Mixed mode fracture propagation by manifold method

The numerical manifold method combined with the virtual crack extension method is proposed to study the mixed mode fracture propagation. The manifold method is a new numerical method, and it provides a unified framework for solving problems dealing with both continuums and jointed materials. This new method can be considered as a generalized finite element method and discontinuous deformation analysis. One of the most innovative features of the method is that it employs both physical mesh and mathematical mesh to formulate the physical problem. These two meshes are separated and independent. They are inter-related through the application of weighting functions. A local mesh refinement and auto-remeshing schemes previously proposed by the authors are adopted in this study. The proposed model is first verified by comparing the numerical stress intensity factors with the benchmark solutions, and the results show satisfactory accuracy. The maximum tangential stress criterion is adopted and the mixed mode fracture propagation problems are then fully investigated. The numerical solutions by the present method agree well with the experimental results.     

Estimation of the fracture front propagation time in structural components

We discuss special features of the fracture front propagation in structural components (variable-thickness disks and plates, internally pressurized thick-walled vessels). The upper and lower bounds of the relative time of the front propagation have been estimated. Some recommendations regarding the application of the results obtained are provided. __________ Translated from Problemy Prochnosti, No. 6, pp. 13–24, November–December, 2007.     

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.     

Ductile fracture initiation and propagation modeling using damage plasticity theory

Ductile fracture is often considered as the consequences of the accumulation of plastic damage. This paper is concerned with the application of a recently developed damage plasticity theory incorporates the pressure sensitivity and the Lode angle dependence into a nonlinear damage rule and the material deterioration. The ductile damaging process is calculated through the so-called “cylindrical decomposition” method. The constitutive equations are discussed and numerically implemented. An experimental and numerical investigation for three-point bending tests is reported for aluminum alloy 2024-T351. Crack initiation and propagation in compact tension specimens are also studied numerically. These simulation results show good agreement with experiments. The present model can successfully predict slant fracture as well as the formation of shear lips.     

一种改进的变步长LMS自适应滤波算法及其仿真

在分析传统的定步长最小均方(LMS)算法、变步长LMS算法的基础上,通过建立误差信号与步长因子之间的新的非线性映射关系,提出新的改进型变步长LMS自适应算法.通过MATLAB仿真分析,证明了该算法具有较好的收敛速度和较小的稳态误差以及较好的时变系统跟踪能力.     

一种新的变步长LMS自适应滤波算法

通过建立步长因子μ与误差信号e之间的非线性关系，提出一种新的基于抽样函数的变步长LMS算法，并进行了计算机仿真．结果表明，该算法除了具有传统LMS算法计算量小、稳定性较好、简单易于实时处理等优点外，其收敛速度、稳定性以及跟踪速度均优于传统固定步长LMS算法抽样函数、SVSI—MS算法。     

一种基于变步长LMS算法的语音增强方法

LMS算法在自适应滤波器中得到广泛应用，但这种方法具有收敛速度慢，对非平稳环境敏感性强，步长需要谨慎选择才能达到收敛和失调的折中等缺点。为了改善非平稳条件下FIR自适应滤波器的性能，文章介绍了一种变步长的LMS算法，这种算法迭代过程中步长在规定的上下限内是关于信噪比的递减函数，用于自适应噪声对消器中去除含噪语音信号中的加性噪声，以解决固定LMS算法中跟踪速度和失调的矛盾。对不同信噪比的含噪语音信号去噪，仿真结果证明该方法优于NLMS(Normalized Least Mean Square)算法，在提高收敛速度的情况下减小了剩余均方误差和失调，但需增加少量的运算量。     

An adaptive hp-version of the finite element method applied to flame propagation problems

This paper describes an adaptive hp-version mesh refinement strategy and its application to the finite element solution of one-dimensional flame propagation problems. The aim is to control the spatial and time discretization errors below a prescribed error tolerance at all time levels. In the algorithm, the optimal time step is first determined in an adaptive manner by considering the variation of the computable error in the reaction zone. Later, the method uses a p-version refinement till the computable a posteriori error is brought down below the tolerance. During the p-version, if the maximum allowable degree of approximation is reached in some elements of the mesh without satisfying the global error tolerance criterion, then conversion from p- to h-version is performed. In the conversion procedure, a gradient based non-uniform h-version refinement has been introduced in the elements of higher degree approximation. In this way, p-version and h-version approaches are used alternately till the a posteriori error criteria are satisfied. The mesh refinement is based on the element error indicators, according to a statistical error equi-distribution procedure. Numerical simulations have been carried out for a linear parabolic problem and premixed flame propagation in one-space dimension. © 1997 John Wiley & Sons, Ltd.     

Plastic,viscoplastic and creep fracture problems with the boundary element method

The present paper shows the applicability of the dual boundary element method to analyse plastic, viscoplastic and creep behaviours in fracture mechanics problems. Several models with a crack, including a square plate, a holed plate and a notched plate, are analysed. Special attention is taken when the discretization of the domain is performed. In fact, for the plasticity and viscoplasticity cases, only the region susceptible to yielding was discretized, whereas the creep case required the discretization of the whole domain. The proposed formulation is presented as an alternative technique to study these kinds of nonlinear problems. Results from the present formulation are compared to those of the well‐established finite element technique, and they are in good agreement. Important fracture mechanic parameters like K_I, K_II, J‐integrals and C‐integrals are also included. In general, the results, for the plastic, viscoplastic and creep cases, exhibit that the highest stress concentrations are in the vicinity of the crack tip and they decrease as the distance from the crack tip is increased.     

A model for short crack propagation in polycrystalline materials

A model for microstructurally short crack propagation in a grain structure of a polycrystalline material is developed. The crack propagation model is based on a crystal plasticity model and a microstructurally short crack propagation model in the spirit of the model by Navarro and de los Rios [A model for short fatigue crack propagation with an interpretation of the short-long crack transition. Fatigue Fract Eng Mater Struct 1987;10:169-86]. Numerical examples, where the combined crystal plasticity and crack propagation model is implemented in a model of a microstructure representing a duplex stainless steel, concludes the paper. Results showing how the misorientation of the crack- and slip-directions between two adjacent austenitic grains influences the crack propagation rate, as the crack propagates across their common grain boundary, are given.     

An indirect boundary element method for three-dimensional dynamic fracture mechanics

Indirect boundary element methods (fictitious load and displacement discontinuity) have been developed for the analysis of three-dimensional elastostatic and elastodynamic fracture mechanics problems. A set of boundary integral equations for fictitious loads and displacement discontinuities have been derived. The stress intensity factors were obtained by the stress equivalent method for static loading. For dynamic loading the problem was studied in Laplace transform space where the numerical calculation procedure, for the stress intensity factor K_I(p), is the same: as that for the static problem. The Durbin inversion method for Laplace transforms was used to obtain the stress intensity factors in the time domain K_I(t). Results of this analysis are presented for a square bar, with either a rectangular or a circular crack, under static and dynamic loads.     

A coupled molecular dynamics and extended finite element method for dynamic crack propagation

A multiscale method is presented which couples a molecular dynamics approach for describing fracture at the crack tip with an extended finite element method for discretizing the remainder of the domain. After recalling the basic equations of molecular dynamics and continuum mechanics, the discretization is discussed for the continuum subdomain where the partition‐of‐unity property of finite element shape functions is used, since in this fashion the crack in the wake of its tip is naturally modelled as a traction‐free discontinuity. Next, the zonal coupling method between the atomistic and continuum models is recapitulated. Finally, it is discussed how the stress has been computed in the atomic subdomain, and a two‐dimensional computation is presented of dynamic fracture using the coupled model. The result shows multiple branching, which is reminiscent of recent results from simulations on dynamic fracture using cohesive‐zone models. Copyright © 2009 John Wiley & Sons, Ltd.     

A precise critical time step formula for the explicit material point method

The material point method (MPM) combines Eulerian method and Lagrangian method and thus both Lagrangian particle position and interaction between neighboring Eulerian grid cells will affect the simulation stability. However, the original critical time step formula in the standard MPM does not reflect the effect of particle position and neighboring cell interaction on stability and overestimates the critical time step so much that the CFL number has to be very small, even smaller than 0.1, to obtain a stable solution at extreme particle positions. Therefore, in many engineering applications, the standard MPM is very expensive due to the small CFL number. In this article, the effect of particle position and neighboring cell interaction on stability of the explicit MPM is studied. An explicit critical time step formula is obtained based on the system eigenvalues in one dimension, and is then extended to two and three dimensions. For extreme deformation problems, the geometric stiffness matrix is taken into consideration which modifies the sound speed of particles in the critical time step formula. Several tests are performed to verify our formula and show a decrease in amount of time steps used for simulation with our formula comparing with the original formula.     

Application of viscoelastic fracture criteria to progressive crack propagation in polymer matrix composites

With the view of comparing local and global viscoelastic fracture criteria, an extension of Christensen's criterion to composite materials with stiff elastic fibers is proposed. Several versions of this criterion are compared with Schapery's local approach of the same phenomenon. The asymptotic version of Christensen's criterion for rapid crack propagation is found suitable for the material investigated, at room temperature. Dissipation in the specimen has two main sources: the undamaged material on one hand, and the damaged material inside the `failure zone' close to the crack tip on the other. The respective roles of these two kinds of dissipation are assessed.     

A hybrid adaptive finite element phase-field method for quasi-static and dynamic brittle fracture

The phase-field approach has unique advantages in describing fracture phenomena, which has received extensive attention in the past decade. Nevertheless, the phase-field modeling of fracture is computationally demanding, due to the high temporal-spatial resolution required for crack tracking. In this contribution, a novel hybrid adaptive finite element phase-field method (ha-PFM) is developed to solve brittle fracture problems under quasi-static and dynamic loading. ha-PFM can dynamically track the propagation of the cracks and adaptively refine the meshes based on a novel crack tip identification strategy. Afterward, the refined meshes in the noncrack progression region are reconverted into coarse meshes. This scheme prominently reduces the computational cost, eg, CPU time and memory usage. Unlike the previous adaptive phase-field method, multilevel hybrid triangular and quadrilateral elements were developed to discretize the computational domain, which eliminates hanging nodes and ensures that the meshes in the vicinity of the crack tip are highly isotropic. Several representative benchmarks containing quasi-static and dynamic fracture were reinvestigated with ha-PFM, and its excellent performance is substantiated by comparison with the standard phase-field method and literature results.     

A variational multiscale method to model crack propagation at finite strains

This contribution presents a hierarchical variational multiscale framework to model propagating discontinuities at finite strains. Thereby the deformation map is decomposed into coarse‐scale and fine‐scale displacements, which results in a decoupled system of coarse‐scale and fine‐scale equations. Both are solved numerically by means of the finite element method whereby crack propagation is taken into account at the fine scale. Growing cracks are numerically handled by the introduction of discontinuous elements. A locality assumption on the fine‐scale solution and an adaptive scheme to resize the fine‐scale domain are introduced and demonstrated to increase the efficiency of the method. Copyright © 2009 John Wiley & Sons, Ltd.     

A p‐adaptive method for electromagnetic wave propagation

The discontinuous Galerkin FEM is used for the numerical solution of the three‐dimensional Maxwell equations. Control of errors in the numerical level for the divergence‐free constraint of the magnetic field can be obtained through the use of divergence‐free vector bases. In this work, the so‐called perfectly hyperbolic formulation of the Maxwell equations is used to retain both divergence‐free magnetic field and in the presence of charges to satisfy the Gauss constraint for the electric field at the numerical level. For both approaches, it is found that higher‐order approximations have favorable effect on the preservation of the divergence constraints and that the perfectly hyperbolic formulations retains these errors to a lower level. It is shown that high‐order accuracy in space and time is achieved in unstructured meshes using implicit time marching. For nonuniform meshes, local resolution refinement is used using p‐type adaptivity to ensure accurate electromagnetic wave propagation. Thus, the potential of the method to reach the required higher resolution in anisotropic meshes and obtain accurate electromagnetic wave propagation with reduced computational effort is demonstrated.     

Application of the RBF meshless method to laminar flame propagation

In this work we explore the applicability of the RBF method to laminar flame propagation modeling. This problem is an interesting challenge for the RBF method since it involves the solution of two coupled nonlinear parabolic equations in temperature and mass fraction. We show the suitability of the method by solving unsteady flame propagation problems in one and two dimensions. We also apply the method to compute the shape of an anchored flame using both equispaced and non-equispaced nodes.     

A local-domain based phase field method for crack propagation

Abstract

The phase field method transfers the crack surface area into a domain integral through a smeared damage parameter. As a result, the fracture energy release rate can be directly incorporated into the energy form without tracking of crack surface, and a minimization approach to optimize the crack shape becomes possible. This nonlocal damage approach requires a fine mesh discretization to capture the damage gradient in the smeared crack zone, which produces a significant degree of freedom problem to optimize over the entire domain. In this research, a local-domain based phase field method is proposed. This method considers only elements in critical regions for energy minimization, whose internal energy are dominant, thus significantly reducing the degree of freedom to be solved. A numerical verification is carried out to demonstrate the feasibility of this approach, and simulations are presented to verify the accuracy of the new method.     

Adaptive crack propagation analysis with the element‐free Galerkin method

In this paper, an adaptive analysis of crack propagation based on the error estimation by the element‐free Galerkin (EFG) method is presented. The adaptivity analysis in quasi‐static crack propagation is achieved by adding and/or removing the nodes along the background integration cells, those are refined or recovered according to the estimated errors. These errors are obtained basically by calculating the difference between the values of the projected stresses and original EFG stresses. To evaluate the performance of the proposed adaptive procedure, the crack propagation behaviour is investigated for several examples. The results of these examples show the efficiency and accuracy of the proposed scheme in crack propagation analysis. Copyright © 2002 John Wiley & Sons, Ltd.