Coarse-grained atomistic simulation of dislocations

This paper presents a new methodology for coarse-grained atomistic simulation of dislocation dynamics. The methodology combines an atomistic formulation of balance equations and a modified finite element method employing rhombohedral-shaped 3D solid elements suitable for fcc crystals. With significantly less degrees of freedom than that of a fully atomistic model and without additional constitutive rules to govern dislocation activities, this new coarse-graining (CG) method is shown to be able to reproduce key phenomena of dislocation dynamics for fcc crystals, including dislocation nucleation and migration, formation of stacking faults and Lomer-Cottrell locks, and splitting of stacking faults, all comparable with fully resolved molecular dynamics simulations. Using a uniform coarse mesh, the CG method is then applied to simulate an initially dislocation-free submicron-sized thin Cu sheet. The results show that the CG simulation has captured the nucleation and migration of large number of dislocations, formation of multiple stacking fault ribbons, and the occurrence of complex dislocation phenomena such as dislocation annihilation, cutting, and passing through the stacking faults. The distinctions of this method from existing coarse-graining or multiscale methods and its potential applications and limitations are also discussed.     

随机有限元方法在断裂分析中的应用

在幂律非线性随机有限元基础上,以单边裂纹板为例给出计算含量钢继裂参数,J（J积分）,δ（裂纹张开位移）,Δ（由裂纹引起的裂纹板上下底面相对位移）,θ（由裂纹引起的裂纹板上下底在相对转角）及其对基本随机变量变化率的方法和分析算例。     

Nodeless variable finite element method for heat transfer analysis by means of flux-based formulation and mesh adaptation

Based on flux-based formulation, a nodeless variable element method is developed to analyze two-dimensional steady-state and transient heat transfer problems. The nodeless variable element employs quadratic interpolation functions to provide higher solution accuracy without necessity to actually generate additional nodes. The flux-based formulation is applied to reduce the complexity in deriving the finite element equations as compared to the conventional finite element method. The solution accuracy is further improved by implementing an adaptive meshing technique to generate finite element mesh that can adapt and move along corresponding to the solution behavior. The technique generates small elements in the regions of steep solution gradients to provide accurate solution, and meanwhile it generates larger elements in the other regions where the solution gradients are slight to reduce the computational time and the computer memory. The effectiveness of the combined procedure is demonstrated by heat transfer problems that have exact solutions. These problems are: (a) a steady-state heat conduction analysis in a square plate subjected to a highly localized surface heating, and (b) a transient heat conduction analysis in a long plate subjected to a moving heat source. The English text was polished by Yunming Chen.     

基于有限断裂法和比例边界有限元法的裂纹扩展模拟

基于有限断裂法和比例边界有限元法提出了一种裂缝开裂过程模拟的数值模型。采用基于有限断裂法的混合断裂准则作为起裂及扩展的判断标准,当最大环向应力和能量释放率同时达到其临界值时,裂缝扩展。结合多边形比例边界有限元法,可以半解析地求解裂尖区域附近的应力场和位移场,在裂尖附近无需富集即可获得高精度的解。计算能量释放率时,只需将裂尖多边形内的裂尖位置局部调整,无需改变整体网格的分布,网格重剖分的工作量降至最少。裂缝扩展步长通过混合断裂准则确定,避免了人为假设的随意性,并可以实现裂缝变步长扩展的模拟,更符合实际情况。通过对四点剪切梁的复合型裂缝扩展过程的模拟,对本文模型进行了验证,并应用于重力坝模型的裂缝扩展模拟,计算结果表明,本文提出的模型简单易行且精度较高。     

Application of scaled boundary finite element method in static and dynamic fracture problems

The scaled boundary finite element method (SBFEM) is a recently developed numerical method combining advantages of both finite element methods (FEM) and boundary element methods (BEM) and with its own special features as well. One of the most prominent advantages is its capability of calculating stress intensity factors (SIFs) directly from the stress solutions whose singularities at crack tips are analytically represented. This advantage is taken in this study to model static and dynamic fracture problems. For static problems, a remeshing algorithm as simple as used in the BEM is developed while retaining the generality and flexibility of the FEM. Fully-automatic modelling of the mixed-mode crack propagation is then realised by combining the remeshing algorithm with a propagation criterion. For dynamic fracture problems, a newly developed series-increasing solution to the SBFEM governing equations in the frequency domain is applied to calculate dynamic SIFs. Three plane problems are modelled. The numerical results show that the SBFEM can accurately predict static and dynamic SIFs, cracking paths and load-displacement curves, using only a fraction of degrees of freedom generally needed by the traditional finite element methods.The project supported by the National Natural Science Foundation of China (50579081) and the Australian Research Council (DP0452681)The English text was polished by Keren Wang.     

Transonic viscous-inviscid interaction by a finite element method

A method is outlined for solving two-dimensional transonic viscous flow problems, in which the velocity vector is split into the gradient of a potential and a rotational component. The approach takes advantage of the fact that for high-Reynolds-number flows the viscous terms of the Navier-Stokes equations are important only in a thin shear layer and therefore solution of the full equations may not be needed everywhere. Most of the flow can be considered inviscid and, neglecting the entropy and vorticity effects, a potential model is a good approximation in the flow core. The rotational part of the flow can then be calculated by solution of the potential, streamfunction and vorticity transport equations. Implementation of the no-slip and no-penetration boundary conditions at the walls provides a simple mechanism for the interaction between the viscous and inviscid solutions and no extra coupling procedures are needed. Results are presented for turbulent transonic internal choked flows.     

Analysis of regular and chaotic dynamics of the Euler-Bernoulli beams using finite difference and finite element methods

Chaotic vibrations of flexible non-linear Euler-Bernoulli beams subjected to harmonic load and with various boundary conditions(symmetric and non-symmetric)are studied in this work.Reliability of the obtained results is verified by the finite difference method(FDM)and the finite element method(FEM)with the Bubnov-Galerkin approximation for various boundary conditions and various dynamic regimes(regular and non-regular).The influence of boundary conditions on the Euler-Bernoulli beams dynamics is studied mainly,dynamic behavior vs.control parameters { ωp,q0 } is reported,and scenarios of the system transition into chaos are illustrated.     

A new finite element method for strain gradient theories and applications to fracture analyses

A new compatible finite element method for strain gradient theories is presented. In the new finite element method, pure displacement derivatives are taken as the fundamental variables. The new numerical method is successfully used to analyze the simple strain gradient problems – the fundamental fracture problems. Through comparing the numerical solutions with the existed exact solutions, the effectiveness of the new finite element method is tested and confirmed. Additionally, an application of the Zienkiewicz–Taylor C1 finite element method to the strain gradient problem is discussed. By using the new finite element method, plane-strain mode I and mode II crack tip fields are calculated based on a constitutive law which is a simple generalization of the conventional J₂ deformation plasticity theory to include strain gradient effects. Three new constitutive parameters enter to characterize the scale over which strain gradient effects become important. During the analysis the general compressible version of Fleck–Hutchinson strain gradient plasticity is adopted. Crack tip solutions, the traction distributions along the plane ahead of the crack tip are calculated. The solutions display the considerable elevation of traction within the zone near the crack tip.     

A finite strain, finite band method for modeling ductile fracture

We present a finite deformation generalization of the finite thickness embedded discontinuity formulation presented in our previous paper [A.E. Huespe, A. Needleman, J. Oliver, P.J. Sánchez, A finite thickness band method for ductile fracture analysis, Int. J. Plasticity 25 (2009) 2349-2365]. In this framework the transition from a weak discontinuity to a strong discontinuity can occur using a single constitutive relation which is of importance in a range of applications, in particular ductile fracture, where localization typically precedes the creation of new free surface. An embedded weak discontinuity is introduced when the loss of ellipticity condition is met. The resulting localized deformation band is given a specified thickness which introduces a length scale thus providing a regularization of the post-localization response. The methodology is illustrated through several example problems emphasizing finite deformation effects including the development of a cup-cone failure in round bar tension.     

Coarse-graining and renormalization of atomistic binding relations and universal macroscopic cohesive behavior

We present two approaches for coarse-graining interplanar potentials and determining the corresponding macroscopic cohesive laws based on energy relaxation and the renormalization group. We analyze the cohesive behavior of a large—but finite—number of interatomic planes and find that the macroscopic cohesive law adopts a universal asymptotic form. The universal form of the macroscopic cohesive law is an attractive fixed point of a suitably-defined renormalization-group transformation.     

Transient thermal shock fracture analysis of functionally graded piezoelectric materials by the extended finite element method

Transient thermal dynamic analysis of stationary cracks in functionally graded piezoelectric materials (FGPMs) based on the extended finite element method (X-FEM) is presented. Both heating and cooling shocks are considered. The material properties are supposed to vary exponentially along specific direction while the crack-faces are assumed to be adiabatic and electrically impermeable. A dynamic X-FEM model is developed in which both Crank–Nicolson and Newmark time integration methods are used for calculating transient responses of thermal and electromechanical fields respectively. The generalized dynamic intensity factors for the thermal stresses and electrical displacements are extracted by using the interaction integral. The accuracy of the developed approach is verified numerically by comparing the calculated results with reference solutions. Numerical examples with mixed-mode crack problems are analyzed. The effects of the crack-length, poling direction, material gradation, etc. on the dynamic intensity factors are investigated. It shows that the transient dynamic crack behaviors under the cooling shock differ from those under the heating shock. The influence of the thermal shock loading on the dynamic intensity factors is significant.     

Streamline upwind finite element method for conjugate heat transfer problems

This paper presents a combined finite element method for solving conjugate heat transfer problems where heat conduction in a solid is coupled with heat convection in viscous fluid flow. The streamline upwind finite element method is used for the analysis of thermal viscous flow in the fluid region, whereas the analysis of heat conduction in solid region is performed by the Galerkin method. The method uses the three-node triangular element with equal-order interpolation functions for all the variables of the velocity components, the pressure and the temperature. The main advantage of the proposed method is to consistently couple heat transfer along the fluid-solid interface. Three test cases, i.e. conjugate Couette flow problem in parallel plate channel, counter-flow in heat exchanger, and conjugate natural convection in a square cavity with a conducting wall, are selected to evaluate the efficiency of the present method. The English text was polished byYunming Chen.     

基于扩展有限元的地质聚合物混凝土断裂过程研究

扩展有限元法是基于常规有限元框架分析裂纹等不连续力学问题的一种有效数值方法,在常规的有限元位移表达式中,增加了能够反映位移不连续性的跳跃函数和渐进缝尖位移场函数来对不连续结构附近的节点自由度进行局部加强。本文介绍了扩展有限元法及粘聚力模型的基本原理,给出了基于扩展有限元法的地质聚合物混凝土断裂过程分析方法。分别采用四种不同的软化曲线对I型缺口地质聚合物混凝土梁从裂纹萌生、扩展直至断裂破坏的全过程进行了模拟,并基于双K断裂准则分析了其断裂韧性。结果表明,Petersson模型与试验结果吻合较好,最后基于模拟结果进一步揭示了断裂过程区的演化过程。     

Site of ductile fracture initiation in cold forging: A finite element model

Ductile fracture occurs due to micro-void nucleation, growth and finally coalescence into micro-crack. In this study a new ductile fracture condition that based on the microscopic phenomena of void nucleation, growth and coalescence was proposed. Using this condition and combining with finite element model to predict the fracture locations in bulk metal forming, the results show that it is in close accordance with observations of some experimental specimens. However, in order to obtaining the high trustiness many experiments have to be carried out.     

A hybrid finite volume/finite element method for incompressible generalized Newtonian fluid flows on unstructured triangular meshes

This paper presents a hybrid finite volume/finite element method for the incompressible generalized Newtonian fluid flow （Power-Law model）. The collocated （i.e. non-staggered） arrangement of variables is used on the unstructured triangular grids, and a fractional step projection method is applied for the velocity-pressure coupling. The cell-centered finite volume method is employed to discretize the momentum equation and the vertex-based finite element for the pressure Poisson equation. The momentum interpolation method is used to suppress unphysical pressure wiggles. Numerical experiments demonstrate that the current hybrid scheme has second order accuracy in both space and time. Results on flows in the lid-driven cavity and between parallel walls for Newtonian and Power-Law models are also in good agreement with the published solutions.     

Determination of stress intensity factors by the finite element discretized symplectic method

When rewriting the governing equations in Hamiltonian form, analytical solutions in the form of symplectic series can be obtained by the method of separation of variable satisfying the crack face conditions. In theory, there exists sufficient number of coefficients of the symplectic series to satisfy any outer boundary conditions. In practice, the matrix relating the coefficients to the outer boundary conditions is ill-conditioned unless the boundary is very simple, e.g., circular. In this paper, a new two-level finite element method using the symplectic series as global functions while using the conventional finite element shape functions as local functions is developed. With the available classical finite elements and symplectic series, the main unknowns are no longer the nodal displacements but are the coefficients of the symplectic series. Since the first few coefficients are the stress intensity factors, post-processing is not required. A number of numerical examples as well as convergence studies are given.     

Enriched goal-oriented error estimation for fracture problems solved by continuum-based shell extended finite element method简

An enriched goal-oriented error estimation method with extended degrees of freedom is developed to estimate the error in the continuum-based shell extended finite element method. It leads to high quality local error bounds in three-dimensional fracture mechanics simulation which involves enrichments to solve the singularity in crack tip. This enriched goal-oriented error estimation gives a chance to evaluate this continuum- based shell extended finite element method simulation. With comparisons of reliability to the stress intensity factor calculation in stretching and bending, the accuracy of the continuum-based shell extended finite element method simulation is evaluated, and the reason of error is discussed.     

Generalized magneto-thermoelasticity in a nonhomogeneous isotropic hollow cylinder using the finite element method

In this paper, we constructed the equations of generalized magneto-thermoelasticity in a perfectly conducting medium. The formulation is applied to generalizations, the Lord–Shulman theory with one relaxation time, and the Green–Lindsay theory with two relaxation times, as well as to the coupled theory. The material of the cylinder is supposed to be nonhomogeneous isotropic both mechanically and thermally. The problem has been solved numerically using a finite element method. Numerical results for the temperature distribution, displacement, radial stress, and hoop stress are represented graphically. The results indicate that the effects of nonhomogeneity, magnetic field, and thermal relaxation times are very pronounced. In the absence of the magnetic field or relaxation times, our results reduce to those of generalized thermoelasticity and/or classical dynamical thermoelasticity, respectively. Results carried out in this paper can be used to design various nonhomogeneous magneto-thermoelastic elements under magnetothermal load to meet special engineering requirements. An erratum to this article can be found at     

Numerical convergence studies of the mixed finite element method for natural convection flow in a fluid-saturated porous medium

This paper describes a numerical approximation scheme for the natural convection (NC) flow in a fluid-saturated porous medium. Our formulation of the problem is based on the mixed finite element method (FEM). Using the so-called consistent splitting scheme, a second-order accuracy in time and in space is verified by the numerical calculation. The resulting flow patterns and heat transfer for different Rayleigh numbers, Darcy numbers and porosities are numerically studied by the proposed scheme.     

A total linearization method for solving viscous free boundary flow problems by the finite element method

In this paper a total linearization method is derived for solving steady viscous free boundary flow problems (including capillary effects) by the finite element method. It is shown that the influence of the geometrical unknown in the totally linearized weak formulation can be expressed in terms of boundary integrals. This means that the implementation of the method is simple. Numerical experiments show that the iterative method gives accurate results and converges very fast.