Showing papers on "Extended finite element method published in 2006"

A method for dynamic crack and shear band propagation with phantom nodes

Abstract: A new method for modelling of arbitrary dynamic crack and shear band propagation is presented. We show that by a rearrangement of the extended finite element basis and the nodal degrees of freedom, the discontinuity can be described by superposed elements and phantom nodes. Cracks are treated by adding phantom nodes and superposing elements on the original mesh. Shear bands are treated by adding phantom degrees of freedom. The proposed method simplifies the treatment of element-by-element crack and shear band propagation in explicit methods. A quadrature method for 4-node quadrilaterals is proposed based on a single quadrature point and hourglass control. The proposed method provides consistent history variables because it does not use a subdomain integration scheme for the discontinuous integrand. Numerical examples for dynamic crack and shear band propagation are provided to demonstrate the effectiveness and robustness of the proposed method. Copyright © 2006 John Wiley & Sons, Ltd.

Stochastic finite element: a non intrusive approach by regression

Abstract: The stochastic finite element method allows to solve stochastic boundary value problems where material properties and loads are random. The method is based on the expansion of the mechanical response onto the so-called polynomial chaos. In this paper, a non intrusive method based on a least-squares minimization procedure is presented. This method is illustrated by the study of the settlement of a foundation. Different analysis are proposed: the computation of the statistical moments of the response, a reliability analysis and a parametric sensitivity analysis.

A Multipoint Flux Mixed Finite Element Method

Abstract: We develop a mixed finite element method for single phase flow in porous media that reduces to cell-centered finite differences on quadrilateral and simplicial grids and performs well for discontinuous full tensor coefficients. Motivated by the multipoint flux approximation method where subedge fluxes are introduced, we consider the lowest order Brezzi-Douglas-Marini (BDM) mixed finite element method. A special quadrature rule is employed that allows for local velocity elimination and leads to a symmetric and positive definite cell-centered system for the pressures. Theoretical and numerical results indicate second-order convergence for pressures at the cell centers and first-order convergence for subedge fluxes. Second-order convergence for edge fluxes is also observed computationally if the grids are sufficiently regular.

Dispersive and Dissipative Properties of Discontinuous Galerkin Finite Element Methods for the Second-Order Wave Equation

Abstract: Discontinuous Galerkin finite element methods (DGFEM) offer certain advantages over standard continuous finite element methods when applied to the spatial discretisation of the acoustic wave equation. For instance, the mass matrix has a block diagonal structure which, used in conjunction with an explicit time stepping scheme, gives an extremely economical scheme for time domain simulation. This feature is ubiquitous and extends to other time-dependent wave problems such as Maxwell's equations. An important consideration in computational wave propagation is the dispersive and dissipative properties of the discretisation scheme in comparison with those of the original system. We investigate these properties for two popular DGFEM schemes: the interior penalty discontinuous Galerkin finite element method applied to the second-order wave equation and a more general family of schemes applied to the corresponding first order system. We show how the analysis of the multi-dimensional case may be reduced to consideration of one-dimensional problems. We derive the dispersion error for various schemes and conjecture on the generalisation to higher order approximation in space

On the elimination of quadrature subcells for discontinuous functions in the eXtended Finite‐Element Method

Abstract: The introduction of discontinuous/non-differentiable functions in the eXtended Finite-Element Method allows to model discontinuities independent of the mesh structure. 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 line is commonly adopted. In the paper, it is shown how standard Gauss quadrature can be used in the elements containing the discontinuity without splitting the elements into subcells or introducing any additional approximation. The technique is illustrated and developed in one, two and three dimensions for crack and material discontinuity problems

The intrinsic XFEM: a method for arbitrary discontinuities without additional unknowns

Abstract: A new method for treating arbitrary discontinuities in a finite element (FE) context is presented. Unlike the standard extended FE method (XFEM), no additional unknowns are introduced at the nodes whose supports are crossed by discontinuities. The method constructs an approximation space consisting of mesh-based, enriched moving least-squares (MLS) functions near discontinuities and standard FE shape functions elsewhere. There is only one shape function per node, and these functions are able to represent known characteristics of the solution such as discontinuities, singularities, etc. The MLS method constructs shape functions based on an intrinsic basis by minimizing a weighted error functional. Thereby, weight functions are involved, and special mesh-based weight functions are proposed in this work. The enrichment is achieved through the intrinsic basis. The method is illustrated for linear elastic examples involving strong and weak discontinuities, and matches optimal rates of convergence even for crack-tip applications. Copyright © 2006 John Wiley & Sons, Ltd.

A Generalized Finite Element Method for polycrystals with discontinuous grain boundaries

Abstract: SUMMARY We present a Generalized Finite Element Method for the analysis of polycrystals with explicit treatment of grain boundaries. Grain boundaries and junctions, understood as loci of possible displacement discontinuity, are inserted into finite elements by exploiting the partition of unity property of finite element shape functions. Consequently, the finite element mesh does not need to conform to the polycrystal topology. The formulation is outlined and a numerical example is presented to demonstrate the potential and accuracy of the approach. The proposed methodology can also be used for branched and intersecting cohesive cracks, and comparisons are made to a related approach (Int. J. Numer. Meth. Engng. 2000; 48:1741). Copyright 2006 John Wiley & Sons, Ltd.

Efficient explicit time stepping for the eXtended Finite Element Method (X-FEM)

Abstract: This paper focuses on the introduction of a lumped mass matrix for enriched elements, which enables one to use a pure explicit formulation in X-FEM applications. A proof of stability for the 1D and 2D cases is given. We show that if one uses this technique, the critical time step does not tend to zero as the support of the discontinuity reaches the boundaries of the elements. We also show that the X-FEM element's critical time step is of the same order as that of the corresponding element without extended degrees of freedom. Copyright © 2006 John Wiley & Sons, Ltd.

Dynamics of Flexible Multibody Systems: Rigid Finite Element Method

Abstract: Homogenous Transformations.- The Rigid Finite Element Method.- Modification of the Rigid Finite Element Method.- Calculations for a Cantilever Beam and Methods of Integrating the Equations of Motion.- Verification of the Method.- Applications.

Improving the accuracy of XFEM crack tip fields using higher order quadrature and statically admissible stress recovery

Abstract: This study is concerned with improving the accuracy of crack tip fields obtained using the extended/generalized finite element method (XFEM). First, the numerical integration of the element stiffness matrices, which guarantees convergence (with quadrature) of not only the regular nodal displacements but also additional degrees of freedom corresponding to the enrichment functions, is studied. As the accuracy of the stresses obtained by direct differentiation of the converged (with quadrature) regular nodal displacements and of the coefficients corresponding to enrichment functions is still not satisfactory, a statically admissible stress recovery (SAR) scheme is introduced. SAR uses basis functions, which meet the equilibrium equations within the domain and the local traction conditions on the boundary, and moving least squares (MLS) to fit the stresses at sampling points (e.g. quadrature points) obtained by the XFEM. Important parameters controlling the accuracy of crack tip fields using the XFEM and SAR, namely the order of quadrature, the number of retained terms in the crack tip asymptotic field, the number of enriched layers and use of arbitrary branch functions, a proper choice of the sampling points in the enriched element and the size of the domain of influence (DOI) of MLS, are investigated. Copyright © 2005 John Wiley & Sons, Ltd.

Introduction to the finite element method in electromagnetics

Abstract: This series lecture is an introduction to the finite element method with applications in electromagnetics. The finite element method is a numerical method that is used to solve boundary-value problems characterized by a partial differential equation and a set of boundary conditions. The geometrical domain of a boundary-value problem is discretized using sub-domain elements, called the finite elements, and the differential equation is applied to a single element after it is brought to a “weak” integro-differential form. A set of shape functions is used to represent the primary unknown variable in the element domain. A set of linear equations is obtained for each element in the discretized domain. A global matrix system is formed after the assembly of all elements. This lecture is divided into two chapters. Chapter 1 describes one-dimensional boundary-value problems with applications to electrostatic problems described by the Poisson's equation. The accuracy of the finite element method is evaluated for linear and higher order elements by computing the numerical error based on two different definitions. Chapter 2 describes two-dimensional boundary-value problems in the areas of electrostatics and electrodynamics (time-harmonic problems). For the second category, an absorbing boundary condition was imposed at the exterior boundary to simulate undisturbed wave propagation toward infinity. Computations of the numerical error were performed in order to evaluate the accuracy and effectiveness of the method in solving electromagnetic problems. Both chapters are accompanied by a number of Matlab codes which can be used by the reader to solve one- and two-dimensional boundary-value problems. These codes can be downloaded from the publisher's URL: www.morganclaypool.com/page/polycarpou This lecture is written primarily for the nonexpert engineer or the undergraduate or graduate student who wants to learn, for the first time, the finite element method with applications to electromagnetics. It is also targeted for research engineers who have knowledge of other numerical techniques and want to familiarize themselves with the finite element method. The lecture begins with the basics of the method, including formulating a boundary-value problem using a weighted-residual method and the Galerkin approach, and continues with imposing all three types of boundary conditions including absorbing boundary conditions. Another important topic of emphasis is the development of shape functions including those of higher order. In simple words, this series lecture provides the reader with all information necessary for someone to apply successfully the finite element method to one- and two-dimensional boundary-value problems in electromagnetics. It is suitable for newcomers in the field of finite elements in electromagnetics.

Crack analysis in orthotropic media using the extended finite element method

Abstract: An extended finite element method has been proposed for modeling crack in orthotropic media To achieve this aim a discontinuous function and two-dimensional asymptotic crack-tip displacement fields are used in a classical finite element approximation enriched with the framework of partition of unity It allows modeling crack by standard finite element method without explicitly defining and re-meshing of surfaces of the crack In this study, fracture properties of the models are defined by the mixed-mode stress intensity factors (SIFs), which are obtained by means of the domain form of the interaction integral (M-integral) Numerical simulations are performed to verify the approach, and the accuracy of the results is discussed by comparison with other numerical or (semi-) analytical methods

A finite element method for poro mechanical modelling of geotechnical problems using local second gradient models

Abstract: In this paper, a new finite element method is described and applied. It is based on a theory developed to model poromechanical problems where the mechanical part is obeying a second gradient theory. The aim of such a work is to properly model the post localized behaviour of soils and rocks saturated with a pore fluid. Beside the development of this new coupled theory, a corresponding finite element method has been developed. The elements used are based on a weak form of the relation between the deformation gradient and the second gradient, using a field of Lagrange multipliers. The global problem is solved by a system of equations where the kinematic variables are fully coupled with the pore pressure. Some numerical experiments showing the effectiveness of the method ends the paper.

A Numerical Approach for Arbitrary Cracks in a Fluid-Saturated Medium

Abstract: A finite element method is proposed that can capture arbitrary discontinuities in a two-phase medium. The discontinuity is described in an exact manner by exploiting the partition-of-unity property of finite element shape functions. The fluid flow away from the discontinuity is modelled in a standard fashion using Darcy’s relation, while at the discontinuity a discrete analogon of Darcy’s relation is proposed. The results of this finite element model are independent of the original discretisation, as is demonstrated by an example of shear banding in a biaxial, plane-strain specimen.

Solution of Fokker-Planck equation by finite element and finite difference methods for nonlinear systems

Abstract: The response of a structural system to white noise excitation (deltacorrelated) constitutes a Markov vector process whose transitional probability density function (TPDF) is governed by both the forward Fokker-Planck and backward Kolmogorov equations. Numerical solution of these equations by finite element and finite difference methods for dynamical systems of engineering interest has been hindered by the problem of dimensionality. In this paper numerical solution of the stationary and transient form of the Fokker-Planck (FP) equation corresponding to two state nonlinear systems is obtained by standard sequential finite element method (FEM) using C0 shape function and Crank-Nicholson time integration scheme. The method is applied to Van-der-Pol and Duffing oscillators providing good agreement between results obtained by it and exact results. An extension of the finite difference discretization scheme developed by Spencer, Bergman and Wojtkiewicz is also presented. This paper presents an extension of the finite difference method for the solution of FP equation up to four dimensions. The difficulties associated in extending these methods to higher dimensional systems are discussed.

Error reduction and convergence for an adaptive mixed finite element method

Abstract: An adaptive mixed finite element method (AMFEM) is designed to guarantee an error reduction, also known as saturation property: after each refinement step, the error for the fine mesh is strictly smaller than the error for the coarse mesh up to oscillation terms. This error reduction property is established here for the Raviart-Thomas finite element method with a reduction factor p < 1 uniformly for the L 2 norm of the flux errors. Our result allows for linear convergence of a proper adaptive mixed finite element algorithm with respect to the number of refinement levels. The adaptive algorithm surprisingly does not require any particular mesh design, unlike the conforming finite element method. The new arguments are a discrete local efficiency and a quasi-orthogonality estimate. The proof does not rely on duality or on regularity.

Analysis of finite element approximations of a phase field model for two-phase fluids

Abstract: This paper studies a phase field model for the mixture of two immiscible and incompressible fluids. The model is described by a nonlinear parabolic system consisting of the nonstationary Stokes equations coupled with the Allen-Cahn equation through an extra phase induced stress term in the Stokes equations and a fluid induced transport term in the Allen-Cahn equation. Both semi-discrete and fully discrete finite element methods are developed for approximating the parabolic system. It is shown that the proposed numerical methods satisfy a discrete energy law which mimics the basic energy law for the phase field model. Error estimates are derived for the semi-discrete method, and the convergence to the phase field model and to its sharp interface limiting model are established for the fully discrete finite element method by making use of the discrete energy law. Numerical experiments are also presented to validate the theory and to show the effectiveness of the combined phase field and finite element approach.

Boundary controllability of a linear semi-discrete 1-D wave equation derived from a mixed finite element method

Abstract: In this article one discusses the controllability of a semi-discrete system obtained by discretizing in space the linear 1-D wave equation with a boundary control at one extremity. It is known that the semi-discrete models obtained with finite difference or the classical finite element method are not uniformly controllable as the discretization parameter h goes to zero (see [8]). Here we introduce a new semi-discrete model based on a mixed finite element method with two different basis functions for the position and velocity. We show that the controls obtained with these semi-discrete systems can be chosen uniformly bounded in L 2(0,T) and in such a way that they converge to the HUM control of the continuous wave equation, i.e. the minimal L 2-norm control. We illustrate the mathematical results with several numerical experiments.