An edge-based smoothed finite element method for visco-elastoplastic analyses of 2D solids using triangular mesh

Transcription

1 Comput Mech (9) 45: 44 DOI.7/s ORIGINAL PAPER An edge-based smoothed finite element method for visco-elastoplastic analyses of D solids using triangular mesh T. Nguyen-Thoi G. R. Liu H. C. Vu-Do H. Nguyen-Xuan Received: 6 March 9 / Accepted: 9 August 9 / Published online: 6 September 9 Springer-Verlag 9 Abstract An edge-based smoothed finite element method (ES-FEM) using triangular elements was recently proposed to improve the accuracy and convergence rate of the existing standard finite element method (FEM) for the elastic solid mechanics problems. In this paper, the ES-FEM is extended to more complicated visco-elastoplastic analyses using the von-mises yield function and the Prandtl Reuss flow rule. The material behavior includes perfect visco-elastoplasticity and visco-elastoplasticity with isotropic and linear kinematic hardening. The formulation shows that the bandwidth of stiffness matrix of the ES-FEM is larger than that of the FEM, and hence the computational cost of the ES-FEM in numerical examples is larger than that of the FEM for the same mesh. However, when the efficiency of computation (computation time for the same accuracy) in terms of a posteriori error estimation is considered, the ES-FEM is more efficient than the FEM. Keywords Numerical methods Edge-based smoothed finite element method (ES-FEM) Finite element method (FEM) Strain smoothing technique Visco-elastoplastic analysis T. Nguyen-Thoi (B) G. R. Liu H. C. Vu-Do Department of Mechanical Engineering, Center for Advanced Computations in Engineering Science (ACES), National University of Singapore, 9 Engineering Drive, Singapore 7576, Singapore ngttrung@hcmuns.edu.vn; thoitrung76@yahoo.com G. R. Liu H. Nguyen-Xuan Singapore-MIT Alliance (SMA), E4-4-, 4 Engineering Drive, Singapore 7576, Singapore T. Nguyen-Thoi H. C. Vu-Do H. Nguyen-Xuan Faculty of Mathematics and Computer Science, University of Science, Vietnam National University-HCM, 7 Nguyen Van Cu, District 5, Ho Chi Minh City, Vietnam Introduction The strain smoothing technique [4] has been proposed to stabilize the solutions in the context of the meshfree method and then applied in the natural element method []. Liu et al. has generalized the gradient (strain) smoothing technique [9] and applied it in the meshfree context to formulate the node-based smoothed point interpolation method (NS-PIM or LC-PIM) [7,8] and the linearly conforming radial point interpolation method (NS-RPIM or LC-RPIM) []. Applying the same idea to FEM, a cell-based smoothed finite element method (SFEM or CS-FEM) [,] and a node-based smoothed finite element method (NS-FEM) [5] have also been formulated. In the CS-FEM, the strain smoothing domains and the integration are performed over the quadrilateral elements, and these smoothing domains can be divided into one or many smoothing cells (SC) on each element, as shown in Fig.. When the number of SC of the elements equals, the CS-FEM solution has the same properties with those of FEM using reduced integration. When SC approaches infinity, the CS-FEM solution approaches to the solution of the standard displacement compatible FEM model []. In practical calculation, using four SCs for each quadrilateral element in the CS-FEM is easy to implement, work well in general and hence advised for all problems. The numerical solution of CS-FEM (SC = 4) is always stable, accurate, much better than FEM, and often very close to the exact solutions. The CS-FEM has been developed for general n-sided polygonal elements (nsfem or ncs-fem) [7], dynamic analyses [6], incompressible materials using selective integration [,5], plate and shell analyses [5,9,4,8,9], and further extended for the extended finite element method (XFEM) to solve fracture mechanics problems in D continuum and plates [].

2 4 Comput Mech (9) 45: 44 Fig. Division of quadrilateral element into the smoothing cells (SCs) in the CS-FEM by connecting the mid-edge-points of opposite edges of SCs. a SC; b SCs;c SCs;d 4SCs; e 8SCs;f 6 SCs y (a) 4 x (b) (c) (d) 4 7 (e) 4 (f) 4 y x 5 : field nodes : added nodes to form the smoothing cells node k cell (k) (k) Γ : field node : central point of n-sided polygonal element : mid-edge point Fig. n-sided polygonal elements and the smoothing cell (shaded area) associated with nodes in the NS-FEM In the NS-FEM, the strain smoothing domains and the integration of the weak form are performed over the cells associated with nodes, and methods can be applied easily to triangular, 4-node quadrilateral, n-sided polygonal elements for D problems and tetrahedral elements for D problems. For n-sided polygonal elements, the cell (k) associated with the node k is created by connecting sequentially the midedge-point to the central points of the surrounding n-sided polygonal elements of the node k as shown in Fig.. When only linear triangular or tetrahedral elements are used, the NS-FEM produces the same results as the method proposed by Dohrmann et al. [8] or to the LC-PIM [7] using linear interpolation. The upper bound property shown in the NS-PIM [6] was also found in the NS-FEM [5]. Then, by combing the NS-FEM, FEM and a scale factor α [, ], a so-called the alpha Finite Element Method (αfem) [] using triangular and tetrahedral elements is proposed to give a nearly exact solution in strain energy. The NS-FEM has been developed for adaptive analysis []. Recently, an edge-based smoothed finite element method (ES-FEM) [4] was also been formulated for static, free and forced vibration analyses in D problems. The ES-FEM uses triangular elements that can be generated automatically for complicated domains. In the ES-FEM, the system stiffness matrix is computed using strains smoothed over the smoothing domains associated with the edges of the triangles. For triangular elements, the smoothing domain (k) associated with the edge k is created by connecting two endpoints of the edge to the centroids of the adjacent elements as shown in Fig.. The numerical results demonstrated that the ES-FEM possesses the following excellent properties: () the numerical results are often found super-convergent and much more accurate than those of FEM using quadrilateral elements with the same sets of nodes; () there are no spurious non-zeros energy modes found and hence the method is also stable and works well for vibration analysis; () the implementation of the method is straightforward and no penalty parameter is used, and the computational efficiency is better than FEM using the same sets of nodes; (4) a novel domain-based selective scheme is proposed leading to a combined ES/NS-FEM model that is immune from volumetric locking and hence works very well for nearly incompressible materials. The ES-FEM has been developed for D piezoelectric [6], plate [7] and primal-dual shakedown analyses []. The idea of the ES-FEM is also quite straightforward to extend for the D problems using tetrahedral elements to give a so-called the face-based smoothed finite element method (FS-FEM) [,]. In this paper, we aim to extend the ES-FEM to even more complicated visco-elastoplastic analyses in the solid

3 Comput Mech (9) 45: 44 5 Fig. Triangular elements and the smoothing domains (shaded areas) associated with edges in the ES-FEM boundary edge m (AB) Γ (m) (lines: AB, BI, IA) (m) A (triangle ABI ) B I C E F H O D inner edge k (DF) (k) Γ (lines: DH, HF, FO, OD) (k) (4-node domain DHFO) G : field node : centroid of triangles (I, O, H) mechanics. In this work, we combine the ES-FEM with the work of Carstensen and Klose [] using the standard FEM in the setting of von-mises conditions and the Prandtl Reuss flow rule. The material behavior includes perfect viscoelastoplasticity and visco-elastoplasticity with isotropic and linear kinematic hardening in a dual model with both displacements and the stresses as the main variables. The numerical procedure, however, eliminates the stress variables and the problem becomes only displacement-dependent and is much easy to deal with. The formulation shows that the bandwidth of stiffness matrix of the ES-FEM is larger than that of FEM, and hence the computational cost of the ES-FEM in numerical examples is larger than that of the FEM. However, when the efficiency of computation (computation time for the same accuracy) in terms of a posteriori error estimation is considered, the ES-FEM is more efficient than the FEM. Dual model of visco-elastoplastic problem using the ES-FEM. Strong form and weak form [] The strong form of the visco-elastoplastic problem which deforms in the interval t [, T ] can be described by equilibrium equation in the domain bounded by Ɣ divσ + b = in () where b (L ()) is the body forces, σ (L ()) is the stress field. The essential and static boundary conditions, respectively, on the Dirichlet boundary Ɣ D and the Neumann boundary Ɣ N are u = w on Ɣ D and σ n = t on Ɣ N () in which u (H ()) is the displacement field; w (H ()) is prescribed surface displacement; t (L (Ɣ N )) is prescribed surface force and n is the unit outward normal matrix. In the context of small strain, the total strain ε(u) = S u, where S u denotes the symmetric part of displacement gradient, is separated into two contributions ε(u) = e(σ ) + p(ξ) () where e(σ ) = C σ is elastic strain tensor; ξ is internal variable and p(ξ) is an irreversible plastic strain in which C is a fourth order tensor of material constants. To describe properly the evolution process for the plastic strain, it is required to define the admissible stresses, a yield function, and an associated flow rule. In this work, we use the von-mises yield function and the Prandtl Reuss flow rule. Let p and ξ be the kinematic variables of the generalized strain P = (p, ξ), and = (σ, α) be the corresponding generalized stress, whereα is the hardening parameter describing internal stresses. We define ϒ to be the admissible stresses set, which is a closed, convex set, containing, and defined by ϒ ={ : () } (4) where is the von-mises yield function which is presented specifically for different visco-elastoplasticity cases as follows: Case a: Perfect visco-elastoplasticity In this case, there is no hardening and the internal variables ξ, α are absent. The von-mises yield function is given simply by (σ ) = dev(σ ) σ Y (5) where σ Y is the yield stress; x is the norm of tensor x and i= computed by x = j= xij ; dev(x) is the deviator tensor of tensor x and defined by

4 6 Comput Mech (9) 45: 44 dev(x) = x tr(x) I (6) in which I is the symmetric unit tensor and tr(x) = i= x ii is the trace operator of tensor x. For the viscosity parameter v>, the Prandtl Reuss flow rule has the form ṗ = v ( dev (σ ) σ Y ) if dev (σ ) >σ Y (7) if dev (σ ) σ Y Case b: Visco-elastoplasticity with isotropic hardening In the case of the isotropic hardening, the problem is characterized by a modulus of hardening H, and α α I (I means Isotropic) becomes a scalar hardening parameter and relates to the scalar internal strain variable ξ by α I = H ξ (8) where H is a positive hardening parameter. The von-mises yield function is given by (σ,α I ) = dev(σ ) σ Y ( + Hα I ) (9) For the viscosity parameter v>, the Prandtl Reuss flow rule has the form In general, the Prandtl Reuss flow rule, with the viscosity parameter v>, has the form [] [ ] ṗ = [ ] σ σ (4) ξ v α α where σ and α are defined as the projections of (σ, α) into the admissible stresses set ϒ. The visco-elastoplastic problem can now be stated generally in a weak formulation with the above-mentioned flow rules as follows: Seek u (H ()) such that u = w on Ɣ D and v (H ()) ={v (H ()) : v = on Ɣ D }, the following equations are satisfied σ (u) : ε(v)d = b vd + t vdɣ (5) Ɣ N [ ] [ ] ṗ ε( u) C σ = = [ ] σ σ (6) ξ ξ( α) v α α where A : B = j,k A jkb jk denotes the scalar products of (symmetric) matrices.. Time discretization scheme A generalized midpoint rule is used as the time-discretization scheme []. In each time step, a spatial problem needs ( ) ( ṗ v + H = σ ) ( ) Y ξ ( ( dev (σ ) +α I H ) ) ( ( σ Y Hσ Y dev (σ ) +α I H ) ) σ Y if dev (σ ) > ( +α I H ) σ Y if dev (σ ) ( +α I H ) σ Y () Case c: Visco-elastoplasticity with linear kinematic hardening In the case of the linear kinematic hardening, the internal stress α α K (K means Kinematic) relates to the internal strain ξ by α K = k ξ () where k is a positive parameter. The von-mises yield function is given by (σ, α K ) = dev(σ ) dev(α K ) σ Y () For the viscosity parameter v>, the Prandtl Reuss flow rule has the form ( ) dev σ α K σy (ṗ ) ( v dev = (σ ) ( ) ) α K if dev σ α K >σy σy ξ ( ) ( ) if dev σ α K σy () to be solved with given variables (u(t), σ (t), α(t)) at time t denoted as (u, σ, α ) and unknowns at time t = t + t denoted as (u, σ, α ). Time derivatives are substituted by backward difference quotients; for instance u is substituted by u ϑ u ϑt where u ϑ = ( ϑ)u + ϑu with / ϑ. The time discrete problem now becomes: Seek u ϑ (H ()) that satisfies u ϑ = w on Ɣ D and for v (H ()) σ (u ϑ ) : ε(v)d = b ϑ vd + t ϑ vdɣ (7) Ɣ N [ ε(uϑ u ) C ] (σ ϑ σ ) = [ ] σ ϑ σ ϑ ϑt ξ(α, t ϑ ) ξ(α, t ) v α ϑ α ϑ (8) where b ϑ = ( ϑ)b + ϑb, t ϑ = ( ϑ) t + ϑ t in which b, t, b and t are body forces and surface forces at time t, t, respectively. Equations (7) and (8) is in fact a dual model that has both stress and displacement as field variables. To solve the set of Eqs. (7) and (8) efficiently, we need to eliminate

5 Comput Mech (9) 45: 44 7 one variable. This can be done by first expressing explicitly the stress σ ϑ in the form of displacement u ϑ using Eq. (8), and then substituting it into Eq. (7). The problem will then becomes only displacement-dependent, and we need to solve the resultant form of Eq. (7).. Analytic expression of the stress tensor Explicit expressions for the stress tensor σ ϑ in different cases of visco-elastoplasticity can be presented briefly as follows []: (a) Perfect visco-elastoplasticity In the elastic phase σ ϑ = C tr (ϑta) I + µdev (ϑta) (9) where A = ε(u ϑ u ) ϑt + C σ ϑt. In the plastic phase, the plastic occurs when dev(ϑta) >βσ Y and σ ϑ = C tr(ϑta)i + (C + C / dev(ϑta) )dev(ϑta). () where C = λ + µ, C = v/(βv + ϑt), C = ϑtσ Y /(βv + ϑt) () in which β = /(µ). (b) Visco-elastoplasticity with isotropic hardening In the elastic phase σ ϑ = C tr(ϑta)i + µdev(ϑta) () In the plastic phase, the plastic occurs when dev(ϑta) >β( + α I H)σ Y and σ ϑ = C tr(ϑta)i + (C /(C dev(ϑta) ) where +C 4 /C )dev(ϑta) () C = λ + µ, C = βv( + H σ Y ) + ϑt( + β H H σ Y ) C = ϑtσ Y ( + α I H), C 4 = H H ϑtσ Y + v( + H σ Y ) (4) in which α I is the initial scalar hardening parameter. (c) Visco-elastoplasticity with linear kinematic hardening In the elastic phase σ ϑ = C tr(ϑta)i + µdev(ϑta) (5) In the plastic phase, the plastic occurs when dev(ϑta βα K ) >βσ Y and σ ϑ = C tr(ϑta)i + (C + C / dev(ϑta βα K ) ) dev(ϑta βα K ) + dev(αk ) (6) where C = λ + µ, C = C = ϑtσ Y ϑt + βϑtk + v/µ ϑtk + v ϑt + βϑtk + v/µ, (7) Now, by replacing the stress σ ϑ described explicitly into Eq. (7), we obtain the only displacement-dependent problem and can apply different numerical methods to solve..4 Discretization in space using the FEM The domain is now discretized into N e elements and N n nodes such that = N e e= e and i j =, i = j. In the discrete version of (7), the spaces V = (H ()) and V = (H ()) are replaced by finite dimensional subspaces V h V and V h V. The discrete problem now becomes: Seek u ϑ V h such that u ϑ = w on Ɣ D and ( ) σ ϑ ε (u ϑ u ) + C σ : ε(v)d = b ϑ vd + Ɣ N t ϑ vdɣ for v V h (8) Let (ϕ,...,ϕ Nn ) be the nodal basis of the finite dimensional space V h, where ϕ i is the independent scalar hat shape function on node satisfying condition Kronecker ϕ i (i) = and ϕ i ( j) =, i = j, then Eq. (8) now becomes: seeking u ϑ V h such that F i = σ ϑ ( ε (u ϑ u ) + C σ ) : ε(ϕ i )d b ϑ ϕ i d Ɣ N t ϑ ϕ i dɣ = (9) for i =,...,N n. F i in Eq. (9) can be written in the sum of a part Q i which depends on u ϑ and a part P i which is independent of u ϑ such as F i (u ϑ ) = Q i (u ϑ ) P i ()

6 8 Comput Mech (9) 45: 44 with Q i (u ϑ ) = Q i = P i = b ϑ ϕ i d +.5 Iterative solution σ ϑ ( ε(u ϑ u ) + C σ ) : ε(ϕ i )d () Ɣ N t ϑ ϕ i dɣ () In order to solve Eq. (9) in this work, Newton Raphson method is used []. In each step of the Newton iterations, the discrete displacement vector u p ϑ expressed in the nodal basis by u p ϑ = N n i= ϕ iu i is determined from iterative solution DF(u p p+ ϑ )uϑ = DF(u p ϑ )u p ϑ F(u p ϑ ) () where DF is in fact the system stiffness matrix whose the local entries are defined as ( )) DF (u p ϑ,,...,u p ϑ,nn rs ( = F r u p ϑ,,...,u p ϑ,n n )/ u p ϑ,s (4) where r, s dof which is the set containing degrees of freedom of all the nodes. To properly apply the Dirichlet boundary conditions for our nonlinear problem, we use the approach of Lagrange multipliers. Combining the Newton iteration () and the set of boundary conditions imposed through Lagrange multipliers λ, the extended system of equations is obtained ( DF(u p ϑ ) GT G )( u p+ ϑ λ ) = ( f w ) (5) with f = DF(u p ϑ )u p ϑ F(u p ϑ ) and G is a matrix created from Dirichlet boundary conditions such that Gu p+ ϑ = w. The extended system of equations (5) can now be solved for u p+ ϑ and λ at each time step. The solving process is iterated until the relative residual F(u p+ ϑ,z,...,u p+ ϑ,z m ) for the m free nodes (z,...,z m ) (where is the set of free nodes) is smaller than a given tolerance or the maximum number of iterations is larger than a prescribed number..6 Discretization in space using the ES-FEM In the ES-FEM, the domain discretization is still based on the triangulation using triangular elements as in the standard FEM, but the integration required in the weak form (9) is performed based on the smoothing domains associated with the edges, and strain smoothing technique [4]is used.in such an integration process, the closed problem domain is divided into N SD = N ed smoothing domains associated with edges such that = N ed k= (k) and (i) ( j) =, i = j, in which N ed is the total number of edges located in the entire problem domain. For triangular elements, the smoothing domain (k) associated with the edge k is created by connecting two endpoints of the edge to centroids of adjacent elements as shown in Fig.. Using the edge-based smoothing domains, smoothed strains ε k can now be obtained using the compatible strains ε h = s u ϑ through the following smoothing operation over domain (k) associated with edge k ε k = ε h (x) k (x)d = s u ϑ (x) k (x)d (6) (k) (k) where k (x) is a given smoothing function that satisfies at least unity property (k) k (x)d = (7) In the ES-FEM [4], we use the simplest local constant smoothing function k (x) = { /A (k) x (k) x / (k) (8) where A (k) is the area of the smoothing domain (k) and is calculated by A (k) = k (N (k) e (k) d = N e (k) A ( e j) (9) j= where N e (k) is the number of elements around the edge = for the boundary edges and N e (k) = for inner edges as shown in Fig. ) and A ( e j) is the area of the jth element around the edge k. In the ES-FEM, the trial function is similar as in the standard FEM with N n u p ϑ = ϕ i u i (4) i= Substituting Eqs. (4) and (8) into(6), the smoothed strain on the domain (k) associated with edge k can be written in the following matrix form of nodal displacements ε k = I (k) dof B I (x k )u I (4) where (k) dof is the set containing degrees of freedom of elements attached to the common edge k (for example, for the boundary edge m as presented in Fig., (k) dof is the set

7 Comput Mech (9) 45: 44 9 containing degrees of freedom of nodes {A, B, C} and the total number of degrees of freedom N (k) dof = 6; for the inner edge k as presented in Fig., (k) dof is the set containing degrees of freedom of nodes {D, E, F, G} and the total number of degrees of freedom N (k) dof = 8) and B I (x k ), that is termed as the smoothed strain matrix on the domain (k),is calculated numerically by an assembly process similarly as in FEM B I (x k ) = N e (k) j) A (k) A( e B j (4) j= where B j is the strain gradient matrix of the jth element around the edge k which has the following form in the D linear elastic problems ϕ j,x B j = ϕ j,y (4) ϕ j,y ϕ j,x Due to the use of the triangular elements with the linear shape functions, the entries of matrix B j are constants, and so are the entries of matrix B I (x k ). Note that with this formulation, only the area and the usual compatible strain gradient matrices B j of triangular elements are needed to calculate the system stiffness matrix for ES-FEM. This formulation is quite straightforward to extend for the D problems using tetrahedral elements [,]. In the discrete version of the visco-elastoplastic problems using the ES-FEM with the smoothed strain (6) used for smoothing domains associated with edges, Eq. (9)now becomes: seeking u ϑ V h such that F i = σ ϑ ( ε (u ϑ u ) + C σ ) : ε(ϕ i )d b ϑ ϕ i d Ɣ N t ϑ ϕ i dɣ = (44) for i =,...,N n, and the local stiffness matrix DF (k) rs in Eq. (4) associated with smoothing domain (k) can be expressed as follows DF (k) rs = F(k) r u p ϑ,s = u p ϑ,s = Q(k) r u p ϑ,s (k) σ ϑ ε k u p ϑ,l ϕ l u l (k) dof + C σ : ε k (ϕ r )d (45) where r, s (k) dof, and Q (k) r = (k) σ ϑ ) ( ε k (u ϑ u ) + C σ : ε k (ϕ r )d (46) The expression σ ϑ ( ε k (u ϑ u ) + C σ ) in Eqs. (45) and (46) now is replaced by σ ϑ written explicitly in Eqs. (9), (), (), (), (5), (6) for different cases of visco-elastoplasticity with just replacing ε by ε k in corresponding positions which give the following results: (a) Visco-elastoplasticity: Q (k) r = A (k) (C tr(ṽ k )tr ( ε k (ϕ r )) + C 4 dev(ṽ k ) : ε k (ϕ r )) (47) DF rs (k) = A (k) (C tr ( ε k (ϕ r )) tr ( ε k (ϕ s )) + C 4 dev ( ε k (ϕ r )) : ε k (ϕ s ) (C 5 ) r dev(ṽ k ) : ε k (ϕ s )) (48) where ṽ k = ε k (u ϑ u ) + C σ and C + C / dev(ṽ k ) if dev(ṽ k ) βσ Y > C 4 = µ else C / dev(ṽ k ) [ ]N (k) dev ( ε k (ϕ r )) : dev(ṽ k ) dof r= if dev(ṽ k ) βσ Y > C 5 = [ ] T (49) else }{{} size of N (k) dof in which C, C, C are determined by (). (b) Visco-elastoplasticity with isotropic hardening Q r (k) = A (k) (C tr(ṽ k )tr ( ε k (ϕ r )) + C 5 dev(ṽ k ) : ε k (ϕ r )) (5) DF rs (k) = A (k) (C tr ( ε k (ϕ r )) tr ( ε k (ϕ s )) + C 5 dev ( ε k (ϕ r )) : ε k (ϕ s ) where (C 6 ) r dev(ṽ k ) : ε k (ϕ s )) (5)

8 Comput Mech (9) 45: 44 Fig. 4 Flow chart to solve the visco-elastoplastic problems using the ES-FEM: part { C / (C C 5 = dev(ṽ k ) ) + C 4 /C if dev(ṽ k ) β ( + α I H) σ Y > µ else C / (C dev(ṽ k ) ) [ dev ( εk (ϕ r )) : dev(ṽ k ) ] N (k) dof if dev(ṽ r= k ) β ( + α I H) σ Y > [ ] C 6 = T else (5) }{{} size of N (k) dof in which C, C, C, C 4 is determined by (4). (c) Visco-elastoplasticity with linear kinematic hardening Q (k) r ( ( ) = A (k) C tr(ṽ k )tr ε k (ϕ r ) ) + C 4 dev(ṽ k ) : ε k (ϕ r ) + c dev(α K ) : ε k(ϕ r ) (5) DF rs (k) = A (k) (C tr ( ε k (ϕ r )) tr ( ε k (ϕ s )) + C 4 dev ( ε k (ϕ r )) : ε k (ϕ s ) (C 5 ) r dev(ṽ k ) : ε k (ϕ s )) (54) where

9 Comput Mech (9) 45: 44 Fig. 5 Flow chart to solve the visco-elastoplastic problems using the ES-FEM: part { C / dev(ṽ k ) + C if dev ( ṽ k βα K ) βσ Y > C 4 = µ else C / dev(ṽ k ) [ dev ( ε k (ϕ r )) : dev(ṽ k ) ] N (k) dof if r= dev (ṽk βα K ) βσy > [ ] T C 5 = else }{{} c = size of N (k) dof { if dev (ṽk βα K ) βσy > else (55) in which C, C, C is determined by (7).

10 Comput Mech (9) 45: 44 Fig. 6 Patches used to calculate stresses of the nodes and the elements of two methods FEM and ES-FEM The patch used to calculate stresses at the node k (FEM) The patch used to calculate stresses at the node k (ES-FEM) k k A C A C The patch used to calculate stresses of the element ABC (FEM) B B The patch used to calculate stresses of the element ABC (ES-FEM) : field node : centroid of triangles y g(t) (,) y g(t).8.6 a (,) x Β(,).4. Α(,) x.8 Fig. 7 Plate with a circular hole subjected to time dependent surface forces g(t) and its quarter model with symmetric conditions imposed on the left and bottom edges.6.4. Applying the Dirichlet boundary conditions and solving the extended system of equations (5) by the ES-FEM are identical to those of the FEM. Recall one disadvantage of the ES-FEM is that the bandwidth of stiffness matrix is larger than that of the FEM, because the number of nodes related to the smoothing domains associated with inner edges is 4, which is larger than that related to the elements. The computational cost of the ES-FEM therefore is larger than that of FEM for the same meshes. We also note that the trial function u ϑ (x) for elements in the ES-FEM is the same as in the standard FEM and therefore the force vector P i in the ES-FEM is computed in the same way as in the FEM. In other words, the ES-FEM changes only the stiffness matrix. Figures 4 and 5 present the.5.5 Fig. 8 A domain discretization using 56 nodes and,4 triangular elements for the plate with a circular hole subjected to time dependent surface forces g(t) flow chart to solve the visco-elastoplastic problems using the ES-FEM. A-posteriori error estimation In order to estimate the accuracy of the ES-FEM compared to the FEM for the visco-elastoplastic problems, in this work

11 Comput Mech (9) 45: 44 Table Number of iterations and the estimated error using the FEM and ES-FEM at various time steps for the plate with hole Step FEM ES-FEM Iterations η h = Rσ h σ h L () σ h L () Iterations η h = Rσ h σ h L () σ h L () (a) 45 4 FEM ES FEM (b).5 FEM ES FEM 5. CPU time (seconds) 5 5 Error estimator η h Degrees of freedom CPU time (seconds) Fig. 9 Comparison of the computational cost and efficiency between the FEM and ES-FEM for a range of meshes at t =. for the plate with hole; a Computational cost; b Computational efficiency we will use the following efficient a-posteriori error estimation [,] Rσ h η h σ h L () = σ h L () ( Ne ( e= e Rσ h σ h) : ( Rσ h σ h) ) ½ d = ( Ne ) ½ (56) e= e σ h : σ h d where Rσ h is a globally continuous recovery stress field derived from the discrete (discontinuous) numerical element stress field σ h. The quantity η h can monitor the local spatial approximation error, and a larger value of η h implies a larger spatial error. For the ES-FEM, when computing the stresses σ h for an element, we can average those stresses of smoothing domains of the element, and the averaged stresses are regarded as the stresses of the element as shown in Fig. 6. Similarly, to calculate numerical stresses σ h (x j ) at a node x j,wesimply average the stresses of smoothing domains associated with the node as shown in Fig. 6. For the FEM, we can regard the stresses at the centroid as the element stresses σ h, while the stresses σ h (x j ) at a node x j are the averaged stresses of those of the elements surrounding the node as shown in Fig. 6. The recovery stress field Rσ h in Eq. (56) for each element in the ES-FEM and the FEM now can be derived from the numerical stresses σ h ( x j ) at the node x j by using the following approximation Rσ h = N j (x) σ h ( ) x j j= (57)

12 4 Comput Mech (9) 45: 44 where N j (x) are the linear shape functions of triangles used in the standard FEM, and σ h (x j ) are stress values at three nodes of the element. In order to evaluate the integrals in Eq. (56) for triangular elements, the mapping procedure using Gauss integration is performed on each element with a summation on all elements. In each element, a proper number of Gauss points depending on the order of the recovery solution Rσ h will be used. 4 Numerical examples In this section, some numerical examples will be presented to demonstrate the properties of the ES-FEM for viscoelastoplastic analyses. To emphasize the superior properties of the present method, the ES-FEM will be compared to the standard FEM by Carstensen and Klose []. 4. Plate with a circular hole: perfect visco-elastoplasticity Figure 7 represents a plate =[, ] [, ] with a central circular hole, radius a = m, subjected to time dependent surface forces g(t) = 5t to outer normal at the top and the bottom edges. The rest of the boundary is free. Due to its symmetry, only the upper right quadrant of the plate is modeled. Symmetric conditions are imposed on the left and bottom edges, and the inner boundary of the hole is traction free. Figure 8 gives a discretization of the domain using 56 nodes and,4 triangular elements. Assuming that the material is perfect viscoplasticity with Young s modulus E = 6, 9, Poisson s ratio v =.9, yield stress σ Y = 45 and the initial data for the stress vector σ is set zero. The solution is computed in the time interval from t = to t =. in uniform steps of time t =.. Using the mesh as shown in Fig. 8, the material remains elastic in six first steps, between t = and t =.8 for both the ES-FEM and FEM as shown in Table, because the number of iterations at these times is. Table also shows that the number of iterations in Newton s method of both ES-FEM and FEM are almost the same, but the estimated errors η h in Eq. (56) of the ES-FEM are more than two times less than those of FEM. In addition, Fig. 9 compares the computational cost and efficiency between the FEM and ES-FEM for a range of meshes at t =.. It is seen that with the same mesh, the computational cost of ES-FEM is larger than that of FEM as shown in Fig. 9a. However, when the efficiency of computation (computation time for the same accuracy) in terms of the error estimator versus computational cost for a range of meshes is considered, the ES-FEM is more efficient than the FEM as shown in Fig. 9b. Figure shows the elastic shear energy density dev(rσ h ) /(4µ) using the FEM and ES-FEM at t =.. The plasticity domain first appears at the corner containing point A(, ) and then at the corner containing point B(, ). The evolution process of the elastic shear energy density dev(rσ h ) /(4µ) can be clearly observed from the ES-FEM results at four different time instances (see Fig. ). Table compares the displacements of points A(, ) and B(, ) using FEM and ES-FEM at different time steps. The results show that the displacements of ES-FEM are larger than those of FEM. This implies that the ES-FEM model can reduce the over-stiffness of the standard FEM model using (a) FEM.5.5 (b) ES-FEM Fig. Elastic shear energy density dev(rσ h ) /(4µ) for plate with hole using the FEM (a) and ES-FEM (b)att =. (mesh with 56 nodes and,4 elements)

13 Comput Mech (9) 45: x (a).5.5 (b) (c).5.5 (d) Fig. Elastic shear energy density dev(rσ h ) /(4µ) obtained using the ES-FEM at some different time steps for the plate with hole; a t =.; b t =.8; c t =.4; d t =. Table Horizontal displacement u A at point A(, ) and vertical displacement v B at point B(, ) using FEM and ES-FEM at various time steps for the plate with hole Step FEM ES-FEM u A ( ) v B ( ) u A ( ) v B ( )

14 6 Comput Mech (9) 45: 44 Elastic strain energy FEM.8 ES FEM Reference ES FEM (87 dofs) Degrees of freedom Fig. Convergence of the elastic strain energy E = σ ϑ : e ϑ d versus the number of degrees of freedom using the FEM and ES-FEM at t =. for the plate with hole triangular elements. This property was found earlier for linear elastic problems by Liu et al. [4]. Figure shows the convergence of the elastic strain energy E = σ ϑ : e ϑ d versus the number of degrees of freedom using the FEM and ES-FEM at t =.. The solution of the ES-FEM using a very fine mesh including 8,7 degrees of freedom is used as reference solution. The results show clearly that the ES-FEM model is much softer and gives more accurate results than the FEM model using triangular elements, especially for the coarse meshes. 4. Cook s membrane problem: visco-elastoplasticity with isotropic hardening This benchmark problem, shown in Fig., refers to a clamped tapered panel subjected to an in-plane shearing load g(t) = e t /4 on the opposite end (x = 48) with vanishing volume force f. Figure also gives a discretization of Fig. Cook s membrane problem and its dicretization using 89 nodes and 5 triangular elements g(t) 6 5 Thickness = 4 4 Table Number of iterations and the estimated error using the FEM and ES-FEM at various time steps for Cook s membrane problem Step FEM ES-FEM Iterations η h = Rσ h σ h L () σ h L () Iterations η h = Rσ h σ h L () σ h L ()

15 Comput Mech (9) 45: 44 7 (a) 8 6 FEM ES FEM (b).4.5 FEM ES FEM CPU time (seconds) Error estimator η h Degrees of freedom CPU time (seconds) Fig. 4 Comparison of the computational cost and efficiency between the FEM and ES-FEM for a range of meshes at t =.5 for the Cook s membrane problem; a Computational cost; b Computational efficiency 6 x x (a) FEM 4 5 (b) ES-FEM Fig. 5 Elastic shear energy density dev(rσ h ) /(4µ) for Cook s membrane problem using the FEM (a) and ES-FEM (b) att =.5 (mesh with 89 nodes, 5 elements) the domain using 89 nodes and 5 triangular elements. Assuming that the material is isotropic hardening viscoplasticity with Young s modulus E =,9, Poisson s ratio v =.4, yield stress σ Y =., H =,, H = ; and the initial data for the stress vector σ and the scalar hardening parameter α I are set zero. The solution is calculated in the time interval from t = to t =.5 in uniform steps of time t =.5. Using the mesh as shown in Fig., the material remains elastic in two first steps, between t = and t =.5 for both ES-FEM and FEM as shown in Table. Table also shows that the number of iterations in Newton s method of both ES-FEM and FEM are almost the same, but the estimated errors η h in Eq. (56) of ES-FEM are about three times less than those of FEM. In addition, Fig. 4 compares the computational cost and efficiency between the FEM and ES-FEM for a range of meshes at t =.5. It is seen that with the same mesh, the computational cost of ES-FEM is larger than that of FEM as shown in Fig.4a. However, when the efficiency of computation (computation time for the same accuracy) in terms of

16 8 Comput Mech (9) 45: 44 x 7 x (a) 4 5 (b) x 6 x (c) 4 5 (d) Fig. 6 Elastic shear energy density dev(rσ h ) /(4µ) using the ES-FEM at some different time steps for Cook s membrane problem; a t =.5; b t =.; c t =.75; d t =.5 the error estimator versus computational cost for a range of meshes is considered, the ES-FEM is more efficient than the FEM as shown in Fig. 4b. Figure 5 shows the elastic shear energy density dev (Rσ h ) /(4µ) using FEM and ES-FEM at t =.5 for the mesh as shown in Fig.. The evolution process of the elastic shear energy density dev(rσ h ) /(4µ) can be clearly observed from the ES-FEM results at four different time instances (see Fig. 6). Table 4 also compares the displacement of point A(48, 6) using the FEM and ES-FEM at different time steps. The results show that the displacements of ES-FEM are larger than those of FEM. This again verifies that the ES-FEM model can reduce the over-stiffness of the standard FEM model using triangular elements. Figure 7 shows the convergence of the elastic strain energy E = σ ϑ : e ϑ d versus the number of degrees of freedom using the FEM and ES-FEM at t =.5. The solution of ES-FEM using a very fine mesh including 6,4 degrees of freedom is used as reference solution. The results again show clearly that the ES-FEM model is much softer and gives more accurate results than the FEM model using triangular elements, especially for the coarse meshes.

17 Comput Mech (9) 45: 44 9 Table 4 Displacement at point A(48, 6) using the FEM and ES-FEM at various time steps for Cook s membrane problem Step FEM ES-FEM u A v A u A v A x Axis-symmetric ring: visco-elastoplasticity with linear kinematic hardening Elastic strain energy FEM. ES FEM Reference ES FEM (64 dofs) Degrees of freedom Fig. 7 Convergence of the elastic strain energy E = σ ϑ : e ϑ d versus the number of degrees of freedom using the FEM and ES-FEM at t =.5 for Cook s membrane problem Figure 8 shows a thick axis-symmetric ring, with internal radius a =, external radius b =, subjected to time depending surface forces g (r,ϕ,t) = te r on inner radius and g (r,ϕ,t) = te r /4 on outer radius with e r = (cos ϕ, sin ϕ). There is no body force. For this problem, the exact solution [] is u (r,ϕ,t) = u r (r, t) e r σ (r,ϕ,t) = σ r (r, t) e r e r + σ ϕ e ϕ e ϕ (58) p (r,ϕ,t) = P r (r, t) ( ) e r e r e ϕ e ϕ where e r = (cos ϕ,sin ϕ),e ϕ = ( sin ϕ,cos ϕ) and t µr ( ψ I (R (t)) r + 4a ) for r R (t) µr t u r (r, t) = µr ψ I (R (t)) ( 4r + 4a ) + ψ I (r) r for r < R (t) µr (59) Fig. 8 The axis-symmetric ring subjected to time dependent surface forces g (t) on inner radius and g (t) on outer radius and its quarter model with symmetric conditions imposed on the left and bottom edges b g (t) y g (t) a r o g (t) ϕ x g (t) o x

18 4 Comput Mech (9) 45: ( aψ I (R (t)) t r 8 σ r (r, t) = t r 8 aψ I (R (t)) +aψ I (r) 4 ) ( r ) r for r R (t) for r < R (t) (6) Fig. 9 Domain discretization of one quarter of the axis-symmetric ring using 56 nodes and,4 triangular elements σ ϕ (r, t) = (rσ r ) r forr R (t) ) P r (r, t) = σ y ( R (aψ + k ) r for r < R (t) ( σ y I (r) = ln r + ( R )) (aψ + k ) r R (6) (6) (6) (a) 9 8 FEM ES FEM (b).5 FEM ES FEM 7. CPU time (seconds) Error estimator η h Degrees of freedom CPU time (seconds) Fig. Comparison of the computational cost and efficiency between the FEM and ES-FEM for a range of meshes at t =. for the axis-symmetric ring problem; a Computational cost; b Computational efficiency Table 5 Number of iterations and the estimated error using the FEM and ES-FEM at various time steps for the axis-symmetric ring problem Step FEM-T ES-FEM Iters e = σ σ h L σ h L η h = Rσ h σ h L σ h L e η h Iters e = σ σ h L σ h L η h = Rσ h σ h L σ h L e η h

19 Comput Mech (9) 45: 44 4 x 7 x (a) FEM.5.5 (b) ES-FEM Fig. Elastic shear energy density dev(rσ h ) /(4µ) for the axis-symmetric ring problem using the FEM (a) and ES-FEM (b) att =. (mesh with 56 nodes and,4 triangular elements) x 8 x (a).5.5 (b) x 7 x (c).5.5 (d) Fig. Evolution of the elastic shear energy density dev(rσ h ) /(4µ) using the ES-FEM at some different time steps for the axis-symmetric ring problem; a t =.4; b t =.; c t =.6; d t =.

20 4 Comput Mech (9) 45: 44 Table 6 Horizontal displacement at point A(, ) and vertical displacement at point B(, ) using FEM and ES-FEM at various time steps for the axis-symmetric ring problem Time FEM ES-FEM Iterations u A v B u A /v B Iterations u A v B u A /v B x 7 x (a) FEM.5.5 (b) ES-FEM Fig. Elastic shear energy density dev(rσ h ) /(4µ) for the axis-symmetric ring problem using the FEM (a) and ES-FEM (b) att =. (mesh with 45 nodes and 64 triangular elements) and a = µ + λ, ψ = µ/(µ + λ). With α = 4aψ/ ((aψ + k )), the radius of the plastic boundary R(t) is the positive root of f (R) = αln R + (α )R α + t (64) σ y Because of the axis-symmetric characteristic of the problem, we only calculate for one quarter of ring as shown in Fig. 8, and symmetric conditions are imposed on the left and bottom edges. Figure 9 gives a discretization of the domain using 56 nodes and,4 triangular elements. The material parameters are given with Young s modulus E = 7,, Poisson s ratio v =., yield stress σ Y =., the hardening parameter is k =, and the initial data for the displacement u, the stress vector σ and the hardening parameter α K are set zero. The solution is calculated in the time interval from t = to t =. in uniform steps of time t =.. Using the mesh as shown in Fig. 9, the material remains elastic in seven first steps, between t = and t =.4 for both ES-FEM and FEM as shown in Table 5. The material becomes plastic at t =.6. Table 5 also shows the exact error e and estimated error η h for the different time steps together with the number of iterations in Newton s method. The results show that the ratio e/η h almost equals for all iterations which implies that the estimated error η h can be regarded as an accurate error estimator. This example therefore asserts the reliability of the estimated error η h used in this work.

21 Comput Mech (9) 45: 44 4 Elastic strain energy 4.88 x FEM 4.76 ES FEM Reference ES FEM (4 dofs) Degrees of freedom Fig. 4 Convergence of the elastic strain energy E = σ ϑ : e ϑ d versus the number of degrees of freedom using the FEM and ES-FEM at t =. for the axis-symmetric ring problem In addition, the results also shows that the number of iterations in Newton s method of both ES-FEM and FEM are almost the same, but the estimated errors η h in Eq. (56)ofthe ES-FEM are about three times less than those of FEM. Further more, Fig. compares the computational cost and efficiency between the FEM and ES-FEM for a range of meshes at t =.. It is seen that with the same mesh, the computational cost of ES-FEM is larger than that of FEM as shown in Fig. a. However, when the efficiency of computation (computation time for the same accuracy) in terms of the error estimator versus computational cost for a range of meshes is considered, the ES-FEM is more efficient than the FEM as shown in Fig. b. Figure shows the elastic shear energy density dev(rσ h ) /(4µ) using FEM and ES-FEM at t =.. The plastic domain first appears at the inner radius and extents toward the outer radius. The evolution process of the elastic shear energy density dev(rσ h ) /(4µ) can be clearly observed from the ES-FEM results at four different time instances (see Fig. ). Table 6 shows the ratio of radial displacements between points A(, ) and B(, ) using the FEM and ES-FEM at various time steps. It is seen that for the symmetric problem, the results of ES-FEM is more symmetric than those of FEM. This finding is also supported by the solution of the elastic shear energy density dev (Rσ h ) /(4µ) shown as contour in Fig. obtained using a coarse mesh with 45 nodes and 64 triangular elements. Figure 4 shows the convergence of the elastic strain energy E = σ ϑ : e ϑ d versus the number of degrees of freedom using the FEM and ES-FEM at t =.. The results again verify that the ES-FEM model is much softer and gives more accurate results than the FEM model using triangular elements, especially for the coarse meshes. 5 Conclusion In this paper, the ES-FEM is extended to more complicated visco-elastoplastic analyses. We combine the ES-FEM using triangular elements with the work of Carstensen and Klose [] in the setting of von-mises conditions and the Prandtl Reuss flow rule, and the material behavior includes perfect visco-elastoplasticity, and visco-elastoplasticity with isotropic and linear kinematic hardening in a dual model, with displacements and the stresses as the main variables. The numerical procedure, however, eliminates the stress variables and the problem becomes only displacement-dependent and is much easy to deal with. The numerical results of ES-FEM using triangular elements show that The displacement results of ES-FEM are larger than those of FEM. The elastic strain energy of ES-FEM is much more accurate than that of FEM. These results show clearly that the ES-FEM model can reduce the overstiffness of the standard FEM model using triangular elements and gives much more accurate results than those of FEM. These properties were demonstrated for the static, free and forced vibration analyses of elastic solids by Liu et al. [4]. The a-posteriori estimated error η h used in this work is shown to be reliable to represent the accurate error estimator. The bandwidth of stiffness matrix of ES-FEM is larger than that of FEM, and hence the computational cost of ES-FEM is larger than that of FEM. However, when the efficiency of computation (computation time for the same accuracy) in terms of the a-posteriori error estimation is considered, the ES-FEM is more efficient than the FEM. For the coarse meshes, the results of ES-FEM are much more accurate than those of FEM. For the axis-symmetric problems, the results of ES-FEM are more symmetric than those of FEM. Acknowledgments This work is partially supported by A*Star, Singapore. It is also partially supported by the Open Research Fund Program of the State Key Laboratory of Advanced Technology of Design and Manufacturing for Vehicle Body, Hunan University, P.R.China under the grant number 495. References. Bordas S, Rabczuk T, Nguyen-Xuan H, Nguyen Vinh P, Natarajan S, Bog T, Do Minh Q, Nguyen Vinh H (9) Strain smoothing in FEM and XFEM. Comput Struct. doi:.6/j.compstruc Carstensen C, Klose R () Elastoviscoplastic finite element analysis in lines of matlab. J Numer Math :57 9

22 44 Comput Mech (9) 45: 44. Carstensen C, Funken SA () Averaging technique for FE-a posteriori error control in elasticity. Part : conforming FEM. Comput Methods Appl Mech Eng 9: Chen JS, Wu CT, Yoon S, You Y () A stabilized conforming nodal integration for Galerkin meshfree method. Int J Numer Methods Eng 5: Cui XY, Liu GR, Li GY, Zhao X, Nguyen-Thoi T, Sun GY (8) A smoothed finite element method (SFEM) for linear and geometrically nonlinear analysis of plates and shells. CMES- Comput Model Eng Sci 8(): Dai KY, Liu GR (7) Free and forced analysis using the smoothed finite element method (SFEM). J Sound Vib : Dai KY, Liu GR, Nguyen-Thoi T (7) An n-sided polygonal smoothed finite element method (nsfem) for solid mechanics. Finite Elements Anal Des 4: Dohrmann CR, Heinstein MW, Jung J, Key SW, Witkowski WR () Node-based uniform strain elements for three-node triangular and four-node tetrahedral meshes. Int J Numer Methods Eng 47: Liu GR (8) A generalized gradient smoothing technique and the smoothed bilinear form for Galerkin formulation of a wide class of computational methods. Int J Comput Methods 5():99 6. Liu GR, Dai KY, Nguyen-Thoi T (7) A smoothed finite element method for mechanics problems. Comput Mech 9: Liu GR, Li Y, Dai KY, Luan MT, Xue W (6) A Linearly conforming radial point interpolation method for solid mechanics problems. Int J Comput Methods (4):4 48. Liu GR, Nguyen-Thoi T, Dai KY, Lam KY (7) Theoretical aspects of the smoothed finite element method (SFEM). Int J Numer Methods Eng 7:9 9. Liu GR, Nguyen-Thoi T, Lam KY (8) A novel alpha finite element method (αfem) for exact solution to mechanics problems using triangular and tetrahedral elements. Comput Methods Appl Mech Eng 97: Liu GR, Nguyen-Thoi T, Lam KY (9) An edge-based smoothed finite element method (ES-FEM) for static, free and forced vibration analyses of solids. J Sound Vib : 5. Liu GR, Nguyen-Thoi T, Nguyen-Xuan H, Lam KY (9) A node-based smoothed finite element method for upper bound solution to solid problems (NS-FEM). Comput Struct 87: Liu GR, Zhang GY (8) Upper bound solution to elasticity problems: a unique property of the linearly conforming point interpolation method (LC-PIM). Int J Numer Methods Eng 74: Liu GR, Zhang GY, Dai KY, Wang YY, Zhong ZH, Li GY, Han X (5) A linearly conforming point interpolation method (LC-PIM) for D solid mechanics problems. Int J Comput Methods (4): Liu GR, Zhang GY, Wang YY, Huang HT, Zhong ZH, Li GY, Han X (7) A linearly conforming point interpolation method (LC-PIM) for three-dimensional elasticity problems. Int J Numer Methods Eng 7: Nguyen-Thanh N, Rabczuk T, Nguyen-Xuan H, Bordas S (8) A smoothed finite element method for shell analysis. Comput Methods Appl Mech Eng 98: Nguyen-Thoi T, Liu GR, Dai KY, Lam KY (7) Selective smoothed finite element method. Tsinghua Sci Technol (5): Nguyen-Thoi T, Liu GR, Lam KY, Zhang GY (9) A face-based smoothed finite element method (FS-FEM) for D linear and nonlinear solid mechanics problems using 4-node tetrahedral elements. Int J Numer Methods Eng 78:4 5. Nguyen-Thoi T, Liu GR, Nguyen-Xuan H, Nguyen-Tran C (9) Adaptive analysis using the node-based smoothed finite element method (NS-FEM). Commun Numer Methods Eng. doi:./ cnm.9. Nguyen-Thoi T, Liu GR, Vu-Do HC, Nguyen-Xuan H (9) A face-based smoothed finite element method (FS-FEM) for D visco-elastoplastic analyses of solid using tetrahedral mesh. Comput Methods Appl Mech Eng. doi:.6/j.cma Nguyen-Van H, Mai-Duy N, Tran-Cong T (8) A smoothed four-node piezoelectric element for analysis of two-dimensional smart structures. CMES-Comput Model Eng Sci ():9 5. Nguyen-Xuan H, Bordas S, Nguyen-Dang H (8) Addressing volumetric locking and instabilities by selective integration in smoothed finite elements. Commun Numer Methods Eng 5: Nguyen-Xuan H, GR Liu, Nguyen-Thoi T, Nguyen-Tran C (9) An edge based smoothed finite element method (ES-FEM) for analysis of two dimensional piezoelectric structures. Smart Mater Struct 8:655 (pp) 7. Nguyen-Xuan H, Liu GR, Thai-Hoang C, Nguyen-Thoi T (9) An edge-based smoothed finite element method with stabilized discrete shear gap technique for analysis of Reissner Mindlin plates. Comput Methods Appl Mech Eng (revised) 8. Nguyen-Xuan H, Rabczuk T, Bordas S, Debongnie JF (8) A smoothed finite element method for plate analysis. Comput Methods Appl Mech Eng 97:84 9. Nguyen-Xuan H, Nguyen-Thoi T (9) A stabilized smoothed finite element method for free vibration analysis of Mindlin Reissner plates. Commun Numer Methods Eng 5: Tran Thanh Ngoc, Liu GR, Nguyen-Xuan H, Nguyen-Thoi T (9) An edge-based smoothed finite element method for primal-dual shakedown analysis of structures. Int J Numer Methods Eng (revised). Yoo JW, Moran B, Chen JS (4) Stabilized conforming nodal integration in the natural-element method. Int J Numer Methods Eng 6:86 89