Showing papers on "Extended finite element method published in 2002"
TL;DR: In this article, an extended finite element method is applied to modeling growth of arbitrary cohesive cracks, which is governed by requiring the stress intensity factors at the tip of the cohesive zone to vanish.
1,395 citations
TL;DR: In this paper, a methodology for solving three-dimensional crack problems with geometries independent of the mesh is described, in which the crack discontinuity is introduced as a Heaviside step function via a partition of unity.
Abstract: A methodology for solving three-dimensional crack problems with geometries that are independent of the mesh is described. The method is based on the extended finite element method, in which the crack discontinuity is introduced as a Heaviside step function via a partition of unity. In addition, branch functions are introduced for all elements containing the crack front. The branch functions include asymptotic near-tip fields that improve the accuracy of the method. The crack geometry is described by two signed distance functions, which in turn can be defined by nodal values. Consequently, no explicit representation of the crack is needed. Examples for three-dimensional elastostatic problems are given and compared to analytic and benchmark solutions. The method is readily extendable to inelastic fracture problems. Copyright © 2002 John Wiley & Sons, Ltd.
652 citations
TL;DR: In this article, a level set method for treating the growth of non-planar 3D cracks is presented, where the crack is defined by two almost-orthogonal level sets (signed distance functions) and the Hamilton-Jacobi equation is used to update the level sets.
Abstract: We present a level set method for treating the growth of non-planar three-dimensional cracks.The crack is defined by two almost-orthogonal level sets (signed distance functions). One of them describes the crack as a two-dimensional surface in a three-dimensional space, and the second is used to describe the one-dimensional crack front, which is the intersection of the two level sets. A Hamilton–Jacobi equation is used to update the level sets. A velocity extension is developed that preserves the old crack surface and can accurately generate the growing surface. The technique is coupled with the extended finite element method which approximates the displacement field with a discontinuous partition of unity. This displacement field is constructed directly in terms of the level sets, so the discretization by finite elements requires no explicit representation of the crack surface. Numerical experiments show the robustness of the method, both in accuracy and in treating cracks with significant changes in topology. Copyright © 2002 John Wiley & Sons, Ltd.
558 citations
TL;DR: In this paper, a stabilized finite element method is proposed to solve the transient Navier-Stokes equations based on the decomposition of the unknowns into resolvable and subgrid scales.
406 citations
TL;DR: An approach for the design of adaptive discontinuous Galerkin finite element methods is applied to physically relevant problems arising in inviscid compressible fluid flows governed by the Euler equations of gas dynamics, providing reliable and efficient error estimation.
402 citations
TL;DR: The elements presented here are the first ones using polynomial shape functions which are known to be stable, and show stability and optimal order approximation.
Abstract: There have been many efforts, dating back four decades, to develop stable mixed finite elements for the stress-displacement formulation of the plane elasticity system. This requires the development of a compatible pair of finite element spaces, one to discretize the space of symmetric tensors in which the stress field is sought, and one to discretize the space of vector fields in which the displacement is sought. Although there are number of well-known mixed finite element pairs known for the analogous problem involving vector fields and scalar fields, the symmetry of the stress field is a substantial additional difficulty, and the elements presented here are the first ones using polynomial shape functions which are known to be stable. We present a family of such pairs of finite element spaces, one for each polynomial degree, beginning with degree two for the stress and degree one for the displacement, and show stability and optimal order approximation. We also analyze some obstructions to the construction of such finite element spaces, which account for the paucity of elements available.
393 citations
TL;DR: Simulations show that using the data at the superconvergence points, the accuracy of the numerical discretization is O(h5/2) in space for smooth subsonic flows, both on structured and on locally refined meshes, and that the space-time adaptation can significantly improve the accuracy and efficiency of the numeric method.
352 citations
TL;DR: In this article, an enriched finite element method for the multi-dimensional Stefan problems is presented, where the standard finite element basis is enriched with a discontinuity in the derivative of the temperature normal to the interface.
Abstract: An enriched finite element method for the multi-dimensional Stefan problems is presented. In this method the standard finite element basis is enriched with a discontinuity in the derivative of the temperature normal to the interface. The approximation can then represent the phase interface and the associated discontinuity in the temperature gradient within an element. The phase interface can be evolved without re-meshing or the use of artificial heat capacity techniques. The interface is described by a level set function that is updated by a stabilized finite element scheme. Several examples are solved by the proposed method to demonstrate the accuracy and robustness of the method. Copyright © 2001 John Wiley & Sons, Ltd.
312 citations
TL;DR: In this paper, a finite element computation of fracture parameters in functionally graded material (FGM) assemblages of arbitrary geometry with stationary cracks is addressed, where the elastic moduli are smooth functions of spatial co-ordinates which are integrated into the element stiffness matrix.
Abstract: This paper is directed towards finite element computation of fracture parameters in functionally graded material (FGM) assemblages of arbitrary geometry with stationary cracks. Graded finite elements are developed where the elastic moduli are smooth functions of spatial co-ordinates which are integrated into the element stiffness matrix. In particular, stress intensity factors for mode I and mixed-mode two-dimensional problems are evaluated and compared through three different approaches tailored for FGMs: path-independent J*k-integral, modified crack-closure integral method, and displacement correlation technique. The accuracy of these methods is discussed based on comparison with available theoretical, experimental or numerical solutions. Copyright © 2001 John Wiley & Sons, Ltd.
310 citations
TL;DR: The scaled boundary finite element method as discussed by the authors is a semi-analytical technique that combines the advantages of the finite element and the boundary element methods with unique properties of its own, such as axisymmetry.
Abstract: The scaled-boundary finite element method is a novel semi-analytical technique, combining the advantages of the finite element and the boundary element methods with unique properties of its own. This paper develops a new virtual work formulation and modal interpretation of the method for elastostatics. This formulation follows a similar procedure to the traditional virtual work derivation of the standard finite element method. As well as making the method more accessible, this approach leads to new techniques for the treatment of body loads, side-face loads and axisymmetry that simplify implementation. The paper fully develops the new formulation, and provides four examples illustrating the versatility, accuracy and efficiency of the scaled boundary finite-element method. Both bounded and unbounded domains are treated, together with problems involving stress singularities.
TL;DR: In this article, the implicit surfaces are defined by radial basis functions and internal features such as material interfaces, sliding interfaces and cracks are treated by enrichment techniques developed in the extended finite element method.
Abstract: A paradigm is developed for generating structured finite element models from solid models by means of implicit surface definitions. The implicit surfaces are defined by radial basis functions. Internal features, such as material interfaces, sliding interfaces and cracks are treated by enrichment techniques developed in the extended finite element method. Methods for integrating the weak form for such models are proposed. These methods simplify the generation of finite element models. Results presented for several examples show that the accuracy of this method is comparable to standard unstructured finite element methods.
TL;DR: In this article, a new interaction energy integral method for the computation of mixed-mode stress intensity factors at the tips of arbitrarily oriented cracks in functionally graded materials is described, where the auxiliary stress and displacement fields are chosen to be the asymptotic near-tip fields for a homogeneous material having the same elastic constants as those found at the crack tip in the functionally graded material.
TL;DR: In this article, the authors consider the approximation properties of finite element spaces on quadrilateral meshes, and demonstrate degradation of the convergence order of these meshes as compared to rectangular meshes for serendipity finite elements and for various mixed and nonconforming finite elements.
Abstract: We consider the approximation properties of finite element spaces on quadrilateral meshes. The finite element spaces are constructed starting with a given finite dimensional space of functions on a square reference element, which is then transformed to a space of functions on each convex quadrilateral element via a bilinear isomorphism of the square onto the element. It is known that for affine isomorphisms, a necessary and sufficient condition for approximation of order r + 1 in L p and order r in W 1 p is that the given space of functions on the reference element contain all polynomial functions of total degree at most r. In the case of bilinear isomorphisms, it is known that the same estimates hold if the function space contains all polynomial functions of separate degree r. We show, by means of a counterexample, that this latter condition is also necessary. As applications, we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for serendipity finite elements and for various mixed and nonconforming finite elements.
TL;DR: It is established by numerical experiments, that most of the proposed finite element methods for Stokes problem or the mixed Poisson's system are not well behaved uniformly in the perturbation parameter, so a new "robust" finite element is introduced which exhibits this property.
Abstract: Finite element methods for a family of systems of singular perturbation problems of a saddle point structure are discussed. The system is approximately a linear Stokes problem when the perturbation parameter is large, while it degenerates to a mixed formulation of Poisson's equation as the perturbation parameter tends to zero. It is established, basically by numerical experiments, that most of the proposed finite element methods for Stokes problem or the mixed Poisson's system are not well behaved uniformly in the perturbation parameter. This is used as the motivation for introducing a new "robust" finite element which exhibits this property.
TL;DR: In this paper, a variational formulation is developed for both linear and non-linear strain-gradient elasticity theories, in which both the displacement and the displacement gradients are used as independent unknowns and their relationship is enforced in an integral sense.
TL;DR: In this paper, a hybrid numerical method for modeling the evolution of sharp phase interfaces on fixed grids is presented, where the temperature field evolves according to classical heat conduction in two subdomains separated by a moving freezing front, and enrichment strategies of the eXtended Finite Element Method (X-FEM) are employed to represent the jump in temperature gradient that governs the velocity of the phase boundary.
Abstract: A hybrid numerical method for modelling the evolution of sharp phase interfaces on fixed grids is presented. We focus attention on two-dimensional solidification problems, where the temperature field evolves according to classical heat conduction in two subdomains separated by a moving freezing front. The enrichment strategies of the eXtended Finite Element Method (X-FEM) are employed to represent the jump in the temperature gradient that governs the velocity of the phase boundary. A new approach with the X-FEM is suggested for this class of problems whereby the partition of unity is constructed with C1(Ω) polynomials and enriched with a C0(Ω) function. This approach leads to jumps in temperature gradient occurring only at the phase boundary, and is shown to significantly improve estimates for the front velocity. Temporal derivatives of the temperature field in the vicinity of the phase front are obtained with a projection that employs discontinuous enrichment. In conjunction with a finer finite difference grid, the Level Set method is used to represent the evolution of the phase interface. An iterative procedure is adopted to satisfy the constraints on the temperature field on the phase boundary. The robustness and utility of the method is demonstrated with several benchmark problems of phase transformation. Copyright © 2002 John Wiley & Sons, Ltd.
TL;DR: In this article, a fast boundary element method for the linear Poisson-Boltzmann equation governing biomolecular electrostatics is presented. But, unlike previous fast boundary elements implementations, the present treatment accommodates finite salt concentrations thus enabling the study of biomolecules under realistic physiological conditions.
Abstract: This article summarizes the development of a fast boundary element method for the linear Poisson-Boltzmann equation governing biomolecular electrostatics. Unlike previous fast boundary element implementations, the present treatment accommodates finite salt concentrations thus enabling the study of biomolecular electrostatics under realistic physiological conditions. This is achieved by using multipole expansions specifically designed for the exponentially decaying Green’s function of the linear Poisson -Boltzmann equation. The particular formulation adopted in the boundary element treatment directly affects the numerical conditioning and thus convergence behavior of the method. Therefore, the formulation and reasons for its choice are first presented. Next, the multipole approximation and its use in the context of a fast boundary element method are described together with the iteration method employed to extract the surface distributions. The method is then subjected to a series of computational tests involving a sphere with interior charges. The purpose of these tests is to assess accuracy and verify the anticipated computational performance trends. Finally, the salt dependence of electrostatic properties of several biomolecular systems (alanine dipeptide, barnase, barstar, and coiled coil tetramer) is examined with the method and the results are compared with finite difference Poisson-Boltzmann codes.
TL;DR: In this paper, a high-order quadrilateral discontinuous spectral element method (DSEM) is proposed to solve electromagnetic scattering problems by approximating Maxwell's equations in the time-domain with a highorder Quadrilateral Discriminative Spectral Element Method.
Abstract: In this paper we solve electromagnetic scattering problems by approximating Maxwell's equations in the time-domain with a high-order quadrilateral discontinuous spectral element method (DSEM). The method is a collocation form of the discontinuous Galerkin method for hyperbolic systems where the solution is approximated by a tensor product Legendre expansion and inner products are replaced with Gauss–Legendre quadratures. To increase flexibility of the method, we use a mortar element method to couple element faces. Mortars provide a means for coupling element faces along which the polynomial orders differ, which allows the flexibility to choose the approximation order within an element by considering only local resolution requirements. Mortars also permit local subdivision of a mesh by connecting element faces that do not share a full side. We present evidence showing that the convergence of the non-conforming approximations is spectral along with examples of their use. Copyright © 2001 John Wiley & Sons, Ltd.
TL;DR: In this article, the theory and application of mimetic finite difference methods for the solution of diffusion problems in strongly heterogeneous anisotropic materials is reviewed and extended for nonorthogonal, nonsmooth, structured and unstructured computational grids.
Abstract: This paper reviews and extends the theory and application of mimetic finite difference methods for the solution of diffusion problems in strongly heterogeneous anisotropic materials. These difference operators satisfy the fundamental identities, conservation laws and theorems of vector and tensor calculus on nonorthogonal, nonsmooth, structured and unstructured computational grids. We provide explicit approximations for equations in two dimensions with discontinuous anisotropic diffusion tensors. We mention the similarities and differences between the new methods and mixed finite element or hybrid mixed finite element methods.
TL;DR: An algebraic multigrid method is presented for the efficient solution of the linear system arising from a finite element discretization of variational problems in H0(curl,Ω).
Abstract: This paper presents an algebraic multigrid method for the efficient solution of the linear system arising from a finite element discretization of variational problems in H0(curl,Ω). The finite element spaces are generated by Nedelec's edge elements.
A coarsening technique is presented, which allows the construction of suitable coarse finite element spaces, corresponding transfer operators and appropriate smoothers. The prolongation operator is designed such that coarse grid kernel functions of the curl-operator are mapped to fine grid kernel functions. Furthermore, coarse grid kernel functions are ‘discrete’ gradients. The smoothers proposed by Hiptmair and Arnold, Falk and Winther are directly used in the algebraic framework.
Numerical studies are presented for 3D problems to show the high efficiency of the proposed technique. Copyright © 2002 John Wiley & Sons, Ltd.
TL;DR: The application of the eXtended finite element method (X-FEM) to thermal problems with moving heat sources and phase boundaries is presented and the ability of the method to capture the highly localized, transient solution in the vicinity of a heat source or material interface is presented.
Abstract: The application of the eXtended finite element method (X-FEM) to thermal problems with moving heat sources and phase boundaries is presented. Of particular interest is the ability of the method to capture the highly localized, transient solution in the vicinity of a heat source or material interface. This is effected through the use of a time-dependent basis formed from the union of traditional shape functions with a set of evolving enrichment functions. The enrichment is constructed through the partition of unity framework, so that the system of equations remains sparse and the resulting approximation is conforming. In this manner, local solutions and arbitrary discontinuities that cannot be represented by the standard shape functions are captured with the enrichment functions. A standard time-projection algorithm is employed to account for the time-dependence of the enrichment, and an iterative strategy is adopted to satisfy local interface conditions. The separation of the approximation into classical shape functions that remain fixed in time and the evolving enrichment leads to a very efficient solution strategy. The robustness and utility of the method is demonstrated with several benchmark problems involving moving heat sources and phase transformations.
TL;DR: In this paper, a displacement-based finite element model for the analysis of steel and concrete composite beams with flexible shear connection is presented, where the stiffness matrix and the fixed-end nodal force vector are directly derived from the "exact" solution of Newmark's differential equation.
01 Jan 2002
TL;DR: A combined method consisting of the mixed finite element method for flow and the discontinuous Galerkin method for transport is introduced for the coupled system of miscible displacement problem.
Abstract: A combined method consisting of the mixed finite element method for flow and the discontinuous Galerkin method for transport is introduced for the coupled system of miscible displacement problem. A “cut-off” operator M is introduced in the discontinuous Galerkin formular in order to make the combined scheme converge. Optimal error estimates in L 2(H 1) for concentration and in L ∞(L 2) for velocity are derived.
TL;DR: In this article, the regularised long wave equation is solved by Galerkin's method using linear space finite elements, which is shown to have good accuracy for small amplitude waves.
TL;DR: A three-dimensional version of the hp-adaptive, mixed finite element (FE) method based on hexahedral elements of possibly variable order for the solution of steady-state Maxwell's equations proposed in Demkowicz and Vardapetyan is presented.
Abstract: This is the second of papers describing implementations of the hp-adaptive, mixed finite element (FE) method for the solution of steady-state Maxwell's equations proposed in Demkowicz and Vardapetyan (Comput. Methods Appl. Mech. Engng. 1998; 152(1–2):103–124). The paper presents a three-dimensional version of the method based on hexahedral elements of possibly variable order. The elements can be subsected (isotropically or anisotropically), and their orders can be enriched, which allows for non-uniform distribution of element sizes h and orders p—the hp adaptation. A few numerical examples illustrate the capability of the method. Copyright © 2001 John Wiley & Sons, Ltd.
TL;DR: In this paper, a displacement and rotation based finite element method for the solution of boundary value problems in linear isotropic Cosserat elasticity is proposed, and the field equations for the problem of plane strain are derived from the three-dimensional theory and expressed in oblique rectilinear coordinates.
TL;DR: In this paper, a hybrid finite volume/element method is analyzed through the computation of creeping flows of viscoelastic fluids in plane 4:1 sharp and rounded-corner contraction geometries.
Abstract: A hybrid finite volume/element method is analysed through the computation of creeping flows of viscoelastic fluids in plane 4:1 sharp and rounded-corner contraction geometries. Simulations are presented for three models: a constant viscosity Oldroyd-B fluid, and Phan-Thien/Tanner (PTT) shear thinning fluids of exponential and linear approximation form. A Taylor–Galerkin/pressure-correction scheme is implemented as the base time-stepping framework. The momentum equations are solved by a finite element method, whilst the constitutive equations are solved by a finite volume approach. Mesh convergence is analysed via refinement around the contraction to capture boundary layers and flow structure. Pressure drop is shown to increase with flow rate for a fixed fluid. For the Oldroyd-B model, singular behaviour is reported in the main stress component as one approaches the corner in the rounded, as with the sharp geometry. Velocity components display an asymptotic trend with a positive slope. Higher values of Weissenberg numbers (We) are reached with these finite volume schemes compared to their finite element counterparts, attributing this to superior accuracy properties.
TL;DR: Theoretical results concerning the existence, stability and convergence of the finite dimensional representation are established and numerical results involving identification of finite dimensional models for both linear and nonlinear infinite dimensional systems are presented.