Showing papers on "Extended finite element method published in 2001"
TL;DR: In this paper, a methodology to model arbitrary holes and material interfaces (inclusions) without meshing the internal boundaries is proposed, which couples the level set method with the extended finite element method (X-FEM).
1,112 citations
TL;DR: In this paper, a model which allows the introduction of displacements jumps to conventional finite elements is developed, where the path of the discontinuity is completely independent of the mesh structure.
Abstract: A model which allows the introduction of displacements jumps to conventional finite elements is developed. The path of the discontinuity is completely independent of the mesh structure. Unlike so-called ‘embedded discontinuity’ models, which are based on incompatible strain modes, there is no restriction on the type of underlying solid finite element that can be used and displacement jumps are continuous across element boundaries. Using finite element shape functions as partitions of unity, the displacement jump across a crack is represented by extra degrees of freedom at existing nodes. To model fracture in quasi-brittle heterogeneous materials, a cohesive crack model is used. Numerical simulations illustrate the ability of the method to objectively simulate fracture with unstructured meshes. Copyright © 2001 John Wiley & Sons, Ltd.
914 citations
Book•
01 Jan 2001
TL;DR: This paper proposes a method for guaranteed a-posteriori error estimation, and Guaranteed a-PosterIORi estimation of the pollution error in the finite element method.
Abstract: Preface 1. Introduction 2. Mathematical formulation of the model problem 3. The finite element method 4. Local behaviour in the finite element method 5. A-posteriori estimation of the error 6. Guaranteed a-posteriori error estimation, and a-posteriori estimation of the pollution error Appendix Index
770 citations
TL;DR: An algorithm which couples the level set method (LSM) with the extended finite element method (X‐FEM) to model crack growth is described, which requires no remeshing as the crack progresses, making the algorithm very efficient.
Abstract: SUMMARYAn algorithm which couples the level set method (LSM) with the extended!nite element method(X-FEM) to model crack growth is described. The level set method is used to represent the cracklocation, including the location of crack tips. The extended!nite element method is used to computethe stress and displacement!elds necessary for determining the rate of crack growth. This combinedmethod requires no remeshing as the crack progresses, making the algorithm very e#cient. Thecombination of these methods has a tremendous potential for a wide range of applications. Numericalexamples are presented to demonstrate the accuracy of the combined methods. Copyright ? 2001John Wiley & Sons, Ltd. KEY WORDS : extended!nite elements method; level set method; crack growth 1. INTRODUCTIONIn this paper, we describe an algorithm where the level set method (LSM) is coupled withthe extended!nite element method (X-FEM) [1–3] to model crack growth. The LSM isa numerical scheme developed by Osher and Sethian [4] to model the motion of interfaces.In the LSM the interface is represented as the zero level set of a function of one higherdimension. The current formulation of the LSM has no provision for modelling free movingendpoints on curves or free moving edges on surfaces. A similar level set representation wasused in Reference [5] to model the evolution of a curve segment. However, unlike the methodpresented here, in Reference [5] the endpoints of the evolving curve segment remain!xed.We present an extension of the LSM for modelling the evolution of an open curve segmentand use this extension to model the growth of a fatigue crack.
747 citations
TL;DR: In this paper, the eXtended Finite Element Method (X-FEM) is used to discretize the equations, allowing for the modeling of cracks whose geometry is independent of the finite element mesh.
546 citations
TL;DR: In this article, a partition of unity finite element method and hp-cloud method for dynamic crack propagation is presented, where the approximation spaces are constructed using a partition-of-unity (PU) and local enrichment functions.
372 citations
TL;DR: It is shown on a variety of problems that the most cost-effective simulations can be obtained using higher-order basis functions when compared with the traditional linear basis.
Abstract: Stabilized finite element methods have been shown to yield robust, accurate numerical solutions to both the compressible and incompressible Navier-Stokes equations for laminar and turbulent flows. The present work focuses on the application of higher-order, hierarchical basis functions to the incompressible Navier-Stokes equations using a stabilized finite element method. It is shown on a variety of problems that the most cost-effective simulations (in terms of CPU time, memory, and disk storage) can be obtained using higher-order basis functions when compared with the traditional linear basis. In addition, algorithms will be presented for the efficient implementation of these methods within the traditional finite element data structures
308 citations
241 citations
TL;DR: The key idea is the treatment of the curvature terms by a variational formulation and in the context of a discontinuous in time space–time element discretization stability in (weak) energy norms can be proved.
Abstract: The instationary Navier–Stokes equations with a free capillary boundary are considered in 2 and 3 space dimensions. A stable finite element discretization is presented. The key idea is the treatment of the curvature terms by a variational formulation. In the context of a discontinuous in time space–time element discretization stability in (weak) energy norms can be proved. Numerical examples in 2 and 3 space dimensions are given.
237 citations
TL;DR: Numerical results are presented that demonstrate the accuracy and efficiency of these methods for the SN equations on three-dimensional unstructured tetrahedral and hexahedral meshes.
Abstract: Discontinuous finite element methods for the SN equations on three-dimensional unstructured tetrahedral and hexahedral meshes are presented. Solution techniques including source iteration and diffu...
TL;DR: In this paper, a non-linearly coupled multi-modal response model is proposed for modeling large deflection beam response involving multiple vibration modes, which can be applied to the case of a homogeneous isotropic beam.
TL;DR: In this paper, a finite element method for the solution of 3D incompressible magnetohydrodynamic (MHD) flows is presented, where a Galerkin-least-squares variational formulation is used allowing equal-order approximations for all unknowns.
TL;DR: In this article, a new method for the simulation of particulate flows, based on the extended finite element method (X-FEM), is described, where particle surfaces need not conform to the finite element boundaries, so that moving particles can be simulated without remeshing.
Abstract: A new method for the simulation of particulate flows, based on the extended finite element method (X-FEM), is described. In this method, the particle surfaces need not conform to the finite element boundaries, so that moving particles can be simulated without remeshing. The near field form of the fluid flow about each particle is built into the finite element basis using a partition of unity enrichment, allowing the simple enforcement of boundary conditions and improved accuracy over other methods on a coarse mesh. We present a weak form of the equations of motion useful for the simulation of freely moving particles, and solve example problems for particles with prescribed and unknown velocities. Copyright © 2001 John Wiley & Sons, Ltd.
TL;DR: In this paper, the reconstruction process is done by Tikhonov regularization with the regularizing terms being the L2-norms of gradients, and is carried out in such a way that the temperature solution of the heat equation matches its fixed time observation and its subregion observation optimally in the L 2-norm sense.
Abstract: Given the measurement of temperature at a fixed time θ>0 and the measurement of temperature in a subregion of the physical domain, we investigate the simultaneous reconstruction of the initial temperature and heat radiative coefficient in a heat conductive system. The stability of the inverse problem is first established, and then the numerical reconstruction is mainly studied. The reconstruction process is done by Tikhonov regularization with the regularizing terms being the L2-norms of gradients, and is carried out in such a way that the temperature solution of the heat equation matches its fixed time observation and its subregion observation optimally in the L2-norm sense. The continuous nonlinear optimization system will be discretized by the piecewise linear finite element method, and the existence of discrete minimizers and convergence of the finite element approximation are shown. The discrete finite element problem is solved by a nonlinear gradient method with an efficient nonlinear multigrid technique for accelerating the reconstruction process. Numerical experiments are given to demonstrate the efficiency of the proposed nonlinear multigrid gradient method for solving the inverse parabolic problem.
TL;DR: In this paper, an analytical proof of the second order convergence of the Multilevel-Newton algorithm is given by authors in the field of non-linear electrical networks, which can be applied in the current context based on the DAE interpretation mentioned above.
Abstract: For the numerical solution of materially non-linear problems like in computational plasticity or viscoplasticity the finite element discretization in space is usually coupled with point-wise defined evolution equations characterizing the material behaviour. The interpretation of such systems as differential–algebraic equations (DAE) allows modern-day integration algorithms from Numerical Mathematics to be efficiently applied. Especially, the application of diagonally implicit Runge–Kutta methods (DIRK) together with a Multilevel-Newton method preserves the algorithmic structure of current finite element implementations which are based on the principle of virtual displacements and on backward Euler schemes for the local time integration. Moreover, the notion of the consistent tangent operator becomes more obvious in this context. The quadratical order of convergence of the Multilevel-Newton algorithm is usually validated by numerical studies. However, an analytical proof of this second order convergence has already been given by authors in the field of non-linear electrical networks. We show that this proof can be applied in the current context based on the DAE interpretation mentioned above. We finally compare the proposed procedure to several well-known stress algorithms and show that the inclusion of a step-size control based on local error estimations merely requires a small extra time-investment. Copyright © 2001 John Wiley & Sons, Ltd.
TL;DR: Some local and parallel discretizations and adaptive finite element algorithms are proposed and analyzed for nonlinear elliptic boundary value problems in both two and three dimensions for finite element solutions on general shape-regular grids.
Abstract: In this paper, some local and parallel discretizations and adaptive finite element algorithms are proposed and analyzed for nonlinear elliptic boundary value problems in both two and three dimensions. The main technique is to use a standard finite element discretization on a coarse grid to approximate low frequencies and then to apply some linearized discretization on a fine grid to correct the resulted residual (which contains mostly high frequencies) by some local/parallel procedures. The theoretical tools for analyzing these methods are some local a priori and a posteriori error estimates for finite element solutions on general shape-regular grids that are also obtained in this paper.
TL;DR: In this paper, an adaptive moving mesh method is developed for the numerical solution of an enthalpy formulation of two-dimensional heat conduction problems with a phase change, where the grid is obtained from a global mapping of the physical to the computational domain which is designed to cluster mesh points around the interface between the two phases of the material.
TL;DR: In this article, a coupled structural-acoustic analysis of finite fluid-filled cylindrical shells is presented using the wave propagation method for uncoupled analysis the natural frequencies of the shell by the present method are compared with the results available in the literature.
TL;DR: In this article, the regularized long wave (RLW) equation is solved numerically by giving a new algorithm based on a kind of space-time least square finite element method, in which a combination of cubic B-splines is used as an approximate function.
TL;DR: In this article, a hybridization of the finite element and boundary integral methods is presented for an efficient and accurate numerical analysis of electromagnetic scattering and radiation problems, which is free of interior resonance and produces a purely sparse system matrix, which can be solved very efficiently.
Abstract: A novel hybridization of the finite element (FE) and boundary integral methods is presented for an efficient and accurate numerical analysis of electromagnetic scattering and radiation problems. The proposed method derives an adaptive numerical absorbing boundary condition (ABC) for the finite element solution based on boundary integral equations. Unlike the standard finite element-boundary integral approach, the proposed method is free of interior resonance and produces a purely sparse system matrix, which can be solved very efficiently. Unlike the traditional finite element-absorbing boundary condition approach, the proposed method uses an arbitrarily shaped truncation boundary placed very close to the scatterer/radiator to minimize the computational domain; and more importantly, the method produces a solution that converges to the true solution of the problem. To demonstrate its great potential, the proposed method is implemented using higher order curvilinear vector elements. A mixed functional is designed to yield both electric and magnetic fields on an integration surface, without numerical differentiation, to be used in the calculation of the adaptive ABC. The required evaluation of boundary integrals is carried out using the multilevel fast multipole algorithm, which greatly reduces both the memory requirement and CPU time. The finite element equations are solved efficiently using the multifrontal algorithm. A mathematical analysis is conducted to study the convergence of the method. Finally, a number of numerical examples are given to illustrate its accuracy and efficiency.
TL;DR: Advanced finite element techniques for the simulation of materials behavior under mechanical loading are reviewed andvantages, limitations and perspectives of different approaches are analyzed.
Abstract: Advanced finite element techniques for the simulation of materials behavior under mechanical loading are reviewed. Advantages, limitations and perspectives of different approaches are analyzed for the simulation of deformation, damage and fracture of mate
TL;DR: In this paper, a simple and e cient algorithm with robust convergence properties is established to determine the real strain and stress of the wrinkled membrane for Hookean materials, and explicit formulas for the internal forces and the tangent sti ness matrix are derived.
Abstract: New results are presented for the nite element analysis of wrinkling in curved elastic membranes undergoing large deformation. Concise continuum level governing equations are derived in which singularities are eliminated. A simple and e cient algorithm with robust convergence properties is established to nd the real strain and stress of the wrinkled membrane for Hookean materials. The continuum theory is implemented into a nite element code. Explicit formulas for the internal forces and the tangent sti ness matrix are derived. Numerical examples are presented that demonstrate the e ectiveness of the new theory for predicting wrinkling in membranes undergoing large deformation. Copyright ? 2001 John Wiley & Sons, Ltd.
TL;DR: In this paper, the methodology of a finite element method (FEM)-based three-dimensional bulk forming modeling program is described using DEFORM™ as an example and discussed on the many considerations such as geometry representations, element selections, volume constancy, equation solvers and meshing methods.
TL;DR: The vector finite element time-domain (VFETD) method is derived, analyzed, and validated and is shown to be stable and to conserve energy and charge for arbitrary hexahedral grids.
Abstract: In this paper the vector finite element time-domain (VFETD) method is derived, analyzed, and validated. The VFETD method uses edge vector finite elements as a basis for the electric field and face vector finite elements as a basis for the magnetic flux density. The Galerkin method is used to convert Maxwell's equations to a coupled system of ordinary differential equations. The leapfrog method is used to advance the fields in time. The method is shown to be stable and to conserve energy and charge for arbitrary hexahedral grids. A numerical dispersion analysis shows the method to be second order accurate on distorted hexahedral grids. Several computational experiments are performed to determine the accuracy and efficiency of the method.
TL;DR: In this paper, a finite element model suited to the nature of the tube spinning has been built and the deformation field of the process has been simulated by the 3D rigid-plastic FEM (finite element method) in this paper.
TL;DR: A finite element method for solving the monochromatic radiation transfer equation including scattering in three dimensions including scattering with adaptivity as well as ordinate parallelization is presented.
Abstract: A finite element method for solving the monochromatic radiation transfer equation including scattering in three dimensions is presented. The algorithm employs unstructured grids which are adaptively refined. Adaptivity as well as ordinate parallelization reduce memory requirements and execution time and make it possible to calculate the radiation field across several length scales for objects with strong opacity gradients. An a posteriori error estimate for one particular quantity is obtained by solving the dual problem. The application to a sample of test problems reveals the properties of the implementation.
TL;DR: In this paper, the history of numerical analysis of partial differential equations is described, starting with the work of Courant, Friedrichs, and Lewy, and proceeding with the development of first finite difference and then finite element methods.
TL;DR: A non-conforming finite element method on a class of anisotropic meshes, namely the Crouzeix-Raviart element, is used on triangles and tetrahedra and for rectangles and prismatic elements a novel set of trial functions is proposed.
Abstract: The paper deals with a non-conforming finite element method on a class of anisotropic meshes. The Crouzeix-Raviart element is used on triangles and tetrahedra. For rectangles and prismatic (pentahedral) elements a novel set of trial functions is proposed. Anisotropic local interpolation error estimates are derived for all these types of element and for functions from classical and weighted Sobolev spaces. The consistency error is estimated for a general differential equation under weak regularity assumptions. As a particular application, an example is investigated where anisotropic finite element meshes are appropriate, namely the Poisson problem in domains with edges. A numerical test is described.
TL;DR: In this article, the partition of unity finite element method (PUFEM) is explored and improved to deal with practical diffraction problems efficiently, and the use of plane waves as an external function space allows an efficient implementation of an approximate exterior nonreflective boundary condition, improving the original proposed by Higdon.
Abstract: The partition of unity finite element method (PUFEM) is explored and improved to deal with practical diffraction problems efficiently. The use of plane waves as an external function space allows an efficient implementation of an approximate exterior non-reflective boundary condition, improving the original proposed by Higdon for general diffraction problems. A ‘virtually’ analytical integration procedure is introduced for multi-dimensional high-frequency problems which exhibits a dramatic decrease in the number of operations for a given error compared with standard integration methods. Suitable conjugate gradient type solvers for the whole range of wavenumbers are used, including such cases in which PUFEM can produce nearly singular matrices caused by ‘round-off’ limits. Copyright © 2001 John Wiley & Sons, Ltd.