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.     

Concurrently coupled atomistic and XFEM models for dislocations and cracks

A method for the modeling of dislocations and cracks by atomistic/continuum models is described. The methodology combines the extended finite element method with the bridging domain method (BDM). The former is used to model crack surfaces and slip planes in the continuum, whereas the BDM is used to link the atomistic models with the continuum. The BDM is an overlapping domain decomposition method in which the atomistic and continuum energies are blended so that their contributions decay to their boundaries on the overlapping subdomain. Compatibility between the continua and atomistic domains is enforced by a continuous Lagrange multiplier field. The methodology allows for simulations with atomistic resolution near crack fronts and dislocation cores while retaining a continuum model in the remaining part of the domain and so a large reduction in the number of atoms is possible. It is applied to the modeling of cracks and dislocations in graphene sheets. Energies and energy distributions compare very well with direct numerical simulations by strictly atomistic models. Copyright © 2008 John Wiley & Sons, Ltd.     

Buckling analysis of carbon nanotubes by a mixed atomistic and continuum model

A mixed atomistic and continuum model is applied to carbon nanotubes, in order to study their buckling behavior. Herein, the term “atomistic” refers to the underlying constitutive model that is formulated on the basis of interatomic potentials, whereas “continuum” means the application of the Cauchy–Born rule, which links the bond vectors before and after deformation via the deformation gradient of the continuum. Because the bond vectors are not infinitesimal and the continuum is modeled as surface, the Cauchy–Born rule has to be appropriately adapted to crystalline sheets. This is done via an exponential mapping in a new and surprisingly simple form such that in the analysis the current configuration has never to be left. The numerical buckling analysis of carbon nanotubes using the mixed atomistic and continuum model is carried out by means of the finite element method. For this purpose, the linearization of the equilibrium equations is provided.     

An h‐hierarchical adaptive procedure for the scaled boundary finite‐element method

The scaled boundary finite‐element method (a novel semi‐analytical method for solving linear partial differential equations) involves the solution of a quadratic eigenproblem, the computational expense of which rises rapidly as the number of degrees of freedom increases. Consequently, it is desirable to use the minimum number of degrees of freedom necessary to achieve the accuracy desired. Stress recovery and error estimation techniques for the method have recently been developed. This paper describes an h‐hierarchical adaptive procedure for the scaled boundary finite‐element method. To allow full advantage to be taken of the ability of the scaled boundary finite‐element method to model stress singularities at the scaling centre, and to avoid discretization of certain adjacent segments of the boundary, a sub‐structuring technique is used. The effectiveness of the procedure is demonstrated through a set of examples. The procedure is compared with a similar h‐hierarchical finite element procedure. Since the error estimators in both cases evaluate the energy norm of the stress error, the computational cost of solutions of similar overall accuracy can be compared directly. The examples include the first reported direct comparison of the computational efficiency of the scaled boundary finite‐element method and the finite element method. The scaled boundary finite‐element method is found to reduce the computational effort considerably. Copyright © 2002 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.     

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.     

Computational method for atomistic homogenization of nanopatterned point defect structures

The development of an approximation method that rigorously averages small‐scale atomistic physics and embeds them in large‐scale mechanics is the principal aim of this work. This paper presents a general computational procedure based on homogenization to average frozen nanoscale atomistics and couple them to the equations of continuum hyperelasticity. The proposed application is to nanopatterned systems in which complex atomic configurations are organized in a repeating periodic array. The finite element method is used to solve the equations at the large scale, but the small‐scale equation is representative of lattice‐statics. The method is predicated on a quasistatic zero‐temperature assumption and, through homogenization, leads to a coupled set of variational equations. The numerical procedure is presented in detail, and 2‐D examples of ultra thin film layers of carbon one atom thick are shown to illustrate its applicability. Homogenization naturally gives rise to an inner displacement term with which point defects are explicitly modelled and their non‐linear interactions with global states of multiaxial strain are studied. Published in 2004 by 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.     

A Frobenius solution to the scaled boundary finite element equations in frequency domain for bounded media

The scaled boundary finite element method (FEM) is a recently developed semi‐analytical numerical approach combining advantages of the FEM and the boundary element method. Although for elastostatics, the governing homogeneous differential equations in the radial co‐ordinate can be solved analytically without much effort, an analytical solution to the non‐homogeneous differential equations in frequency domain for elastodynamics has so far only been obtained by a rather tedious series‐expansion procedure. This paper develops a much simpler procedure to obtain such an analytical solution by increasing the number of power series in the solution until the required accuracy is achieved. The procedure is applied to an extensive study of the steady‐state frequency response of a square plate subjected to harmonic excitation. Comparison of the results with those obtained using ABAQUS shows that the new method is as accurate as a detailed finite element model in calculating steady‐state responses for a wide range of frequencies using only a fraction of the degrees of freedom required in the latter. Copyright © 2006 John Wiley & Sons, Ltd.     

A damping boundary condition for coupled atomistic–continuum simulations

Many atomistic–continuum coupling techniques employ an overlapping subdomain to suppress spurious wave reflections. In this paper, we propose the imposition of a new damping condition on the overlapping subdomain to enhance the capability of such methods in eliminating spurious wave reflections. In this technique, the total displacements of the atoms in the overlapping subdomain are decomposed into fine and coarse scales. The fine scale displacements represent the oscillations which cannot be resolved by the continuum mesh and must be eliminated to avoid the artificial reflections. This is achieved by modifying the equations of motion of the fine scale displacements to include a damping term. The flexibility of the proposed technique is verified by applying it to the bridging scale method and bridging domain method. Numerical simulations of one- and two-dimensional problems demonstrate the effectiveness of the technique in enhancing the elimination of the spurious wave reflections in coupled atomistic–continuum techniques.     

The scaled boundary finite element method—alias consistent infinitesimal finite element cell method—for diffusion

The scaled boundary finite element method, alias the consistent infinitesimal finite element cell method, is developed starting from the diffusion equation. Only the boundary of the medium is discretized with surface finite elements yielding a reduction of the spatial dimension by one. No fundamental solution is necessary, and thus no singular integrals need to be evaluated. Essential and natural boundary conditions on surfaces and conditions on interfaces between different materials are enforced exactly without any discretization. The solution of the function in the radial direction is analytical. This method is thus exact in the radial direction and converges to the exact solution in the finite element sense in the circumferential directions. The semi‐analytical solution inside the domain leads to an efficient procedure to calculate singularities accurately without discretization in the vicinity of the singular point. For a bounded medium symmetric steady‐state stiffness and mass matrices with respect to the degrees of freedom on the boundary result without any additional assumption. Copyright © 1999 John Wiley & Sons, Ltd.     

An embedded statistical method for coupling molecular dynamics and finite element analyses

The coupling of molecular dynamics (MD) simulations with finite element methods (FEM) yields computationally efficient models that link fundamental material processes at the atomistic level with continuum field responses at higher length scales. The theoretical challenge involves developing a seamless connection along an interface between two inherently different simulation frameworks. Various specialized methods have been developed to solve particular classes of problems. Many of these methods link the kinematics of individual MD atoms with finite element (FE) nodes at their common interface, necessarily requiring that the FE mesh be refined to atomic resolution. Some of these coupling approaches also require simulations to be carried out at 0 K and restrict modelling to two‐dimensional material domains due to difficulties in simulating full three‐dimensional material processes. In the present work, a new approach to MD–FEM coupling is developed based on a restatement of the standard boundary value problem used to define a coupled domain. The method replaces a direct linkage of individual MD atoms and FE nodes with a statistical averaging of atomistic displacements in local atomic volumes associated with each FE node in an interface region. The FEM and MD computational systems are effectively independent and communicate only through an iterative update of their boundary conditions. Thus, the method lends itself for use with any FEM or MD code. With the use of statistical averages of the atomistic quantities to couple the two computational schemes, the developed approach is referred to as an embedded statistical coupling method (ESCM). ESCM provides an enhanced coupling methodology that is inherently applicable to three‐dimensional domains, avoids discretization of the continuum model to atomic scale resolution, and permits finite temperature states to be applied. Published in 2009 by John Wiley & Sons, Ltd.     

Discrete and continuum methods for numerical simulations of non‐linear wave propagation in discontinuous media

The main objective of this paper is to develop a methodology to model dynamic loading of various discontinuous media. Two methods for modeling non‐linear waves in a media with multiple discontinuities are considered. The first one, a discrete method, is based on the Simple Common Plane contact algorithm. This method can be applied both to compliant contacts characterized by finite thickness and elastic moduli (such as joints in geomechanics) as well as to non‐compliant frictional contacts traditionally described by the slide lines in finite element/finite difference codes. The second one, a continuum method, assumes that the contacts are not compliant and can be modeled as one or several weakness planes cutting through the elements of the computational mesh. Both discrete and continuum methods described in the paper can be applied to derive equivalent continuum properties for media with multiple discontinuities. An example of such application for a randomly jointed media is given in the paper. Copyright © 2010 John Wiley & Sons, Ltd.     

Combined atomistic simulation and continuum mechanics: Size-dependent behavior of atomistic simulation for brittle fracture in bcc-iron

The size-dependent behavior of atomistic simulation for brittle fracture in bcc-iron is studied using combined continuum-atomistic method. The result of displacement distribution at the crack tip indicates the radius of discrete region at the crack tip is about 120 Å, and the truncated distance of continuum elastic field is about 120 Å away from the crack tip. Further investigations of energy and atomic structure show that the elastic field of continuum mechanics cannot affect the crack tip processes effectively through the common region if the diameter of the atomistic region is smaller than 300 Å. We assume that the range of the system size that can bridge continuum with discrete region is at the scale of about 300 Å.     

The atomistic‐continuum hybrid taxonomy and the hybrid‐hybrid approach

The hybrid taxonomy — a means of characterizing different atomistic‐continuum methods on the basis of the type of information exchanged between the atomistic and the continuum solver — is introduced. The formulation of the taxonomy raises a new hybrid possibility, called a ‘hybrid‐hybrid’ method. Some examples of hybrid‐hybrid simulations for dense fluids are discussed and validated against full molecular dynamics results. Copyright © 2014 John Wiley & Sons, Ltd.     

Approximate imposition of boundary conditions in immersed boundary methods

We analyze several possibilities to prescribe boundary conditions in the context of immersed boundary methods. As basic approximation technique we consider the finite element method with a mesh that does not match the boundary of the computational domain, and therefore Dirichlet boundary conditions need to be prescribed in an approximate way. As starting variational approach we consider Nitsche's methods, and we then move to two options that yield non‐symmetric problems but that turned out to be robust and efficient. The essential idea is to use the degrees of freedom of certain nodes of the finite element mesh to minimize the difference between the exact and the approximated boundary condition. Copyright © 2009 John Wiley & Sons, Ltd.     

Proper orthogonal decomposition and modal analysis for acceleration of transient FEM thermal analysis

A method of reducing the number of degrees of freedom and the overall computing times in finite element method (FEM) has been devised. The technique is valid for linear problems and arbitrary temporal variation of boundary conditions. At the first stage of the method standard FEM time stepping procedure is invoked. The temperature fields obtained for the first few time steps undergo statistical analysis yielding an optimal set of globally defined trial and weighting functions for the Galerkin solution of the problem at hand. Simple matrix manipulations applied to the original FEM system produce a set of ordinary differential equations of a dimensionality greatly reduced when compared with the original FEM formulation. Using the concept of modal analysis the set is then solved analytically. Treatment of non‐homogeneous initial conditions, time‐dependent boundary conditions and controlling the error introduced by the reduction of the degrees of freedom are discussed. Several numerical examples are included for validation of the approach. Copyright © 2004 John Wiley & Sons, Ltd.     

Continuum metrics for deformation and microrotation from atomistic simulations: Application to grain boundaries

Motivated by a desire to incorporate micro- and nanoscale deformation mechanisms into continuum mechanical models of material behavior, we apply recently developed volume-averaged metrics to the results of atomistic simulations to investigate deformation and microrotation in the vicinity of grain boundaries. Three-dimensional bicrystalline structures are employed to study the inelastic deformation behavior under uniaxial tension and simple shear at a temperature of 10 K. Each bicrystal is constructed by molecular statics followed by thermal equilibration under NPT using an embedded atom method potential for copper. Strain is imposed in each simulation cell at a constant 10⁹ s⁻¹ strain rate applied perpendicular and parallel to the grain boundary plane for tension and shear, respectively. A variety of grain boundary deformation mechanisms arise and the resulting deformation and microrotation fields are examined. We also include an analysis showing how microrotation varies as a function of distance from the grain boundary with increasing strain for different grain boundary deformation mechanisms. This work demonstrates that critical interface behavior can be extracted from atomistic simulations using volume-averaged metrics, offering a potential avenue for translating fundamental information to continuum theories of grain boundary deformation in polycrystalline materials.     

A coupling atomistic–continuum approach for modeling mechanical behavior of nano-crystalline structures

In this article, a novel approach is presented for the concurrent coupling of continuum–atomistic model in the nano-mechanical behavior of atomic structures. The study is focused on the static concurrent multi-scale simulation, which is able to effectively capture the surface effects intrinsic in the molecular mechanics modeling. The Hamiltonian approach is applied to combine the continuum and molecular models with the same weight in the overlapping domain. A Lagrange-multiplier method is employed over the overlapping domain for coupling the continuum nodal displacement with the atomic lattice deformation. A multiple-step algorithm is developed to decouple the solution process in the atomic and continuum domains. The mass and stiffness matrices of continuum domain are computed based on the linear bridging map of the atomic lattice displacement, laid underneath the continuum grid to the element displacements. Numerical simulation results present that the stress and displacement contours of the presented coupling method are in good agreement with those obtained from the molecular mechanics simulation.     

Finite element implementation of a kinetic model of dynamic strain aging in aluminum–magnesium alloys

Soare and Curtin (Acta Mater. 2008; 56 :4091–4101, 4046–4061) have recently developed a model of dynamic strain aging in solute‐strengthened alloys. Their constitutive law describes time‐dependent solute strengthening using rate equations that can be calibrated using atomistic simulations. In this paper, their material model is incorporated into a continuum finite element simulation, with a view to completing a multi‐scale method for predicting the formability of solute‐strengthened alloys. The Soare–Curtin model is first re‐formulated as a state‐variable constitutive law, which is suitable for finite element computations. An efficient numerical procedure is then developed to track the strength distribution of aging mobile and forest dislocations in the solid during deformation. The method is tested by simulating the behavior of a 3D aluminum–magnesium alloy tensile specimen subjected to uniaxial loading at constant nominal strain rate. The model predicts the influence of strain rate on the steady‐state flow stress of Al–Mg alloys, but no Portevin–Le Châtelier bands or serrated flow were observed in any of our simulations, and the influence of strain rate on tensile ductility is not predicted correctly. The reasons for this behavior and possible resolutions are discussed. Copyright © 2010 John Wiley & Sons, Ltd.