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See detailXFEM modelling of delamination in composite materials
Cahill, L.; Natarajan, S.; McCarthy, C. et al

Scientific Conference (2010)

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See detailModelling of crack growth behaviour in composite materials using the extended finite element method (XFEM)
Cahill, L.; Natarajan, S.; McCarthy, C. et al

Scientific Conference (2010)

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See detailAn element nodal force-based large increment method for elastoplasticity
Long, Danbin; Guo, Zaoyang; Liu, Xila et al

in AIP Conference Proceedings (2010), 1233(PART 1), 1401-1405

This paper presents a new method for establishing the basic equations in the novel force-based large increment method (LIM) for continuum elastoplastic problems. In LIM, unlike traditional displacement ... [more ▼]

This paper presents a new method for establishing the basic equations in the novel force-based large increment method (LIM) for continuum elastoplastic problems. In LIM, unlike traditional displacement methods, the (generalised) elemental force variables are adopted as system unknowns. The equilibrium equations can then be obtained directly at every nodal degree of freedom without physical equations (i.e., constitutive equations) involved. The generalised inverse of the non-square system of equations is employed to obtain the set of solutions of the non-square matrix equations directly. A conjugate gradient procedure is then used to find the correct solution from this set of solutions by optimising the compatibility of the solution based on the fact that the correct solution should also satisfy the constitutive equations and the compatibility equations. In this paper, the generalised elemental force variables are defined based on the element nodal forces. The LIM framework is therefore successfully applied to elements based on this definition. The efficiency and accuracy of the LIM are illustrated with a few benchmark problems and the results are compared with the analytical solution and the conventional displacement-based finite element method. [less ▲]

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See detailA simple and robust three-dimensional cracking-particle method without enrichment
Rabczuk, T.; Zi, G.; Bordas, Stéphane UL et al

in Computer Methods in Applied Mechanics & Engineering (2010), 199(37-40), 2437-2455

A new robust and efficient approach for modeling discrete cracks in meshfree methods is described. The method is motivated by the cracking-particle method (Rabczuk T., Belytschko T., International Journal ... [more ▼]

A new robust and efficient approach for modeling discrete cracks in meshfree methods is described. The method is motivated by the cracking-particle method (Rabczuk T., Belytschko T., International Journal for Numerical Methods in Engineering, 2004) where the crack is modeled by a set of cracked segments. However, in contrast to the above mentioned paper, we do not introduce additional unknowns in the variational formulation to capture the displacement discontinuity. Instead, the crack is modeled by splitting particles located on opposite sides of the associated crack segments and we make use of the visibility method in order to describe the crack kinematics. We apply this method to several two- and three-dimensional problems in statics and dynamics and show through several numerical examples that the method does not show any "mesh" orientation bias. © 2010 Elsevier B.V. [less ▲]

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See detailAn alternative alpha finite element method (AαFEM) for free and forced structural vibration using triangular meshes
Nguyen-Thanh, N.; Rabczuk, Timon; Nguyen-Xuan, H. et al

in Journal of Computational and Applied Mathematics (2010), 233(9), 2112-2135

An alternative alpha finite element method (AαFEM) using triangular elements is proposed that significantly improves the accuracy of the standard triangular finite elements and provides a superconvergent ... [more ▼]

An alternative alpha finite element method (AαFEM) using triangular elements is proposed that significantly improves the accuracy of the standard triangular finite elements and provides a superconvergent solution in the energy norm for the static analysis of two-dimensional solid mechanics problems. In the AαFEM, the piecewise constant strain field of linear triangular FEM models is enhanced by additional strain terms with an adjustable parameter α which results in an effectively softer stiffness formulation compared to a linear triangular element. The element is further extended to the free and forced vibration analyses of solids. Several numerical examples show that the AαFEM achieves high reliability compared to other existing elements in the literature. [less ▲]

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See detailAnalysis of thermoelastic waves in a two-dimensional functionally graded materials domain by the Meshless Local Petrov-Galerkin (MLPG) method
Ahmad Akbari, R.; Bagri, Akbar; Bordas, Stéphane UL et al

in Computer Modeling in Engineering & Sciences (2010), 65(1), 27-74

This contribution focuses on the simulation of two-dimensional elastic wave propagation in functionally graded solids and structures. Gradient volume fractions of the constituent materials are assumed to ... [more ▼]

This contribution focuses on the simulation of two-dimensional elastic wave propagation in functionally graded solids and structures. Gradient volume fractions of the constituent materials are assumed to obey the power law function of position in only one direction and the effective mechanical properties of the material are determined by the Mori–Tanaka scheme. The investigations are carried out by extending a meshless method known as the Meshless Local Petrov-Galerkin (MLPG) method which is a truly meshless approach to thermo-elastic wave propagation. Simulations are carried out for rectangular domains under transient thermal loading. To investigate the effect of material composition on the dynamic response of functionally graded materials, a metal/ceramic (Aluminum (Al) and Alumina (Al2O3) are considered as ceramic and metal constituents) composite is considered for which the transient thermal field, dynamic displacement and stress fields are reported for different material distributions. [less ▲]

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See detailEnriched finite elements (XFEM) for multi-crack growth simulations in orthotropic materials
Cahill, L.; Natarajan, S.; McCarthy, C. et al

Scientific Conference (2010)

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See detailA cell-based smoothed finite element method for kinematic limit analysis
Le, Canh. V.; Nguyen-Xuan, H.; Askes, H. et al

in International Journal for Numerical Methods in Engineering (2010), 83(12), 1651-1674

This paper presents a new numerical procedure for kinematic limit analysis problems, which incorporates the cell-based smoothed finite element method with second-order cone programming. The application of ... [more ▼]

This paper presents a new numerical procedure for kinematic limit analysis problems, which incorporates the cell-based smoothed finite element method with second-order cone programming. The application of a strain smoothing technique to the standard displacement finite element both rules out volumetric locking and also results in an efficient method that can provide accurate solutions with minimal computational effort. The non-smooth optimization problem is formulated as a problem of minimizing a sum of Euclidean norms, ensuring that the resulting optimization problem can be solved by an efficient second-order cone programming algorithm. Plane stress and plane strain problems governed by the von Mises criterion are considered, but extensions to problems with other yield criteria having a similar conic quadratic form or 3D problems can be envisaged. [less ▲]

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See detailA node-based smoothed finite element method with stabilized discrete shear gap technique for analysis of Reissner-Mindlin plates
Nguyen-Xuan, H.; Rabczuk, T.; Nguyen-Thanh, N. et al

in Computational Mechanics (2010), 46(5), 679-701

In this paper, a node-based smoothed finite element method (NS-FEM) using 3-node triangular elements is formulated for static, free vibration and buckling analyses of Reissner-Mindlin plates. The discrete ... [more ▼]

In this paper, a node-based smoothed finite element method (NS-FEM) using 3-node triangular elements is formulated for static, free vibration and buckling analyses of Reissner-Mindlin plates. The discrete weak form of the NS-FEM is obtained based on the strain smoothing technique over smoothing domains associated with the nodes of the elements. The discrete shear gap (DSG) method together with a stabilization technique is incorporated into the NS-FEM to eliminate transverse shear locking and to maintain stability of the present formulation.Aso-called node-based smoothed stabilized discrete shear gap method (NS-DSG) is then proposed. Several numerical examples are used to illustrate the accuracy and effectiveness of the present method. [less ▲]

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See detailStrain smoothing in FEM and XFEM
Bordas, Stéphane UL; Rabczuk, T.; Hung, N.-X. et al

in Computers & Structures (2010), 88(23-24), 1419-1443

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 ... [more ▼]

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. © 2008 Elsevier Ltd. All rights reserved. [less ▲]

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See detailNumerical integration over arbitrary surfaces in partition of unity finite elements
Natarajan, Sundararajan; dal Pont, Stefano; Hung, Nguyen-Xuan et al

Scientific Conference (2009, September)

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See detailThe smoothed extended finite element method for strong discontinuities
Natarajan, S.; Bordas, Stéphane UL; Rabczuk, Timon

Scientific Conference (2009, June)

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See detailOn the Smoothed eXtended Finite Element Method for Continuum
Natarajan, Sundararajan; Bordas, Stéphane UL; Rabczuk, Timon et al

Scientific Conference (2009, April)

In this paper, we combine the strain smoothing technique proposed by Liu et al [1] coined as the smoothed finite element method (SFEM) to partition of unity methods, namely the extended finite element ... [more ▼]

In this paper, we combine the strain smoothing technique proposed by Liu et al [1] coined as the smoothed finite element method (SFEM) to partition of unity methods, namely the extended finite element method (XFEM) [2] to give birth to the smoothed extended finite element method (SmXFEM) [3]. SmXFEM shares properties both with the SFEM and the XFEM. The proposed method eliminates the need to compute and integrate the derivatives of shape functions (which are singular at the tip for linear elastic fracture mechanics). The need for isoparametric mapping is eliminated because the integration is done along the boundary of the finite element or smoothing cells, which allows elements of arbitrary shape. We present numerical results for various differential equations that have singularity or steep gradient at the boundary. The method is verified on several examples and comparisons are made to the conventional XFEM. [less ▲]

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See detailA novel numerical integration technique over arbitrary polygons
Natarajan, Sundararajan; Mahapatra, D Roy; Bordas, Stéphane UL et al

Scientific Conference (2009, April)

In this paper, a new numerical integration technique [1] on arbitrary polygons is presented. The polygonal do- main is mapped conformally to the unit disk using Schwarz-Christoffel mapping [2] and a ... [more ▼]

In this paper, a new numerical integration technique [1] on arbitrary polygons is presented. The polygonal do- main is mapped conformally to the unit disk using Schwarz-Christoffel mapping [2] and a midpoint quadrature rule defined on the unit circle is used. This method eliminates the need for a two level isoparametric mapping usuall required [3]. Moreover the positivity of the Jacobian is guaranteed. We present numerical results for a few benchmark problems in the context of polygonal finite elements that show the effectiveness of the method. [less ▲]

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See detailInfluence of the microstructure on the stress state of solder joints dusing thermal cycling
Menk, A.; Bordas, Stéphane UL

in Proceedings of 10th International Conference on Thermal, Mechanical and Multi-Physics Simulation and Experiments in Microelectronics and Microsystems, EuroSimE 2009 (2009)

The lifetime of a solder joint is mainly determined by its creep behaviour. Creep arises from the stresses inside the solder joints as a consequence of the thermomechanical mismatch of the board and the ... [more ▼]

The lifetime of a solder joint is mainly determined by its creep behaviour. Creep arises from the stresses inside the solder joints as a consequence of the thermomechanical mismatch of the board and the substrate. The stress state is heavily influenced by the anisotropy of tin. To understand the damage process in solder joints, the influence of the anisotropic microstructure must be understood. In this paper the influence of different grain sizes, shapes and orientations on the stress state is evaluated, based on numerical experiments. [less ▲]

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See detailAPPLICATION OF EXTENDED ELEMENT-FREE GALERKIN METHOD TO MULTIPLE FLAWS UNDER BRITTLE FRACTURE CONDITIONS
RABCZUK, T.; BEZENSEK, B.; Bordas, Stéphane UL

in PROCEEDINGS OF THE ASME PRESSURE VESSELS AND PIPING CONFERENCE - 2008, VOL 6, PT A AND B (2009)

The extended element-free Galerkin (XEFG) method incorporates cracks through partition of unity enrichment of the standard basis functions. Discontinuous functions are added to capture the jump through ... [more ▼]

The extended element-free Galerkin (XEFG) method incorporates cracks through partition of unity enrichment of the standard basis functions. Discontinuous functions are added to capture the jump through the crack faces and near-front enrichment is added to capture the asymptotic behaviour in the vicinity of the crack fronts. Depending on the material behaviour, these functions can be of various type. The method can treat initiation, growth and coalescence of cracks seamlessly in both linear elastic and non-linear settings. The method is a powerful tool for modelling and studying crack paths, which are a central feature in the assessment of multiple flaws.The method is applied to the problem of multiple non-aligned flaws in a ferritic plate under cleavage failure. Fracture paths from two nonaligned notches in a plate are modelled. Based on the observations of crack paths the critical flaw alignment distance is established for nonaligned through-wall flaws. [less ▲]

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See detailNumerical integration over arbitrary polygonal domains based on Schwarz-Christoffel conformal mapping
Natarajan, S.; Bordas, Stéphane UL; Roy mahapatra, D.

in International Journal for Numerical Methods in Engineering (2009), 80(1), 103-134

This paper presents a new numerical integration technique on arbitrary polygonal domains. The polygonal domain is mapped conformally to the unit disk using Schwarz-Christoffel mapping and a midpoint ... [more ▼]

This paper presents a new numerical integration technique on arbitrary polygonal domains. The polygonal domain is mapped conformally to the unit disk using Schwarz-Christoffel mapping and a midpoint quadrature rule defined on this unit disk is used. This method eliminates the need for a two-level isoparametric mapping usually required. Moreover, the positivity of the Jacobian is guaranteed. Numerical results presented for a few benchmark problems in the context of polygonal finite elements show that the proposed method yields accurate results. © 2009 John Wiley & Sons, Ltd. [less ▲]

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See detailAddressing volumetric locking and instabilities by selective integration in smoothed finite elements
Hung, Nguyen-Xuan; Bordas, Stéphane UL; Hung, Nguyen-Dang

in Communications in Numerical Methods in Engineering (2009), 25(1), 19-34

This paper promotes the development of a novel family of finite elements with smoothed strains, offering remarkable properties. In the smoothed finite element method (FEM), elements are divided into ... [more ▼]

This paper promotes the development of a novel family of finite elements with smoothed strains, offering remarkable properties. In the smoothed finite element method (FEM), elements are divided into subcells. The strain at a point is defined as a weighted average of the standard strain field over a representative domain. This yields superconvergent stresses, both in regular and singular settings, as well as increased accuracy, with slightly lower computational cost than the standard FEM. The one-subcell version that does not exhibit volumetric locking yields more accurate stresses but less accurate displacements and is equivalent to a quasi-equilibrium FEM. It is also subject to instabilities. In the limit where the number of subcells goes to infinity, the standard FEM is recovered, which yields more accurate displacements and less accurate stresses. The specific contribution of this paper is to show that expressing the volumetric part of the strain field using a one-subcell formulation is sufficient to get rid of volumetric locking and increase the displacement accuracy compared with the standard FEM when the single subcell version is used to express both the volumetric and deviatoric parts of the strain. Selective integration also alleviates instabilities associated with the single subcell element, which are due to rank deficiency. Numerical examples on various compressible and incompressible linear elastic test cases show that high accuracy is retained compared with the standard FEM without increasing computational cost. [less ▲]

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See detailA posteriori error estimation for extended finite elements by an extended global recovery
Duflot, M.; Bordas, Stéphane UL

in International Journal for Numerical Methods in Engineering (2008), 76(8), 1123-1138

This contribution presents an extended global derivative recovery for enriched finite element methods (FEMs), such as the extended FEM along with an associated error indicator. Owing to its simplicity ... [more ▼]

This contribution presents an extended global derivative recovery for enriched finite element methods (FEMs), such as the extended FEM along with an associated error indicator. Owing to its simplicity, the proposed scheme is ideally suited to industrial applications. The procedure is based on global minimization of the L2 norm of the difference between the raw strain field (C-1) and the recovered (C0) strain field. The methodology engineered in this paper extends the ideas of Oden and Brauchli (Int. J. Numer. Meth. Engng 1971; 3) and Hinton and Campbell (Int. J. Numer. Meth. Engng 1974; 8) by enriching the approximation used for the construction of the recovered derivatives (strains) with the gradients of the functions employed to enrich the approximation employed for the primal unknown (displacements). We show linear elastic fracture mechanics examples, both in simple two-dimensional settings, and for a three-dimensional structure. Numerically, we show that the effectivity index of the proposed indicator converges to unity upon mesh refinement. Consequently, the approximate error converges to the exact error, indicating that the error indicator is valid. Additionally, the numerical examples suggest a novel adaptive strategy for enriched approximations in which the dimensions of the enrichment zone are first increased, before standard h- and p-adaptivities are applied; we suggest to coin this methodology e-adaptivity. Copyright © 2008 John Wiley & Sons, Ltd. [less ▲]

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See detailA simple error estimator for extended finite elements
Bordas, Stéphane UL; Duflot, Marc; Le, Phong

in Communications in Numerical Methods in Engineering (2008), 24(11), 961-971

This short communication presents the idea of an a posteriori error estimate for enriched (extended) finite elements (XFEM). The enhanced strain field against which the XFEM strains are compared, is ... [more ▼]

This short communication presents the idea of an a posteriori error estimate for enriched (extended) finite elements (XFEM). The enhanced strain field against which the XFEM strains are compared, is computed through extended moving least-squares smoothing constructed using the diffraction method to preserve the discontinuity. The error estimator is the L2 norm of the difference of the XFEM strain with the enhanced strain. We prove the concept of the proposed method on a 1D example with a singular solution and a 2D fracture mechanics example and conclude with some future work based on our paradigm. [less ▲]

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