Haim Waisman  Ted Belytschko 《International journal for numerical methods in engineering》2008,73(12):1671-1692

An adaptive method within the extended finite element method (XFEM) framework which adapts the enrichment function locally to the physics of a problem, as opposed to polynomial or mesh refinement, is presented. The method minimizes a local residual and determines the parameters of the enrichment function. We consider an energy form and a ‘strong’ form of the residual as error measures to drive the algorithm. Numerical examples for boundary layers and solid mechanics problems illustrate that the procedure converges. Moreover, when only the character of the solution is known, a good approximation is obtained in the area of interest. It is also shown that the method can be used to determine the order of singularities in solutions. Copyright © 2007 John Wiley & Sons, Ltd.  相似文献   

    

Roy H. Stogner  Graham F. Carey  Bruce T. Murray 《International journal for numerical methods in engineering》2008,76(5):636-661

A variational formulation and C¹ finite element scheme with adaptive mesh refinement and coarsening are developed for phase‐separation processes described by the Cahn–Hilliard diffuse interface model of transport in a mixture or alloy. The adaptive scheme is guided by a Laplacian jump indicator based on the corresponding term arising from the weak formulation of the fourth‐order non‐linear problem, and is implemented in a parallel solution framework. It is then applied to resolve complex evolving interfacial solution behavior for 2D and 3D simulations of the classic spinodal decomposition problem from a random initial mixture and to other phase‐transformation applications of interest. Simulation results and adaptive performance are discussed. The scheme permits efficient, robust multiscale resolution and interface characterization. Copyright © 2008 John Wiley & Sons, Ltd.  相似文献   

    

Stéphane Valance  René de Borst  Julien Réthoré  Michel Coret 《International journal for numerical methods in engineering》2008,76(10):1513-1527

Level set methods have recently gained much popularity to capture discontinuities, including their possible propagation. Typically, the partial differential equations that arise in level set methods, in particular the Hamilton–Jacobi equation, are solved by finite difference methods. However, finite difference methods are less suited for irregular domains. Moreover, it seems slightly awkward to use finite differences for the capturing of a discontinuity, while in a subsequent stress analysis finite elements are normally used. For this reason, we here present a finite element approach to solving the governing equations of level set methods. After a review of the governing equations, the initialization of the level sets, the discretization on a finite domain, and the stabilization of the resulting finite element method will be discussed. Special attention will be given to the proper treatment of the internal boundary condition, which is achieved by exploiting the partition‐of‐unity property of finite element shape functions. Finally, a quantitative analysis including accuracy analysis is given for a one‐dimensional example and a qualitative example is given for a two‐dimensional case with a curved discontinuity. Copyright © 2008 John Wiley & Sons, Ltd.  相似文献   

    

Michael A. Scott  Michael J. Borden  Clemens V. Verhoosel  Thomas W. Sederberg  Thomas J. R. Hughes 《International journal for numerical methods in engineering》2011,88(2):126-156

We develop finite element data structures for T‐splines based on Bézier extraction generalizing our previous work for NURBS. As in traditional finite element analysis, the extracted Bézier elements are defined in terms of a fixed set of polynomial basis functions, the so‐called Bernstein basis. The Bézier elements may be processed in the same way as in a standard finite element computer program, utilizing exactly the same data processing arrays. In fact, only the shape function subroutine needs to be modified while all other aspects of a finite element program remain the same. A byproduct of the extraction process is the element extraction operator. This operator localizes the topological and global smoothness information to the element level, and represents a canonical treatment of T‐junctions, referred to as ‘hanging nodes’ in finite element analysis and a fundamental feature of T‐splines. A detailed example is presented to illustrate the ideas. Copyright © 2011 John Wiley & Sons, Ltd.  相似文献   

    

J. F. Mougaard  P. N. Poulsen  L. O. Nielsen 《International journal for numerical methods in engineering》2011,85(13):1667-1686

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G. Venturini  J. Z. Yang  M. Ortiz  J. E. Marsden 《International journal for numerical methods in engineering》2011,88(6):586-611

This paper is concerned with the classical problem of wave propagation in discrete models of nonuniform spatial resolution. We develop a new class of Replica Time Integrators (RTIs) that permit the two‐way transmission of thermal phonons across mesh interfaces. This two‐way transmissibility is accomplished by representing the state of the coarse regions by means of replica ensembles, consisting of collections of identical copies of the coarse regions. In dimension d, RTIs afford an O(n^d) speed‐up factor in sequential mode, and O(n^{d + 1}) in parallel, over regions that are coarsened n‐fold. In this work, we restrict ourselves to the solution of the 3d continuous wave equation, for both linear and non‐linear materials. By a combination of phase‐error analysis and numerical testing, we show that RTIs are convergent and result in exact two‐way transmissibility at the Courant–Friedrichs–Lewy limit for any angle of incidence. In this limit, RTIs allow step waves and high‐frequency harmonics to cross mesh interfaces in both directions without internal reflections or appreciable loss or addition of energy. The possible connections of RTIs with discrete‐to‐continuum approaches and, in particular, with the transition between molecular dynamics and continuum thermodynamics are also pointed to by way of future outlook. Copyright © 2011 John Wiley & Sons, Ltd.  相似文献   

    

A. Gajo  R. Denzer 《International journal for numerical methods in engineering》2011,85(13):1705-1736

Two finite element formulations are proposed to analyse the dynamic conditions of saturated porous media at large strains with compressible solid and fluid constituents. Unlike similar works published in the literature, the proposed formulations are based on a recently proposed hyperelastic framework in which the compressibility of the solid and fluid constituents is fully taken into account when geometrical non‐linear effects are relevant on both micro‐ and macroscales. The first formulation leads to a three‐field finite element method (FEM), which is suitable for analysing high‐frequency dynamic problems, whereas the second is a simplification of the first, leading to a two‐field FEM, in which some inertial effects of the pore fluid are disregarded, hence the second formulation is suitable for studying low‐frequency problems. A fully Lagrangian approach is considered, hence all terms are expressed with reference to the material setting; the balance equations for the pore fluid are also expressed in terms of the chemical potential and the mass flux of the pore fluid in order to take the compressibility of the fluid into account. To improve the numerical response in the case of wave propagation, a discontinuous Galerkin FEM in the time domain is applied to the three‐field formulation. The results are compared with analytical and semi‐analytical solutions, highlighting the different effects of the discontinuous Galerkin method on the longitudinal waves of the first and second kind. Copyright © 2010 John Wiley & Sons, Ltd.  相似文献   

    

K. Krabbenhoft  A. V. Lyamin  S. W. Sloan  P. Wriggers 《International journal for numerical methods in engineering》2007,69(3):592-626

The problem of small‐deformation, rate‐independent elastoplasticity is treated using convex programming theory and algorithms. A finite‐step variational formulation is first derived after which the relevant potential is discretized in space and subsequently viewed as the Lagrangian associated with a convex mathematical program. Next, an algorithm, based on the classical primal–dual interior point method, is developed. Several key modifications to the conventional implementation of this algorithm are made to fully exploit the nature of the common elastoplastic boundary value problem. The resulting method is compared to state‐of‐the‐art elastoplastic procedures for which both similarities and differences are found. Finally, a number of examples are solved, demonstrating the capabilities of the algorithm when applied to standard perfect plasticity, hardening multisurface plasticity, and problems involving softening. Copyright © 2006 John Wiley & Sons, Ltd.  相似文献   

    

Vijayaraghavan Rajagopal  Jae‐Hoon Chung  David Bullivant  Poul M. F. Nielsen  Martyn P. Nash 《International journal for numerical methods in engineering》2007,72(12):1434-1451

There are a number of situations where the deformed configuration of a body is known and it is necessary to determine the reference state. Previous methods developed to calculate the reference state involve the formulation of the finite elasticity equations in terms of the deformed configuration. This paper demonstrates that the undeformed reference state can be accurately determined from a deformed configuration and the associated loading conditions, by using conventional finite elasticity balance equations together with a solution procedure that treats the reference configuration as the unknowns. The mathematical theory behind the solution method is described, validated with an analytical solution, and verified using experimental studies on gel phantoms. The practical utility of this method is then demonstrated in the field of breast biomechanics. Copyright © 2007 John Wiley & Sons, Ltd.  相似文献   

    

Alberto Salvadori 《International journal for numerical methods in engineering》2007,72(6):722-743

The present work deals with the application of the traction hypersingular boundary integral equation to approximate the stress tensor along the boundary, by making use of unit vectors that differ from the normal at the boundary. It is proved that special care is required in the evaluation of the free terms, in order to avoid insidious and unexpected errors. Copyright © 2007 John Wiley & Sons, Ltd.  相似文献