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Showing papers on "Extended finite element method published in 2014"


Book
23 Sep 2014
TL;DR: Finite Element Solution of Boundary Value Problems: Theory and Computation as mentioned in this paper provides a thorough, balanced introduction to both the theoretical and the computational aspects of the finite element method for solving boundary value problems for partial differential equations.
Abstract: Finite Element Solution of Boundary Value Problems: Theory and Computation provides a thorough, balanced introduction to both the theoretical and the computational aspects of the finite element method for solving boundary value problems for partial differential equations. Although significant advances have been made in the finite element method since this book first appeared in 1984, the basics have remained the same, and this classic, well-written text explains these basics and prepares the reader for more advanced study. Useful as both a reference and a textbook, complete with examples and exercies, it remains as relevant today as it was when originally published. This book is written for advanced undergraduate and graduate students in the areas of numerical analysis, mathematics, and computer science, as well as for theoretically inclined practitioners in engineering and the physical science.

506 citations


Journal ArticleDOI
TL;DR: In this article, a weak Galerkin (WG) method is introduced and analyzed for the second order elliptic equation formulated as a system of two first order linear equations.
Abstract: . A new weak Galerkin (WG) method is introduced and analyzed for the second order elliptic equation formulated as a system of two first order linear equations. This method, called WG-MFEM, is designed by using discontinuous piecewise polynomials on finite element partitions with arbitrary shape of polygons/polyhedra. The WG-MFEM is capable of providing very accurate numerical approximations for both the primary and flux variables. Allowing the use of discontinuous approximating functions on arbitrary shape of polygons/polyhedra makes the method highly flexible in practical computation. Optimal order error estimates in both discrete H and L norms are established for the corresponding weak Galerkin mixed finite element solutions.

440 citations


Journal ArticleDOI
TL;DR: PerMIX is an object oriented open-source effort written primarily in Fortran 2003 standard with Fortran/C++ interfaces to a number of other libraries such as LAMMPS, ABAQUS, LS-DYNA and GMSH for multiscale modeling and simulation of fracture in solids.
Abstract: We present an open-source software framework called PERMIX for multiscale modeling and simulation of fracture in solids. The framework is an object oriented open-source effort written primarily in Fortran 2003 standard with Fortran/C++ interfaces to a number of other libraries such as LAMMPS, ABAQUS, LS-DYNA and GMSH. Fracture on the continuum level is modeled by the extended finite element method (XFEM). Using several novel or state of the art methods, the piece software handles semi-concurrent multiscale methods as well as concurrent multiscale methods for fracture, coupling two continuum domains or atomistic domains to continuum domains, respectively. The efficiency of our open-source software is shown through several simulations including a 3D crack modeling in clay nanocomposites, a semi-concurrent FE-FE coupling, a 3D Arlequin multiscale example and an MD-XFEM coupling for dynamic crack propagation.

430 citations


Journal ArticleDOI
TL;DR: In this paper, a local generalized finite element basis for elliptic problems with heterogeneous and highly varying coefficients was constructed, which does not rely on regularity of the solution or scale separation in the coefficient.
Abstract: This paper constructs a local generalized finite element basis for elliptic problems with heterogeneous and highly varying coefficients. The basis functions are solutions of local problems on vertex patches. The error of the corresponding generalized finite element method decays exponentially with respect to the number of layers of elements in the patches. Hence, on a uniform mesh of size $ H$, patches of diameter $ H\log (1/H)$ are sufficient to preserve a linear rate of convergence in $ H$ without pre-asymptotic or resonance effects. The analysis does not rely on regularity of the solution or scale separation in the coefficient. This result motivates new and justifies old classes of variational multiscale methods. - See more at: http://www.ams.org/journals/mcom/2014-83-290/S0025-5718-2014-02868-8/#sthash.z2CCFXIg.dpuf

424 citations


Journal ArticleDOI
TL;DR: In this paper, a phase-field model for fracture in Kirchoff-love thin shells using the local maximum-entropy (LME) mesh-free method is presented, which does not require an explicit representation and tracking, which is advantage over techniques as the extended finite element method that requires tracking of the crack paths.

274 citations



BookDOI
01 Jan 2014
TL;DR: The Babuska-Brezzi Theory and Raviart-Thomas Spaces as discussed by the authors have been applied to the Mixed Finite Element Methods (MFFM) method for space partitioning.
Abstract: Introduction.- The Babuska-Brezzi Theory.- Raviart-Thomas Spaces.- Mixed Finite Element Methods.

187 citations


Journal ArticleDOI
TL;DR: In this paper, the authors developed a method based on local maximum entropy shape functions together with enrichment functions used in partition of unity methods to discretize problems in linear elastic fracture mechanics and obtained improved accuracy relative to the standard extended finite element method at a comparable computational cost.
Abstract: In this paper, we develop a method based on local maximum entropy shape functions together with enrichment functions used in partition of unity methods to discretize problems in linear elastic fracture mechanics. We obtain improved accuracy relative to the standard extended finite element method at a comparable computational cost. In addition, we keep the advantages of the LME shape functions, such as smoothness and non-negativity. We show numerically that optimal convergence (same as in FEM) for energy norm and stress intensity factors can be obtained through the use of geometric (fixed area) enrichment with no special treatment of the nodes near the crack such as blending or shifting.

181 citations


Journal ArticleDOI
TL;DR: This work considers the case of a network of intersecting fractures, with the aim of deriving physically consistent and effective interface conditions to impose at the intersection between fractures, using the extended finite element method (XFEM).
Abstract: Subsurface flows are influenced by the presence of faults and large fractures which act as preferential paths or barriers for the flow. In literature models were proposed to handle fractures in a porous medium as objects of codimension 1. In this work we consider the case of a network of intersecting fractures, with the aim of deriving physically consistent and effective interface conditions to impose at the intersection between fractures. This new model accounts for the angle between fractures at the intersections and allows for jumps of pressure across intersections. This fact permits to describe the flow when fractures are characterized by different properties more accurately with respect to other models that impose pressure continuity. The main mathematical properties of the model, derived in the two-dimensional setting, are analyzed. As concerns the numerical discretization we allow the grids of the fractures to be independent, thus in general non-matching at the intersection, by means of the extended finite element method (XFEM). This increases the flexibility of the method in the case of complex geometries characterized by a high number of fractures.

170 citations


Journal ArticleDOI
TL;DR: In this article, an extension of the classical smeared (or regularized) approach to fracture is proposed, which can be seen as an intermediate proposition between purely cohesive formulations and the smeared modeling.

166 citations


Journal ArticleDOI
TL;DR: This paper first presents an overview of previous developments on conforming polygonal and polyhedral finite elements, and then appeals to the exact decomposition in the VEM to obtain a robust and efficient generalized barycentric coordinate-based Galerkin method on polygonnal andpolyhedral elements.
Abstract: Generalized barycentric coordinates such as Wachspress and mean value coordinates have been used in polygonal and polyhedral finite element methods. Recently, mimetic finite difference schemes were cast within a variational framework, and a consistent and stable finite element method on arbitrary polygonal meshes was devised. The method was coined as the virtual element method (VEM), since it did not require the explicit construction of basis functions. This advance provides a more in-depth understanding of mimetic schemes, and also endows polygonal-based Galerkin methods with greater flexibility than three-node and four-node finite element methods. In the VEM, a projection operator is used to realize the decomposition of the stiffness matrix into two terms: a consistent matrix that is known, and a stability matrix that must be positive semi-definite and which is only required to scale like the consistent matrix. In this paper, we first present an overview of previous developments on conforming polygonal and polyhedral finite elements, and then appeal to the exact decomposition in the VEM to obtain a robust and efficient generalized barycentric coordinate-based Galerkin method on polygonal and polyhedral elements. The consistent matrix of the VEM is adopted, and numerical quadrature with generalized barycentric coordinates is used to compute the stability matrix. This facilitates post-processing of field variables and visualization in the VEM, and on the other hand, provides a means to exactly satisfy the patch test with efficient numerical integration in polygonal and polyhedral finite elements. We present numerical examples that demonstrate the sound accuracy and performance of the proposed method. For Poisson problems in ℝ2 and ℝ3, we establish that linearly complete generalized barycentric interpolants deliver optimal rates of convergence in the L2-norm and the H1-seminorm.

Journal ArticleDOI
TL;DR: The proposed floating node method is particularly suited for modelling weak and cohesive discontinuities and for the representation of complex crack networks and can model multiple plies and interfaces of a composite laminate, and both matrix crack and delamination, within a user-defined element.

Journal ArticleDOI
TL;DR: A numerical theory based on the mixed finite element method for a time-fractional fourth-order partial differential equation (PDE) is presented and an a priori error result in H^1-norm for the scalar unknown u also is proved.

Journal ArticleDOI
TL;DR: In this article, the authors present simulations of 3D non-planar fracture propagation using an adaptive generalized FEM, which greatly facilitates the discretization of complex 3D fractures, as finite element faces are not required to fit the crack surfaces.
Abstract: SUMMARY Hydraulic fracturing is the method of choice to enhance reservoir permeability and well efficiency for extraction of shale gas. Multi-stranded non-planar hydraulic fractures are often observed in stimulation sites. Non-planar fractures propagating from wellbores inclined from the direction of maximum horizontal stress have also been reported. The pressure required to propagate non-planar fractures is in general higher than in the case of planar fractures. Current computational methods for the simulation of hydraulic fractures generally assume single, symmetric, and planar crack geometries. In order to better understand hydraulic fracturing in complex-layered naturally fractured reservoirs, fully 3D models need to be developed. In this paper, we present simulations of 3D non-planar fracture propagation using an adaptive generalized FEM. This method greatly facilitates the discretization of complex 3D fractures, as finite element faces are not required to fit the crack surfaces. A solution strategy for fully automatic propagation of arbitrary 3D cracks is presented. The fracture surface on which pressure is applied is also automatically updated at each step. An efficient technique to numerically integrate boundary conditions on crack surfaces is also proposed and implemented. Strongly graded localized refinement and analytical asymptotic expansions are used as enrichment functions in the neighborhood of fracture fronts to increase the computational accuracy and efficiency of the method. Stress intensity factors with pressure on crack faces are extracted using the contour integral method. Various non-planar crack geometries are investigated to demonstrate the robustness and flexibility of the proposed simulation methodology. Copyright © 2014 John Wiley & Sons, Ltd.

Journal ArticleDOI
TL;DR: In this paper, a dynamic X-FEM model is developed in which both Crank-Nicolson and Newmark time integration methods are used for calculating transient responses of thermal and electromechanical fields respectively.

Journal ArticleDOI
TL;DR: The results suggest that the proposed preconditioning scheme leads to condition numbers similar to that of a body-fitted mesh using the traditional finite element method without loss of solution accuracy.
Abstract: The extended finite element method (XFEM) is an approach for solving problems with non-smooth solutions, which arise from geometric features such as cracks, holes, and material inclusions. In the XFEM, the approximate solution is locally enriched to capture the discontinuities without requiring a mesh which conforms to the geometric features. One drawback of the XFEM is that an ill-conditioned system of equations results when the ratio of volumes on either side of the interface in an element is small. Such interface configurations are often unavoidable, in particular for moving interface problems on fixed meshes. In general, the ill-conditioning reduces the performance of iterative linear solvers and impedes the convergence of solvers for nonlinear problems. This paper studies the XFEM with a Heaviside enrichment strategy for solving problems with stationary and moving material interfaces. A generalized formulation of the XFEM is combined with the level set method to implicitly define the embedded interface geometry. In order to avoid the ill-conditioning, a simple and efficient scheme based on a geometric preconditioner and constraining degrees of freedom to zero for small intersections is proposed. The geometric preconditioner is computed from the nodal basis functions, and therefore may be constructed prior to building the system of equations. This feature and the low-cost of constructing the preconditioning matrix makes it well suited for nonlinear problems with fixed and moving interfaces. It is shown by numerical examples that the proposed preconditioning scheme performs well for discontinuous problems and $$C^0$$C0-continuous problems with both the stabilized Lagrange and Nitsche methods for enforcing the continuity constraint at the interface. Numerical examples are presented which compare the condition number and solution error with and without the proposed preconditioning scheme. The results suggest that the proposed preconditioning scheme leads to condition numbers similar to that of a body-fitted mesh using the traditional finite element method without loss of solution accuracy.

Journal ArticleDOI
TL;DR: In this article, a scaled boundary polygon formulation is presented to model elasto-plastic material responses in structures, where the stiffness matrix and residual load vector are matrix power integrals that can be evaluated analytically even when a strain singularity is present.

Book
21 Jan 2014
TL;DR: The Babuska-Brezzi Theory and Raviart-Thomas Spaces as discussed by the authors have been applied to the Mixed Finite Element Methods (MFFM) method for space partitioning.
Abstract: Introduction.- The Babuska-Brezzi Theory.- Raviart-Thomas Spaces.- Mixed Finite Element Methods.

Journal ArticleDOI
TL;DR: In this article, the static and dynamic analyses of multi-layered plates with discontinuities were performed using the strong formulation finite element method, and the numerical results in terms of natural frequencies and maximum deflections were compared to literature and to the same results obtained with a finite element code.
Abstract: This paper deals with the static and dynamic analyses of multi-layered plates with discontinuities. The two-dimensional first-order shear deformation theory is used to derive the fundamental system of equations in terms of generalized displacements. The fundamental set, with its boundary conditions, is solved in its strong form. A new method termed strong formulation finite element method is considered in the present paper to solve this kind of plates. This numerical methodology is the cohesion of derivative evaluation of partial differential systems of equations and a domain sub-division. The numerical results in terms of natural frequencies and maximum deflections are compared to literature and to the same results obtained with a finite element code. The stability, accuracy and reliability of the present methodology is shown through several numerical applications.

Journal ArticleDOI
TL;DR: In this article, the spectral cell method is proposed to combine the finite cell method with the spectral element method for the analysis of wave propagation phenomena, which is referred to as the spectralcell method.
Abstract: SUMMARY An accurate and efficient simulation of wave propagation phenomena plays an important role in different engineering disciplines. In structural health monitoring, for example, ultrasonic guided waves are used to detect and localize damage and to assess the structural integrity of the component part under consideration. Because of the complexity of real structures, the numerical simulation of structural health monitoring systems is a computationally demanding task. Therefore, to facilitate the analysis of wave propagation phenomena, the authors propose to combine the finite cell method with the spectral element method. The ensuing novel method is referred to as the spectral cell method. Because it does not rely on body-fitted meshes, the resulting approach eliminates all discretization difficulties encountered in conventional finite element methods. Moreover, with the aid of mass lumping, it paves the way for the use of explicit time-integration algorithms. In the first part of the paper, we show that using a lumped mass matrix instead of the consistent one has no detrimental effect on the accuracy of the spectral element method. We introduce the spectral cell method in the second part, showing that, when applied to wave propagation analysis, the spectral cell method yields results comparable with other standard higher order finite element approaches.Copyright © 2014 John Wiley & Sons, Ltd.

Journal ArticleDOI
TL;DR: A new stabilized XFEM based fixed-grid approach for the transient incompressible Navier-Stokes equations using cut elements is developed, which is much more accurate and less sensitive to the location of the interface.

Journal ArticleDOI
TL;DR: In this paper, the authors proposed a new technique for solving free vibration problems of composite arbitrarily shaped membranes by using Generalized Differential Quadrature Finite Element Method (GDQFEM).

Journal ArticleDOI
TL;DR: In this article, the authors describe discretisations of the shallow-water equations on the sphere using the framework of finite element exterior calculus, which are extensions of the mimetic finite difference framework presented in Ringler (2010).

Journal ArticleDOI
TL;DR: In this article, a mixed finite element method for the two-dimensional Biot's consolidation model of poroelasticity is proposed, which uses the total stress tensor and fluid flux as primary unknown variables as well as the displacement and pore pressure.
Abstract: In this article, we propose a mixed finite element method for the two-dimensional Biot's consolidation model of poroelasticity. The new mixed formulation presented herein uses the total stress tensor and fluid flux as primary unknown variables as well as the displacement and pore pressure. This method is based on coupling two mixed finite element methods for each subproblem: the standard mixed finite element method for the flow subproblem and the Hellinger–Reissner formulation for the mechanical subproblem. Optimal a-priori error estimates are proved for both semidiscrete and fully discrete problems when the Raviart–Thomas space for the flow problem and the Arnold–Winther space for the elasticity problem are used. In particular, optimality in the stress, displacement, and pressure has been proved in when the constrained-specific storage coefficient is strictly positive and in the weaker norm when is nonnegative. We also present some of our numerical results.Copyright © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1189–1210, 2014

Journal ArticleDOI
TL;DR: A full three-dimensional FEA for the prediction of railway ground-borne vibrations and investigations with commercial FEA software ABAQUS are presented, with the development of an external meshing tool so as to automatically define the infinite elements at the model boundary.

Journal ArticleDOI
TL;DR: In this paper, a crack growth simulation is presented in saturated porous media using the extended finite element method, where the mass balance equation of fluid phase and the momentum balance of bulk and fluid phases are employed to obtain the fully coupled set of equations in the framework of $$u{-}p$$ formulation.
Abstract: In this paper, the crack growth simulation is presented in saturated porous media using the extended finite element method. The mass balance equation of fluid phase and the momentum balance of bulk and fluid phases are employed to obtain the fully coupled set of equations in the framework of $$u{-}p$$ formulation. The fluid flow within the fracture is modeled using the Darcy law, in which the fracture permeability is assumed according to the well-known cubic law. The spatial discritization is performed using the extended finite element method, the time domain discritization is performed based on the generalized Newmark scheme, and the non-linear system of equations is solved using the Newton–Raphson iterative procedure. In the context of the X-FEM, the discontinuity in the displacement field is modeled by enhancing the standard piecewise polynomial basis with the Heaviside and crack-tip asymptotic functions, and the discontinuity in the fluid flow normal to the fracture is modeled by enhancing the pressure approximation field with the modified level-set function, which is commonly used for weak discontinuities. Two alternative computational algorithms are employed to compute the interfacial forces due to fluid pressure exerted on the fracture faces based on a ‘partitioned solution algorithm’ and a ‘time-dependent constant pressure algorithm’ that are mostly applicable to impermeable media, and the results are compared with the coupling X-FEM model. Finally, several benchmark problems are solved numerically to illustrate the performance of the X-FEM method for hydraulic fracture propagation in saturated porous media.

Journal ArticleDOI
TL;DR: The method uses the edge-based vector basis functions, which automatically enforce the divergence free conditions for electric and magnetic fields, which is effective in modeling the seafloor bathymetry using hexahedral mesh.

Journal ArticleDOI
TL;DR: In this article, a generalized Heaviside enrichment strategy is presented that adapts the set of enrichment functions to the material layout and consistently interpolates the state variable fields, bypassing the limitations of the traditional approach.
Abstract: This paper studies level set topology optimization of structures predicting the structural response by the eXtended Finite Element Method (XFEM). In contrast to Ersatz material approaches, the XFEM represents the geometry in the mechanical model by crisp boundaries. The traditional XFEM approach augments the approximation of the state variable fields with a fixed set of enrichment functions. For complex material layouts with small geometric features, this strategy may result in interpolation errors and non-physical coupling between disconnected material domains. These defects can lead to numerical instabilities in the optimized material layout, similar to checker-board patterns found in density methods. In this paper, a generalized Heaviside enrichment strategy is presented that adapts the set of enrichment functions to the material layout and consistently interpolates the state variable fields, bypassing the limitations of the traditional approach. This XFEM formulation is embedded into a level set topology optimization framework and studied with "material-void" and "material-material" design problems, optimizing the compliance via a mathematical programming method. The numerical results suggest that the generalized formulation of the XFEM resolves numerical instabilities, but regularization techniques are still required to control the optimized geometry. It is observed that constraining the perimeter effectively eliminates the emergence of small geometric features. In contrast, smoothing the level set field does not provide a reliable geometry control but mainly improves the convergence rate of the optimization process.

Journal ArticleDOI
TL;DR: Weak Galerkin (WG) finite element methods were used in this paper to approximate weak partial derivatives and their approximations for a class of discontinuous functions defined on a finite element partition of the domain consisting of general polygons or polyhedra.
Abstract: This paper presents a new and efficient numerical algorithm for the biharmonic equation by using weak Galerkin (WG) finite element methods. The WG finite element scheme is based on a variational form of the biharmonic equation that is equivalent to the usual H 2 -semi norm. Weak partial derivatives and their approximations, called discrete weak partial derivatives, are introduced for a class of discontinuous functions defined on a finite element partition of the domain consisting of general polygons or polyhedra. The discrete weak partial derivatives serve as building blocks for the WG finite element method. The resulting matrix from the WG method is symmetric, positive definite, and parameter free. An error estimate of optimal order is derived in an H 2 -equivalent norm for the WG finite element solutions. Error estimates in the usual L 2 norm are established, yielding optimal order of convergence for all the WG finite element algorithms except the one corresponding to the lowest order (i.e., piecewise quadratic elements). Some numerical experiments are presented to illustrate the efficiency and accuracy of the numerical scheme.

Journal ArticleDOI
TL;DR: In this article, the failure analysis using explicit finite element method is performed to predict the failure properties and burst strengths of aluminum-carbon fiber/epoxy composite cylindrical laminate structures in terms of three composite pressure vessels with different geometry sizes.
Abstract: Based on continuum damage mechanics, the progressive failure analysis using explicit finite element method is performed to predict the failure properties and burst strengths of aluminum–carbon fiber/epoxy composite cylindrical laminate structures in terms of three composite pressure vessels with different geometry sizes. The failure analysis employs the Hashin damage initiation criterion and the fracture energy-based damage evolution law for composite layers. The numerical convergence problem is solved by introducing viscous damping effect into finite element equations for strain softening phenomenon. Effects of the calculation time and mesh sizes on the failure properties of composite laminates are explored. In addition, the predicted failure strengths of composite laminates using explicit finite element analysis are also compared with those by experiments and implicit finite element analysis.