Adaptive atomistic‐to‐continuum modeling of propagating defects

An adaptive atomistic‐to‐continuum method is presented for modeling the propagation of material defects. This method extends the bridging domain method to allow the atomic domain to dynamically conform to the evolving defect regions during a simulation, without introducing spurious oscillations and without requiring mesh refinement. The atomic domain expands as defects approach the bridging domain method coupling domain by fine graining nearby finite elements into equivalent atomistic subdomains. Additional algorithms coarse grain portions of the atomic domain to the continuum scale, reducing the degrees of freedom, when the atomic displacements in a subdomain can be approximated by FEM or extended FEM elements to within a certain homogeneity tolerance. The extended FEM approximations are created by fitting the broken inter‐atomic bonds of fractured surfaces and dislocation slip planes. Because atomic degrees of freedom are maintained only where needed for each timestep, the solution retains the advantages of multiscale modeling, with a reduced computational cost compared with other multiscale methods. Copyright © 2012 John Wiley & Sons, Ltd.     

Seamless coupling of molecular dynamics and material point method via smoothed molecular dynamics

A concurrent multiscale method coupling molecular dynamics (MD) and continuum‐based material point method (MPM) is proposed. Seamless coupling is realized by utilizing smoothed molecular dynamics (SMD) method. One set of background mesh is used in SMD method. Atomic equations of motion are assembled onto mesh nodes, and atomic variables are updated with nodal increments. SMD allows much larger time step size than MD critical time step size but keeps nice global accuracy. SMD is similar to MD except for the mapping process between background mesh nodes and atoms. SMD and MPM share the feature using the background mesh to solve momentum equations and to update variables. So bridging MD and MPM via SMD is straightforward and concise. A recently proposed transition scheme based on frequency decomposition is adopted to suppress phonon reflection at MD‐SMD interface. The nodal equations in SMD–MPM interface region have contributions from both atoms and material points, which ensure the consistency between SMD region and MPM region. A multiple‐time‐step scheme is adopted for high efficiency. Numerical examples including wave propagation, bending, and crack propagation validate the proposed method, and the results show nice accuracy. The computational cost is greatly saved compared with pure MD computation. Copyright © 2017 John Wiley & Sons, Ltd.     

Multiscale coupling using a finite element framework at finite temperature

This paper presents the formulation and application of a multiscale methodology that couples three domains using a finite element framework. The proposed method efficiently models atomistic systems by decomposing the system into continuum, bridging, and atomistic domains. The atomistic and bridging domains are solved using a combined finite element–molecular mechanics simulation where the system is discretized into atom/nodal centric elements based on the atomic scale finite element method. Coupling between the atomistic domain and continuum domain is performed through the bridging cells, which contain locally formulated atoms whose displacements are mapped to the nodes of the bridging cell elements. The method implements a temperature‐dependent potential for finite temperature simulations. Validation and demonstration of the methodology are provided through three case studies: displacement in a one‐dimensional chain, stress around nanoscale voids, and fracture. From these studies differences between multiscale and fully atomistic simulations were very small with the simulation time of the proposed methodology being approximately a tenth of the time of the fully atomistic model. Copyright © 2012 John Wiley & Sons, Ltd.     

Elastic crack propagation model for crystalline solids using a self-consistent coupled atomistic–continuum framework

Deformation and failure processes of crystalline materials are governed by complex phenomena at multiple scales. It is necessary to couple these scales for physics-based modeling of these phenomena, while overcoming limitations of modeling at individual scales. To address this issue, this paper develops self-consistent elastic constitutive and crack propagation relations of crystalline materials containing atomic scale cracks, from observations made in a concurrent multi-scale simulation system coupling atomistic and continuum domain models. The concurrent multi-scale model incorporates a finite temperature atomistic region containing the crack, a continuum region represented by a self-consistent crystal elasticity constitutive model, and a handshaking interphase region. Atomistic modeling is done by the molecular dynamics code LAMMPS, while continuum modeling is conducted by the finite element method. For single crystal nickel a nonlinear and nonlocal crystal elasticity constitutive relation is derived, consistent with the atomic potential function. An efficient, staggered solution scheme with parallel implementation is designed for the coupled problem. The atomistic–continuum coupling is achieved by enforcing geometric compatibility and force equilibrium in the interphase region. Quantitative analyses of the crack propagation process focuses on size dependence, strain energy release rate, crack propagation rate and degradation of the local stiffness. The self-consistent constitutive and crack propagation relations, derived from the concurrent model simulation results are validated by comparing results from the concurrent and full FE models. Excellent accuracy and enhanced efficiency are observed in comparison with pure MD and concurrent model results.     

A continuum‐to‐atomistic bridging domain method for composite lattices

The bridging domain method is an overlapping domain decomposition approach for coupling finite element continuum models and molecular mechanics models. In this method, the total energy is decomposed into atomistic and continuum parts by complementary weight functions applied to each part of the energy in the coupling domain. To enforce compatibility, the motions of the coupled atoms are constrained by the continuum displacement field using Lagrange multipliers. For composite lattices, this approach is suboptimal because the internal modes of the lattice are suppressed by the homogeneous continuum displacement field in the coupling region. To overcome this difficulty, we present a relaxed bridging domain method. In this method, the atom set is divided into primary and secondary atoms; the relative motions between them are often called the internal modes. Only the primary atoms are constrained in the coupling region, which succeed in allowing these internal modes to fully relax. Several one‐ and two‐dimensional examples are presented, which demonstrate improved accuracy over the standard bridging domain method. Copyright © 2009 John Wiley & Sons, Ltd.     

A concurrent atomistic and continuum coupling method with applications to thermo‐mechanical problems

We present a novel method to couple molecular dynamics with finite elements at finite temperatures using spatial filters. The mismatch in the dispersion relations between continuum and atomistic models leads, at finite temperature, to unwanted mesh vibrations, which are illustrated using a standard least square coupling formulation. We propose the use of spatial filters with the least square minimization to selectively damp the unwanted mesh vibrations. Then, we extend the idea of selective damping of wavelength modes to couple atomistic and continuum models at finite temperatures. The restitution force from the generalized Langevin equation is modified to perform a two‐way thermal coupling between the two models. Three different numerical examples are shown to validate the proposed coupling formulation in two‐dimensional space. Finally, the method is applied to a high‐speed impact simulation. Copyright © 2013 John Wiley & Sons, Ltd.     

Multiscale modeling of intergranular fracture in aluminum: constitutive relation for interface debonding

Intergranular fracture is a dominant mode of failure in ultrafine grained materials. In the present study, the atomistic mechanisms of grain-boundary debonding during intergranular fracture in aluminum are modeled using a coupled molecular dynamics—finite element simulation. Using a statistical mechanics approach, a cohesive-zone law in the form of a traction–displacement constitutive relationship, characterizing the load transfer across the plane of a growing edge crack, is extracted from atomistic simulations and then recast in a form suitable for inclusion within a continuum finite element model. The cohesive-zone law derived by the presented technique is free of finite size effects and is statistically representative for describing the interfacial debonding of a grain boundary (GB) interface examined at atomic length scales. By incorporating the cohesive-zone law in cohesive-zone finite elements, the debonding of a GB interface can be simulated in a coupled continuum–atomistic model, in which a crack starts in the continuum environment, smoothly penetrates the continuum–atomistic interface, and continues its propagation in the atomistic environment. This study is a step toward relating atomistically derived decohesion laws to macroscopic predictions of fracture and constructing multiscale models for nanocrystalline and ultrafine grained materials.     

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 conformal decomposition finite element method for arbitrary discontinuities on moving interfaces

Interface capturing methods using enriched finite element formulations are well suited for solving multimaterial transport problems that contain weak or strong discontinuities. The conformal decomposition FEM decomposes multimaterial elements of a non‐conforming background mesh into sub‐elements that conform to material interfaces captured using a level set method. As the interface evolves, interfacial nodes move, and background nodes may change material. The present work describes approaches for handling moving interfaces in the context of the conformal decomposition FEM for both weakly and strongly discontinuous fields. Dynamic discretization methods using extrapolation and moving mesh approaches are considered and developed with first‐order and second‐order time integration methods. The moving mesh approach is demonstrated to be a stable method that preserves both weak and strong discontinuities on a variety of one‐dimensional and two‐dimensional test problems, while achieving the expected second‐order error convergence rate in space and time. Copyright © 2014 John Wiley & Sons, Ltd.     

Modeling of fracture behavior in polymer composites using concurrent multi-scale coupling approach

ABSTRACT

Embedded statistical coupling method was originally developed to provide computational efficiency, to decrease coupling complexities, and to avoid the need to discretize the continuum model to atomic scale resolution in concurrent multi-scale modeling. An embedded statistical coupling method scheme is relatively easy to implement within a conventional finite element method code and has been tested in standard solid lattice structures. However, this method encounters difficulties when being implemented for amorphous materials like polymers, due to the fact that they lack specific ordered lattice structure and atoms may not be covalently bonded with each other, which are the requirements of common coupling schemes. Therefore, a new approach needs to be developed to resolve this problem. In this article, details of a modified embedded statistical coupling method approach for atomistic-continuum coupling developed to perform simulations of macroscale crack growth in polymers is presented. The presence of the continuum domain surrounding the molecular dynamics region allows for the application of far field loading, and prevents stress wave reflections from the external boundary impinging back on the crack tip. In our approach, a material point method, which is a meshless particle-in-cell method based on an arbitrary Euler-Lagrange scheme and has been proven to have good performance in large deformation problems, is used to model the continuum domain. It is concurrently coupled with molecular dynamics, a widely used method in atomistic simulations, using a so-called handshake region. Anchor points, the equilibrium positions of the constrained particles, which are designed to transmit displacements and forces between nanoscale and macroscale model, are defined in the handshake region. A concurrently coupled material point method-molecular dynamics simulation of crack propagation inside a polymer is performed to verify this new coupling approach, thereby providing a better understanding of the fracture mechanisms at the nanoscale to predict the macro-scale fracture toughness of a polymer system. Results are presented for concurrently coupled simulation of crack initiation and crack propagation in a di-functional cross-linked thermoset polymer, EPON 862.     

A robust nano-mechanics approach for tensile and modal analysis using atomistic–continuum mechanics method

Nanoscale engineering has been developing rapidly. However, experimental investigations at the nanoscale level are very difficult to conduct. This research seeks to employ the same model to investigate an atomic-scale structure for tensile and modal analyses, based on atomistic–continuum mechanics (ACM) and a finite element method (FEM). The ACM transfers an originally discrete atomic structure into an equilibrium continuum model using atomistic–continuum transfer elements. All interatomic forces, described by the empirical potential functions, can be transferred into springs to form the atomic structure. The spring network models were also widely utilized in FEM based nano-structure studies. Thus, this paper attempts to explore ACM using three examples including silicon, carbon nanotube, and copper. All of the results are validated by bulk properties or literature.     

Selective damping method for the weak‐Arlequin coupling of molecular dynamics and finite element method

An artificial damping force is introduced in the weak coupling between the molecular dynamics (MD) and finite element (FE) models, to reduce the reflection of the high‐frequency motion that cannot be transmitted from the MD domain to the FE domain. We take advantage of the orthogonal property of the decomposed velocity in the weak coupling method and apply the damping force only to the high‐frequency part, therefore minimizing its effect on the low‐frequency part, which can be transmitted into the FE domain. The effectiveness of the damping method will be demonstrated by 1D numerical examples with linear force field applied to the atomistic model. In addition, we emphasize the importance of using the Arlequin energy interpolation, which is usually ignored in the weak coupling literature. Non‐uniform rational basis spline functions have been used to interpolate the MD data for the weak coupling method, and the influence of changing the number and order of basis functions on the interpolation accuracy has been investigated numerically. For this work, we restrict our discussion to mechanical problems only, involving only mechanical energy terms (e.g., strain potential and kinetic energy). Copyright © 2013 John Wiley & Sons, Ltd.     

Molecular dynamics simulation of asphalt-aggregate interface adhesion strength with moisture effect

This study developed an atomistic simulation framework based on the classical molecular dynamics (MD) method to study the moisture-induced damage at the asphalt-aggregate interface. The interface adhesion strength of the asphalt–quartz system was predicted using MD simulation for the first time. The interface stress-separation curve under tension that was obtained from MD simulation resembles the failure behaviour measured from the pull-off strength conducted at the macroscopic scale. The results show that the presence of moisture at the asphalt–quartz interface significantly reduces the interface adhesion strength. The interface failure process is affected by the chemical compositions of asphalt. The interface adhesion strength decreases as the moisture content increases or the temperature increases. It was found that the atomistic model size (number of atoms) and the loading rate in MD simulation have considerable effects on the predicted interface adhesion strength. The findings from MD simulation provide fundamental understanding of material failure at the atomistic scale that cannot be observed at the normal experimental testing environment for asphalt materials. The MD simulation results can be potentially calibrated and utilised as inputs for higher scale micromechanical models to predict bulk mechanical responses of asphalt mixtures.     

Transient heat conduction analysis in a piecewise homogeneous domain by a coupled boundary and finite element method

A coupled finite element–boundary element analysis method for the solution of transient two‐dimensional heat conduction equations involving dissimilar materials and geometric discontinuities is developed. Along the interfaces between different material regions of the domain, temperature continuity and energy balance are enforced directly. Also, a special algorithm is implemented in the boundary element method (BEM) to treat the existence of corners of arbitrary angles along the boundary of the domain. Unknown interface fluxes are expressed in terms of unknown interface temperatures by using the boundary element method for each material region of the domain. Energy balance and temperature continuity are used for the solution of unknown interface temperatures leading to a complete set of boundary conditions in each region, thus allowing the solution of the remaining unknown boundary quantities. The concepts developed for the BEM formulation of a domain with dissimilar regions is employed in the finite element–boundary element coupling procedure. Along the common boundaries of FEM–BEM regions, fluxes from specific BEM regions are expressed in terms of common boundary (interface) temperatures, then integrated and lumped at the nodal points of the common FEM–BEM boundary so that they are treated as boundary conditions in the analysis of finite element method (FEM) regions along the common FEM–BEM boundary. Copyright © 2002 John Wiley & Sons, Ltd.     

Non‐reflecting boundary conditions for atomistic,continuum and coupled atomistic/continuum simulations

We present a method to numerically calculate a non‐reflecting boundary condition which is applicable to atomistic, continuum and coupled multiscale atomistic/continuum simulations. The method is based on the assumption that the forces near the domain boundary can be well represented as a linear function of the displacements, and utilizes standard Laplace and Fourier transform techniques to eliminate the unnecessary degrees of freedom. The eliminated degrees of freedom are accounted for in a time‐history kernel that can be calculated for arbitrary crystal lattices and interatomic potentials, or regular finite element meshes using an automated numerical procedure. The new theoretical developments presented in this work allow the application of the method to non‐nearest neighbour atomic interactions; it is also demonstrated that the identical procedure can be used for finite element and mesh‐free simulations. We illustrate the effectiveness of the method on a one‐dimensional model problem, and calculate the time‐history kernel for FCC gold using the embedded atom method (EAM). Copyright © 2005 John Wiley & Sons, Ltd.     

A bridging domain and strain computation method for coupled atomistic–continuum modelling of solids

We present a multiscale method that couples atomistic models with continuum mechanics. The method is based on an overlapping domain‐decomposition scheme. Constraints are imposed by a Lagrange multiplier method to enforce displacement compatibility in the overlapping subdomain in which atomistic and continuum representations overlap. An efficient version of the method is developed for cases where the continuum can be modelled as a linear elastic material. An iterative scheme is utilized to optimize the coupled configuration. Conditions for the regularity of the constrained matrices are determined. A method for computing strain in atomistic models and handshake domains is formulated based on a moving least‐square approximation which includes both extensional and angle‐bending terms. It is shown that this method exactly computes the linear strain field. Applications to the fracture of defected single‐layer atomic sheets and nanotubes are given. Copyright © 2006 John Wiley & Sons, Ltd.     

A coupling of multi‐zone curved Galerkin BEM with finite elements for independently modelled sub‐domains with non‐matching nodes in elasticity

When the different parts of a structure are modelled independently by BEM or FEM methods, it is sometimes necessary to put the parts together without remeshing of the nodes along the part interfaces. Frequently the nodes do not match along the interface. In this work, the symmetric Galerkin multi‐zone curved boundary element is a fully symmetric formulation and is the method used for the boundary element part. For BEM–FEM coupling it is then necessary to interpolate the tractions in‐between the non‐matching nodes for the FEM part. Finally, the coupling is achieved by transforming the finite element domains to equivalent boundary element domains in a block symmetric formulation. This system is then coupled with a boundary element domain with non‐matching nodes in‐between. Copyright © 2004 John Wiley & Sons, Ltd.     

A locking‐free selective smoothed finite element method using tetrahedral and triangular elements with adaptive mesh rezoning for large deformation problems

A novel finite element (FE) formulation with adaptive mesh rezoning for large deformation problems is proposed. The proposed method takes the advantage of the selective smoothed FE method (S‐FEM), which has been recently developed as a locking‐free FE formulation with strain smoothing technique. We adopt the selective face‐based smoothed/node‐based smoothed FEM (FS/NS‐FEM‐T4) and edge‐based smoothed/node‐based smoothed FEM (ES/NS‐FEM‐T3) basically but modify them partly so that our method can handle any kind of material constitutive models other than elastic models. We also present an adaptive mesh rezoning method specialized for our S‐FEM formulation with material constitutive models in total form. Because of the modification of the selective S‐FEMs and specialization of adaptive mesh rezoning, our method is locking‐free for severely large deformation problems even with the use of tetrahedral and triangular meshes. The formulation details for static implicit analysis and several examples of analysis of the proposed method are presented in this paper to demonstrate its efficiency. Copyright © 2014 John Wiley & Sons, Ltd.     

FE/FMBE coupling to model fluid–structure interaction

In this paper, a finite element (FE)/fast multipole boundary element (FMBE)‐coupling method is presented for modeling fluid–structure interaction problems numerically. Vibrating structures are assumed to consist of elastic or sound absorbing materials. An FE method (FEM) is used for this part of the solution. This structural sub‐domain is embedded in a homogeneous fluid. The case where the boundary of the structural sub‐domain has a very complex geometry is of special interest. In this case, the BE method (BEM) is a more suitable numerical tool than FEM to account for the sound propagation in the homogeneous fluid. The efficiency of the BEM is increased by using FMBEM. The BE‐surface mesh required is directly generated by the FE‐mesh used to discretize the structural sub‐domain and the absorbing material. This FE/FMBE‐coupling method makes it possible to predict the effects of arbitrarily shaped absorbing materials and vibrating structures on the sound field in the surrounding fluid numerically. The coupling method proposed is used to study the acoustic behavior of the lining of an anechoic chamber and that of an entire anechoic chamber in the low‐frequency range. The numerical results obtained are compared with the experimental data. Copyright © 2008 John Wiley & Sons, Ltd.     

A simple dynamical scale‐coupling method for concurrent simulation of hybridized atomistic/coarse‐grained‐particle system

We propose a simple method for dynamical coupling of two sub‐systems with different characteristic scales described with different theoretical models, such as the fine‐scale sub‐system with the atomistic model (AM) such as the empirical inter‐atomic potential and the coarse‐scale sub‐system with the coarse‐grained particle (CGP) method, in a concurrent hybrid simulation scheme. Naive coupling of the different‐scale sub‐systems results in reflection of high wavenumber waves at the interface because of the differences in the phonon Brillouin‐zone and in the dispersion relation. To solve the problem, the present scale‐coupling method introduces (virtual) extra atoms and particles for the AM and the CGP sub‐systems, respectively, beyond the atom–particle interface, and uses the extra atoms and the particles to mutually transfer information of the waves between the two sub‐systems and to suppress the artificial reflection of the incident wave in the whole wavenumber range. As the algorithm in the present scale‐coupling method is local in time and space, it is applicable to hybrid systems with any interface shape at low computation and memory requirement. Accuracy of the present scale‐coupling method is compared with that of the existing methods for a simple model system. The hybrid AM‐CGP simulation of indentation of a graphene nano‐drum using the present scale‐coupling method is performed to demonstrate its accuracy and usefulness through its comparison with the fully atomistic results. Copyright © 2010 John Wiley & Sons, Ltd.