Mass lumping strategies for X‐FEM explicit dynamics: Application to crack propagation

This paper deals with the numerical modelling of cracks in the dynamic case using the extended finite element method. More precisely, we are interested in explicit algorithms. We prove that by using a specific lumping technique, the critical time step is exactly the same as if no crack were present. This somewhat improves a previous result for which the critical time step was reduced by a factor of square root of 2 from the case with no crack. The new lumping technique is obtained by using a lumping strategy initially developed to handle elements containing voids. To be precise, the results obtained are valid only when the crack is modelled by the Heaviside enrichment. Note also that the resulting lumped matrix is block diagonal (blocks of size 2 × 2). For constant strain elements (linear simplex elements) the critical time step is not modified when the element is cut. Thanks to the lumped mass matrix, the critical time step never tends to zero. Moreover, the lumping techniques conserve kinetic energy for rigid motions. In addition, tensile stress waves do not propagate through the discontinuity. Hence, the lumping techniques create neither error on kinetic energy conservation for rigid motions nor wave propagation through the crack. Both these techniques will be used in a numerical experiment. Copyright © 2007 John Wiley & Sons, Ltd.     

Level set X‐FEM non‐matching meshes: application to dynamic crack propagation in elastic–plastic media

This paper develops two aspects improving crack propagation modelling with the X‐FEM method. On the one hand, it explains how one can use at the same time a regular structured mesh for a precise and efficient level set update and an unstructured irregular one for the mechanical model. On the other hand, a new numerical scheme based on the X‐FEM method is proposed for dynamic elastic–plastic situations. The simulation results are compared with two experiments on PMMA for which crack speed and crack path are provided. Copyright © 2006 John Wiley & Sons, Ltd.     

Nonconforming rotated Q1 tetrahedral element with explicit time stepping for elastodynamics

In this paper, we apply a rotated bilinear tetrahedral element to elastodynamics in . This element performs superior to the constant strain element in bending and, unlike the conforming linear strain tetrahedron, allows for row‐sum lumping of the mass matrix. We study the effect of different choices of approximation (pointwise continuity versus edge average continuity) as well as lumping versus consistent mass in the setting of eigenvibrations. We also use the element in combination with the leapfrog method for time domain computations and make numerical comparisons with the constant strain and linear strain tetrahedra. Copyright © 2012 John Wiley & Sons, Ltd.     

On the numerical stability and mass‐lumping schemes for explicit enriched meshfree methods

Meshfree methods (MMs) such as the element free Galerkin (EFG)method have gained popularity because of some advantages over other numerical methods such as the finite element method (FEM). A group of problems that have attracted a great deal of attention from the EFG method community includes the treatment of large deformations and dealing with strong discontinuities such as cracks. One efficient solution to model cracks is adding special enrichment functions to the standard shape functions such as extended FEM, within the FEM context, and the cracking particles method, based on EFG method. It is well known that explicit time integration in dynamic applications is conditionally stable. Furthermore, in enriched methods, the critical time step may tend to very small values leading to computationally expensive simulations. In this work, we study the stability of enriched MMs and propose two mass‐lumping strategies. Then we show that the critical time step for enriched MMs based on lumped mass matrices is of the same order as the critical time step of MMs without enrichment. Moreover, we show that, in contrast to extended FEM, even with a consistent mass matrix, the critical time step does not vanish even when the crack directly crosses a node. Copyright © 2011 John Wiley & Sons, Ltd.     

Application of the X‐FEM to the fracture of piezoelectric materials

This paper presents an application of the extended finite element method (X‐FEM) to the analysis of fracture in piezoelectric materials. These materials are increasingly used in actuators and sensors. New applications can be found as constituents of smart composites for adaptive electromechanical structures. Under in service loading, phenomena of crack initiation and propagation may occur due to high electromechanical field concentrations. In the past few years, the X‐FEM has been applied mostly to model cracks in structural materials. The present paper focuses at first on the definition of new enrichment functions suitable for cracks in piezoelectric structures. At second, generalized domain integrals are used for the determination of crack tip parameters. The approach is based on specific asymptotic crack tip solutions, derived for piezoelectric materials. We present convergence results in the energy norm and for the stress intensity factors, in various settings. Copyright © 2008 John Wiley & Sons, Ltd.     

扩展有限元法在裂纹扩展问题中的应用

扩展有限元法(Extended finite element method,XFEM)是近几年发展起来的数值方法,属于传统有限元法的扩展,具有区别于传统有限元法的优点,在求解不连续断裂问题上具有更高的精度及效率。本文针对影响裂纹扩展的主要因素进行探讨,继而介绍扩展有限元的基本原理,并对其在裂纹扩展中的应用进行综述,同时对该方法的下一步研究进行了展望。     

The effects of element shape on the critical time step in explicit time integrators for elasto‐dynamics

In this paper, the effects of element shape on the critical time step are investigated. The common rule‐of‐thumb, used in practice, is that the critical time step is set by the shortest distance within an element divided by the dilatational (compressive) wave speed, with a modest safety factor. For regularly shaped elements, many analytical solutions for the critical time step are available, but this paper focusses on distorted element shapes. The main purpose is to verify whether element distortion adversely affects the critical time step or not. Two types of element distortion will be considered, namely aspect ratio distortion and angular distortion, and two particular elements will be studied: four‐noded bilinear quadrilaterals and three‐noded linear triangles. The maximum eigenfrequencies of the distorted elements are determined and compared to those of the corresponding undistorted elements. The critical time steps obtained from single element calculations are also compared to those from calculations based on finite element patches with multiple elements. Copyright © 2014 John Wiley & Sons, Ltd.     

An energy‐conserving scheme for dynamic crack growth using the eXtended finite element method

This paper proposes a generalization of the eXtended finite element method (X‐FEM) to model dynamic fracture and time‐dependent problems from a more general point of view, and gives a proof of the stability of the numerical scheme in the linear case. First, we study the stability conditions of Newmark‐type schemes for problems with evolving discretizations. We prove that the proposed enrichment strategy satisfies these conditions and also ensures energy conservation. Using this approach, as the crack propagates, the enrichment can evolve with no occurrence of instability or uncontrolled energy transfer. Then, we present a technique based on Lagrangian conservation for the estimation of dynamic stress intensity factors for arbitrary 2D cracks. The results presented for several applications are accurate for stationary or moving cracks. Copyright © 2005 John Wiley & Sons, Ltd.     

Selective mass scaling for explicit finite element analyses

Due to their inherent lack of convergence problems explicit finite element techniques are widely used for analysing non‐linear mechanical processes. In many such processes the energy content in the high frequency domain is small. By focusing an artificial mass scaling on this domain, the critical time step may be increased substantially without significantly affecting the low frequency behaviour. This is what we refer to as selective mass scaling. Two methods for selective mass scaling are introduced in this work. The proposed methods are based on non‐diagonal mass matrices that scale down the eigenfrequencies of the system. The applicability of the methods is illustrated in two example models where the critical time step is increased by up to 30 times its original size. Copyright © 2005 John Wiley & Sons, Ltd.     

Interface problems with quadratic X‐FEM: design of a stable multiplier space and error analysis

The aim of this paper is to propose a procedure to accurately compute curved interfaces problems within the extended finite element method and with quadratic elements. It is dedicated to gradient discontinuous problems, which cover the case of bimaterials as the main application. We focus on the use of Lagrange multipliers to enforce adherence at the interface, which makes this strategy applicable to cohesive laws or unilateral contact. Convergence then occurs under the condition that a discrete inf‐sup condition is passed. A dedicated P1 multiplier space intended for use with P2 displacements is introduced. Analytical proof that it passes the inf‐sup condition is presented in the two‐dimensional case. Under the assumption that this inf‐sup condition holds, a priori error estimates are derived for linear or quadratic elements as functions of the curved interface resolution and of the interpolation properties of the discrete Lagrange multipliers space. The estimates are successfully checked against several numerical experiments: disparities, when they occur, are explained in the literature. Besides, the new multiplier space is able to produce quadratic convergence from P2 displacements and quadratic geometry resolution. Copyright © 2014 John Wiley & Sons, Ltd.     

Conservation properties of a time FE method—part II: Time‐stepping schemes for non‐linear elastodynamics

In the present paper one‐step implicit integration algorithms for non‐linear elastodynamics are developed. The discretization process rests on Galerkin methods in space and time. In particular, the continuous Galerkin method applied to the Hamiltonian formulation of semidiscrete non‐linear elastodynamics lies at the heart of the time‐stepping schemes. Algorithmic conservation of energy and angular momentum are shown to be closely related to quadrature formulas that are required for the calculation of time integrals. We newly introduce the ‘assumed strain method in time’ which enables the design of energy–momentum conserving schemes and which can be interpreted as temporal counterpart of the well‐established assumed strain method for finite elements in space. The numerical examples deal with quasi‐rigid motion as well as large‐strain motion. Copyright © 2001 John Wiley & Sons, Ltd.     

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

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. Copyright © 2005 John Wiley & Sons, Ltd.     

Multi‐model Arlequin method for transient structural dynamics with explicit time integration

This paper addresses the explicit time integration for solving multi‐model structural dynamics by the Arlequin method. Our study focuses on the stability of the central difference scheme in the Arlequin framework. Although the Arlequin coupling matrices can introduce a weak instability, the time integrator remains stable as long as the initial kinematic conditions of both models agree on the coupling zone. After showing that the Arlequin weights have an adverse impact on the critical time step, we present two approaches to circumvent this issue. Computational tests confirm that the two approaches effectively preserve a feasible critical time step and show the efficiency of the Arlequin method for structural explicit dynamic simulations. Copyright © 2017 John Wiley & Sons, Ltd.     

An explicit time integration method for structural dynamics using septuple B‐spline functions

In this paper, an explicit time integration method with three parameters is proposed for structural dynamics using periodic septuple B‐spline interpolation polynomial functions. In this way, by use of septuple B‐splines, the authors have proceeded to solve the DE of motion governing a single DOF system, and later, the presented method has been generalized for a multiple DOF system. In the proposed method, a direct recursive formula for response of the system was formulated on the basis of septuple B‐spline interpolation approximation. In terms of the specific requirements of this proposed method, two initialization approaches are given for initial calculation. One is called direct initialization, and the other is indirect initialization. The stability analysis of the proposed method illustrates that, by use of adjustable parameters, a high‐frequency response can be damped out without inducing excessive algorithmic damping in important low frequency modes. The computational accuracy and efficiency of the proposed method is demonstrated with three numerical examples, and the results from the proposed method are compared with those from some of the existent numerical methods, such as the Newmark and Wilson‐ θ methods. The compared results show that the proposed method has high accuracy with low time consumption. Copyright © 2013 John Wiley & Sons, Ltd.     

修正结构有限元模型的一种方法

本文讨论了一个寻找结构有限元模型的误差源和对它进行修正的方法.为了减少待修正模型的未知参数数目,将质量、刚度矩阵描述为设计变量的函数.在此基础上,先用少量测得的静位移数据修正刚变矩阵;然后用少数几阶实验的固有频率、固有振型数据修正质量矩阵.为了避免测量旋转自由度的静位移和固有振型分量的困难,还讨论了聚缩模型的修正问题.最后,文中附有算例及结果.     

Efficient implementation of an explicit partitioned shear and longitudinal wave propagation algorithm

The paper complements and extends the previous works on partitioned explicit wave propagation analysis methods, which were presented for discontinuous wave propagation problems in solids. An efficient implementation of the partitioned explicit wave propagation analysis methods is introduced. The present implementation achieves about 25% overall computational effort compared with the previous implementation with the same accuracy. The present algorithm tracks, with different integration time step sizes in accordance with their different wave speeds, the propagation fronts of longitudinal and shear waves. This is accomplished by integrating separately the element‐by‐element partitioned longitud inal and shear equations of motion. The state vectors (displacements, velocity and accelerations) of the longitudinal and shear components are reconciled at the end of each time step. The reconciliation procedure does not require any system parameters such as material properties, density, unlike conventional artificial viscosity methods. Numerical examples are presented as applied to linear and non‐linear wave propagation problems, which demonstrate high‐fidelity wavefront tracking ability of the present method, and compared with existing conventional wave propagation analysis methods. Copyright © 2015 John Wiley & Sons, Ltd.     

Element‐local level set method for three‐dimensional dynamic crack growth

An approximate level set method for three‐dimensional crack propagation is presented. In this method, the discontinuity surface in each cracked element is defined by element‐local level sets (ELLSs). The local level sets are generated by a fitting procedure that meets the fracture directionality and its continuity with the adjacent element crack surfaces in a least‐square sense. A simple iterative procedure is introduced to improve the consistency of the generated element crack surface with those of the adjacent cracked elements. The discrete discontinuity is treated by the phantom node method which is a simplified version of the extended finite element method (XFEM). The ELLS method and the phantom node technology are combined for the solution of dynamic fracture problems. Numerical examples for three‐dimensional dynamic crack propagation are provided to demonstrate the effectiveness and robustness of the proposed method. Copyright © 2009 John Wiley & Sons, Ltd.     

Numerical Prediction of Dynamically Propagating and Branching Cracks Using Moving Finite Element Method

Phenomena of dynamic crack branching are investigated numerically from a macroscopic point of view. Repetitive branching phenomena, interaction of cracks after bifurcation and their stability, bifurcation into two and three branches were the objectives of this research. For the analysis of dynamic crack branching, recently we developed moving finite element method based on Delaunay automatic triangulation [Nishioka, Furutuka, Tchouikov and Fujimoto (2002)]. In this study this method was extended to be applicable for complicated crack branching phenomena, such as bifurcation of the propagating crack into more than two branches, multiple crack bifurcation and so on. The switching method of the path independent dynamic J integral, which was developed for the case of simple two cracks branching phenomena, demonstrated it's excellent applicability also for the case of complicated crack branching. The simulation results are discussed with consideration to the experimental findings.     

hp‐Generalized FEM and crack surface representation for non‐planar 3‐D cracks

A high‐order generalized finite element method (GFEM) for non‐planar three‐dimensional crack surfaces is presented. Discontinuous p‐hierarchical enrichment functions are applied to strongly graded tetrahedral meshes automatically created around crack fronts. The GFEM is able to model a crack arbitrarily located within a finite element (FE) mesh and thus the proposed method allows fully automated fracture analysis using an existing FE discretization without cracks. We also propose a crack surface representation that is independent of the underlying GFEM discretization and controlled only by the physics of the problem. The representation preserves continuity of the crack surface while being able to represent non‐planar, non‐smooth, crack surfaces inside of elements of any size. The proposed representation also provides support for the implementation of accurate, robust, and computationally efficient numerical integration of the weak form over elements cut by the crack surface. Numerical simulations using the proposed GFEM show high convergence rates of extracted stress intensity factors along non‐planar curved crack fronts and the robustness of the method. Copyright © 2008 John Wiley & Sons, Ltd.