DSpace at VNU: A novel singular node-based smoothed finite element method (NS-FEM) for upper bound solutions of fracture problems

DSpace at VNU: A novel singular node-based smoothed finite element method (NS-FEM) for upper bound solutions of fracture...

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int J Numer Meth Engng 2010; 83:1466–1497 Published online 15 March 2010 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/nme.2868 A novel singular node-based smoothed finite element method (NS-FEM) for upper bound solutions of fracture problems G R Liu1,2 , L Chen1, ∗, † , T Nguyen-Thoi1,3 , K Y Zeng4 and G Y Zhang2 for Advanced Computations in Engineering Science (ACES), Department of Mechanical Engineering, National University of Singapore, Engineering Drive 1, Singapore 117576, Singapore Singapore-MIT Alliance (SMA), E4-04-10, Engineering Drive 3, Singapore 117576, Singapore Faculty of Mathematics and Computer Science, University of Science, Vietnam National University–HCM, Hanoi, Vietnam Department of Mechanical Engineering, National University of Singapore, Singapore 117576, Singapore Center SUMMARY It is well known that the lower bound to exact solutions in linear fracture problems can be easily obtained by the displacement compatible finite element method (FEM) together with the singular crack tip elements It is, however, much more difficult to obtain the upper bound solutions for these problems This paper aims to formulate a novel singular node-based smoothed finite element method (NS-FEM) to obtain the upper bound solutions for fracture problems In the present singular NS-FEM, the calculation of the system stiffness matrix is performed using the strain smoothing technique over the smoothing domains (SDs) associated with nodes, which leads to the line integrations using only the shape function values along the boundaries of the SDs A five-node singular crack tip element is used within the framework of NS-FEM to construct singular shape functions via direct point interpolation with proper order of fractional basis The mix-mode stress intensity factors are evaluated using the domain forms of the interaction integrals The upper bound solutions of the present singular NS-FEM are demonstrated via benchmark examples for a wide range of material combinations and boundary conditions Copyright q 2010 John Wiley & Sons, Ltd Received August 2009; Revised January 2010; Accepted 15 January 2010 KEY WORDS: numerical methods; meshfree method; upper bound; crack; stress intensity factor; J -integral; energy release rate; NS-FEM; singularity INTRODUCTION In fracture analyses, it is important to evaluate fracture parameters, such as the stress intensity factors (SIFs) or the energy release rate (J -integral), which is the measure of the intensity of the ∗ Correspondence to: L Chen, Center for Advanced Computations in Engineering Science (ACES), Department of Mechanical Engineering, National University of Singapore, Engineering Drive 1, Singapore 117576, Singapore † E-mail: g0700888@nus.edu.sg Copyright q 2010 John Wiley & Sons, Ltd NS-FEM FOR UPPER BOUND SOLUTIONS OF FRACTURE PROBLEMS 1467 crack tip fields [1] In addition, the upper and lower bound analyses [2] or the so-called dual analyses [3] for evaluation of fracture parameters have been an important means for safety and reliability assessments of structural properties In practice, to implement these analyses, two numerical models are usually used: one gives a lower bound and the other gives an upper bound of the unknown exact solution The most popular models giving a lower bound solution are the displacement compatible finite element method (FEM) models, which are widely used in solving complicated engineering problems The models that give an upper bound can be one of the following models: (1) the stress equilibrium FEM model [4]; (2) the recovery models using a statically admissible stress field from displacement FEM solution [5, 6]; and (3) the hybrid equilibrium FEM models [7] These three models, however, are known to have the following two common disadvantages: (1) the formulation and numerical implementation are computationally complicated and expensive and (2) there exist spurious modes in the hybrid models or the spurious modes often occur due to the simple fact that tractions cannot be equilibrated by the stress approximation field Owing to these drawbacks, these three models are not yet widely used in practical applications, and are still very much confined in the area of academic research In the linear√fracture mechanics, the stresses and strains near the crack tip are singular: ij ∼ √ 1/ r , εij ∼ 1/ r , (where r is the radial distance from the crack tip) [1] To capture the singularity in the vicinity of the crack tip, the numerical simulation of cracks can be carried out with several different numerical approaches, such as FEM [8–10] and meshless methods [11–15] When the displacement compatible FEM is used, the eight-node quarter-point element or √ the six-node quarterpoint element (collapsed quadrilateral) is often adopted to model the inverse r stress singularity [8–10] However, to ensure that the singular elements are compatible with other standard elements, quadratic elements are required for the whole domain Otherwise, transition elements [16, 17] are needed to bridge between the crack tip elements and the standard elements Recently, Belytschko and Moes developed so-called extended finite element method (XFEM) to model arbitrary discontinuities in meshes [18, 19] This method allows the crack to be arbitrarily aligned within the mesh, and thus crack propagation simulations can be carried out without remeshing [20] Moreover, this extension exploits the partition of unity property of finite elements, which allows local enrichment functions to be easily incorporated into a finite element approximation while still preserving the classical displacement variational setting [21] However, the enrichment is only partial in the elements at the edge of the enriched subdomain, and consequently, the partitions-of-unity in the original XFEM is lost in the ‘transition’ zones, and hence ‘blending’ elements need to be used in these transition zones [22] More recently, Liu et al has generalized the gradient (strain) smoothing technique [23–25] and applied it in the FEM context to formulate a cell-based smoothed finite element method (SFEM or CS-FEM) [26–28] In the CS-FEM, cell-based strain smoothing technique is incorporated to the standard FEM formulation to reduce the over stiffness of the compatible FEM model The CS-FEM has been developed for general n-sided polygonal elements (nCS-FEM) [29] dynamic analyses [30], incompressible materials using selective integration [31, 32], plate and shell analyses [33–36], and further extended for the XFEM to solve fracture mechanics problems in 2D continuum and plates [37] To further reduce the stiffness, a node-based smoothed finite element (NS-FEM) [38, 39] has been formulated using the smoothing domains (SDs) associated with nodes Liu et al [38–42] has shown that the NS-FEM has the very important property of producing upper-bound solutions, which offers a very practical means to bound the solutions from both above and below for complicated engineering problems, as long as a displacement FEM model can be built Such bounds are obtained Copyright q 2010 John Wiley & Sons, Ltd Int J Numer Meth Engng 2010; 83:1466–1497 DOI: 10.1002/nme 1468 G R LIU ET AL using only one set of mesh and without knowing the exact solution of the problem The development in this new direction originated from the recently discovered NS-PIM [43, 44] using the simple point interpolation method (PIM) In the NS-FEM, the computation of the system stiffness matrix is performed using the strain smoothing technique over the SDs associated with nodes, which leads to the line integrations using the shape function values directly along the boundaries of the SDs Exploiting this special property of line integration, we now can further develop the NS-FEM for fracture analyses by computing √ the system stiffness matrix directly from the special basis shape functions which create the r displacement field, and thus obtain a proper singular stress field in the vicinity of the crack tip In this paper, a singular NS-FEM is formulated to obtain the upper bound solutions for the mix-mode cracks Four schemes of SDs around the crack tip have been proposed based on the triangular elements to model the singularity In addition, a five-node singular element is used within the framework of NS-FEM to construct singular shape functions for the SDs connected to the crack tip In our singular NS-FEM, the displacement field is at least (complete) linearly consistent, and the enrichment near the crack tip is on top of the complete linear field Therefore, the partitionof-unity property as well as the linear consistency property are both ensured throughout the entire problem domain, ensuring the stability and convergence of the solution Using the singular NS-FEM together with the singular FEM, we can now have a systematical way to numerically obtain both upper and lower bounds of fracture parameters to crack problems Intensive benchmark numerical examples for a wide range of material combinations and boundary conditions will be presented to demonstrate the interesting properties of the proposed method BASIC EQUATIONS Consider a 2D static elasticity problem governed by the equilibrium equation in the domain bounded by ( = u + t ; u ∩ t = 0) as: LTd r+b = in (1) where Ld is a matrix of differential operator defined as: ⎡ ⎤ * ⎢ *x ⎥ ⎢ ⎥ ⎢ *⎥ ⎢ ⎥ Ld = ⎢ *y ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ * *⎦ *y rT = { ∈ R2 (2) *x bT = {b xx yy xy } is the vector of stresses, x b y } is the vector of body force applied in the problem domain The stresses relate the strains via the generalized Hook’s law: r = De where D is the matrix of material constants and eT = {εxx e = Ld u where u = {u x u y Copyright q }T (3) εyy xy } is the vector of strains given by: (4) is the vector of the displacement 2010 John Wiley & Sons, Ltd Int J Numer Meth Engng 2010; 83:1466–1497 DOI: 10.1002/nme NS-FEM FOR UPPER BOUND SOLUTIONS OF FRACTURE PROBLEMS 1469 The essential boundary condition is given by: u=u on (5) u where u is the vector of the prescribed displacements In this paper, for simplicity of discussion, we only consider the force-driving problems with the homogeneous essential boundary condition: u=0 on (6) u The natural boundary condition is given by: LTn r = t on (7) t where t is the vector of prescribed tractions on t , and LTn is the matrix of unit outward normal which can be expressed as: ⎤ ⎡ nx ⎥ ⎢ Ln = ⎣ n y ⎦ (8) ny nx BRIEF ON THE SINGULAR FEM 3.1 Basic formulation The domain is first discretized into Ne of non-overlapping and non-gap elements and Nn nodes, Ne e e e such that = m=1 m and i ∩ j = 0, ∀i = j Then, the approximation of displacement field for a 2D static elasticity problem is given by: uh (x) = i∈n en Ni (x)di (9) where n en is the set of nodes of the element containing x, di = [dxi dyi ]T is the vector of nodal displacements, respectively, in x-axis and y-axis, and Ni is a matrix of shape functions Ni (x) = Ni (x) 0 Ni (x) (10) in which Ni (x) is the shape function for node i Using Equations (4) and (9), the compatible strain of FEM approximation is given by: eh (x) = Ld uh (x) (11) The standard Galerkin weak form for the FEM now can be described as: Find uh ∈ (H10 ( ))2 such that ( eh (uh ))T Deh (uh ) d − ( u h )T b d − ( uh )T t d = 0, ∀ uh ∈ (H10 ( ))2 (12) t where (H10 ( ))2 denotes the Sobolev space of functions with square integrable derivatives in and with vanishing values on u Copyright q 2010 John Wiley & Sons, Ltd Int J Numer Meth Engng 2010; 83:1466–1497 DOI: 10.1002/nme 1470 G R LIU ET AL By substituting the approximations uh in Equation (9) into Equations (11) and (12), and invoking the arbitrariness of virtual nodal displacements, we obtain the standard discretized algebraic system of equations: Kd = f (13) Here, K is the system stiffness matrix of FEM that is assembled using: Kij = Ne m=1 Kiej,m = Ne e m m=1 BiT DB j d where Bi (x) is the compatible strain gradient matrix at node i and computed by ⎡ ⎤ *Ni (x) ⎢ *x ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ *N (x) i ⎢ ⎥ Bi (x) = Ld Ni (x) = ⎢ *y ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ *Ni (x) *Ni (x) ⎥ ⎣ ⎦ *y *x (14) (15) In Equation (12), f is the vector of nodal forces at the unconstrained nodes and is assembled using: f= NT (x)b d + NT (x)t d (16) t 3.2 Singular element A fundamental issue in modeling fracture mechanics problems is to simulate the singularity of stress field near the crack tip In order to capture the singularity expressed as r −1/2 accurately without using so many elements around the crack tip, the singular element is generally incorporated into the standard FEM The theory and application of the different kinds of singular elements are well documented in [8–10] The most popular singular element is the eight-node quarter-point element or the six-node quarter-point element (collapsed quadrilateral) These quarter-point quadratic elements shift the corresponding mid-nodes to the quarter-point position as shown in Figure However, to ensure that the singular elements are compatible with other standard elements, the quadratic elements are required even for domains far away from the crack tip, which significantly increases the computational cost Otherwise, transition elements [16, 17] are needed to bridge between the crack tip elements and the standard elements THE IDEA OF SINGULAR NS-FEM Detailed formulations of the NS-FEM have been proposed in the previous work [38] Here, we mainly focus on the construction of a singular field near the crack tip using a basic mesh for three-node linear triangular elements Copyright q 2010 John Wiley & Sons, Ltd Int J Numer Meth Engng 2010; 83:1466–1497 DOI: 10.1002/nme NS-FEM FOR UPPER BOUND SOLUTIONS OF FRACTURE PROBLEMS 1471 6 crack tip l/4 l/4 crack tip l l Figure The schematic of the eight-node and six-node quarter-point singular elements k s Γk s Ωk crack tip s Ω tip Field node Centroid of triangle Mid-edge-point Figure Construction of node-based strain smoothing domains 4.1 Brief on the NS-FEM In the NS-FEM, the domain is discretized using elements, as in the FEM However, we not use the compatible strains but the strains ‘smoothed’ over a set of non-overlap no-gap SDs Ns Nn s s s associated with nodes, such that = k=1 k and i ∩ j = 0, ∀i = j, in which Nn is the total number of nodes in the element mesh In this case, the number of SDs are the same as the number of nodes: Ns = Nn The strain smoothing technique [25] is used to generate a modified strain field using the node-based SDs and the assumed displacement field constructed using the element mesh For the triangular elements, the SD sk for node k is created by connecting sequentially the mid-edge-points and the centroids of the surrounding triangles of the node as shown in Figure Copyright q 2010 John Wiley & Sons, Ltd Int J Numer Meth Engng 2010; 83:1466–1497 DOI: 10.1002/nme 1472 G R LIU ET AL Using the node-based SDs, smoothed strains can be obtained using the compatible strains through the following smoothing operation over domain sk associated with the node k: e¯ k = Ask s k Ld uh (x) d = Ask s k Ln uh (x) d (17) where Ask = s d is the area of the SD sk and sk is the boundary of the SD sk k Substituting Equation (9) into Equation (17), the smoothed strain can be written in the following matrix form of nodal displacements e¯ k = i∈n sk ¯ i (xk )d¯ i B (18) ¯ i (xk ) is termed as the smoothed strain where n sk is the set of nodes associated the SD sk and B gradient matrix that is calculated by: ⎤ ⎡ b¯ix (xk ) ⎥ ⎢ ¯ i (xk ) = ⎢ ¯iy (xk )⎥ B (19) b ⎦ ⎣ b¯iy (xk ) b¯ix (xk ) where b¯ih (xk ), h = x, y, is computed by: b¯ih (xk ) = s Ak s k n h (x)Ni (x) d (20) Using the Gauss integration along the segments of boundary b¯ih = s Ak Nseg Ngau m=1 n=1 wm,n Ni (xm,n )n h (xm,n ) s k, we have: (h = x, y) (21) where Nseg is the number of segments of the boundary sk , Ngau is the number of Gauss points used in each segment, wm,n is the corresponding weight of Gauss points, n h is the outward unit normal corresponding to each segment on the SD boundary, and xm,n is the n-th Gaussian point on the m-th segment of the boundary sk Because the NS-FEM is variationally consistent as proven (when the solutions are sought in the (H10 ( ))2 space) in [24], the assumed displacement uh and the smoothed strains e¯ satisfy the smoothed Galerkin weak form: (¯e(uh ))T D(¯e(uh )) d − ( u h )T b d − ( u h )T t d = (22) t Substituting the approximated displacements in Equation (9) and the smoothed strains from Equation (17) into the smoothed Galerkin weak form yields the following system of equations: ¯ d¯ = f K (23) ¯ are then assembled by: where f is computed similarly by Equation (16) and the stiffness matrix K ¯ ij = K Ns k=1 Copyright q ¯s = K ij,k Ns k=1 2010 John Wiley & Sons, Ltd s k ¯ iT DB¯ j d = B Ns k=1 ¯ iT DB¯ j As B k (24) Int J Numer Meth Engng 2010; 83:1466–1497 DOI: 10.1002/nme NS-FEM FOR UPPER BOUND SOLUTIONS OF FRACTURE PROBLEMS 1473 crack tip Figure Scheme of smoothing domains around the crack tip 4.2 Smoothing domains around the crack tip It is well known that the FEM model using displacement-based compatible shape functions will provide a stiffening effect to the exact model, and give the lower bound of the solution in strain energy On the other hand, the strain smoothing operation used in an S-FEM will provide a softening effect to the compatible FEM model and give the larger solution in strain energy than that of the compatible FEM model Therefore, the battle between the softening and stiffening effects will determine the bound properties and accuracy of the proposed numerical method Liu et al [38–42] have found that the softening effect depends on the number of elements associated with SD The more the elements that participate in a SD, the more the softening effect becomes The NS-FEM produces the upper-bound solutions, and is found in general to be overly soft [40], due to many elements participating in the node-based strain smoothing operation More importantly, it is noticed that only one SD stip around the crack tip cannot adequately capture the singularity of the stresses as shown in Figure Therefore, the proper schemes of SDs around the crack tip should be used to reduce the softening effect and capture the singularity In the present singular NS-FEM, we propose four schemes for triangular-element-based SDs around the crack tip, as shown in Figure • Scheme In Scheme 1, only one layer of SD (from SD(1) to SD(7) ) around the crack tip is used as shown in Figure Note that there are two kinds of SDs: inner and boundary ones, and each SD is created based on the edge connected directly to the crack tip For the inner one, each SD is created by connecting sequentially the following points: (1) the crack tip; (2) the centroid of one adjacent singular element of the edge; (3) the mid-edge-point; (4) the centroid of another adjacent singular element; and return to (1) the crack tip For example, the SD(2) filled with the blue shadow in Figure is created by connecting sequentially #A, #C, #D, #E, and #A Copyright q 2010 John Wiley & Sons, Ltd Int J Numer Meth Engng 2010; 83:1466–1497 DOI: 10.1002/nme 1474 G R LIU ET AL SD SD (6) (5) SD (7) A SD (4) (1) B SD SD (2) SD (3) C F D E (13) SD crack tip (14) A SD SD (12) (11) SD G SD (8) SD (10) (9) SD H K I J Figure Scheme of smoothing domains around the crack tip For the boundary one, each SD is created by connecting sequentially the following points: (1) the crack tip; (2) the mid-edge-point; (3) the centroid of adjacent singular element; and return to (1) the crack tip For example, the SD(1) filled with the red shadow in Figure is created by connecting sequentially #A, #B, #C, and #A • Scheme Scheme contains two layers of SDs at the crack tip as illustrated in Figure Similar to the above, each layer of SD is also based on the edge connected to the crack tip, and includes two kinds: inner and boundary SDs (i) For the layer of SDs far away the crack tip (from SD(1) to SD(7) : Each inner SD is constructed by connecting (1) the centroid of one adjacent singular element of the edge; (2) the mid-edge-point; (3) the centroid of another adjacent singular element; (4) the 1/8-centroid-point of the second adjacent singular element; (5) the 1/8-edge-point; (6) the 1/8centroid-point of the first adjacent singular element; and back to (1) the centroid of the first adjacent singular element For example, the SD(2) filled with the blue shadow is created by connecting sequentially #C, #D, #E, #J, #I, #H, and #C Each boundary SD is created by connecting (1) the mid-edge-point; (2) the centroid of the adjacent singular element; (3) the 1/8-centroid-point of the adjacent singular element; (4) the Copyright q 2010 John Wiley & Sons, Ltd Int J Numer Meth Engng 2010; 83:1466–1497 DOI: 10.1002/nme NS-FEM FOR UPPER BOUND SOLUTIONS OF FRACTURE PROBLEMS 1475 crack tip Figure Scheme of smoothing domains around the crack tip 1/8-edge-point; and back to (1) the mid-edge-point For example, the SD(1) filled with the red shadow in Figure is created by connecting sequentially #B, #C, #H, #G, and #B (ii) For the layer of SDs connected directly to the crack tip (from SD(8) to SD(4) ): Each SD is the left part of the region of the SD in Scheme subtracting the region of the corresponding SD in the layer far away from the crack tip defined in Scheme For instance, the inner SD(8) filled with the black shadow is constructed by connecting sequentially #A, #H, #I, #J, and #A, and the boundary SD(7) filled with the green shadow is constructed by connecting sequentially #A, #G, #H, and #A in Figure • Scheme As given in Figure 5, one layer of SD (from SD(1) to SD(6) ) based on the singular element is used around the crack tip in Scheme The SD is just one-third of the region of the singular element connected to the crack tip For example, the SD(2) filled with the blue shadow in Figure is constructed by the node set containing #A, #D, #E, and #F • Scheme In Scheme 4, two layers of SDs are constructed near the crack tip based on the singular elements as illustrated in Figure (i) For the layer of SDs far away from the crack tip (from SD(1) to SD(6) ), each one is created by connecting (1) the mid-edge-point of one edge; (2) the centroid; (3) the mid-edge-point of another edge; (4) the 1/8-edge-point of the second edge; (5) the 1/8-centroid-point; (6) the 1/8-edge-point of the first edge; and back to (1) the mid-edge-point of first edge For instance, the SD(2) with the blue shadow is created by connecting sequentially #D, #E, #F, #K, #J, #I, and #D as shown in Figure (ii) For the layer of SDs connected directly to the crack tip (from SD(7) to SD(2 )), each one is the left part of the one-third region of the singular element connected to the crack tip Therefore, the SD(8) with the black shadow in Figure is constructed by the node set containing #A, #I, #J, and #K Copyright q 2010 John Wiley & Sons, Ltd Int J Numer Meth Engng 2010; 83:1466–1497 DOI: 10.1002/nme NS-FEM FOR UPPER BOUND SOLUTIONS OF FRACTURE PROBLEMS 1483 where W= (2) (1) (2) (1) *u i ε − xj ij ik ik *x − (1) (2) *u i ij *x (47) NUMERICAL IMPLEMENTATION The numerical procedure for the singular NS-FEM is outlined as follows: (1) Divide the problem domain into a set of elements and obtain information about node coordinates and element connectivity (2) Create the normal SDs using the rule given in Section 4.1, and crack tip SDs using the schemes of SD around the crack tip discussed in Section 4.2 (3) Loop over SDs (a) Determine the outward unit normal, the proper number of Gauss points for each boundary segment of the SDs; ¯ i (xk ) by using Equation (10) for normal (b) Calculate the smoothed strain gradient matrix B SDs, and using Equation (37) for crack tip SD; ¯ s and load vector of the current smoothing (c) Evaluate the smoothed stiffness matrix K i j,k domain; ¯ and (d) Assemble the contribution of the current SD to form the system stiffness matrix K force vector (4) (5) (6) (7) Implement essential boundary conditions Solve the linear system of equations to obtain the nodal displacements Evaluate strains and stresses at locations of interest Calculate the fracture parameters including the J -integral (energy release rate), and SIFs of K as well as K NUMERICAL EXAMPLES 7.1 Edge crack in an isotropic material plate A benchmark problem, edge crack in an isotropic material plate loaded by tension and shear, is first analyzed The material parameters are Young’s modulus, E = 3×107 Pa and Poisson’s ratio v = 0.3, and plane strain conditions are assumed For the tension case, a plate with dimension mm×2 mm is loaded at the top edge with = 1.0 Pa and for the shear case, the dimension mm×16 mm with crack length a = 3.5 mm, and a shear of = 1.0 Pa is applied to the top edge The displacements along the y-axis are fixed at the bottom edge and the plate is clamped at the bottom left corner The geometry, loading, and boundary conditions are shown in Figure 10 The exact solution of K for the tension case is given by [1]: K 1exact = C Copyright q 2010 John Wiley & Sons, Ltd √ √ a = 1.6118 Pa mm (48) Int J Numer Meth Engng 2010; 83:1466–1497 DOI: 10.1002/nme 1484 G R LIU ET AL = 1.0 a H = 16.0 H = 2.0 = 1.0 a = 3.5 b = 7.0 b = 1.0 (a) (b) Figure 10 (a) Plate with edge crack under tension and (b) plate with edge crack under shear where the crack length is a = 0.3 mm and C is a finite-geometry correction factor: C = 1.12−0.231 a a +10.55 b b −21.72 a b +30.39 The exact mixed mode SIFs for the shear case are given in [1]: √ √ K 1exact = 34.0 Pa mm, K 2exact = 4.55 Pa mm a b (49) (50) Four discretizations with uniform nodes as a/ h: (4.0, 6.0, 8.0, and 10.0), are used for the present singular NS-FEM, where h is the mesh spacing A sample mesh (a/ h = 8.0) in the vicinity of crack tip is shown in Figure 11 The domain radius of rd =rk h e is set by parameter rk = 3.0 For comparison, four models are also computed using standard FEM, singular FEM, and standard NS-FEM The strain energy and the error in energy norm are, respectively, defined as: E( ) = eT De d (51) ref 1/2 ee = |E (num ) − E( )| (52) The relative error of fracture parameters is given by: e= F P num − F P ref ×100% F P ref (53) where the superscript ‘ref’ denotes the exact or reference solution and ‘num’ denotes numerical solution obtained using a numerical method From Equation (53) it is clear that the negative relative error means that the numerical solution is smaller than the exact value, and vice versa Copyright q 2010 John Wiley & Sons, Ltd Int J Numer Meth Engng 2010; 83:1466–1497 DOI: 10.1002/nme NS-FEM FOR UPPER BOUND SOLUTIONS OF FRACTURE PROBLEMS 1485 a Figure 11 Meshes in the vicinity of the crack (a/ h = 8.0) 7.1.1 Bound property of solutions Figures 12 and 13, respectively, show the convergence status of the strain energy and K against the increase of Degree of Freedom (DOF) for the tension problem The reference solution of the strain energy is calculated using the singular FEM with a very fine mesh (23 488 nodes) It can be clearly observed that for both the strain energy and K , the computed values of the FEM and singular FEM models are always smaller than the exact solutions; on the contrary, the computed values of the NS-FEM and singular NS-FEM models are always bigger than the exact solutions The results confirm that the singular NS-FEM provides upper bound solutions The figures also show that with the increase of DOF, the strain energy and K of the singular FEM models and the singular NS-FEM models converge to the exact solutions from below and above, respectively All of these clearly show the very important fact that we can now bound the exact solution from both sides The SIFs and strain energy for each model are also calculated for the shear case As shown in Figures 14–16, the SIFs and strain energy of the singular FEM models are always smaller than the exact solutions and converge from below with the increase of DOF On the contrary, the SIFs and strain energy of the singular NS-FEM models are always bigger than the exact solutions and converge from above The results confirm again that the singular NS-FEM provides an upper bound, and thus we can bound the exact fracture parameters from both sides 7.1.2 Effect of the schemes of smoothing domains The computed SIFs and strain energy by the standard NS-FEM and the singular NS-FEM with four different schemes of SDs around the crack tip given in Section 4.2 are compared in this study From Figures 14–16, it is seen that the results of the singular NS-FEM using four schemes are closer to the exact values, compared to those of the standard NS-FEM It is also noted that the singular NS-FEM using Scheme (singular NS-FEM (4)) provides the best accuracy in the strain energy and SIFs with respect to three other schemes of SDs around the crack tip These results agree well with the analysis of four different schemes in Section 4.2 Thus, all the following studies are conducted by the singular NS-FEM (4) and termed as the singular NS-FEM, unless stated otherwise 7.1.3 Convergence rate study Figures 17 and 18 compare, respectively, the convergence rate in terms of the error in the strain energy norm and K for different numerical methods In this Copyright q 2010 John Wiley & Sons, Ltd Int J Numer Meth Engng 2010; 83:1466–1497 DOI: 10.1002/nme 1486 G R LIU ET AL x 10-8 3.7 3.65 3.6 Strain Energy 3.55 3.5 3.45 NS-FEM-T3 Singular NS-FEM-T3 (1) Singular NS-FEM-T3 (2) Singular NS-FEM-T3 (3) Singular NS-FEM-T3 (4) Singular FEM-T6 FEM-T3 Reference solu 3.4 3.35 3.3 3.25 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000 DOF Figure 12 Convergence of the strain energy for the problem of plate with edge crack under remote tension 1.1 1.05 K1 /K 1exact 0.95 NS-FEM-T3 Singular NS-FEM-T3 (1) Singular NS-FEM-T3 (2) Singular NS-FEM-T3 (3) Singular NS-FEM-T3 (4) Singular FEM-T6 FEM-T3 Reference solu 0.9 0.85 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000 DOF Figure 13 Convergence of the normalized K for the problem of plate with edge crack under remote tension comparison, the tension case is considered It can be easily seen that the convergence rate of both the error in energy norm and the relative error of K for the singular NS-FEM models is higher than that of the standard FEM or NS-FEM models Moreover, the convergence rate of K is about R = 0.84 which is much higher than the strain energy with about R = 0.5 This is true for all the numerical methods used Copyright q 2010 John Wiley & Sons, Ltd Int J Numer Meth Engng 2010; 83:1466–1497 DOI: 10.1002/nme NS-FEM FOR UPPER BOUND SOLUTIONS OF FRACTURE PROBLEMS 9.6 1487 x 10-5 9.4 9.2 Strain Energy 8.8 8.6 8.4 NS-FEM-T3 Singular NS-FEM-T3 (1) Singular NS-FEM-T3 (2) Singular NS-FEM-T3 (3) Singular NS-FEM-T3 (4) Singular FEM-T6 FEM-T3 Reference solu 8.2 7.8 7.6 1000 2000 3000 4000 5000 DOF 6000 7000 8000 9000 10000 Figure 14 Convergence of the strain energy for the problem of plate with edge crack under remote shear 1.2 NS-FEM-T3 Singular NS-FEM-T3 (1) Singular NS-FEM-T3 (2) Singular NS-FEM-T3 (3) Singular NS-FEM-T3 (4) Singular FEM-T6 FEM-T3 Reference solu 1.15 1.1 K1/K1exact 1.05 0.95 0.9 0.85 0.8 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 DOF Figure 15 Convergence of the normalized K for the problem of plate with edge crack under remote shear 7.1.4 Influence of the number of Gauss points Table I lists the results of the study of the influence of the number of Gauss points along one segment of SDs on the SIFs, energy release rate G and strain energy In this study, the mesh with a/ h = 8.0 is used It can be seen that when fewer Gauss Copyright q 2010 John Wiley & Sons, Ltd Int J Numer Meth Engng 2010; 83:1466–1497 DOI: 10.1002/nme 1488 G R LIU ET AL 1.25 NS-FEM-T3 Singular NS-FEM-T3 (1) Singular NS-FEM-T3 (2) Singular NS-FEM-T3 (3) Singular NS-FEM-T3 (4) Singular FEM-T6 FEM-T3 Reference solu 1.2 K2 / K2 exact 1.15 1.1 1.05 0.95 0.9 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 DOF Figure 16 Convergence of the normalized K for the problem of plate with edge crack under remote shear -3.2 -3.25 -3.3 NS-FEM-T3 R=0.46 Singular NS-FEM-T3 (1) R=0.53 Singular NS-FEM-T3 (2) R=0.52 Singular NS-FEM-T3 (3) R=0.54 Singular NS-FEM-T3 (4) R=0.55 FEM-T3 R=0.43 Log10 (ee) -3.35 -3.4 -3.45 -3.5 -3.55 -3.6 -3.65 -3.7 -0.85 -0.8 -0.75 -0.7 -0.65 -0.6 Log10 (h) -0.55 -0.5 -0.45 -0.4 Figure 17 Convergence rate in term of energy error norm for the problem of plate with edge crack under remote tension points are used, higher values are obtained for the strain energy and the SIFs When more than five Gauss points are used, the strain energy and the SIFs have very little change Thus, all the models discussed later use five Gauss points along one segment of the SD Copyright q 2010 John Wiley & Sons, Ltd Int J Numer Meth Engng 2010; 83:1466–1497 DOI: 10.1002/nme 1489 NS-FEM FOR UPPER BOUND SOLUTIONS OF FRACTURE PROBLEMS NS-FEM-T3 R=0.79 Singular NS-FEM-T3 (1) R=0.84 Singular NS-FEM-T3 (2) R=0.84 Singular NS-FEM-T3 (3) R=0.84 Singular NS-FEM-T3 (4) R=0.85 FEM-T3 R=0.79 -0.9 -1 -1.1 Log10 (eK1) -1.2 -1.3 -1.4 -1.5 -1.6 -1.7 -1.8 -1.9 -0.85 -0.8 -0.75 -0.7 -0.65 -0.6 -0.55 -0.5 -0.45 -0.4 Log10 (h) Figure 18 Convergence rate in term of K for the problem of plate with edge crack under remote tension Table I Edge crack in one isotropic material plate: the number of Gauss points effects Tension Ngau E ( ) (10−8 ) 3.9119 3.9076 3.9074 3.9073 K /K (% Error) 1.0151 1.0130 1.0129 1.0128 (1.5) (1.3) (1.3) (1.3) Shear E ( ) (10−5 ) 8.9332 8.9096 8.9086 8.9084 K /K (% Error) 1.0221 1.0176 1.0174 1.0174 (2.2) (1.8) (1.7) (1.7) K /K (% Error) 1.0086 1.0072 1.0068 1.0068 (0.86) (0.72) (0.68) (0.68) 7.1.5 Domain independence study In this study, we consider several different domain sizes for the interaction integrals described in Section 5, and the results are given in Table II We can easily observe the domain independence of the SIFs with parameters rk >3 for all the models used 7.2 Center crack in an infinite bi-material plate The problem of an interface crack between two different elastic semi-infinite planes is then ∞ studied The exact solution to this problem under remote traction t = ∞ 22 +i 12 was obtained by Rice [47] The solution for K and K at the right crack tip is [47, 48]: √ ∞ K C = K +i K = ( ∞ a(2a)−iε (54) 22 +i 12 )(1+2iε) We consider the case of pure tension remote loading In the computation, only half the specimen is considered with the appropriate displacement constraint due to symmetry (see Figure 19) Copyright q 2010 John Wiley & Sons, Ltd Int J Numer Meth Engng 2010; 83:1466–1497 DOI: 10.1002/nme 1490 G R LIU ET AL Table II Domain independence study for the cracks in one isotropic material Tension Mesh (nodes) rk a/ h = 4.0 4 a/ h = 8.0 Shear K /K (% Error) 1.0243 1.0248 1.0251 1.0126 1.0129 1.0130 1.0130 K /K (% Error) (2.4) (2.5) (2.5) (1.2) (1.3) (1.3) (1.3) 1.0385 1.0389 1.0390 1.0171 1.0175 1.0175 1.0175 (3.8) (3.9) (3.9) (1.7) (1.8) (1.8) (1.8) K /K (% Error) 1.0217 1.0167 1.0190 1.0116 1.0068 1.0084 1.0087 (2.17) (1.67) (1.90) (1.16) (0.68) (0.84) (0.87) ux = Material L ux = a Material L w σ 22 Figure 19 Center crack with bi-materials under remote tension (half model) The right edge is constrained in x-direction to remove the edge singularity [47] The factors K and J0 are used to normalize the SIFs and J -integral, respectively K0 = ∞√ 22 a, J0 = ( ∞ )2 22 a E1 (55) where 2a is the crack length The material constants used in the numerical computation are: E = 10 GPa, E /E = 22, v1 = 0.3 and v2 = 0.2571, and plane strain conditions are assumed The exact solutions from Equation (54) are: K1 = 1.008, K0 Copyright q 2010 John Wiley & Sons, Ltd K2 = 0.1097, K0 J = 1.4358 J0 (56) Int J Numer Meth Engng 2010; 83:1466–1497 DOI: 10.1002/nme 1491 NS-FEM FOR UPPER BOUND SOLUTIONS OF FRACTURE PROBLEMS Table III Center crack with bi-materials under remote tension: comparison of stress intensity factors and energy release rate using FEM, singular FEM, NS-FEM, and singular NS-FEM Exact solution Mesh (a/ h) 4.0 (% Error) K /K = 1.008 FEM Sin FEM NS-FEM Sin NS-FEM 0.9834 1.0051 1.0252 1.0166 (−2.4) (−0.3) (1.7) (0.8) K /K = 0.1097 FEM Sin FEM NS-FEM Sin NS-FEM 0.1192 (8.6) 0.1106 (0.9) 0.0801 (−27.0) 0.0912 (−16.9) J/J0 = 1.4358 FEM Sin FEM NS-FEM Sin NS-FEM 1.3699 1.4273 1.4765 1.4546 (−4.6) (−0.6) (2.8) (1.3) 6.0 (% Error) 8.0 (% Error) 10.0 (% Error) 12.0 (% Error) 0.9903 (−1.8) 1.0060 (−0.2) 1.0205 (1.2) 1.0127(0.4) 0.9939 1.0061 1.0189 1.0116 (−1.4) (−0.2) (1.0) (0.3) 0.9959 (−1.2) 1.0063 (−0.2) 1.0179 (0.9) 1.0105(0.2) 0.9974 1.0064 1.0161 1.0094 (−1.1) (−0.2) (0.8) (0.1) 0.1141 (4.6) 0.1105 (0.9) 0.0924 (−15.8) 0.0981 (−10.6) 0.1122 0.1101 0.0979 0.1037 (2.3) (0.3) (−10.8) (−5.5) 0.1111 0.1099 0.1023 0.1086 (1.3) (0.2) (−6.7) (−1.0) 0.1105 0.1097 0.1044 0.1091 (0.7) (0.02) (−4.8) (−0.5) 1.3964 1.4301 1.4627 1.4438 (−2.7) (−0.4) (1.8) (0.5) 1.4018 1.4306 1.4611 1.4421 (−2.4) (−0.3) (1.7) (0.4) 1.4053 1.4306 1.4550 1.4401 (−2.1) (−0.3) (1.3) (0.3) 1.3872 1.4296 1.4659 1.4453 (−3.4) (−0.4) (2.1) (0.7) The crack dimension is selected as a = mm As the exact solution is for the infinite domain problem, the sample size W/a = 30 is used in all models to avoid the effect of finite size Five structured meshes with a/ h: (4.0, 6.0, 8.0, 10.0, and 12.0) are used All the studies are conducted using the domain radius parameter rk = unless stated otherwise In Table III, the comparison of the SIFs and the energy release rate using different numerical methods (FEM, singular FEM, NS-FEM, and singular NS-FEM) are presented The corresponding figures are plotted in Figures 20–22 From the table and figures, it can be found again that the K and J of the singular FEM are not more than the exact solutions (the negative relative error) and converge from below, the corresponding values of the singular NS-FEM are not less than the exact ones (the positive relative error) and converge from above On the contrary, the K of the singular FEM models converges from above, and the K of the singular NS-FEM models converges from below (see Figure 21) However, we can easily observe that the computed values of the singular FEM and the singular NS-FEM models converge from different directions, regardless of any fracture parameters considered Thus, we can also bound the exact solutions from two sides The domain independence study is also conducted, and the results for different choices of domain sizes are given in Table IV Different from the cracks in an isotropic material media, domain independence of the SIFs is only realized for rk >5 for interface cracks This may be due to the fact that the SDs along the interface between two isotropic materials are divided into two parts 7.3 Film/substrate system by the four-point bending test The third example is a film/substrate system with the four-point bending test Owing to symmetry, one half of the specimen is used in the computation The specimen dimensions, crack orientation, loading and the displacement boundary conditions are given in Figure 23 The thickness of the film is h f and that of the substrate is h s , with the total thickness denoted by h t E f and vf are used to denote Young’s modulus and Poisson’s ratio of film E s and vs are the corresponding properties for the substrate Copyright q 2010 John Wiley & Sons, Ltd Int J Numer Meth Engng 2010; 83:1466–1497 DOI: 10.1002/nme 1492 G R LIU ET AL 1.04 NS-FEM-T3 Singular NS-FEM-T3 Singular FEM-T6 1.03 FEM-T3 Reference solu K1 / K0 1.02 1.01 0.99 0.98 2000 4000 6000 8000 DOF 10000 12000 14000 16000 Figure 20 Convergence of the normalized K for center crack with bi-materials under tension 0.12 0.115 0.11 K2 / K0 0.105 0.1 0.095 0.09 NS-FEM-T3 Singular NS-FEM-T3 Singular FEM-T6 FEM-T3 Reference solu 0.085 0.08 2000 4000 6000 8000 DOF 10000 12000 14000 16000 Figure 21 Convergence of the normalized K for center crack with bi-materials under tension When the interface crack length significantly exceeds the thickness of the film, steady state conditions are reached and the energy release rate stabilizes to a constant value, G ss , the steady state energy [49]: G ss = Copyright q 3(1−vs2 )P L 2E s b2 h 3t ht hs − 2010 John Wiley & Sons, Ltd hf ht + hs ht +3 hfhs h 2t hf hs + ht ht (57) Int J Numer Meth Engng 2010; 83:1466–1497 DOI: 10.1002/nme 1493 NS-FEM FOR UPPER BOUND SOLUTIONS OF FRACTURE PROBLEMS 1.48 1.46 J-intergral / J0 1.44 1.42 1.4 NS-FEM-T3 Singular NS-FEM-T3 1.38 Singular FEM-T6 FEM-T3 Reference solu 1.36 2000 4000 6000 8000 10000 12000 14000 16000 DOF Figure 22 Convergence of the normalized J -integral for center crack with bi-materials under tension Table IV Center crack bi-materials under remote tension: domain independence study Mesh rk a/ h = 6.0 6 a/ h = 8.0 K /K (% Error) 1.0097 1.0124 1.0127 1.0129 1.0073 1.0106 1.0116 1.0121 K /K (% Error) 0.1230 (12.1) 0.1070 (−2.5) 0.0981 (−10.6) 0.0952 (−13.1) 0.1328 (21.1) 0.1126 (2.5) 0.1037 (−5.5) 0.1010 (−7.9) (0.2) (0.4) (0.5) (0.5) (−0.1) (0.3) (0.4) (0.4) 1.4445 1.4470 1.4453 1.4451 1.4413 1.4437 1.4438 1.4444 (0.6) (0.8) (0.6) (0.6) (0.4) (0.5) (0.6) (0.6) P / 2b L hf G/G (% Error) Film: a Interface: G hs Substrate: D Figure 23 Schematic diagram of film/substrate system by four-point bending test (half model) Copyright q 2010 John Wiley & Sons, Ltd Int J Numer Meth Engng 2010; 83:1466–1497 DOI: 10.1002/nme 1494 G R LIU ET AL Table V Film/substrate system by four-point bending test: comparison of stress intensity factors and energy release rate using FEM, singular FEM, NS-FEM, and singular NS-FEM under the same triangular mesh with h t / h = 6.0* Method K /K K /K FEM Sin FEM NS-FEM Sin NS-FEM 0.9386 0.9580 0.9970 0.9882 1.2832 1.2970 1.3123 1.3061 G/G (% Error) 43.81 43.57 42.77 42.89 1.3142 1.3531 1.4123 1.3779 (−3.6) (−0.7) (3.5) (1.1) *The exact energy release rate which equals to the J-integral from Equation (1) is 1.3632 where b is the depth of film/substrate system and = is defined as: E s (1−vf2 ) E f (1−vs2 ) (58) The phase angle is a measure of the relative proportion of shear to normal tractions at a characteristic distance l ahead of the crack tip It is defined through the relation [47] Kl iε = |K|ei (59) The phase angle is an important parameter in the characterization of interfacial fracture toughness Here, the characteristic length l given in Equation (59) is taken as the total thickness of film/substrate system h t : = tan−1 Im[Kh iε t ] (60) Re[Kh iε t ] In addition, we choose the factors K and G , respectively, to normalize the SIFs and the energy release rate which equals to J -integral K0 = PL , 3/2 bht G0 = (1−vs2 )P L E s b2 h 3t (61) In the numerical model, the depth b is taken to be unity, h f = 1, h t / h f = 10, L/D = 2.5, D/ h t = 5, and a/ h t = Thus, the problem domain is L ×h t = 125×10 and a = 30, where the units of all the geometry sizes are mm The material parameters are E s = 10 GPa, E f /E s = 10, and vf = vs = 0.3 The mesh with h t / h = 6.0 and the domain radius parameter rk = are used Based on this, the exact normalized steady state energy release rate is 1.3632 from Equation (57) Comparison of energy release rates obtained by FEM, singular FEM, NS-FEM, and singular NS-FEM are presented in Table V The results for K , K , and are also indicated for completeness From the results, the singular NS-FEM is found again to produce an upper bound solution in the energy release rate, whereas the singular FEM produces the lower bound to energy release rate Then, the fixed total thickness h t = 10 is used and the thickness ratio h f / h t is varied from 0.1 to 0.5 In addition, we varied the combinations of material properties of film and substrate Table VI lists the steady state energy release rate for different thickness ratio and different material combinations Again, it is observed that all the results by the singular NS-FEM give the positive relative errors in energy release rate with the corresponding exact solutions, which contrast with Copyright q 2010 John Wiley & Sons, Ltd Int J Numer Meth Engng 2010; 83:1466–1497 DOI: 10.1002/nme 1495 NS-FEM FOR UPPER BOUND SOLUTIONS OF FRACTURE PROBLEMS Table VI Film/substrate system by four-point bending test: effect of elastic modulus ratio and thickness ratio Singular FEM Ef Es hf h s +h f 10 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 0.1 Singular NS-FEM G/G (% Error) 43.25 41.93 38.42 34.84 31.96 63.90 57.85 53.25 49.55 46.66 1.3511 2.2626 3.6969 6.2754 11.3451 0.0801 0.3022 0.8959 2.4518 6.4706 (−0.7) (−0.6) (−0.6) (−0.7) (−0.8) (−0.8) (−0.6) (−0.4) (−0.4) (−0.6) G/G (% Error) 42.89 42.00 38.85 35.34 32.32 61.98 55.72 51.14 47.43 44.70 1.3779 2.3234 3.7875 6.4264 11.6746 0.0822 0.3112 0.9290 2.4976 6.6522 (1.1) (1.9) (1.7) (1.6) (1.9) (1.8) (2.2) (2.5) (1.3) (1.8) Exact solution G/G 1.3632 2.2793 3.7254 6.3249 11.4523 0.0807 0.3043 0.9010 2.4655 6.5228 the results by the singular FEM It means again that the singular NS-FEM produces an upper bound solution, whereas the singular FEM produces a lower bound CONCLUSION In this work, a novel singular NS-FEM is formulated to obtain the upper solutions for fracture problems with a wide range of material combinations and boundary conditions To model the singularity at the crack tip, four schemes of SDs around the crack tip have been proposed based on the triangle elements A five-node singular element is also used within the framework of NS-FEM to construct the singular shape functions Through the formulation and numerical examples, some conclusions can be drawn as follows: In our singular NS-FEM, the displacement field is at least (complete) linearly consistent, and the enrichment near the crack tip is on top of the complete linear field Therefore, the partition-of-unity property as well as the linear consistency property is ensured throughout the entire problem domain, ensuring the stability and the convergence of the solution In terms of the J -integral or the energy release rate, the singular NS-FEM produces an upper bound solution, whereas the singular FEM produces the lower bound The computed values of the singular FEM and the singular NS-FEM models converge from different directions for any fracture parameters Therefore, we can bound the exact solution from both sides Compared to the standard NS-FEM, the computed SIFs and J -integral of the singular NSFEM with the schemes of SDs around the crack tip are much closer to the exact values In the Gauss integrations for computing the stiffness matrix, at least five Gauss points should be used along one segment of the crack tip SD for the five-node singular element to ensure the accuracy of results Domain independence of the SIFs is realized by the domain radius of rd =rk h e with parameter rk >3 for the cracks in an isotropic material, and the corresponding parameter changes to rk >5 for the interface cracks between two dissimilar isotropic materials Copyright q 2010 John Wiley & Sons, Ltd Int J Numer Meth Engng 2010; 83:1466–1497 DOI: 10.1002/nme 1496 G R LIU ET AL In the singular NS-FEM, the system stiffness matrix are computed directly using only the shape functions themselves and no explicit √ analytical form of shape functions, e.g the derivatives of the singular shape functions (1/ r ), is required Thus, it can be easily incorporated into the existing codes The NS-FEM uses triangular mesh and is designed for simplicity, adaptivity and the essential features for linear completeness with proper enrichment for capturing singular field The implementation of the present method is very straightforward All the numerical operations can be automated as long as a triangular mesh is available REFERENCES Anderson TL Fracture Mechanics: Fundamentals and Applications CRC Press: Boca Raton, 1995 Sze KY, Wu CC, Huang YQ Upper and lower bounds for evaluation of nonlinear fracture parameters Engineering Fracture Mechanics 1999; 62:185–201 Wu CC, Xiao QZ, Yagawa G Dual analysis for path integrals and bounds for crack parameter International Journal of Solids and Structures 1998; 35:1635–1652 Veubeke BF Displacement and equilibrium models in the finite element method In Stress Analysis, Zienkiewicz OC, Holister GS (eds) Wiley: London, 1965 Ladeveze P, Pelle JP, Rougeot P Error estimation and mesh optimization for classical finite elements Engineering Computation 1991; 8:69–80 Coorevits P, Ladeveze P, Pelle JP An automatic procedure with a control of accuracy for finite element analysis in 2D elasticity Computer Methods in Applied Mechanics and Engineering 1995; 121:91–121 Almeida JPM, Freitas JAT Alternative approach to the formulation of hybrid equilibrium finite elements Computers and Structures 1991; 40:1043–1047 Henshell RD, Shaw KG Crack tip finite elements are unnecessary International Journal for Numerical Methods in Engineering 1975; 9:495–507 Barsoum RS On the use of isoparametric finite elements in linear fracture mechanics International Journal for Numerical Methods in Engineering 1976; 10:551–564 10 Barsoum RS Triangular quarter-point elements as elastic and perfectly plastic crack tip elements International Journal for Numerical Methods in Engineering 1977; 11:85–98 11 Belytschko T, Lu YY, Gu L Element free Galerkin methods International Journal for Numerical Methods in Engineering 1994; 37(2):229–256 12 Belytschko T, Krongauz Y, Organ D, Fleming M, Krysl P Meshless methods: an overview and recent developments Computer Methods in Applied Mechanics and Engineering 1996; 139:3–47 13 Rabczuk T, Belytschko T A three dimensional large deformation meshfree method for arbitrary evolving cracks Computer Methods in Applied Mechanics and Engineering 2007; 196(29–30):2777–2799 14 Rabczuk T, Zi G A meshfree method based on the local partition of unity for cohesive cracks Computational Mechanics 2007; 39(6):743–760 15 Rabczuk T, Belytschko T, Xiao SP Stable particle methods based on Lagrangian kernels Computer Methods in Applied Mechanics and Engineering 2004; 193(12–14):1035–1063 16 Lynn PP, Ingraffea AR Transition elements to be used with quarter-point crack-tip elements International Journal for Numerical Methods in Engineering 1980; 15:1427–1445 17 Liu GR, Quek SS The Finite Element Method: A Practical Course Butterworth Heinemann: Oxford, 2002 18 Belytschko T, Black T Elastic crack growth in finite elements with minimal remeshing International Journal for Numerical Methods in Engineering 1999; 45(5):601–620 19 Moes N, Dolbow J, Belytschko T A finite element method for crack growth without remeshing International Journal for Numerical Methods in Engineering 1999; 46(1):131–150 20 Song JH, Areias PMA, Belytschko T A method for dynamic crack and shear band propagation with phantom nodes International Journal for Numerical Methods in Engineering 2006; 67:868–893 21 Belytschko T, Moes N, Usui S, Parimi C Arbitrary discontinuities in finite elements International Journal for Numerical Methods in Engineering 2001; 50(4):993–1013 22 Chessa J, Wang H, Belytschko On the construction of blending elements for local partition of unity enriched finite elements International Journal for Numerical Methods in Engineering 2003; 57:1015–1038 Copyright q 2010 John Wiley & Sons, Ltd Int J Numer Meth Engng 2010; 83:1466–1497 DOI: 10.1002/nme NS-FEM FOR UPPER BOUND SOLUTIONS OF FRACTURE PROBLEMS 1497 23 Liu GR A generalized gradient smoothing technique and the smoothed bilinear form for Galerkin formulation of a wide class of computational methods International Journal of Computational Methods 2008; 5(2):199–236 24 Liu GR A G space theory and weakened weak (W2) form for a unified formulation of compatible and incompatible methods: part I theory and part II application International Journal for Numerical Methods in Engineering 2009; 87:1093–1156 25 Chen JS, Wu CT, Yoon Y A stabilized conforming nodal integration for Galerkin mesh-free methods International Journal for Numerical Methods in Engineering 2001; 50:435–466 26 Liu GR, Dai KY, Nguyen TT A smoothed finite element method for mechanics problems Computational Mechanics 2007; 39:859–877 27 Liu GR, Nguyen TT, Dai KY, Lam KY Theoretical aspects of the smoothed finite element method (SFEM) International Journal for Numerical Methods in Engineering 2007; 71:902–930 28 Liu GR, Nguyen-Thoi T, Nguyen-Xuan H, Dai KY, Lam KY On the essence and the evaluation of the shape functions for the smoothed finite element method (SFEM) (Letter to Editor) International Journal for Numerical Methods in Engineering 2009; 77:1863–1869 29 Dai KY, Liu GR, Nguyen TT An n-sided polygonal smoothed finite element method (nSFEM) for solid mechanics Finite Elements in Analysis and Design 2007; 43:847–860 30 Dai KY, Liu GR Free and forced analysis using the smoothed finite element method (SFEM) Journal of Sound and Vibration 2007; 301:803–820 31 Nguyen-Thoi T, Liu GR, Dai KY, Lam KY Selective smoothed finite element method Tsinghua Science and Technology 2007; 12(5):497–508 32 Nguyen-Xuan H, Bordas S, Nguyen-Dang H Addressing volumetric locking and instabilities by selective integration in smoothed finite elements Communications in Numerical Methods and Engineering 2008; 25:19–34 33 Nguyen-Xuan H, Nguyen-Thoi T A stabilized smoothed finite element method for free vibration analysis of Mindlin–Reissner plates Communications in Numerical Methods in Engineering 2009; 25:882–906 34 Nguyen-Thanh N, Rabczuk T, Nguyen-Xuan H, Bordas S A smoothed finite element method for shell analysis Computer Methods in Applied Mechanics and Engineering 2008; 198:165–177 35 Nguyen-Van H, Mai-Duy N, Tran-Cong T A smoothed four-node piezoelectric element for analysis of two-dimensional smart structures Computer Modeling in Engineering and Sciences 2008; 23(3):209–222 36 Nguyen-Xuan H, Rabczuk T, Bordas S, Debongnie JF A smoothed finite element method for plate analysis Computer Methods in Applied Mechanics and Engineering 2008; 197:1184–1203 37 Bordas S, Rabczuk T, Nguyen-Xuan H, Nguyen-Vinh P, Natarajan S, Bog T, Do-Minh Q, Nguyen-Vinh H Strain smoothing in FEM and XFEM Computers and Structures 2009; DOI: 10.1016/j.compstruc.2008.07.006 38 Liu GR, Nguyen-Thoi T, Lam KY A node-based smoothed finite element method for upper bound solution to solid problems (NS-FEM) Computers and Structures 2008; 87:14–26 39 Nguyen-Thoi T, Liu GR, Nguyen-Xuan H Additional properties of the node-based smoothed finite element method (NS-FEM) for solid mechanics problems International Journal of Computational Methods 2009; 6:633–666 40 Liu GR, Zhang GY Upper bound solution to elasticity problems: a unique property of the linearly conforming point interpolation method (LC-PIM) International Journal for Numerical Methods in Engineering 2008; 74:1128–1161 41 Chen L, Nguyen-Xuan H, Nguyen-Thoi T, Zeng KY, Wu SC Assessment of smoothed point interpolation methods for elastic mechanics Communications in Numerical Methods in Engineering 2009; DOI: 10.1002/cnm.1251 42 Nguyen-Thoi T, Liu GR, Nguyen-Xuan H, Nguyen-Tran C Adaptive analysis using the node-based smoothed finite element method (NS-FEM) Communications in Numerical Methods in Engineering 2009; DOI: 10.1002/cnm.1291 43 Liu GR, Zhang GY, Dai KY, Wang YY, Zhong ZH, Li GY, Han X A linearly conforming point interpolation method (LC-PIM) for 2D solid mechanics problems International Journal of Computational Methods 2005; 2(4):645–665 44 Liu GR, Gu YT A point interpolation method for two-dimensional solids International Journal for Numerical Methods in Engineering 2001; 50:937–951 45 Moran B, Shih CF Crack tip and associated domain integrals from momentum and energy balance Engineering Fracture Mechanics 1987; 27(6):615–641 46 Li FZ, Shih CF, Needleman A A comparison of methods for calculating energy release rates Engineering Fracture Mechanics 1985; 21(2):405–421 47 Rice JR Elastic fracture mechanics concepts for interfacial cracks Journal of Applied Mechanics 1988; 55:98–103 48 Rice JR, Sih GC Plane problems of cracks in dissimilar media Journal of Applied Mechanics 1965; 32:418–423 49 Charalambides PG, Lund J, Evans AG, McMeeking RM A test specimen for determining the fracture resistance of bimaterial interfaces Journal of Applied Mechanics 1989; 56:77–82 Copyright q 2010 John Wiley & Sons, Ltd Int J Numer Meth Engng 2010; 83:1466–1497 DOI: 10.1002/nme ... bounds for evaluation of nonlinear fracture parameters Engineering Fracture Mechanics 1999; 62:185–201 Wu CC, Xiao QZ, Yagawa G Dual analysis for path integrals and bounds for crack parameter... we can now have a systematical way to numerically obtain both upper and lower bounds of fracture parameters to crack problems Intensive benchmark numerical examples for a wide range of material... FOR UPPER BOUND SOLUTIONS OF FRACTURE PROBLEMS 1467 crack tip fields [1] In addition, the upper and lower bound analyses [2] or the so-called dual analyses [3] for evaluation of fracture parameters