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See detailWeakening the tight coupling between geometry and simulation in isogeometric analysis: from sub- and super- geometric analysis to Geometry Independent Field approximaTion (GIFT)
Atroshchenko, Elena; Tomar, Satyendra UL; Xu, Gang et al

in International Journal for Numerical Methods in Engineering (2018)

This paper presents an approach to generalize the concept of isogeometric analysis (IGA) by allowing different spaces for parameterization of the computational domain and for approximation of the solution ... [more ▼]

This paper presents an approach to generalize the concept of isogeometric analysis (IGA) by allowing different spaces for parameterization of the computational domain and for approximation of the solution field. The method inherits the main advantage of isogeometric analysis, i.e. preserves the original, exact CAD geometry (for example, given by NURBS), but allows pairing it with an approximation space which is more suitable/flexible for analysis, for example, T-splines, LR-splines, (truncated) hierarchical B-splines, and PHT-splines. This generalization offers the advantage of adaptive local refinement without the need to re-parameterize the domain, and therefore without weakening the link with the CAD model. We demonstrate the use of the method with different choices of the geometry and field splines, and show that, despite the failure of the standard patch test, the optimum convergence rate is achieved for non-nested spaces. [less ▲]

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See detailAn adaptive variational Quasicontinuum methodology for lattice networks with localized damage
Rokos, Ondrej; Peerlings, Ron; Zeman, Jan et al

in International Journal for Numerical Methods in Engineering (2017), 112(2),

Lattice networks with dissipative interactions can be used to describe the mechanics of discrete meso‐structures of materials such as 3D‐printed structures and foams. This contribution deals with the ... [more ▼]

Lattice networks with dissipative interactions can be used to describe the mechanics of discrete meso‐structures of materials such as 3D‐printed structures and foams. This contribution deals with the crack initiation and propagation in such materials and focuses on an adaptive multiscale approach that captures the spatially evolving fracture. Lattice networks naturally incorporate non‐locality, large deformations and dissipative mechanisms taking place inside fracture zones. Because the physically relevant length scales are significantly larger than those of individual interactions, discrete models are computationally expensive. The Quasicontinuum (QC) method is a multiscale approach specifically constructed for discrete models. This method reduces the computational cost by fully resolving the underlying lattice only in regions of interest, while coarsening elsewhere. In this contribution, the (variational) QC is applied to damageable lattices for engineering‐scale predictions. To deal with the spatially evolving fracture zone, an adaptive scheme is proposed. Implications induced by the adaptive procedure are discussed from the energy‐consistency point of view, and theoretical considerations are demonstrated on two examples. The first one serves as a proof of concept, illustrates the consistency of the adaptive schemes and presents errors in energies. The second one demonstrates the performance of the adaptive QC scheme for a more complex problem. [less ▲]

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See detailTime and frequency domain analysis of piezoelectric energy harvesters by monolithic finite element modeling
Ravi, Srivathsan UL; Zilian, Andreas UL

in International Journal for Numerical Methods in Engineering (2017)

The successful design of piezoelectric energy harvesting devices relies upon the identification of optimal geometrical and material configurations to maximize the power output for a specific band of ... [more ▼]

The successful design of piezoelectric energy harvesting devices relies upon the identification of optimal geometrical and material configurations to maximize the power output for a specific band of excitation frequencies. Extendable predictive models and associated approximate solution methods are essential for analysis of a wide variety of future advanced energy harvesting devices involving more complex geometries and material distributions. Based on a holistic continuum mechanics modeling approach to the multi-physics energy harvesting problem, this article proposes a monolithic numerical solution scheme using a mixed-hybrid 3-dimensional finite element formulation of the coupled governing equations for analysis in time and frequency domain. The weak form of the electromechanical/circuit system uses velocities and potential rate within the piezoelectric structure, free boundary charge on the electrodes, and potential at the level of the generic electric circuit as global degrees of freedom. The approximation of stress and dielectric displacement follows the work by Pian, Sze, and Pan. Results obtained with the proposed model are compared with analytical results for the reduced-order model of a cantilevered bimorph harvester with tip mass reported in the literature. The flexibility of the method is demonstrated by studying the influence of partial electrode coverage on the generated power output. [less ▲]

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See detailGuaranteed error bounds in homogenisation: an optimum stochastic approach to preserve the numerical separation of scales
Paladim, Daniel-Alves; de Almeida, José Paulo Baptista; Bordas, Stéphane UL et al

in International Journal for Numerical Methods in Engineering (2017), 110(2), 103132

This paper proposes a new methodology to guarantee the accuracy of the homogenisation schemes that are traditionally employed to approximate the solution of PDEs with random, fast evolving diffusion ... [more ▼]

This paper proposes a new methodology to guarantee the accuracy of the homogenisation schemes that are traditionally employed to approximate the solution of PDEs with random, fast evolving diffusion coefficients. We typically consider linear elliptic diffusion problems in randomly packed particulate composites. Our work extends the pioneering work presented in [26,32] in order to bound the error in the expectation and second moment of quantities of interest, without ever solving the fine-scale, intractable stochastic problem. The most attractive feature of our approach is that the error bounds are computed without any integration of the fine-scale features. Our computations are purely macroscopic, deterministic, and remain tractable even for small scale ratios. The second contribution of the paper is an alternative derivation of modelling error bounds through the Prager-Synge hypercircle theorem. We show that this approach allows us to fully characterise and optimally tighten the interval in which predicted quantities of interest are guaranteed to lie. We interpret our optimum result as an extension of Reuss-Voigt approaches, which are classically used to estimate the homogenised diffusion coefficients of composites, to the estimation of macroscopic engineering quantities of interest. Finally, we make use of these derivations to obtain an efficient procedure for multiscale model verification and adaptation. [less ▲]

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See detailA fully smoothed XFEM for analysis of axisymmetric problems with weak discontinuities
Wan, Detao; Hu, Dean; Natarajan, Sundararajan et al

in International Journal for Numerical Methods in Engineering (2017), 110(3), 203-226

In this paper, we propose a fully smoothed extended finite element method (SmXFEM) for axisymmetric problems with weak discontinuities. The salient feature of the proposed approach is that all the terms ... [more ▼]

In this paper, we propose a fully smoothed extended finite element method (SmXFEM) for axisymmetric problems with weak discontinuities. The salient feature of the proposed approach is that all the terms in the stiffness and mass matrixes can be computed by smoothing technique. This is accomplished by combining the Green’s divergence theorem with the evaluation of indefinite integral based on smoothing technique, which is used to transform the domain integral into boundary integral. The proposed technique completely eliminates the need for isoparametric mapping and the computing of Jacobian matrix even for the mass matrix. When employed over the enriched elements, the proposed technique does not require sub-triangulation for the purpose of numerical integration. The accuracy and convergence properties of the proposed technique are demonstrated with a few problems in elastostatics and elastodynamics with weak discontinuities. It can be seen that the proposed technique yields stable and accurate solutions and is less sensitive to mesh distortion. [less ▲]

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See detailStable 3D XFEM/vector-level sets for non-planar 3D crack propagation and comparison of enrichment schemes
Agathos, Konstantinos UL; Ventura, Giulio; Chatzi, Eleni et al

in International Journal for Numerical Methods in Engineering (2017)

We present a three-dimensional (3D) vector level set method coupled to a recently developed stable extended finite element method (XFEM). We further investigate a new enrichment approach for XFEM adopting ... [more ▼]

We present a three-dimensional (3D) vector level set method coupled to a recently developed stable extended finite element method (XFEM). We further investigate a new enrichment approach for XFEM adopting discontinuous linear enrichment functions in place of the asymptotic near-tip functions. Through the vector level set method, level set values for propagating cracks are obtained via simple geometrical operations, eliminating the need for solution of differential evolution equations. The first XFEM variant ensures optimal convergence rates by means of geometrical enrichment, i.e., the use of enriched elements in a fixed volume around the crack front, without giving rise to conditioning problems. The linear enrichment approach significantly simplifies implementation and reduces the computational cost associated with numerical integration. The two dicretization schemes are tested for different benchmark problems, and their combination to the vector level set method is verified for non-planar crack propagation problems. [less ▲]

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See detailImplementation of regularized isogeometric boundary element methods for gradient-based shape optimization in two-dimensional linear elasticity
Haojie, Lian; Pierre, Kerfriden; Bordas, Stéphane UL

in International Journal for Numerical Methods in Engineering (2015)

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See detailVirtual and smoothed finite elements: A connection and its application to polygonal/polyhedral finite element methods
Natarajan, Sundararajan; Bordas, Stéphane UL; Ooi, Ean Tat

in International Journal for Numerical Methods in Engineering (2015), 104(13), 1173-1199

We show both theoretically and numerically a connection between the smoothed finite element method (SFEM) and the virtual element method and use this approach to derive stable, cheap and optimally ... [more ▼]

We show both theoretically and numerically a connection between the smoothed finite element method (SFEM) and the virtual element method and use this approach to derive stable, cheap and optimally convergent polyhedral FEM.We show that the stiffness matrix computed with one subcell SFEM is identical to the consistency term of the virtual element method, irrespective of the topology of the element, as long as the shape functions vary linearly on the boundary. Using this connection, we propose a new stable approach to strain smoothing for polygonal/polyhedral elements where, instead of using sub-triangulations, we are able to use one single polygonal/polyhedral subcell for each element while maintaining stability. For a similar number of degrees of freedom, the proposed approach is more accurate than the conventional SFEM with triangular subcells. The time to compute the stiffness matrix scales with the O.dof s/1:1 in case of the conventional polygonal FEM, while it scales as O.dof s/0:7 in the proposed approach. The accuracy and the convergence properties of the SFEM are studied with a few benchmark problems in 2D and 3D linear elasticity. [less ▲]

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See detailCertification of projection-based reduced order modelling in computational homogenisation by the Constitutive Relation Error
Kerfriden, Pierre; Ródenas, Juan-José; Bordas, Stéphane UL

in International Journal for Numerical Methods in Engineering (2014), 97(6), 395-422

In this paper, we propose upper and lower error bounding techniques for reduced order modelling applied to the computational homogenisation of random composites. The upper bound relies on the construction ... [more ▼]

In this paper, we propose upper and lower error bounding techniques for reduced order modelling applied to the computational homogenisation of random composites. The upper bound relies on the construction of a reduced model for the stress field. Upon ensuring that the reduced stress satisfies the equilibrium in the nite element sense, the desired bounding property is obtained. The lower bound is obtained by defining a hierarchical enriched reduced model for the displacement. We show that the sharpness of both error estimates can be seamlessly controlled by adapting the parameters of the corresponding reduced order model. [less ▲]

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See detailCertification of projection-based reduced order modelling in computational homogenisation by the constitutive relation error
Kerfriden, P.; Ródenas, J. J.; Bordas, Stéphane UL

in International Journal for Numerical Methods in Engineering (2013)

SUMMARY: In this paper, we propose upper and lower error bounding techniques for reduced order modelling applied to the computational homogenisation of random composites. The upper bound relies on the ... [more ▼]

SUMMARY: In this paper, we propose upper and lower error bounding techniques for reduced order modelling applied to the computational homogenisation of random composites. The upper bound relies on the construction of a reduced model for the stress field. Upon ensuring that the reduced stress satisfies the equilibrium in the finite element sense, the desired bounding property is obtained. The lower bound is obtained by defining a hierarchical enriched reduced model for the displacement. We show that the sharpness of both error estimates can be seamlessly controlled by adapting the parameters of the corresponding reduced order model. © 2013 John Wiley & Sons, Ltd. [less ▲]

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See detailExtended space-time finite elements for landslide dynamics
Pasenow, F.; Zilian, Andreas UL; Dinkler, D.

in International Journal for Numerical Methods in Engineering (2013), 93(3), 329-354

The paper introduces a methodology for numerical simulation of landslides experiencing considerable deformations and topological changes. Within an interface capturing approach, all interfaces are ... [more ▼]

The paper introduces a methodology for numerical simulation of landslides experiencing considerable deformations and topological changes. Within an interface capturing approach, all interfaces are implicitly described by specifically defined level-set functions allowing arbitrarily evolving complex topologies. The transient interface evolution is obtained by solving the level-set equation driven by the physical velocity field for all three level-set functions in a block Jacobi approach. The three boundary-coupled fluid-like continua involved are modeled as incompressible, governed by a generalized non-Newtonian material law taking into account capillary pressure at moving fluid-fluid interfaces. The weighted residual formulation of the level-set equations and the fluid equations is discretized with finite elements in space and time using velocity and pressure as unknown variables. Non-smooth solution characteristics are represented by enriched approximations to fluid velocity (weak discontinuity) and fluid pressure (strong discontinuity). Special attention is given to the construction of enriched approximations for elements containing evolving triple junctions. Numerical examples of three-fluid tank sloshing and air-water-liquefied soil landslides demonstrate the potential and applicability of the method in geotechnical investigations. © 2012 John Wiley & Sons, Ltd. [less ▲]

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See detailLocal/global model order reduction strategy for the simulation of quasi-brittle fracture
Kerfriden, P.; Passieux, J. C.; Bordas, Stéphane UL

in International Journal for Numerical Methods in Engineering (2012), 89(2), 154-179

This paper proposes a novel technique to reduce the computational burden associated with the simulation of localized failure. The proposed methodology affords the simulation of damage initiation and ... [more ▼]

This paper proposes a novel technique to reduce the computational burden associated with the simulation of localized failure. The proposed methodology affords the simulation of damage initiation and propagation while concentrating the computational effort where it is most needed, that is, in the localization zones. To do so, a local/global technique is devised where the global (slave) problem (far from the zones undergoing severe damage and cracking) is solved for in a reduced space computed by the classical proper orthogonal decomposition while the local (master) degrees of freedom (associated with the part of the structure where most of the damage is taking place) are fully resolved. Both domains are coupled through a local/global technique. This method circumvents the difficulties associated with model order reduction for the simulation of highly nonlinear mechanical failure and offers an alternative or complementary approach to the development of multiscale fracture simulators. © 2011 John Wiley & Sons, Ltd. [less ▲]

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See detailA quasicontinuum methodology for multiscale analyses of discrete microstructural models
Beex, Lars UL; Peerlings, Ron; Geers, Marc

in International Journal for Numerical Methods in Engineering (2011), 87(7), 701-718

Many studies in different research fields use lattice models to investigate the mechanical behavior of materials. Full lattice calculations are often performed to determine the influence of localized ... [more ▼]

Many studies in different research fields use lattice models to investigate the mechanical behavior of materials. Full lattice calculations are often performed to determine the influence of localized microscale phenomena on large-scale responses but they are usually computationally expensive. In this study the quasicontinuum (QC) method (Phil. Mag. A 1996; 73:1529–1563) is extended towards lattice models that employ discrete elements, such as trusses and beams. The QC method is a multiscale approach that uses a triangulation to interpolate the lattice model in regions with small fluctuations in the deformation field, while in regions of high interest the exact lattice model is obtained by refining the triangulation to the internal spacing of the lattice. Interpolation ensures that the number of unknowns is reduced while summation ensures that only a selective part of the underlying lattice model must be visited to construct the governing equations. As the QC method has so far only been applied to atomic lattice models, the existing summation procedures have been revisited for structural lattice models containing discrete elements. This has led to a new QC method that makes use of the characteristic structure of the considered truss network. The proposed QC method is, to the best of the authors’ knowledge, the only QC method that does not need any correction at the interface between the interpolated and the fully resolved region and at the same time gives exact results unlike the cluster QC methods. In its present formulation, the proposed QC method can only be used for lattice models containing nearest neighbor interactions, but with some minor adaptations it can also be used for lattices with next-nearest neighbor interactions such as atomic lattices. [less ▲]

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See detailA rheological interface model and its space-time finite element formulation for fluid-structure interaction
Legay, A.; Zilian, Andreas UL; Janssen, C.

in International Journal for Numerical Methods in Engineering (2011), 86(6), 667-687

This contribution discusses extended physical interface models for fluid-structure interaction problems and investigates their phenomenological effects on the behavior of coupled systems by numerical ... [more ▼]

This contribution discusses extended physical interface models for fluid-structure interaction problems and investigates their phenomenological effects on the behavior of coupled systems by numerical simulation. Besides the various types of friction at the fluid-structure interface the most interesting phenomena are related to effects due to additional interface stiffness and damping. The paper introduces extended models at the fluid-structure interface on the basis of rheological devices (Hooke, Newton, Kelvin, Maxwell, Zener). The interface is decomposed into a Lagrangian layer for the solid-like part and an Eulerian layer for the fluid-like part. The mechanical model for fluid-structure interaction is based on the equations of rigid body dynamics for the structural part and the incompressible Navier-Stokes equations for viscous flow. The resulting weighted residual form uses the interface velocity and interface tractions in both layers in addition to the field variables for fluid and structure. The weak formulation of the whole coupled system is discretized using space-time finite elements with a discontinuous Galerkin method for time-integration leading to a monolithic algebraic system. The deforming fluid domain is taken into account by deformable space-time finite elements and a pseudo-structure approach for mesh motion. The sensitivity of coupled systems to modification of the interface model and its parameters is investigated by numerical simulation of flow induced vibrations of a spring supported fluid-immersed cylinder. It is shown that the presented rheological interface model allows to influence flow-induced vibrations. © 2010 John Wiley & Sons, Ltd. [less ▲]

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See detailExtended Finite Element Method
Fries, T.-P.; Zilian, Andreas UL; Moës, N.

in International Journal for Numerical Methods in Engineering (2011), 86(4-5), 403

[No abstract available]

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See detailA robust preconditioning technique for the extended finite element method
Menk, A.; Bordas, Stéphane UL

in International Journal for Numerical Methods in Engineering (2011), 85(13), 1609-1632

The extended finite element method enhances the approximation properties of the finite element space by using additional enrichment functions. But the resulting stiffness matrices can become ill ... [more ▼]

The extended finite element method enhances the approximation properties of the finite element space by using additional enrichment functions. But the resulting stiffness matrices can become ill-conditioned. In that case iterative solvers need a large number of iterations to obtain an acceptable solution. In this paper a procedure is described to obtain stiffness matrices whose condition number is close to the one of the finite element matrices without any enrichments. A domain decomposition is employed and the algorithm is very well suited for parallel computations. The method was tested in numerical experiments to show its effectiveness. The experiments have been conducted for structures containing cracks and material interfaces. We show that the corresponding enrichments can result in arbitrarily ill-conditioned matrices. The method proposed here, however, provides well-conditioned matrices and can be applied to any sort of enrichment. The complexity of this approach and its relation to the domain decomposition is discussed. Computation times have been measured for a structure containing multiple cracks. For this structure the computation times could be decreased by a factor of 2. © 2010 John Wiley & Sons, Ltd. [less ▲]

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See detailAccurate fracture modelling using meshless methods, the visibility criterion and level sets: Formulation and 2D modelling
Zhuang, Xiaoying; Augarde, Charles; Bordas, Stéphane UL

in International Journal for Numerical Methods in Engineering (2011), 86(2), 249-268

Fracture modelling using numerical methods is well-advanced in 2D using techniques such as the extended finite element method (XFEM). The use of meshless methods for these problems lags somewhat behind ... [more ▼]

Fracture modelling using numerical methods is well-advanced in 2D using techniques such as the extended finite element method (XFEM). The use of meshless methods for these problems lags somewhat behind, but the potential benefits of no meshing (particularly in 3D) prompt continued research into their development. In methods where the crack face is not explicitly modelled (as the edge of an element for instance), two procedures are instead used to associate the displacement jump with the crack surface: the visibility criterion and the diffraction method. The visibility criterion is simple to implement and efficient to compute, especially with the help of level set coordinates. However, spurious discontinuities have been reported around crack tips using the visibility criterion, whereas implementing the diffraction method in 3D is much more complicated than the visibility criterion. In this paper, a tying procedure is proposed to remove the difficulty with the visibility criterion so that crack tip closure can be ensured while the advantages of the visibility criterion can be preserved. The formulation is based on the use of level set coordinates and the element-free Galerkin method, and is generally applicable for single or multiple crack problems in 2D or 3D. The paper explains the formulation and provides verification of the method against a number of 2D crack problems. [less ▲]

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See detailOn the performance of strain smoothing for quadratic and enriched finite element approximations (XFEM/GFEM/PUFEM)
Bordas, Stéphane UL; Natarajan, S.; Kerfriden, P. et al

in International Journal for Numerical Methods in Engineering (2011), 86(4-5), 637-666

By using the strain smoothing technique proposed by Chen et al. (Comput. Mech. 2000; 25:137-156) for meshless methods in the context of the finite element method (FEM), Liu et al. (Comput. Mech. 2007; 39 ... [more ▼]

By using the strain smoothing technique proposed by Chen et al. (Comput. Mech. 2000; 25:137-156) for meshless methods in the context of the finite element method (FEM), Liu et al. (Comput. Mech. 2007; 39(6):859-877) developed the Smoothed FEM (SFEM). Although the SFEM is not yet well understood mathematically, numerical experiments point to potentially useful features of this particularly simple modification of the FEM. To date, the SFEM has only been investigated for bilinear and Wachspress approximations and is limited to linear reproducing conditions. The goal of this paper is to extend the strain smoothing to higher order elements and to investigate numerically in which condition strain smoothing is beneficial to accuracy and convergence of enriched finite element approximations. We focus on three widely used enrichment schemes, namely: (a) weak discontinuities; (b) strong discontinuities; (c) near-tip linear elastic fracture mechanics functions. The main conclusion is that strain smoothing in enriched approximation is only beneficial when the enrichment functions are polynomial (cases (a) and (b)), but that non-polynomial enrichment of type (c) lead to inferior methods compared to the standard enriched FEM (e.g. XFEM). © 2011 John Wiley & Sons, Ltd. [less ▲]

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See detailOn the approximation in the smoothed finite element method (SFEM)
Bordas, Stéphane UL; Natarajan, S.

in International Journal for Numerical Methods in Engineering (2010), 81(5), 660-670

This letter aims at resolving the issues raised in the recent short communication (Int. J. Numer. Meth. Engng 2008; 76(8):1285-1295. DOI: 10.1002/nme.2460) and answered by (Int. J. Numer. Meth. Engng 2009 ... [more ▼]

This letter aims at resolving the issues raised in the recent short communication (Int. J. Numer. Meth. Engng 2008; 76(8):1285-1295. DOI: 10.1002/nme.2460) and answered by (Int. J. Numer. Meth. Engng 2009; DOI: 10.1002/nme.2587) by proposing a systematic approximation scheme based on non-mapped shape functions, which both allows to fully exploit the unique advantages of the smoothed finite element method (SFEM) (Comput. Mech. 2007; 39(6):859-877. DOI: 10.1007/s00466-006-0075-4; Commun. Numer. Meth. Engng 2009; 25(1):19-34. DOI: 10.1002/cnm.1098; Int. J. Numer. Meth. Engng 2007; 71(8):902-930; Comput. Meth. Appl. Mech. Engng 2008; 198(2):165-177. DOI: 10.1016/j.cma.2008.05.029; Comput. Meth. Appl. Mech. Engng 2007; submitted; Int. J. Numer. Meth. Engng 2008; 74(2):175-208. DOI: 10.1002/nme.2146; Comput. Meth. Appl. Mech. Engng 2008; 197 (13-16):1184-1203. DOI: 10.1016/j.cma.2007.10.008) and resolve the existence, linearity and positivity deficiencies pointed out in (Int. J. Numer. Meth. Engng 2008; 76(8):1285-1295). We show that Wachspress interpolants (A Rational Basis for Function Approximation. Academic Press, Inc.: New York, 1975) computed in the physical coordinate system are very well suited to the SFEM, especially when elements are heavily distorted (obtuse interior angles). The proposed approximation leads to results that are almost identical to those of the SFEM initially proposed in (Comput. Mech. 2007; 39(6):859-877. DOI: 10.1007/s00466-006-0075-4). These results suggest that the proposed approximation scheme forms a strong and rigorous basis for the construction of SFEMs. © 2009 John Wiley & Sons, Ltd. [less ▲]

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See detailIntegrating strong and weak discontinuities without integration subcells and example applications in an XFEM/GFEM framework
Natarajan, S.; Roy Mahapatra, D.; Bordas, Stéphane UL

in International Journal for Numerical Methods in Engineering (2010), 83(3), 269-294

Partition of unity methods, such as the extended finite element method, allows discontinuities to be simulated independently of the mesh (Int. J. Numer. Meth. Engng. 1999; 45:601-620). This eliminates the ... [more ▼]

Partition of unity methods, such as the extended finite element method, allows discontinuities to be simulated independently of the mesh (Int. J. Numer. Meth. Engng. 1999; 45:601-620). This eliminates the need for the mesh to be aligned with the discontinuity or cumbersome re-meshing, as the discontinuity evolves. However, to compute the stiffness matrix of the elements intersected by the discontinuity, a subdivision of the elements into quadrature subcells aligned with the discontinuity is commonly adopted. In this paper, we use a simple integration technique, proposed for polygonal domains (Int. J. Numer. Meth. Engng 2009; 80(1):103-134. DOI: 10.1002/nme.2589) to suppress the need for element subdivision. Numerical results presented for a few benchmark problems in the context of linear elastic fracture mechanics and a multi-material problem show that the proposed method yields accurate results. Owing to its simplicity, the proposed integration technique can be easily integrated in any existing code. © 2010 John Wiley & Sons, Ltd. [less ▲]

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