Multiscale coupling of molecular dynamics and peridynamics

We propose a multiscale computational model to couple molecular dynamics and peridynamics. The multiscale coupling model is based on a previously developed multiscale micromorphic molecular dynamics (MMMD) theory, which has three dynamics equations at three different scales, namely, microscale, mesoscale, and macroscale. In the proposed multiscale coupling approach, we divide the simulation domain into atomistic region and macroscale region. Molecular dynamics is used to simulate atom motions in atomistic region, and peridynamics is used to simulate macroscale material point motions in macroscale region, and both methods are nonlocal particle methods. A transition zone is introduced as a messenger to pass the information between the two regions or scales. We employ the “supercell” developed in the MMMD theory as the transition element, which is named as the adaptive multiscale element due to its ability of passing information from different scales, because the adaptive multiscale element can realize both top-down and bottom-up communications. We introduce the Cauchy–Born rule based stress evaluation into state-based peridynamics formulation to formulate atomistic-enriched constitutive relations. To mitigate the issue of wave reflection on the interface, a filter is constructed by switching on and off the MMMD dynamic equations at different scales. Benchmark tests of one-dimensional (1-D) and two-dimensional (2-D) wave propagations from atomistic region to macro region are presented. The mechanical wave can transit through the interface smoothly without spurious wave deflections, and the filtering process is proven to be efficient.     

Implication of scaling hierarchy associated with nonequilibrium: field and particulate

The advent of nanotechnology has necessitated a better understanding of how material microstructure changes at the atomic level would affect the macroscopic properties that control the performance. Such a challenge has uncovered many phenomena that were not previously understood and taken for granted. Among them are the basic foundation of dislocation theories which are now known to be inadequate. Simplifying assumptions invoked at the macroscale may not be applicable at the micro- and/or nanoscale. There are implications of scaling hierrachy associated with inhomegeneity and nonequilibrium of physical systems. What is taken to be homogeneous and equilibrium at the macroscale may not be so when the physical size of the material is reduced to microns. These fundamental issues cannot be dispensed at will for the sake of convenience because they could alter the outcome of predictions. Even more unsatisfying is the lack of consistency in modeling physical systems. This could translate to the inability for identifying the relevant manufacturing parameters and rendering the end product unpractical because of high cost. Advanced composite and ceramic materials are cases in point.Discussed are potential pitfalls for applying models at both the atomic and continuum levels. No encouragement is made to unravel the truth of nature. Let it be partiuclates, a smooth continuum or a combination of both. The present trend of development in scaling tends to seek for different characteristic lengths of material microstructures with or without the influence of time effects. Much will be learned from atomistic simulation models to show how results could differ as boundary conditions and scales are changed. Quantum mechanics, continuum and cosmological models provide evidence that no general approach is in sight. Of immediate interest is perhaps the establishment of greater precision in terminology so as to better communicate results involving multiscale physical events.     

与非平衡问题相关的尺度效应:场与微粒

纳米技术的出现,使我们有必要更好地了解,在原子水平上材料微结构的变化是如何影响和控制着材料的宏观性能.这一挑战涉及到许多以前不曾考虑和不曾了解的现象.其中,位错理论的基础现在知道是有问题的.宏观尺度下采用的简化假设,也许不能用于微观和纳米尺度.尺度效应的含义,涉及到物理系统的非均质和非平衡特性.宏观尺度下的均匀与平衡特性,在材料的物理尺度减少到微米量级时就不再保持了.这些基本观点不能够为了方便而随意到处使用,因为这会改变预测的结果.更令人不满的是在建立物理模型时缺乏一致性.由此产生的问题是在确定制造过程中的有关参数时无能为力,导致由于成本过高而不切实际的终端产品.先进的复合材料和陶瓷材料就存在这样的问题.本文将要讨论的是在原子尺度与连续介质尺度下应用理论模型时存在的潜在问题,而不是去揭示自然的真相.主要讨论微粒,均匀连续介质或者两者的结合.尺度效应问题当前的发展趋势,趋向于在有或者没有时间效应的情况下寻找材料微结构的不同特征尺寸.从原子模拟模型中将了解到许多情况,原子模拟计算将揭示计算结果如何随着边界条件和尺度变化而不同.量子力学,连续介质力学和宇宙模型证明,没有普遍适用的方法.当前的主要兴趣也许是针对多尺度物理问题在技术上建立更高的精度,以得到一个更好的表达结果.      

Coupling the finite element method and molecular dynamics in the framework of the heterogeneous multiscale method for quasi-static isothermal problems

Multiscale models are designed to handle problems with different length scales and time scales in a suitable and efficient manner. Such problems include inelastic deformation or failure of materials. In particular, hierarchical multiscale methods are computationally powerful as no direct coupling between the scales is given. This paper proposes a hierarchical two-scale setting appropriate for isothermal quasi-static problems: a macroscale treated by continuum mechanics and the finite element method and a microscale modelled by a canonical ensemble of statistical mechanics solved with molecular dynamics. This model will be implemented into the framework of the heterogeneous multiscale method. The focus is laid on an efficient coupling of the macro- and micro-solvers. An iterative solution algorithm presents the macroscopic solver, which invokes for each iteration an atomistic computation. As the microscopic computation is considered to be very time consuming, two optimisation strategies are proposed. Firstly, the macroscopic solver is chosen to reduce the number of required iterations to a minimum. Secondly, the number of time steps used for the time average on the microscale will be increased with each iteration. As a result, the molecular dynamics cell will be allowed to reach its state of thermodynamic equilibrium only in the last macroscopic iteration step. In the preceding iteration steps, the molecular dynamics cell will reach a state close to equilibrium by using considerably fewer microscopic time steps. This adapted number of microsteps will result in an accelerated algorithm (aFE-MD-HMM) obtaining the same accuracy of results at significantly reduced computational cost. Numerical examples demonstrate the performance of the proposed scheme.     

Shape sensitivity analysis for non-differentiable performance measures in multiscale crack propagation simulation using regression hybrid method

In this paper, we present a regression hybrid method that calculates shape sensitivity coe?cients for multiscale crack propagation problems with performance measures that are non-differentiable in numerical implementation. These measures are crack propagation speed (or crack speed) defined at atomistic level obtained by solving coupled atomistic/continuum structures using the bridging scale method (BSM). The major contributions of this paper are: first, by analyzing the characteristics of the performance measures of crack speed in design space, this paper verifies for the first time that these measures are theoretically continuous and differentiable with respect to design variables, and as a result, the sensitivity coe?cients exist in theory; second, to overcome the non-differentiability of the performance measures in numerical computation due to the finite size of integration time step, this paper proposes a regression hybrid method that calculates the shape sensitivity coe?cients of crack speed through polynomial regression analysis based on the sensitivity of atomic responses, which is calculated through analytical shape design sensitivity analysis (DSA). And finally, the proposed method supports for 3D crack propagation problems with periodic boundary condition in one direction. A nano-beam example is used to demonstrate numerically the feasibility, accuracy, and e?ciency of the proposed method.     

The continuum elastic and atomistic viewpoints on the formation volume and strain energy of a point defect

We discuss the roles of continuum linear elasticity and atomistic calculations in determining the formation volume and the strain energy of formation of a point defect in a crystal. Our considerations bear special relevance to defect formation under stress. The elasticity treatment is based on the Green's function solution for a center of contraction or expansion in an anisotropic solid. It makes possible the precise definition of a formation volume tensor and leads to an extension of Eshelby's [Proc. R. Soc. London Ser. A 241 (1226), 376] result for the work done by an external stress during the transformation of a continuum inclusion. Parameters necessary for a complete continuum calculation of elastic fields around a point defect are obtained by comparing with an atomistic solution in the far field. However, an elasticity result makes it possible to test the validity of the formation volume that is obtained via atomistic calculations under various boundary conditions. It also yields the correction term for formation volume calculated under these boundary conditions. Using two types of boundary conditions commonly employed in atomistic calculations, a comparison is also made of the strain energies of formation predicted by continuum elasticity and atomistic calculations. The limitations of the continuum linear elastic treatment are revealed by comparing with atomistic calculations of the formation volume and strain energies of small crystals enclosing point defects.     

Multiscale constitutive modeling and numerical simulation of fabric material

A multiscale model for a fabric material is introduced. The model is based on the assumption that on the macroscale the fabric behaves as a continuum membrane, while on the microscale the properties of the microstructure are accounted for by a constitutive law derived by modeling a pair of overlapping crimped yarns as extensible elasticae. A two-scale finite element method is devised to solve selected boundary-value problems.     

Coupled quantum–continuum analysis of crack tip processes in aluminum

A concurrent multiscale method is presented that couples a quantum mechanically governed atomistic domain to a continuum domain. The approach is general in that it is applicable to a wide range of quantum and continuum material modeling methodologies. It also provides quantifiable and controllable coupling errors via a force-based-coupling strategy. The applications presented here utilize an atomistic region that is governed by Kohn–Sham density functional theory and a continuum region governed by linear elasticity with discrete dislocation capabilities. As a validation we compute the core structure of a screw dislocation in aluminum and compare to previously published results. Then we investigate two crack orientations in aluminum and predict the critical load at which crack propagation and crack tip dislocation nucleation occurs. We compute critical loads with both LDA and GGA exchange correlation functionals and compare our results to popular empirical potentials in the context of classical continuum models. Overall this work aims to lay a foundation for future quantum mechanics-based investigations of crack tip processes involving Al alloys and impurity elements.     

An atomistically validated continuum model for strain relaxation and misfit dislocation formation

In this paper, molecular dynamics (MD) calculations have been used to examine the physics behind continuum models of misfit dislocation formation and to assess the limitations and consequences of approximations made within these models. Without compromising the physics of misfit dislocations below a surface, our MD calculations consider arrays of dislocation dipoles constituting a mirror imaged “surface”. This allows use of periodic boundary conditions to create a direct correspondence between atomistic and continuum representations of dislocations, which would be difficult to achieve with free surfaces. Additionally, by using long-time averages of system properties, we have essentially reduced the errors of atomistic simulations of large systems to “zero”. This enables us to deterministically compare atomistic and continuum calculations. Our work results in a robust approach that uses atomistic simulation to accurately calculate dislocation core radius and energy without the continuum boundary conditions typically assumed in the past, and the novel insight that continuum misfit dislocation models can be inaccurate when incorrect definitions of dislocation spacing and Burgers vector in lattice-mismatched systems are used. We show that when these insights are properly incorporated into the continuum model, the resulting energy density expression of the lattice-mismatched systems is essentially indistinguishable from the MD results.     

Multiscale modeling of crack initiation and propagation at the nanoscale

Fracture occurs on multiple interacting length scales; atoms separate on the atomic scale while plasticity develops on the microscale. A dynamic multiscale approach (CADD: coupled atomistics and discrete dislocations) is employed to investigate an edge-cracked specimen of single-crystal nickel, Ni, (brittle failure) and aluminum, Al, (ductile failure) subjected to mode-I loading. The dynamic model couples continuum finite elements to a fully atomistic region, with key advantages such as the ability to accommodate discrete dislocations in the continuum region and an algorithm for automatically detecting dislocations as they move from the atomistic region to the continuum region and then correctly “converting” the atomistic dislocations into discrete dislocations, or vice-versa. An ad hoc computational technique is also applied to dissipate localized waves formed during crack advance in the atomistic zone, whereby an embedded damping zone at the atomistic/continuum interface effectively eliminates the spurious reflection of high-frequency phonons, while allowing low-frequency phonons to pass into the continuum region.The simulations accurately capture the essential physics of the crack propagation in a Ni specimen at different temperatures, including the formation of nano-voids and the sudden acceleration of the crack tip to a velocity close to the material Rayleigh wave speed. The nanoscale brittle fracture happens through the crack growth in the form of nano-void nucleation, growth and coalescence ahead of the crack tip, and as such resembles fracture at the microscale. When the crack tip behaves in a ductile manner, the crack does not advance rapidly after the pre-opening process but is blunted by dislocation generation from its tip. The effect of temperature on crack speed is found to be perceptible in both ductile and brittle specimens.     

Evolution of nanoscale defects to planar cracks in a brittle solid

Fracture of a solid is a highly multiscale process that associates atomic scale bond breaking with macroscopic crack propagation, and the process can be dramatically influenced by the presence of defects in materials. In a nanomaterial, defect formation energy decreases with the reduction of material size, and therefore, the role of defects in crack formation and subsequent crack growth in such materials may not be understood from the classical laws of fracture mechanism. In this study, we investigated the crack formation process of a defective (with missing atoms) nanostructured material (NaCl) using a series of molecular dynamics (MD) simulations. It was demonstrated that simple defects in the form of several missing atoms in the material could develop into a planar crack. Subsequently, MD simulations on failures of nanosized NaCl with pre-defined planar atomistic cracks of two different lengths under prescribed tensile displacement loads were performed. These failure loads were then applied on the equivalent continuum models, separately, to evaluate the associated fracture toughness values using the finite element analysis. For small cracks, the fracture toughness thus obtained is cracksize dependent and the corresponding critical energy release rate is significantly smaller than Griffith’s theoretical value. Explanation for this discrepancy between LEFM and the atomistic model was attempted.     

Computational multiscale methods for granular materials

The fine-scale heterogeneity of granular material is characterized by its polydisperse microstructure with randomness and no periodicity. To predict the mechanical response of the material as the microstructure evolves, it is demonstrated to develop computational multiscale methods using discrete particle assembly-Cosserat continuum modeling in micro- and macro- scales, respectively. The computational homogenization method and the bridge scale method along the concurrent scale linking approach are briefly introduced. Based on the weak form of the Hu-Washizu variational principle, the mixed finite element procedure of gradient Cosserat continuum in the frame of the second-order homogenization scheme is developed. The meso-mechanically informed anisotropic damage of effective Cosserat continuum is characterized and identified and the microscopic mechanisms of macroscopic damage phenomenon are revealed.     

A multiscale method for dislocation nucleation and seamlessly passing scale boundaries

In the concurrent multiscale analysis, it is difficult to have truly seamless transition between the atomistic and continuum scale. This situation is even worse when defects pass through the boundary between different scales. For example, there is a lack of effective methods to handle the dislocation passing through scale boundaries which is important to investigate plasticity at the nanoscale. In this paper, the generalized particle (GP) method proposed by the first author is further developed so that a seamless transition and dislocation passing between different scales can be realized. Specifically, the linkage between different scales is through material neighbor-link cells (NLC) with scale duality. This indicates that material elements can be high-scale particles through a lumping process and can also be atoms via decomposition depending on the needs of the simulation. At the interface, the information transfer from bottom scale-up or from top scale-down is through the particles or atoms in the NLC. They are with the same material structure, all possess nonlocal constitutive behavior; thus, the smooth transition at the interface between different scales can be attained and validated to avoid non-physical responses. To save degrees of freedom, atoms are lumped together into a generalized particle in the domain in which the deformation gradient is near homogeneous. On the other hand, when defects such as dislocations in the atomistic domain are near the particle domain, the particles along dislocation propagation path and its surrounding region will be decomposed into atoms so dislocations can freely pass through the scale boundary and propagate inside the model just as it propagates in the deformed atomistic crystal structure. The method is verified first for seamless transition of variables at the scale boundary by a one-dimensional model and then verified for dislocation nucleation and propagation passing through scale boundaries in two cases, one is near the free surface and the other is inside of the copper nanowire. All the validations are through comparisons with fully atomistic analyses under same conditions. The comparison is satisfactory.     

Computation of head–disk interface gap micro flowfields using DSMC and continuum–atomistic hybrid methods

This paper discusses computational modeling of micro flow in the head–disk interface (HDI) gap using the direct simulation Monte Carlo (DSMC) method. Modeling considerations are discussed in detail both for a stand‐alone DSMC computation and for the case of a hybrid continuum–atomistic simulation that couples the Navier–Stokes (NS) equation to a DSMC solver. The impact of the number of particles and number of cells on the accuracy of a DSMC simulation of the HDI gap is investigated both for two‐ and three‐dimensional configurations. An appropriate implicit boundary treatment method for modeling inflow and outflow boundaries is used in this work for a three‐dimensional DSMC micro flow simulation. As the flow outside the slider is in the continuum regime, a hybrid continuum–atomistic method based on the Schwarz alternating method is used to couple the DSMC model in the slider bearing region to the flow outside the slider modeled by NS equation. Schwarz coupling is done in two dimensions by taking overlap regions along two directions and the Chapman–Enskog distribution is employed for imposing the boundary condition from the continuum region to the DSMC region. Converged hybrid flow solutions are obtained in about five iterations and the hybrid DSMC–NS solutions show good agreement with the exact solutions in the entire domain considered. An investigation on the impact of the size of the overlap region on the convergence behavior of the Schwarz method indicates that the hybrid coupling by the Schwarz method is weakly dependent on the size of the overlap region. However, the use of a finite overlap region will facilitate the exchange of boundary conditions as the hybrid solution has been found to diverge in the absence of an overlap region for coupling the two models. Copyright © 2009 John Wiley & Sons, Ltd.     

准连续体方法研究进展

准连续体方法是一种将连续介质方法和原子模拟相耦合的多尺度模拟方法。该方法巧妙地将分子静力学与有限元法相结合,在变形梯度较小的规则区域采用代表性原子作为计算点,以此为节点形成有限元网格,其周围其他原子的位置通过插值得到。代表性原子的引入,使系统原子势能的计算大大简化。自适应手段的应用,保证了材料缺陷核心附近原子尺度的细节描述。与其他纳米力学计算方法相比,准连续体方法具有计算精度高,计算规模大等优点。本文主要介绍了准连续体方法的基本原理及发展状况,归纳并总结了这一些优点及今后的发展方向。     

On a formulation for a multiscale atomistic-continuum homogenization method

The homogenization method is used as a framework for developing a multiscale system of equations involving atoms at zero temperature at the small scale and continuum mechanics at the very large scale. The Tersoff–Brenner Type II potential [Physical Review Letters 61(25) (1988) 2879; Physical Review B 42 (15) (1990) 9458] is employed to model the atomic interactions while hyperelasticity governs the continuum. A quasistatic assumption is used together with the Cauchy–Born approximation to enforce the gross deformation of the continuum on the positions of the atoms. The two-scale homogenization method establishes coupled self-consistent variational equations in which the information at the atomistic scale, formulated in terms of the Lagrangian stiffness tensor, concurrently feeds the material information to the continuum equations. Analytical results for a one dimensional molecular wire and numerical experiments for a two dimensional graphene sheet demonstrate the method and its applicability.     

Unconstrained and Cauchy-Born-constrained atomistic systems: deformational and configurational mechanics

In this contribution, the deformational and configurational mechanics of (elastic) discrete atomistic systems in relation to their continuum counterparts are considered for the quasi-static case. Thereby, we firstly investigate the basic unconstrained case in the sense of lattice statics as a reference. Based on these results, we consider two Cauchy-Born-type constraints that (locally) describe the change of the position of atoms (between the material and the spatial configuration) in terms of either a linear or a quadratic map, respectively. Insertion of these kinematic constraints into the variation of the total potential energy of the unconstrained case renders eventually Cauchy-Born-type definitions for atomistic stresses and hyperstresses, for both deformational and configurational cases. In the continuum limit, these are the relevant continuum stresses and hyperstresses contributing to the local force balances of first- (classical) and second-order (non-classical) gradient continua (here first- and second-order gradient refers to the highest gradient of the deformation map characterizing the kinematics). It is emphasized that the atomistic and the Cauchy-Born-type configurational quantities represent novel and unexpected contributions. Among other things, they may be useful in assessing the singular or non-singular character of deformation fields at crack tips and comparing numerical estimates resulting from atomistic simulations with analytical predictions resulting from solutions of related boundary value problems for gradient continua.     

An atomistic–continuum hybrid scheme for numerical simulation of droplet spreading on a solid surface

We present an atomistic–continuum hybrid method to investigate spreading dynamics of drops on solid surfaces. The Navier–Stokes equations are solved by the finite-volume method in a continuum domain comprised of the main body of the drop, and atomistic molecular dynamics simulations are used in a particle domain in the vicinity of the contact line. The spatial coupling between the continuum and particle domains is achieved through constrained dynamics of flux continuities in an overlap domain.