DSpace at VNU: Strain smoothing in FEM and XFEM

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Computers and Structures 88 (2010) 1419–1443 Contents lists available at ScienceDirect Computers and Structures journal homepage: www.elsevier.com/locate/compstruc Strain smoothing in FEM and XFEM Stéphane P.A Bordas a,*, Timon Rabczuk b, Nguyen-Xuan Hung c, Vinh Phu Nguyen d, Sundararajan Natarajan e, Tino Bog b, Do Minh Quan c, Nguyen Vinh Hiep c a Department of Civil Engineering, University of Glasgow, G12 8LT Scotland, UK Department of Mechanical Engineering, University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand c Division of Computational Mechanics, Department of Mathematics and Informatics, University of Natural Sciences, VNU-HCM, 227 Nguyen Van Cu, Viet Nam d Delft University of Technology, Faculty of Civil Engineering and Geosciences, Stevinweg 1, 2628 CN Delft, The Netherlands e GE Aviation, India Technology Center, Bangalore, India b a r t i c l e i n f o Article history: Received 30 September 2007 Accepted 22 July 2008 Available online 23 February 2009 Keywords: Smoothed finite element SFEM Extended finite element method XFEM Convergence Accuracy a b s t r a c t We present in this paper recent achievements realised on the application of strain smoothing in finite elements and propose suitable extensions to problems with discontinuities and singularities The numerical results indicate that for 2D and 3D continuum, locking can be avoided New plate and shell formulations that avoid both shear and membrane locking are also briefly reviewed The principle is then extended to partition of unity enrichment to simplify numerical integration of discontinuous approximations in the extended finite element method Examples are presented to test the new elements for problems involving cracks in linear elastic continua and cracked plates In the latter case, the proposed formulation suppresses locking and yields elements which behave very well, even in the thin plate limit Two important features of the set of elements presented are their insensitivity to mesh distortion and a lower computational cost than standard finite elements for the same accuracy These elements are easily implemented in existing codes since they only require the modification of the discretized gradient operator, B Ó 2008 Elsevier Ltd All rights reserved Introduction In this introduction, we first review the motivation for the use of strain smoothing (SS) in a finite element context, discuss salient features of the method before describing its theoretical foundations We then briefly examine the extension of the strain smoothing techniques to plate and shell formulations and elasto-plasticity Finally, the extension of strain smoothing to partition of unity finite elements is outlined 1.1 Inspiration In some meshfree methods,1 the derivatives of the shape functions vanish at the nodes, leading to spurious modes To avoid their evaluation at the nodes, Chen in 2001 [22] introduced stabilized conforming nodal integration, later extended by Yoo and Moran to the natural element method (NEM) [107] This permits to fulfill linear consistency in Galerkin approximations The idea behind stabilized conforming nodal integration is to use a strain measure calculated as the spatial average of the standard,2 strain field This meshfree concept, Liu extended to the context of finite elements in the seminal publication on the smoothed finite element method [53].3 To so, the elements are divided into smoothing cells over which the strain is smoothed By the divergence theorem, interior integration on the smoothing cells is transformed into boundary integration Depending on the number of smoothing cells (nc) used in an element, as we shall see below, the formulation offers a range of different properties 1.2 Basic properties of the smoothed finite element method (SFEM) The major features of the smoothed finite element method (SFEM) are: * Corresponding author Tel.: +44 0141 330 4075; fax: +44 0141 330 4557 E-mail addresses: stephane.bordas@alumni.northwestern.edu, stephane.bordas@ gmail.com (S.P.A Bordas) See [63] for a very recent review and details on key implementation aspects as well as [38] for an excellent overview and classification of mesfhree methods 0045-7949/$ - see front matter Ó 2008 Elsevier Ltd All rights reserved doi:10.1016/j.compstruc.2008.07.006 By standard, we mean the symmetric gradient of the displacements The same concept had been utilized in the framework of the natural element method [93] (NEM) in [107] 1420 S.P.A Bordas et al / Computers and Structures 88 (2010) 1419–1443 (1) Insensitivity to mesh distortion (absence of isoparametric mapping) (2) Derivatives of the shape functions are not required (3) Lower computational cost than FEM for the same accuracy level (4) Insensitivity to locking for low numbers of subcells (5) Sound variational basis, but lack of a priori error estimators (6) Flexibility, offering elements ranging from the standard FEM to quasi-equilibrium FEM, within a single framework (7) Rank deficiency for the one subcell version (8) The one subcell version is equivalent to a quasi-equilibrium method, therefore, the gradient (stress) accuracy and convergence rate are increased For an infinite number of subcells, the method becomes equivalent to the standard finite element method, therefore, the accuracy on the primary field variable (displacements) is increased (9) Possibility to construct arbitrary polygonal elements In what follows, after reviewing the basics of the SFEM, we show examples in 2D and 3D linear elasticity, 2D elasto-plasticity, plate and shell formulations and couple the method with partition of unity enrichment [56,5]4 to solve fracture problems in continua and plates Let us first review the basic results on SFEM and its theoretical foundation In what follows, nc denotes the number of subcells in a smoothed finite element Fig Stress field in a smoothed finite element The stress is constant over each smoothing cell, but discontinuous across cells On the contrary, the displacement field is continuous within the element 1.3 Theoretical foundations In the first paper of 2006, Liu and his group proposed an application of strain smoothing to the finite element method, and called the resulting technique smoothed finite element method – SFEM [53] Soon after, two papers proposed mathematical bases for the SFEM in the form of a multi-field Hu–Washizu variational principle [52,65] The idea behind the SFEM is to split a finite element into subcells over which the gradients (strains) are smoothed To help picture the situation, Fig shows a possible state of stress within a smoothed four-noded quadrilateral divided into four subcells (nc = 4) Remark Note that the stresses are discontinuous across the cell walls, but the displacement remains continuous because the shape functions of the underlying finite element are used to define the displacement field throughout the element Each subcell plays a similar role to Gauß points in the standard FEM In non-linear computations (Section 4.3 and [66]), the cells carry the internal variables 1.3.1 Influence of the number of subcells It was soon found [52] that the SFEM solution has different properties for different numbers of subcells For nc 21; ỵ1ẵ, the solution is bounded by the FEM solution with reduced integration and a quasi-equilibrium element (nc = 1) and the standard displacement FEM solution ðnc ! ỵ1ị In [65] and [66], it is shown that the one subcell version of the four-noded quadrilateral (Q4) SFEM is equivalent to a quasi-equilibrium element This equivalence explains the superconvergence properties observed in the energy norm (H1) [53,52,65] as well as the increased stress (gradient) accuracy of the one subcell SFEM Additionally [65,66], the dual We choose here the extended finite element method, XFEM [11,57] nature of the one subcell version, which can overestimate the energy5 explains the zero-energy modes often observed for the one subcell element [52] Liu shows very clearly in [52] that for a bilinear four node quadrilateral element, the SFEM is variationally consistent only for one or an infinite number of subcells (Fig 2), but always energetically consistent The compliance of the resulting stiffness matrix increases with the number of subcells, as the stress error, total energy and sensitivity to volumetric locking On the contrary, the displacement error decreases with an increasing number of subcells [52,65], see Fig In order to obtain both accurate displacements and avoid volumetric locking selective reduced integration was extended to SFEM, where one subcell is used for the volumetric part of the deformation, and nc > subcells are used for the deviatoric part This permits to both benefit from the insensitivity to locking of the one subcell formulation, while retaining higher displacement accuracy through the use of multiple subcells for the deviatoric deformation [43,62] Reasons for the superconvergent behavior of the one subcell Q4 SFEM in the presence of singularities are linked to that of reduced integration in standard FEM [109], to which it is equivalent In [65], the well-known L-shape problem and a simple crack problem were solved for various numbers of subcells The numerical results showed that for the linear elastic crack problem, the convergence rate attained by the one subcell version reaches 1.0 in the energy (H1) norm, as opposed to the theoretical rate of 12 A summary of the properties outlined above are shown in Figs and The one subcell element is only a quasi-equilibrium element, not a full equilibrium element Therefore, there is no guarantee that the energy is overestimated for all cases To date, only a proof of the equivalence of the one subcell four node quadrilateral element is available, from Ref [66], and numerical experiments indicate that ‘‘most of the time”, this overestimation property is verified To fully conclude on this point, more theoretical and numerical work is required S.P.A Bordas et al / Computers and Structures 88 (2010) 1419–1443 1421 Fig Summary of properties of the SFEM The ++ signify greater accuracy, and the ÀÀ lesser accuracy See [65,52] for details 1.3.2 Computational cost The computational cost of the method was studied in [53,67,65], which showed that thanks to the lack of isoparametric mapping (no Jacobian inversion needs to be performed), the SFEM is slightly more efficient than the standard FEM, although this difference can be considered quite minor (6% for the four-subcell element to 20% less expensive for the one subcell element6) The following section discusses how strain smoothing can be successfully applied to solve plate and shell structures, material non-linearities and be useful on polygonal meshes 1.4 Extension to plates, shells, elasto-plasticity and polygonal meshes The smoothed FEM was extended to plate and shell analyses in [67,64], plasticity in [66] and polygonal meshes [30] 1.4.1 Plates and shells In plate and shell analyses, curvature smoothing, introduced by Wang and Chen [98] removes shear locking from Mindlin–Reissner As all computational cost results based on numerical trials, these figures must be taken cautiously since they are implementation-dependent A careful algorithmic complexity analysis should be carried out in order to be fully conclusive plate formulations, which is due to the inability of the approximation functions to reproduce the Kirchhoff mode, and the inability of the method to satisfy pure bending exactness (BE) – this is the analogue of the linear exactness/consistency mentioned above for continuum meshfree approximations as seen in [22] Other techniques to remove locking in meshfree plate discretizations are given in [84,40,47,48,51,73] Recent work on cracking shells is also of interest [70] By extending curvature smoothing in the context of MITC4SFEM Mindlin–Reissner plate elements, we show in [67,64] that locking is suppressed in the thin plate/shell limit, and that very distorted meshes yield much more accurate results than other reference elements from the literature 1.4.2 Polygonal meshes Liu’s team recently proposed an extension of the SFEM to arbitrary polygons (nSFEM) with success [30] This extension contributes to research on polygonal approximation schemes, for which interest has been steadily increasing in the past few years See for instance the very interesting work on polygonal interpolation presented in Refs [3,4,89,61,92,91,88,90] and their extension to partition of unity enrichment in the very recent article by Sukumar’s group [94] 1422 S.P.A Bordas et al / Computers and Structures 88 (2010) 1419–1443 Fig Scaling of the displacement, stress error, stiffness, stability and total energy depending on the number of subcells See [65,52] for details Very recently, SFEM and polygonal approximation schemes (Laplace and Wachspress) were coupled to obtain versatile smoothed polygonal finite elements, by Natarajan and Bordas [59] and smoothed extended finite elements on polygons in [60] 1.5 Strain smoothing and partition of unity enrichment We propose in this paper the extension of SFEM to partition of unity enrichment Partition of unity enrichment proposed by Melenk and Babuška [56] and Babuška and Melenk [5] has led to quite a revolution in computational mechanics by allowing the exact reproduction of arbitrary functions within subsets of the computational domain This led to two streams of research:  The generalized finite element method (GFEM) proposed in the thesis of Melenk [55] and the papers [85,86]  The extended finite element method (XFEM) of Belytschko and Black [11] and Moës et al [57] is the first to propose the use of discontinuous enrichment It is not in the scope of this paper to review recent advances in partition of unity methods, and the interested reader is referred to the literature, for instance [21] and Bordas and Legay [17] For the purpose of this communication, it suffices to say that extended finite element methods still suffer from some difficulties: (1) when the approximation is discontinuous or non-polynomial in an element special care must be taken for numerical integration; (2) the low order of continuity of the solution leads to poor accuracy of the derivatives close to regions of high gradient, such as crack fronts [106] which motivated very recent work on adaptivity for GFEM [6,87] and XFEM by Bordas and Duflot [16], Bordas et al [20] as well as [77] The methods that we propose in this paper aim at alleviating some of these difficulties by7: (1) Decreasing the complexity of sub-integration in XFEM by integrating over the boundary of the elements split by discontinuities (material interfaces, cracks): Figs and Note the introduction by Ventura [96] of a technique to avoid integration cells based on replacing non polynomial functions by ‘‘equivalent” polynomials However, this technique, We consider only four-noded quadrilaterals in this paper, although this is not an intrinsic limitation S.P.A Bordas et al / Computers and Structures 88 (2010) 1419–1443 1423 Fig Integration in an element with a straight discontinuity (left) Standard decomposition of an element for integration of a discontinuous weak form for XFEM: Gauß points are introduced within each (dotted lines) triangle to ensure proper integration of the discontinuous displacement field (right) Absence of decomposition allowed by the strain smoothing technique Fig Integration in an element where a discontinuity changes directions (left) Standard decomposition of an element for integration of a discontinuous weak form for XFEM: Gauß points are introduced within each (dotted lines) triangle to ensure proper integration of the discontinuous displacement field (right) Absence of decomposition allowed by the strain smoothing technique in its initial form, still has limitations Strain smoothing provides an elegant solution to this problem by transforming interior integration into boundary integration (2) Avoiding the need of integrating the singular functions present in the XFEM stiffness matrix in linear elastic fracture mechanics With strain smoothing, the derivatives of the shape functions are no longer necessary, hence the 1/r term does not appear (3) Improving the accuracy of the stress field in the vicinity of the crack fronts It was seen earlier that the one subcell strain smoothing in FEM leads to higher accuracy for the gradient 1424 S.P.A Bordas et al / Computers and Structures 88 (2010) 1419–1443 (4) Increase accuracy of the stress intensity factors, through more accurate stress fields and a new smoothed domain integral (Sm J) technique under development (5) Improve the behavior of XFEM in incompressible settings leveraging the locking-free properties of the one subcell SFEM elements and/or selective cell-wise smoothing [43,62] (6) Increase the efficiency of XFEM, and decrease its sensibility to mesh distortion thanks to the absence of isoparametric mapping enabled by strain smoothing 1.6 Outline Section recalls the basics behind the SFEM, without going into much detail, since this is described lengthily in other papers [53,52] Section provides definitions and nomenclature In Section 4, the SFEM is applied successfully to quasi-incompressible elasticity and elasto-plasticity Key results for plates and shell formulations are given in Section Extension to three dimensions (3D) are proposed in Section The basic theory behind the coupling of strain smoothing with partition of unity enrichment is developed in Section and results are shown for cracked continuum The method is named the ‘‘smoothed extended finite element method (SmXFEM)” Section presents preliminary results on the extension of the smoothed XFEM to arbitrary cracks in plates Conclusions, future work, and open problems are presented in Section Fig Calculation of the smoothed discretized gradient operator Basics of strain smoothing in FEM The strain smoothing method (SSM) was proposed in [22] where the strain is written as the divergence of a spatial average of the standard (compatible) strain field – i.e symmetric gradient of the displacement field Elements are divided into subcells, as shown in Figs 1, and The strain field ~ehij , used to compute the stiffness matrix is computed by a weighted average of the standard strain field ehij At a point xC in an element Xh , ~ehij xC ị ẳ Z Xh ehij xịUx xC ịdx ð1Þ where U is a smoothing function that generally satisfies the following properties [107] (see Fig 7) U P and Uẳ in XC AC Z Xh Uxịdx ẳ 2ị and U ẳ elsewhere 3ị To use Eq (1), the subcell containing point xC must first be located in order to compute the correct value of the weight function U The discretized strain field is computed, through the so-called e defined by (see Fig smoothed discretized gradient operator B, for a schematic representation of the construction) e C xC ịq ~eh xC ị ẳ B 4ị where the q are unknown displacements coefficients defined at the nodes of the finite element, as usual The smoothed element stiffness matrix for element e is computed by the sum of the contributions of the subcells (Fig 6)8 Fig The weight function is defined, for each subcell as constant equal to the inverse of the area of the subcell and zero elsewhere This permits transforming the domain integral into a boundary integral over the boundary of the subcell If a single subcell is used, integration over the boundary of the finite element is recovered This function has been used in all published work on the SFEM, to date This figure shows the weight function, U used for subcell X1 See Eqs (2) and (3) h The subcells XC form a partition of the element X ee ¼ K nc Z X C¼1 XC e T DB e C dX ¼ B C nc X C¼1 e T DB eC B C Z XC dX ¼ nc X e T DB e C AC B C ð5Þ C¼1 where nc is the number of the smoothing cells of the element e C is constant over each XC and The strain–displacement matrix B is of the following form h eC ¼ B e C1 B e C2 B e C3 B e C4 B i ð6Þ 1425 S.P.A Bordas et al / Computers and Structures 88 (2010) 1419–1443 e CI where for all shape functions I f1; ; 4g, the  submatrix B represents the contribution to the strain–displacement matrix associated with shape function I and cell C and writes (see Fig 6) 8I f1; 2; ; 4g; 8C f1; 2; ncg e CI ¼ B AC Z nT ðxÞNI ðxÞdS ¼ SC Z SC nx 40 ny ny 5ðxÞNI ðxÞdS nx ð7Þ or, since Eq (7) is computed on the boundary of XC : NI ðxGb Þnx nb X B e B CI ðxC Þ ¼ @ AC b¼1 NI ðxGb Þny CC NI ðxGb Þny Alb G NI ðxb Þnx ð8Þ Fig Mesh with element distortion of varying magnitude a C where xGb and lb are the center point (Gauß point) and the length of CCb , respectively Table Element nomenclature k represents the number of smoothing cells Remark Continuum SCkQ4 [53,52,65]  The resulting matrix is symmetric positive definite, and sparse  No isoparametric mapping is required, since integration is performed on the boundary of the smoothing cells In the case of bilinear shape functions, one integration point on each edge is sufficient  The shape functions need to be computed along the edges of the smoothing cells, this is done by simple linear interpolation of the underlying elements’ shape functions, as detailed in [65,66]  The Hu–Washizu variational principle is an adequate formalism for the SFEM, this is detailed in [52,65] The equivalence with an assumed stress formulation is also proved in [65]  If the shape functions of the underlying element are linear (twonoded bar, three-noded triangle, four-noded tetrahedron), SFEM coincides exactly with FEM, see [52,65]  The one subcell version of the four-noded quadrilateral (Q4) SFEM is equivalent to an underintegrated Q4 [52] and to a quasi-equilibrium element [66] It is therefore insensitive to locking, but exhibits zero-energy modes, while enabling greater stress accuracy, as mentioned in the introduction and shown in the numerical examples below SCkH8 Plate Shell MISCk [67] MISTk [64] nomenclature used is given in Table for convenience and ease of reading The following section is concerned with applications to linear incompressible elasticity and elasto-plasticity Applications to incompressible elasticity and elasto-plasticity 4.1 Driven cavity flow We tackle the problem of the driven cavity flow Three boundaries shown in Fig are clamped and a unit tangential velocity distribution (horizontal) is prescribed along the upper boundary The material parameters are Young’s modulus E = 105 and Poisson’s ratio m = 0.49999 The solution to this problem is the steady-state solution of the Stokes flow problem in a driven cavity [28] Figs 10 and 11 show the velocity and pressure results for a 40  40 mesh of quadrilateral elements Due to incompressibility, Definitions 3.1 Mesh distortion In this paper, the influence of mesh distortion will be studied repeatedly We give here the basic notations The idea is to distort the mesh in a way which would naturally occur in real situations such as shear banding and large deformations The distortion equation is such that the nodal position is perturbed by a random amount in both coordinate directions, as follows: x0 ẳ x ỵ r c aDx 9aị y0 ẳ y ỵ rc aDy 9bị where r c is a random number between À1.0 and 1.0, a is the magnitude of the distortion and Dx; Dy are initial regular element sizes in the x- and y-directions, respectively (see [65] for more details) Fig exemplifies element distortion for different magnitudes of a Note that the higher a, the more pronounced the distortion 3.2 Element nomenclature We will study several elements in the rest of this paper We concentrate on four-noded quadrilateral elements (Q4) for all two-dimensional examples and eight-noded hexahedral elements (H8) for the three-dimensional examples A summary of the Fig Driven cavity and an example of coarse mesh 1426 S.P.A Bordas et al / Computers and Structures 88 (2010) 1419–1443 selective integration must be used for the standard Q4 element [54] For comparison, besides the SFEM with a single subcell presented above and denoted here by one subcell, a division of the element into two subcells, three subcells and four subcells are also considered From the analysis, we remark that the stabilization of the velocity field shown in Fig 10 is obtained with all four elements 1.2 QEE(1−subcell) SSM(2−subcell) SSM(3−subcell) SSM(4−subcell) SIMQ4 0.8 u velocity 0.6 0.4 0.2 −0.2 −0.4 0.2 0.4 0.6 y coordinate 0.8 Remark Note that the single cell element gives the best pressure results Fig 11 indicates that while the others display strong pressure dissipation at boundaries, the presented method can reproduce well the sinusoidal-like features of the solution This is related to the fact (see [43,62]) that using the one subcell method for the volumetric (pressure) part of the strain is sufficient to enforce the incompressibility constraint while remaining stable In Refs [43,62], this technique is named cell-wise selective integration ‘‘Selective”, in this case refers to ‘‘one” or ‘‘several” cell integration 0.2 4.2 Cook’s membrane QEE(1−subcell) SSM(2−subcell) SSM(3−subcell) SSM(4−subcell) SIMQ4 0.15 0.1 This benchmark problem in Ref [29], shown in Fig 12, refers to a clamped tapered panel is subjected to an in-plane shearing load, resulting in deformation that is dominated by a bending elastic response Assuming plane strain conditions, Young’s modulus and Poisson’s ratio m = 0.4999 or 0.4999999, Fig 13 plots the vertical displacement at the right top corner It shows that the FEM Q4 element provides poor results while the other elements based on strain smoothing formulations remain reliable for nearly incompressible materials Of course, selective reduced integration also stabilizes the Q4 formulation v velocity 0.05 −0.05 −0.1 −0.15 4.3 Thick elasto-plastic cylinder under uniform constant pressure −0.2 0.2 0.4 0.6 x coordinate 0.8 Fig 10 Variation of the velocity fields: (a) the horizontal velocity (u) in the ydirection at x = 0.5; (b) the vertical velocity (v) in the x-direction at y = 0.5 QEE stands for quasi-equilibrium element, i.e the one subcell Q4; SSM stands for strain smoothing method; SIM stands for selective integration method 5 x 10 QEE(1−subcell) SSM(2−subcell) SSM(3−subcell) SSM(4−subcell) SIMQ4 Consider a thick cylinder under pressure, the material is assumed to follow von Mises plasticity Hill [41] shows the exact solution for this problem The yield stress is ry ¼ 24, Poisson’s ratio m = 0.49999, and the internal pressure p is varied between and 20 We denote the plastic radius, i.e the value of r below which the cylinder undergoes plastic deformation by c The return mapping algorithm [82] combined with four-noded smooth finite elements is employed to simulate the plastic behavior Here we evaluate the development of the plastic domain in the case of isotropic hardening plasticity Fig 14 presents elastic–plastic results for the thick cylinder in three cases: (a) perfectly plastic material, H = 0, (b) hardening case, Pressure −1 −2 −3 −4 −5 0.2 0.4 0.6 x coordinate 0.8 Fig 11 The pressure distributions along y = 0.5 QEE stands for quasi-equilibrium element, i.e the one subcell Q4; SSM stands for strain smoothing method; SIM stands for selective integration method Fig 12 Cook’s membrane and initial mesh S.P.A Bordas et al / Computers and Structures 88 (2010) 1419–1443 field is smoothed in a continuum SFEM formulation [53], in a plate SFEM formulation, the curvature is smoothed, using the same techniques as outlined above (see Eq (1)) (2) Approximating the shear strains with interpolation functions independent of the bending term (namely the Mixed Interpolation of Tensorial Components: MITC4 [8]) 10 Top corner vertical displacement v Q4 SIM 1−Subcell 2−Subcell 3−Subcell 4−Subcell Remark Note that the curvature smoothing need only be used on the shear term Remark As shown below, the elements obtained are free of shear locking However, this is not surprising, since they are based on the MITC4 elements, which serve this purpose very well The major advantage of the method, as shall be seen in the numerical experiments, is to remain accurate for very distorted meshes, which is not the case of the MITC4 element, due to the need for an isoparametric mapping 10 20 30 40 50 Number of elements per side 60 1427 70 10 Definition We will call MISCk the elements with k smoothing cells, resulting from Mixed Interpolation with Smoothing Top corner vertical displacement v We only present a simple plate bending example since the MISCk elements are already studied in detail in Ref [67] to which the reader is referred to for more details Q4 SIM 1−Subcell 2−Subcell 3−Subcell 4−Subcell H = E/3 and (c) H = 2E/3 The numerical results show that the development of the plastic domain slows down with increasing hardening factor, H Also, the proposed method agrees well with the exact solution [41] and other published results [68,27] 5.1.2 Clamped plate subjected to a center point load A clamped plate subjected to a center point load F = 16.3527 with geometry and material parameters: length L = 100, thickness t = 1, Young’s modulus E = 104, Poisson’s ratio m = 0.3 Only a quarter of the plate is modelled with a mesh of  elements To study the effect of mesh distortion on the results, the position of interior nodes is perturbed In the results below, a large value for the distortion parameter a signifies more distortion (see Eq (9a) for node movement equation and typical meshes in Fig 8) For regular meshes, the MISCk results are only slightly more accurate than both the Q4 element with selective reduced integration and the MITC4 element For extremely distorted meshes (large a’s), our elements behave significantly better and conserve their convergence property Here, the MISC1 element gives the best result However, this element contains two zero-energy modes In simple problems, these hourglass modes can be automatically eliminated by the boundary conditions However, this is not in general the case Otherwise, the MISC2, MISC3 and MISC4 elements retain a sufficient rank and give good results compared to other elements in the literature, as shown in Fig 15 and Table Application to plates and shells 5.2 Extensions to shell elements 5.1 Mindlin–Reissner plates The reader is referred to Nhon et al [64] for more details on the formulation The basic idea is to use smoothing for the bending and membrane parts of the shell stiffness to construct the element stiffness matrix, while leaving the shear stiffness unchanged The underlying finite element formulation is again based on the MITC4 of Bathe and Dvorkin [36,9] Particularly clear overviews can be found, e.g, in the textbooks [108,7], among other reference texts 10 20 30 40 50 Number of elements per side 60 70 Fig 13 Vertical displacement at the corner top of Cook’s membrane problem: (a) m = 0.4999 and (b) m = 0.4999999 SIM stands for selective integration method (for Q4) Note that all SFEMs underestimate the energy except for the one subcell element Also, all SFEM solutions are bounded by the displacement FEM on one side and the exact solution on the other 5.1.1 Extension of SFEM to plates Similarly to the continuum case, smoothing is used to stabilize the formulation In plate formulations, the stiffness matrix K can be decomposed in a bending part Kb and a shear deformation part Ks as follows – see [67] for a detailed derivation and the definition of the discretized differentiation operators Bb and Bs and material matrices Db and Ds – K¼ Z Z ðBb ịT Db Bb dX ỵ Bs ịT Ds Bs dX : Xe Xe |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} Kb ð10Þ Ks For the purpose of this paper, it is sufficient to understand that the SFEM can be generalized to plate formulations by (1) The curvature smoothing method, proposed by Chen et al [22] to the first term of Eq (10) In other words, as the strain Definition We will call MISTk the shell elements with k smoothing cells, resulting from Mixed Interpolation with Smoothing Consider a cylindrical shell with rigid end diaphragms subjected to a point load at the center of the cylindrical surface as shown in Fig 16 Only one eighth of the cylinder shown in Fig 16 is modeled, by symmetry The expected deflection under a concentrated unit load is 1.8425  10À5 [76] The problem is discretized with N  N MITC4 or MISTk elements and the influence of mesh distortion is studied The meshes used are shown in Fig 16 1428 S.P.A Bordas et al / Computers and Structures 88 (2010) 1419–1443 18 19 Theory(H=0) QEE(H=0) QEE(H=7000) QEE(H=14000) 17 16 17 16 Hoop stress σθ(r) Hoop stress σθ(r) 15 14 13 12 15 14 13 11 12 10 11 10 Theory(H=0) QEE(H=0) QEE(H=7000) QEE(H=14000) 18 1.2 1.4 1.6 Radial distance r 1.8 1.2 1.4 1.6 Radial distance r 1.8 24 22 Hoop stress σθ(r) 20 18 16 14 Theory(H=0) QEE(H=0) QEE(H=7000) QEE(H=14000) 12 10 1.2 1.4 1.6 Radial distance r 1.8 Fig 14 View of the hoop stress for the perfectly plastic behavior: (a) p = 12; (b) p = 14; (c) p = 18 QEE stands for quasi-equilibrium element (see [66] where it is proven equivalent to the one subcell SFEM for Q4’s) SIM stands for selective integration Q4 Figs 17 and 18 illustrate the convergence of the displacement at the center point and the strain energy, respectively, for the MITC4 element and our MISTk elements for regular meshes The MISTk elements are slightly more accurate than the MITC4 element In Table 3, the normalized displacement at the center point of the smoothed element are compared to other elements in the literature Note that the MISTk elements are always more accurate than the elements compared with The error in the strain energy is summarized in Table The advantage of the MISTk elements becomes more relevant for distorted meshes, see Figs 19, 20 and Tables 5, 6, where it is clear that the smoothed elements are significantly more accurate than the elements compared with for distorted meshes 0.26 CRB1 CRB2 S1 S4R DKQ MITC4 MISC1 MISC2 MISC3 MISC4 Exact Central deflection wc/(pL /100D) 0.24 0.22 0.2 0.18 0.16 0.14 0.12 Three-dimensional smoothed finite elements 0.1 0.08 −1.5 −1 −0.5 0.5 Distortion parameter 1.5 Fig 15 Comparison with other plate elements through the center deflection with mesh distortion, a No distortion is associated with a distortion parameter equal zero The smoothed elements, MISCk are all quite insensitive to mesh distortion, with an approximate relative scatter of 15% around the solution obtained with a regular mesh, for the worst element (MISC4) See also Table 6.1 Extension to three dimensions As shown in Fig 21, the idea behind the SFEM in 3D is exactly identical to that in 2D The calculation of the shape functions on the boundary of the subcells is shown in Fig 22 It can be seen from this figure that when the shape functions have to be evaluated at the Gauß points located on the faces of the subcells, they are eval- 1429 S.P.A Bordas et al / Computers and Structures 88 (2010) 1419–1443 Table The central deflection wc =pL4 =100Dị; D ẳ Et3 =121 m2 ị with mesh distortion, a for thin clamped plate subjected to uniform load p See Eq (9a) for the node movement equation CRB1 [100] CRB2 [100] S1 [100] S4R [1] DKQ [46] MITC4 MISC1 MISC2 MISC3 MISC4 Exact solution a = À1.249 a = À1.00 a = À0.5 a = 0.00 a = 0.5 a = 1.00 a = 1.249 0.1381 0.2423 0.1105 0.1337 0.1694 0.0973 0.1187 0.1151 0.1126 0.1113 0.1265 0.1390 0.1935 0.1160 0.1369 0.1658 0.1032 0.1198 0.1164 0.1144 0.1130 0.1265 0.1247 0.1284 0.1209 0.1354 0.1543 0.1133 0.1241 0.1207 0.1189 0.1174 0.1265 0.1212 0.1212 0.1211 0.1295 0.1460 0.1211 0.1302 0.1266 0.1249 0.1233 0.1265 0.1347 0.1331 0.1165 0.1234 0.1418 0.1245 0.1361 0.1323 0.1305 0.1287 0.1265 0.1347 0.1647 0.1059 0.1192 0.1427 0.1189 0.1377 0.1331 0.1309 0.1288 0.1265 0.1249 0.1947 0.0975 0.1180 0.1398 0.1087 0.1347 0.1287 0.1260 0.1227 0.1265 Fig 16 Pinched cylinder with diaphragm boundary conditions (P = 1; R = 300; L = 600; t = 3; m = 0.3; E =  107) uated by linear interpolation between the eight nodes of the element Figs 23 and 24 show the integration points located on the faces of the integration cells As in 2D, the smoothed element stiffness matrix is the sum over the subcells of the contribution from each subcell, which is constant: ee ¼ K nc Z X C¼1 XC e T DB e C dX ¼ B C nc X Cẳ1 e T DB eCVC B C 11ị e C is constant over each XC and is The strain–displacement matrix B of the following form h eC ¼ B e C1 B e C2 B e C3 B ÁÁÁ e C8 B i ð12Þ e CI where for all shape functions I f1; ; 8g, the  submatrix B represents the contribution to the strain–displacement matrix associated with shape function I and cell C and writes 1430 S.P.A Bordas et al / Computers and Structures 88 (2010) 1419–1443 Table The strain energy under the load for regular meshes Mesh number MITC4 Present elements 4Â4 8Â8 12  12 16  16 20  20 24  24 8.4675eÀ7 1.6958eÀ6 1.9937eÀ6 2.1196eÀ6 2.1836eÀ6 2.2210eÀ6 Normalized deflection w 0.9 Exact MITC4 MIST1 MIST2 MIST4 0.8 0.7 0.6 MIST1 MIST2 MIST4 1.0837eÀ6 1.8462eÀ6 2.0891eÀ6 2.1837eÀ6 2.2296eÀ6 2.2556eÀ6 1.0078eÀ6 1.7970eÀ6 2.0579eÀ6 2.1630eÀ6 2.2147eÀ6 2.2444eÀ6 8.8394eÀ7 1.7230eÀ6 2.0118eÀ6 2.1320eÀ6 2.1926eÀ6 2.2278eÀ6 0.5 0.4 0.3 10 14 12 16 Index mesh N 18 20 22 24 Fig 17 Regular meshes: convergence of deflection under the load −6 2.4 x 10 2.2 MITC4 MIST1 MIST2 MIST4 Strain energy 1.8 Fig 19 The convergence of deflection for the set of distorted meshes Larger values of a signify larger amounts of distortion 1.6 1.4 −6 2.4 1.2 2.2 10 12 14 16 Index mesh N 18 20 22 24 Fig 18 Regular meshes: the convergence of strain energy under the load 8I f1; 2; ; 8g; 8C f1; 2; ncg e CI ¼ B VC Z SC MITC4(s=0.5) MIST2(s=0.1) MIST2(s=0.2) MIST2(s=0.3) MIST2(s=0.4) MIST2(s=0.5) 1.8 1.6 1.4 1.2 nx 0 60 Z 60 T n ðxÞNI xịdS ẳ SC ny 40 ny nz 07 7 nz 7ðxÞNI ðxÞdS 07 7 ny nz nx nx Strain energy 0.8 x 10 ð13Þ 0.8 10 12 14 16 Index mesh N 18 20 22 24 or, since Eq (13) is computed on the boundary of XC and one Gauß point is sufficient for an exact integration: Fig 20 The convergence of strain energy for the set of distorted meshes Notice that the standard finite element formulation (MITC4) does not capture the correct strain energy for severely distorted meshes (a = 0.5), whereas the two-subcell MIST element (MIST2) provides distortion-independent solutions Table Normal displacement under the load for regular meshes Table Normal displacement under the load for distorted meshes Higher values of a signify higher distortion Mesh 4Â4 8Â8 12  12 16  16 20  20 24  24 MITC4 0.3712 0.7434 0.8740 0.9292 0.9573 0.9737 Mixed [83] 0.399 0.763 0.935 - QPH [14] 0.370 0.740 0.930 - SRI [42] 0.373 0.747 0.935 - Mesh number Present elements MIST1 MIST2 MIST4 0.4751 0.8094 0.9159 0.9574 0.9774 0.9889 0.4418 0.7878 0.9022 0.9483 0.9709 0.9840 0.3875 0.7554 0.8820 0.9347 0.9612 0.9767 4Â4 8Â8 12  12 16  16 20  20 24  24 MITC4 (a = 0.5) 0.3539 0.6950 0.7402 0.8488 0.8960 0.8718 MIST2 a = 0.1 a = 0.2 a = 0.3 a = 0.4 a = 0.5 0.4370 0.7777 0.8941 0.9397 0.9614 0.9746 0.4342 0.7786 0.8938 0.9394 0.9631 0.9739 0.4331 0.7839 0.8945 0.9344 0.9586 0.9764 0.4261 0.7803 0.8959 0.9402 0.9628 0.9755 0.4398 0.7860 0.8930 0.9350 0.9601 0.9672 1431 S.P.A Bordas et al / Computers and Structures 88 (2010) 1419–1443 Table The strain energy under the load for various levels of mesh distorsion Higher values of a signify higher distortion Mesh number MITC4 (a = 0.5) 4Â4 8Â8 12  12 16  16 20  20 24  24 8.1512eÀ7 1.6007eÀ6 1.7047eÀ6 1.9549eÀ6 2.0636eÀ6 2.0078eÀ6 MIST2 a = 0.1 a = 0.2 a = 0.3 a = 0.4 a = 0.5 1.0065eÀ6 1.7911eÀ6 2.0591eÀ6 2.1642eÀ6 2.2142eÀ6 2.2445eÀ6 1.0001eÀ6 1.7932eÀ6 2.0585eÀ6 2.1636eÀ6 2.2182eÀ6 2.2431eÀ6 9.9738eÀ7 1.8054eÀ6 2.0601eÀ6 2.1521eÀ6 2.2077eÀ6 2.2488eÀ6 9.8127eÀ7 1.7971eÀ6 2.0634eÀ6 2.1654eÀ6 2.2175eÀ6 2.2466eÀ6 1.0129eÀ6 1.8102eÀ6 2.0567eÀ6 2.1534eÀ6 2.2113eÀ6 2.2276eÀ6 Fig 23 Smoothing cells in 3D Fig 21 Example of a smoothed finite element in 3D with its smoothing cells NI ðxGb Þnx B B nb B X B e CI xC ị ẳ B B G V C b¼1 B B N I ðxb Þny B @ NI ðxGb Þnz N I ðxGb Þny NI ðxGb Þnx NI ðxGb Þnz 0 C C C NI ðxGb Þnz C CACb C C C G NI ðxb Þny A NI ðxGb Þnx ð14Þ where xGb and ACb are the center point (Gauß point) and the area of face SCb , respectively Inserting Eqs (14) and (12) into Eq (11) yields a matrix expression for the smoothed stiffness matrix which can easily be implemented Definition We will call SCkH8 the eight-noded hexahedral elements with k smoothing cells, resulting from Mixed Interpolation with Smoothing Fig 22 The value of the shape functions at the vertices of the subcells The list of numbers represents the value of the shape functions at the vertex in question, i.e ðN ; N ; N ; N ; N ; N ; N ; N Þ 1432 S.P.A Bordas et al / Computers and Structures 88 (2010) 1419–1443 Fig 24 Smoothing cells of the element with the indices of vertices and surfaces 6.2 A beveled cantilever beam Deformed 3D mesh Remark These three-dimensional results are only preliminary, and more detailed studies are under way Of particular interest is the development of stabilized formulations for eight-noded hexahedral elements based on the combination of the one- and foursubcell elements 0.9 0.8 0.7 0.6 y axis A beveled cantilever beam subjected to a uniform unit load is considered as an example, as shown in Fig 25 The material properties are: Young’s modulus E = 1, Poisson’s ratio m = 0.25 In this problem, the exact solution is unknown Therefore, the reference strain energy is taken as U = 0.305686 from Ref [2] Additionally, we also estimate the exact solution by Richardson’s extrapolation [75] This value is 0.303698 which is 0.65% away from the reference solution of Moitinho De Almeida and Almeida Pereira [2] (see Figs 26 and 27): Figs 28 and 29 illustrate the convergence of strain energy and the rate of convergence of the strain energy, respectively Results are shown also in Table It can be seen that all smooth finite element solutions: SCkH8 for k = 1, 2, 4, are in good agreement with the exact energy and converge faster than the standard eightnoded hexahedral element, H8 (2   quadrature is used in the H8 element) 0.5 0.4 0.3 0.2 0.1 0 0.2 −0.2 0.4 0.6 x axis −0.4 0.8 z axis Fig 26 Undeformed and deformed configurations Strain smoothing for extended finite element methods 7.1 The basic idea Fig 25 A beveled cantilever beam under a uniform load We show in this part of the paper how the incorporation of strain smoothing into the extended finite element method (XFEM) can be implemented XFEM was introduced by Belytschko’s group in 1999 [11,13] and permits arbitrary functions to be incorporated in a FE approximation, which leads to greater flexibility in modelling moving boundary problems XFEM was used successfully for crack propagation and other fields in computational physics [23,25,24,26,33] Recently, open source XFEM codes were released to help the development of the method [21] and numerical implementation and efficiency aspects were studied [35] XFEM is quite a robust and popular method which is now used for industrial problems [18,19,105,104,103,102] and under implementation by leading computational software companies However, some areas for improvement remain, and strain smoothing may be a suitable remedy to: S.P.A Bordas et al / Computers and Structures 88 (2010) 1419–1443 1433 Fig 27 Displacement in the y-direction and von Mises stress field for regular meshes  Simplify integration of discontinuous functions by replacing domain integration by boundary integration  Inherit the interesting properties of SFEM shown above In particular, the suppression of volumetric locking for incompressible materials [31,50]  Increase stress and stress intensity factor accuracy This is also the subject of recent work on derivative recovery [106] and a posteriori error estimation in XFEM [20,16,34] Duality techniques such as those proposed in [81,80,78,79,69] are promising tools to devise goal-oriented error estimates where the stress intensity factor (SIF) is the quantity of interest  Improve the accuracy of the domain integral calculations through smoothing In XFEM, discontinuous and near-tip, non-polynomial enrichment is commonly used for fracture mechanics Therefore, smoothing must now be performed on discontinuous and non-polynomial approximations As we will see, this has important implications Let us take the case of linear elastic fracture mechanics (LEFM) In this case, the first enrichment function, H, is discontinuous through the crack, and four near-tip, asymptotic fields noted fBa g16a64 , spanning the near-tip displacement fields (Westergaard’s 1434 S.P.A Bordas et al / Computers and Structures 88 (2010) 1419–1443 10 −0.45 10 Strain energy set Nf belong to the elements that contain a crack tip These nodes are enriched with the Heaviside and near-tip (branch functions) fields, respectively In this paper, the value of the Heaviside function is constant in each subcell (including its boundary) and equals the sign of the signed distance function from the center of the cell, to the crack: Reference Estimated sol H8 SC1H8 SC2H8 SC4H8 SC8H8 0.42 0.48 10 Hxị ẳ ỵ1 if the center of the subcell is above the crack face À1 if the center of the subcell is below the crack face ð16Þ −0.51 10 For two-dimensional elasticity, using the usual polar coordinates (r, h), defined by the crack (Fig 31, e.g., [37,17]), the branch functions are given by −0.54 10 0.57 Ba r; hịị16a64 ẳ 10 10 10 Number of DOF 10 pffiffiffi Note that, in (17), r sin 2h, is discontinuous across the crack face The element stiffness matrix writes as the sum of the contributions from each of the nc subcells of the element e ee ¼ K −0.4 H8 SC1H8 SC2H8 SC4H8 SC8H8 −0.6 & ' pffiffiffi pffiffiffi h pffiffiffi h pffiffiffi h h r sin ; r cos ; r sin sin h; r cos sin h 2 2 ð17Þ Fig 28 The convergence of strain energy Note that the one subcell version overestimates the energy nc Z X C¼1 XC e T DB e C dX B C ð18Þ where C f1; 2; ; ncg is the number of the subcell XC e C in Eq (18) are constants over each subAll entries in matrix B cell XC – each of these entries are line integrals calculated along the boundaries of the subcells The stiffness matrix in Eq (18) can therefore be rewritten as −0.8 ee ¼ K −1 nc X e T DB eC B C C¼1 10 log (Error in energy norm) & Z dX ¼ XC nc X e T DB e C AC B C 19ị Cẳ1 R where AC ¼ XC 1dX is the area of the subcell XC , e C remains to be explicitly written, which is the topic of the folB lowing section −1.2 −1.4 7.2 The strain smoothing method for XFEM −2 −1.8 −1.6 −1.4 h −1.2 −1 −0.8 In the following, let us restrict to a single enrichment function, w, for simplicity Our goal is to write the smoothed discretized Fig 29 Rate of convergence of strain energy e C , for a subcell C in a given element strain–displacement matrix B The approximation used is identical to the standard XFEM and reads Table The strain energy for regular meshes uh xị ẳ Meshes Dofs H8 SC1H8 SC2H8 SC4H8 SC8H8 2Â2Â1 4Â4Â2 8Â8Â4 16  16  20  20  10 30 160 1008 7072 13,440 0.2652 0.2864 0.2970 0.3012 0.3019 0.3916 0.3173 0.3067 0.3044 0.3041 0.3081 0.3017 0.3023 0.3030 0.3032 0.2787 0.2920 0.2990 0.3019 0.3024 0.2774 0.2915 0.2988 0.3018 0.3023 solution), are added to the approximation These functions are now well-known and recalled below for clarity In XFEM, enrichment is extrinsic and resolved through additional degrees of freedom The enriched approximation takes the form [11,13,17] uh xị ẳ X I2Nfem N I xịqI ỵ X J2Nc N J xịHxịaJ ỵ X K2Nf NK xị X a Ba xịbK aẳ1 ð15Þ where aJ and bK are nodal degrees of freedom corresponding to the Heaviside function H and the near-tip functions, fBa g16a64 Nodes in set Nc are such that their support is split by the crack and nodes in X NI xịqI ỵ X NJ xịwxịaJ 20ị J2Nenr I2Nfem Nfem and Nenr are the set of standard nodes and enriched nodes, respectively The NI ’s are the finite element shape functions of standard nodes and the NJ ’s, the finite element shape function associated with enriched nodes In this paper, these functions are identical, i.e the same partition of unity is used for the standard and enriched part, but this is not necessary Nodal subtraction is now commonly used in enriched FEMs [13,21], and will be used throughout this paper9: uh xị ẳ X I2Nfem NI xịqI ỵ X NJ xịẵwxị wxJ ịaJ 21ị J2Nenr Nodal subtraction ensures that the approximation uh evaluated at an arbitrary node xI equals the value of the unknown coefficient at this node, i.e uI see, e.g., [17] for details The reader will note the very recent papers by Belytschko’s group [44,97] and Fries [39], which provide an improved enrichment scheme to avoid blending errors reported in the literature (e.g [23,45]) 1435 S.P.A Bordas et al / Computers and Structures 88 (2010) 1419–1443 Let us now derive the approximation for the strains, from the displacement approximation equation (21) Denote by eh ¼ ehij the discretized, enriched, strain field deduced by differentiation of Eq (21) X eh xị ẳ BIfem xịqI ỵ X BJenr xịaJ ẳ ẵBfem kBenr ẵq J2Nenr I2Nfem 22ị Note that we have replaced the dx ¼ dxdy by dX, which are equivalent Using the divergence theorem, and noting n ¼ ðnx ; ny Þ the outward unit normal to the smoothing cell XC and CC ¼ oXC , we obtain: eC B Ifem h e xị ẳ ẵBxfem ẵq The Bxfem matrix in Eq (22) includes two terms Bfem and Benr corresponding to the standard nodes (fem) and enriched nodes (enr) The Bfem term contains the first derivatives of the standard finite element shape functions: oN I ox Bfem ¼6 40 oNI oy oNI oy oNI ox 7 oẵN wxịwx ịị 6 Benr ¼ J ox oẵN I wxịwxJ ịị 7: oy oẵNJ wxịwxJ Þފ oy o½NJ ðwðxÞÀwðxJ Þފ ox Z h ij ðxÞ e X Uðx À xC Þdx ð25Þ where U is a smoothing function defined exactly as in the SFEM (see Fig 7) and recalled for convenience Z U P and & Ux xC ị ẳ Uxịdx ẳ ð26aÞ X 1=AC ; x XC 0; x R XC Z eC q Bxfem qUðx À xC ÞdX ¼ B xfem Xh Z Bxfem ðxÞdx ð27Þ ð28Þ Remark 10 At this point, it is easy to see that the smoothed eC symmetric discretized gradient operator, B xfem is constant (integral over the fixed spatial domain XC of the standard (compatible) symmetric discretized gradient operator Bxfem) eC eC The B xfem matrix in Eq (28) also includes two terms B fem and C e B enr corresponding to the standard nodes (fem) and enriched nodes e C term associated with node I in the ele(enr) The submatrix of B fem ment X reads: eC ¼ B Ifem AC XC e C , the submatrix associated Performing the same operations for B enr with node J writes eC ¼ B Jenr AC Z XC 6 oN I ox 60 oNI oy oNI oy oNI ox 7dX o½N J wxịwxJ ịị ox oẵN J wxịwxJ Þފ 7d oy o½N J ðwðxÞÀwðxJ Þފ oy oẵN J wxịwxJ ịị ox X 31aị Using the divergence theorem to transform area integration into line integration, we obtain: eC ¼ B Jenr AC Z CC nx ẵNJ wxị wxJ ịị ny ½NJ ðwðxÞ À wðxJ Þފ ny ½NJ wxị wxJ ịị 5dC nx ẵNJ wxị wxJ Þފ ð31bÞ eC The smoothed enriched stiffness matrix for subcell C, K xfem is computed by eC ¼ K xfem Z XC e T DB e C dX ¼ B e T DB e C AC B C C ð32Þ eC  B e C , AC is the area of the subcell We used the result of where B xfem e C out of the integral sign Remark 10 to take B ee The smoothed enriched element stiffness matrix K xfem is the C e sum of the K xfem , for all subcells, C ee ¼ K nc X e T DB eC B C Z C¼1 XC dX ¼ nc X e T DB e C AC B C 33ị Cẳ1 7.3 Nomenclature XC Z 30ị nx NI where nc is the number of smoothing subcells of the element eC where the smoothed matrix B xfem in Eq (27) is defined by eC ¼ B xfem AC ny NI ð26bÞ Substituting Eq (22) into Eq (25) and using Eqs (26b), (23) and (24), we obtain10 ~eh xC ị ẳ ny NI 5dC nx NI ð24Þ Now that the XFEM strain–displacement operators have been derived, their smoothed counterparts can be deduced as follows Let us write the smoothed strain field at an arbitrary point xC (this is identical to Eq (1)): ~ehij xC ị ẳ XC 23ị The Benr term is composed of the first derivatives of the product of the finite element shape functions with the enrichment function: J Z ẳ AC 29ị 10 Because the weight function, U is only non-zero on cell XC , the integral over the element X boils down to the integral over XC Definition 11 As in standard XFEM, there are four types of element as illustrated in Fig 30: Tip elements either contain the tip, or are within a fixed distance, renr of the tip, if geometrical enrichment is used [10,49] All nodes belonging to a tip element are enriched with the near-tip fields of Eq (17) Split elements are elements completely cut by the crack Their nodes are enriched with the discontinuous function H of Eq (16) Tip-blending elements are elements neighboring tip elements They are such that some of their nodes are enriched with the near-tip fields and others are not enriched at all Split-blending elements are elements neighboring split elements They are such that some of their nodes are enriched with the discontinuous function, H, and others are not enriched at all Standard elements are elements that are in neither of the above categories None of their nodes are enriched 7.4 Numerical validation of the smoothed extended finite element method This section only shows preliminary results of the smoothed XFEM for the calculation of 2D mixed mode stress intensity factors Let us first give the assumptions and numerical parameters that are used in the computations 1436 S.P.A Bordas et al / Computers and Structures 88 (2010) 1419–1443 Fig 30 Element categories in SmXFEM Note that sub-elements does not carry any degrees of freedom and are solely used for smoothing and numerical integration of the J integral [74,58], as is now usual in the XFEM literature (e.g see recent listing [21]) We used 16 Gauß points for all elements in the domain used to calculate the J integral 7.4.1.4 Material models Unless otherwise stated, in the following examples, the material is linear elastic with properties: Young’s modulus E = 103 and Poisson’s ratio m = 0.3 Plane strain conditions are assumed throughout Let us now examine the case of a mode I crack in 2D Fig 31 Illustration of normal and tangential coordinates for the crack In the above case, the jump function at the center of the subcell is H(x) = À1 7.4.2 Infinite plate under tension Consider an infinite plate containing a straight crack of length a and loaded by a remote uniform stress field r Along ABCD the closed form solution in terms of polar coordinates in a reference frame (r, h) centered at the crack tip is   h h 3h À sin sin 2 r   KI h h 3h ry r; hị ẳ p cos ỵ sin sin 2 r KI h h 3h rxy r; hị ẳ p sin cos cos 2 r K 7.4.1 Preliminaries, assumptions and parameters We will use four-noded quadrilaterals, that we note Q4 and three-noded triangular elements, which we note T3 The goal of the following numerical examples is to show preliminary comparisons between the standard XFEM to the smoothed XFEM presented above Let us first define the integration rules that will be employed 7.4.1.1 Numerical integration for XFEM Since the approximation differs from element to element, different integration rules are used For XFEM built on Q4 elements, we chose the following Gauß quadrature rules: Standard elements: four Gauß points Tip-blending elements: 16 Gauß points Split-blending elements: three Gauß points for each subelement Split elements: three Gauß points for each sub-element Tip elements: 13 Gauß points for each sub-element 7.4.1.2 Numerical integration for the smoothed XFEM  Standard elements: one subcell with one Gauß point per edge  Tip-blending elements: eight subcells with one Gauß points per edge  Split-blending elements: one subcell with one Gauß point per edge  Split elements: three subcells with one Gauß point per edge for each part of the split element (total of six subcells)  Tip elements: eight subcells with one Gauß point per edge 7.4.1.3 Numerical integration for the domain integrals The calculation of the stress intensity factors is done using the domain form rx r; hị ẳ pI cos 34aị 34bị 34cị The closed form near-tip displacement eld is: 21 ỵ mị p 2p 21 ỵ mị uy r; hị ẳ p 2p ux r; hị ẳ  K I p h r cos À 2m À cos2 E  K I pffiffiffi h r cos À 2m À cos2 E  h  h ð35aÞ 35bị p In the two previous expressions K I ẳ r pa denotes the stress intensity factor, m is Poisson’s ratio and E is Young’s modulus All simulations are performed with a = 100 mm and r = 104 N/mm2 on a square mesh with sides of length 10 mm The geometry, loading and boundary conditions are shown in Fig 32 The displacements of nodes on the bottom, right and top edges are prescribed by Eq (35a) A structured 50  50 mesh was used for the comparative study For the standard XFEM two types of elements were chosen, viz., three-noded triangular element (T3) and four-noded quadrilateral element (Q4) and for the smoothed XFEM four-noded quadrilateral elements were used The domain size is characterized by the local mesh spacing at the crack tip hlocal : r d ¼ r k hlocal , where rk is a scalar multiple and hlocal is the square root of the area of the tip element The normalized stress intensity factors for various domain sizes are given in Table and indicate that the smoothed XFEM leads to slightly more accurate stress intensity factors than XFEM(Q4) and are almost identical to those obtained from XFEM(T3) For smaller domain sizes, the computed SIFs are less accurate than for larger domain sizes, which is a usual behavior since for small domain 1437 S.P.A Bordas et al / Computers and Structures 88 (2010) 1419–1443 σ =1.0 L=2.0 a σ =1.0 H=1.0 Fig 33 Plate with edge crack under tension Fig 32 Infinite cracked plate under remote tension: geometry and loads rk ¼ J domain size size of tip element h XFEM(T3) XFEM(Q4) Smoothed XFEM rd local 2.5 1.0040 0.9738 1.0159 0.9943 0.9741 0.9993 3.5 0.9933 0.9741 0.9993 4.0 4.5 0.9934 0.9743 0.9934 0.9938 0.9741 0.9949 sizes, the influence of the (less accurate) stresses in the vicinity of the crack tip is larger For a sufficiently large domain, the J integral is virtually independent of the integration domain It is also seen that structured triangular meshes (XFEM(T3)) and the smoothed XFEM converge faster with increase in domain size as compared to the structured Q4 mesh (XFEM(Q4)) 7.4.3 Edge crack under tension A plate of dimension  is loaded by a tension r = over the top edge The displacements along the y-axis is fixed at the bottom right corner and the plate is clamped at the bottom left corner The geometry, loading and boundary conditions are shown in Fig 33 The reference mode I SIF is given by KI ¼ F a pffiffiffiffiffiffi r pa b where a is the crack length, b is the plate width and F À Á ical function given as for ba 0:6 : F a a2 a3 ¼ 1:12 0:231 ỵ 10:55 21:72 b b b b a4 ỵ 30:39 b Number of nodes 288 1152 3444 4608 5000 Crack size 0.1 0.2 0.3 0.4 0.5 1.0721 0.9715 1.0554 0.9762 0.9622 1.0027 0.9567 1.0302 1.0061 0.9710 1.0058 0.9119 1.0194 1.0466 0.9781 0.9554 1.0478 1.0087 0.9658 0.9886 0.8885 0.9603 0.9864 0.9904 0.9916 τ=1.0 a=3.5 H=16.0 Table Normalized stress intensity factor K I , for infinite plate under tension, exact K I ¼ 1:7725  105 Table Normalized stress intensity factor K I calculated using the smoothed XFEM for an edge crack under tension for different crack sizes ð36Þ Àa Á b is an empir- L=7.0 a ð37Þ Table shows the stress intensity factor for various discretizations and crack sizes Note that even for the coarsest meshes, the error is Fig 34 Plate with edge crack under shear 1438 S.P.A Bordas et al / Computers and Structures 88 (2010) 1419–1443 Table 10 Normalized stress intensity factors for and edge crack under shear for different mesh sizes, exact K I ¼ 34:0 and K II ¼ 4:55 Table 12 Normalized stress intensity factor K II for edge crack with shear force, exact K II ¼ 4:55 Influence of the size of the J integral domain: see also Table Number of nodes KI K II r d =hlocal 2.5 3.5 4.0 288 1152 3444 4608 5000 0.8921 0.9634 0.9894 0.9929 0.9941 0.9302 0.9630 0.9924 0.9968 0.9980 XFEM Smoothed XFEM 0.9815 0.9825 0.9796 1.0054 0.9788 1.0289 0.9850 1.0138 Table 11 Normalized stress intensity factor K I for edge crack with shear force, exact K I ¼ 34:00 Influence of the size of the J integral domain: see also Table r d =hlocal 2.5 3.5 4.0 XFEM Smoothed XFEM 0.9645 0.9952 0.9641 0.9983 0.9664 0.9961 0.9592 0.9927 around a few percent and decreases with mesh refinement although the convergence is not absolutely monotonic 7.4.4 Edge crack under shear stress For this problem, a plate is clamped at the bottom edge and loaded by a shear traction s = over the top edge The material parameters are Young’s modulus, E =  107 and Poisson’s ratio m = 0.25 The geometry, loading and boundary conditions are shown in Fig 34 The reference mixed mode stress intensity factors are given in [99]: Fig 35 Branch functions spanning the near tip crack field (a) Set of near tip functions for transverse displacement (b) Set of near tip functions for rotations S.P.A Bordas et al / Computers and Structures 88 (2010) 1419–1443 K I ¼ 34:0; K II ẳ 4:55 38ị In this example, mesh renement and domain independence of the J integral were studied on the mixed mode SIFs Table 10 gives the SIFs for various levels of mesh refinement It is evident that with mesh refinement, the computed SIFs converge to the reference mixed mode SIFs given in [99] A structured mesh of 50  50 was used for the comparative study Table 11 and 12 gives the SIF for various domain integral radii We note that the SIFs yielded by both the smoothed XFEM and the XFEM are very accurate and that the smoothed XFEM provides slightly more accurate mode I SIFs Smoothed extended finite elements for arbitrary cracks in plates In this section, we show how curvature smoothing coupled with partition of unity enrichment can produce a plate element capable of cracking which is significantly more accurate than formerly proposed elements in the thin plate limit and when the mesh is distorted These results are preliminary and a more detailed exposé is presented in a forthcoming paper where mixed mode cracking is also tackled 8.1 Rationale 1439 GL xị ẳ GL xị GL xI ịị 39aị F L xị ẳ ðF L ðxÞ À F L ðxI ÞÞ ð39bÞ These sets are not only responsible for closing the crack at the tip (Gl & F L ! for r ! 0), they also introduce analytical information into the approximate solution For GL only the functions propor3 tional to r are used Fig 35 shows the two sets, which are given as fGI ðr; hÞg  &        ' h h 3h 3h 3 3 ; r2 cos ; r sin ; r cos r sin 2 2 ð40aÞ & fF I ðr; hÞg        pffiffiffi h pffiffiffi h pffiffiffi h ; r cos ; r sin sinðhÞ; r sin 2   ' pffiffiffi h sinðhÞ Â r cos ð40bÞ 8.3 Material parameters and assumptions For all calculations isotropic material properties are assumed with a Young’s modulus of E = 200 GPa and Poisson’s ratio of m = 0.3 The examples cover calculations of convergence, distortion sensitivity and behavior for different geometries of the bending moment intensity factor In this section, results of preliminary studies on the coupling of the smoothed curvature plate formulation of Section 1.4 (see also [67,64] for details) with partition of unity enrichment (Section 7) are shown The long term goal is to devise a formulation which does not break down in the thin plate limit and whereby plate cracking can be simulated based on highly distorted meshes The extended finite element formulation is extended to the MISC element (see [67] and Section 1.4.1) and tested against analytical data from work carried out by Boduroglu and Erdogan [15] The results are compared to those obtained by enriched formulations of the standard Q4 plate element as well as the XFEM MITC4 element similar to that proposed by Dolbow et al [32] 8.2 Enrichment functions The discontinuous part of the enrichment, to account for the jump, is defined as in Eq (16) Additionally, near tip functions GL and F L are the sets of shifted near tip functions y d l x a 2b M0 t Fig 36 Rectangular plate containing a single internal crack subjected to a far field moment M0 Fig 37 Displacement configuration for symmetrically modeled plate with a center crack 1440 S.P.A Bordas et al / Computers and Structures 88 (2010) 1419–1443 Below, three enriched plate formulations are considered:  The enriched Q4 plate element uses the basic FE formulation for the Mindlin–Reissner plate theory, i.e uses the standard gradient operators (plain shape function derivatives) for both bending and shear part  The enriched MITC element interpolates the shear stress using collocation points at the element boundaries whilst still using domain integration  The enriched MISC element was described in [67] and Section 1.4.1 and relies on curvature smoothing 8.4 Problem description The calculations are carried out for a plate containing a double symmetric single crack, see Fig 36 Due to the plate symmetry only half of the plate has to be modelled The plate is subjected to a constant distributed far field moment To account for the far field moment character of the external loading the plate dimensions have been chosen as width 2b = and length l = The loading is purely mode I, therefore only the bending moment intensity factor K I will be computed using the contour integral The intensity factor will be normalized, following the common approach of referred works: K Inorm ¼ KI pffiffiffi M0 a ð41Þ 8.5 Displacement, moment and shear force results Fig 37 shows typical displacements for the given configuration It has to be noted that they are displayed using a cubic spline interpolation of the whole nodal solution, so that no real crack opening will show The discontinuities can be identified however in the rotation plots bx and by Figs 38 and 39 are qualitative displays of internal bending moment and shear force distributions, respectively 8.6 Convergence The convergence behavior of the MISC formulation is analyzed using a uniform mesh with square elements Fig 40 shows two Fig 38 Internal bending moments in a symmetrically modeled plate with a center crack Fig 39 Internal shear forces in a symmetrically modeled plate with a center crack 1441 S.P.A Bordas et al / Computers and Structures 88 (2010) 1419–1443 0.14 MISC MITC4 Q4 Relative error of K1 0.12 0.1 0.08 0.06 0.04 0.1 0.2 0.4 0.3 Size of Element — h 0.5 0.6 Fig 42 The relative error of K I with influence of mesh distortion Fig 40 Two different mesh discretization using elements of square dimensions: (a) 69 elements and (b) 3289 elements meshes with different levels of discretization Additional geometry parameters are half crack length a = 0.5 and width to thickness ratio bt ¼ The analytical solution for this example can be found in Ref [15] The MISC formulation shows superior convergence over the MITC4 and Q4 formulations, see Table 13 and Fig 41 Table 13 Numerical values for convergence of relative error in K I for a plate with a crack of length a = 0.5 and thickness t = L/4 h is the element size h MISC MITC4 Q4 0.1250 0.0625 0.0417 0.0313 0.0208 0.0179 0.0156 0.0139 0.0125 0.7672 0.7949 0.8059 0.8116 0.8175 0.8192 0.8205 0.8215 0.8223 0.7375 0.7764 0.7888 0.7951 0.8020 0.8041 0.8057 0.8070 0.8080 0.6535 0.7525 0.7736 0.7895 0.7997 0.8024 0.8045 0.8061 0.8073 0.2 MISC MITC4 Q4 a MISC MITC4 Q4 0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 0.36 0.40 0.44 0.48 0.52 0.56 0.60 0.8110 0.8093 0.8074 0.8053 0.8031 0.7999 0.7970 0.7991 0.7903 0.7975 0.7953 0.7928 0.7941 0.7955 0.7969 0.7945 0.7979 0.7940 0.7939 0.7846 0.7838 0.7680 0.7540 0.7430 0.7123 0.7704 0.7627 0.7541 0.7534 0.7381 0.7775 0.7213 0.7894 0.7886 0.7865 0.7855 0.7842 0.7821 0.7793 0.7777 0.7774 0.7768 0.7754 0.7736 0.7728 0.7686 0.7654 0.7653 8.7 Influence of element distortion The influence of element distortion is analyzed with the same geometry parameters that were used in the convergence analysis The distorted meshes are generated by disarranging the interior nodes of an uniform mesh made up of 4509 elements The distortion equation is given as in Eq (9a) and recalled here for convenience: x0 ẳ x ỵ r c aDx y0 ẳ y ỵ r c aDy 0.15 Relative error of K1 Table 14 Numerical values of K I under the influence of the distortion parameters ð42aÞ ð42bÞ where rc is a random number between À1.0 and 1.0, s ½0; 0:6Š is magnitude of distortion and dx; dy are initial regular element sizes in the x- and y-directions, respectively [65] Fig exemplifies element distortion for different magnitudes of a Fig 42 and Table 14 show that the MISC formulation is less sensitive to mesh distortion than the MITC4 and Q4 formulations 0.1 0.05 Conclusions and prospects —0.05 10-1 Size of Element — h Fig 41 Convergence of relative error in K I for a plate with a crack of length a = 0.5 and thickness t = L/4 We reviewed in this paper the theoretical and numerical results obtained to date on the smoothed finite element method (SFEM) We showed through brief numerical examples that strain smoothing in finite elements can produce superconvergence, alleviate shortcomings related to mesh distortion, locking in incompressible as well as plate and shell formulations, allow polygonal 1442 S.P.A Bordas et al / Computers and Structures 88 (2010) 1419–1443 meshes and contour integration and that the SFEM applies to linear and non-linear problems equally, in two- and three-dimensional settings We also presented in this paper how to extend strain smoothing to the extended finite element method, to obtain the smoothed extended finite element method Through contour integration, the new method simplifies the integration of discontinuous approximations and suppresses the need to integrate singular functions numerically when the Westergaard [101] solution is introduced in the approximation Preliminary results on the calculation of 2D stress intensity factors are encouraging Further 3D tests and precise mathematical analysis are under way An important issue which remains unresolved concerns the numerical integration in smoothed XFEM when non-polynomial functions are used In Sm(X)FEM, domain integration is changed into boundary integration by the divergence theorem; the derivatives of the shape functions are therefore not needed Consequently, when Westergaard fields are added to the FE space, there is no need to integrate singular functions, as in standard XFEM However, the functions to be integrated remain non-polynomial, and optimized one-dimensional integration techniques for these functions are promising routes to increased accuracy The stability and robustness and low computational cost (up to 20% less than FEM11) of this formulation has the potential to lead to fruitful research paths in computational mechanics In particular, strain smoothing appears like a possible route to generate FE based methods midway between expensive and smooth meshfree methods and cheap, but mesh distortion and locking sensitive extended finite element methods Indeed, the insensitivity of the smoothed XFEM to mesh distortion and locking could make the method a competitive alternative to solve complex 3D problems including multiple propagating 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context of MITC4SFEM Mindlin–Reissner plate elements, we show in [67,64] that locking is suppressed in the thin plate/shell... the smoothing cell XC and CC ¼ oXC , we obtain: eC B Ifem h e xị ẳ ẵBxfem ẵq The Bxfem matrix in Eq (22) includes two terms Bfem and Benr corresponding to the standard nodes (fem) and enriched