause the volumetric locking (note that in limit analysis problems, the locking problem accurs due to a large number of incompressibility constraints is enforced) Collapse multipliers and associated errors for different values of Nf , the number of divisions under the footing edge, are 1340004-11 3rd Reading January 16, 2013 15:57 WSPC/0219-8762 196-IJCM 1340004 C V Le et al Table The punch problem: collapse multiplier using regular nodal layout Nf (total number of elements) Models (160) (640) (2560) ES-FEM Error (%) CPU-Mosek (s)∗ 5.3247 3.56 0.43 5.2251 1.62 2.46 5.1814 0.77 10.29 ∗ Time 16 (10240) 5.1511 0.38 108.9 taken to solve on a 2.8 GHz Pentium PC Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF WARSAW on 01/23/15 For personal use only Table Collapse load multiplier compared with previously obtained solutions Approach Authors Collapse multiplier Error (%) Kinematic da Silva and Antao [2007] Sloan and Kleeman [1995] Makrodimopoulos et al [2006a] Le et al [2010d] Present method 5.264 5.210 5.148 5.143 5.151 +2.38 +1.33 +0.12 +0.03 +0.18 Analytical solution Prandtl [1920] 2+π — Mixed formulation Capsoni and Corradi [1997] 5.240 +1.91 Static Makrodimopoulos et al [2006b] 5.141 −0.01 shown in Table It can be observed that accurate solutions can be obtained using the present method Table compares solutions obtained using the present method with upper and lower bound solutions that have previously been reported in the literature The present results are shown to be competitive with those obtained by other methods Considering previously obtained upper bound solutions, the present method provides lower (more accurate) upper bound solutions than in da Silva and Antao [2007] and Sloan and Kleeman [1995] The solution of the finest mesh used here is slightly higher than the best upper bound in Makrodimopoulos et al [2006a] However, the number of elements was employed here is far smaller than that was used in Makrodimopoulos et al [2006a], 10,240 T3 elements compared with 18,719 six-node triangle elements, which was generated with reduced element size close to the footing In comparison with solution obtained in Le et al [2010d] using CS-FEM1, the best upper solution found here is also higher (with the same mesh size) However, the proposed method can provide stable solutions while CSFEM1 does not guarantee, especially in shakedown analysis problems in which linear equations need to be solved to obtain fictitious elastic stresses The collapse mechanism and pattern of plastic energy dissipation are shown in Figs and 8, respectively The pattern of plastic energy dissipation looks very similar to the shear band pattern presented in Rabczuk et al [2007] 1340004-12 3rd Reading January 16, 2013 15:57 WSPC/0219-8762 196-IJCM 1340004 Computation of Limit Load Using Edge-Based Smoothed Finite Element Method Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF WARSAW on 01/23/15 For personal use only Fig Prandtl problem: collapse mechanism Fig (Color online) Prandtl problem: plastic dissipation distribution Conclusion A numerical limit analysis procedure that uses the ES-FEM and SOCP has been proposed Advantages of applying the ES-FEM to limit analysis problems are that the size of optimization problem is reduced and accurate and stable solutions can be obtained with minimal computational effort Moreover, with the use of smoothed strains volumetric locking problem can be removed while ensuring that the incompressibility condition holds everywhere in the problem domain Numerical examples were presented to show that the proposed procedure is able to solve large-scale problems in engineering practice Acknowledgments The first author acknowledges the support of Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No 107.02-2011.01 References Andersen, K D., Christiansen, E and Overton, M L [1998] “Computing limit loads by minimizing a sum of norms,” SIAM J Sci Comput 19, 1046–1062 Belytschko, T [1972] “Plane stress 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Method Eng 46, 1185–1202 Ciria, H., Peraire, J and Bonet, J [2008] “Mesh adaptive computation of upper and lower bounds in limit analysis,” Int J Numer Method Eng 75, 899–944 Gaydon, F A and McCrum, A W [1954] “A theoretical investigation of the yieldpoint loading of a square plate with a central circular hole,” J Mech Phys Solid 2, 156–169 Gross-Weege, J [1997] “On the numerical assessment of the safety factor of elasto-plastic structures under variable loading,” Int J Mech Sci 39, 417–433 Heitzer, M [1999] Traglast- und Einspielanalyse zur Bewertung der Sicherheit passiver Komponenten Berichte des Forschungszentrums Julich, Jul-3704, Dissertation, RWTH Aachen, Germany Le, C V., Gilbert, M and Askes, H [2009] “Limit analysis of plates using the EFG method and second-order cone programming,” Int J Numer Method Eng 78, 1532–1552 Le, C V., Askes, H and Gilbert, M [2010a] “Adaptive Element-Free Galerkin method applied to the limit analysis of plates,” Comput Method Appl Mech Eng 199, 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Computation of Limit Load Using Edge-Based Smoothed Finite Element Method ˙ = 1}, the collapse load multiplier λ+ can be Upon defining C = {u˙ ∈ Y | F (u) determined by the following mathematical programming. .. Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF WARSAW on 01/23/15 For personal use only Computation of Limit Load Using Edge-Based Smoothed Finite Element Method. .. 196-IJCM 1340004 Computation of Limit Load Using Edge-Based Smoothed Finite Element Method Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF WARSAW on 01/23/15