Molecular dynamics‐smoothed molecular dynamics (MD‐SMD) adaptive coupling method with seamless transition

Smoothed molecular dynamics (SMD) method is a recently proposed efficient molecular simulation method by introducing one set of background mesh and mapping process into molecular dynamics (MD) flow chart. SMD can sharply enlarge MD time step size while maintaining global accuracy. MD‐SMD coupling method was proposed to improve the capability to describe local atom disorders. The coupling method is greatly improved in this paper in two essential aspects. Firstly, a transition scheme is proposed to avoid artificial wave reflection at the interface of MD and SMD regions. The new transition scheme has simple formulation and high efficiency, and the wave reflection can be well suppressed. Secondly, an adaptive scheme is proposed to automatically identify the regions requiring MD simulation. Two adaptive criteria, the centro‐symmetry parameter criterion and the displacement criterion, are also proposed. It is found that both the two criteria can achieve good accuracy but the efficiency of the displacement criterion is much better. The coupling method does not demand reduction in mesh size near the interface, and a multiple time stepping scheme is adopted to ensure high efficiency. Numerical results including wave propagation, nano‐indentation, and crack propagation validate the method and show nice accuracy. Copyright © 2016 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.     

Adaptive smoothed molecular dynamics for multiscale modeling

In the smoothed molecular dynamics (SMD), the high frequency modes are eliminated from the motion of atoms to enlarge the time step significantly. In some situations, however, rearrangements or atoms disorder may occur. Hence, it is desirable to use MD in localized regions to capture the interesting high frequency motion, while use SMD elsewhere to save the computational cost. In this paper, an adaptive smoothed molecular dynamics (ASMD) is developed. During the simulation process, if the high frequency motions of atoms are dominant in a region, the background grid in the region is refined hierarchically until it is able to capture the high frequency motion of the atoms.     

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.     

Parallel computing of high‐speed compressible flows using a node‐based finite‐element method

An efficient parallel computing method for high‐speed compressible flows is presented. The numerical analysis of flows with shocks requires very fine computational grids and grid generation requires a great deal of time. In the proposed method, all computational procedures, from the mesh generation to the solution of a system of equations, can be performed seamlessly in parallel in terms of nodes. Local finite‐element mesh is generated robustly around each node, even for severe boundary shapes such as cracks. The algorithm and the data structure of finite‐element calculation are based on nodes, and parallel computing is realized by dividing a system of equations by the row of the global coefficient matrix. The inter‐processor communication is minimized by renumbering the nodal identification number using ParMETIS. The numerical scheme for high‐speed compressible flows is based on the two‐step Taylor–Galerkin method. The proposed method is implemented on distributed memory systems, such as an Alpha PC cluster, and a parallel supercomputer, Hitachi SR8000. The performance of the method is illustrated by the computation of supersonic flows over a forward facing step. The numerical examples show that crisp shocks are effectively computed on multiprocessors at high efficiency. Copyright © 2003 John Wiley & Sons, Ltd.     

Matching boundary conditions for lattice dynamics

We design a class of accurate and efficient absorbing boundary conditions for molecular dynamics simulations of crystalline solids. In one space dimension, the proposed matching boundary conditions take the form of a linear constraint of displacement and velocity at atoms near the boundary, where the coefficients are determined by matching the dispersion relation with a minimal number of atoms involved. Bearing the nice features of compactness, locality, and high efficiency, the matching boundary conditions are then extended to treat the out‐of‐plane wave problems in the square lattice. We construct multidirectional absorbing boundary conditions via operator multiplications. Reflection coefficient analysis and numerical studies verify their effectiveness for spurious reflection suppression in all directions. Compact and local in both space and time, they are directly applicable to nonlinear lattices and multiscale simulations. Copyright © 2012 John Wiley & Sons, Ltd.     

An explicit discontinuous Galerkin method for non‐linear solid dynamics: Formulation,parallel implementation and scalability properties

An explicit‐dynamics spatially discontinuous Galerkin (DG) formulation for non‐linear solid dynamics is proposed and implemented for parallel computation. DG methods have particular appeal in problems involving complex material response, e.g. non‐local behavior and failure, as, even in the presence of discontinuities, they provide a rigorous means of ensuring both consistency and stability. In the proposed method, these are guaranteed: the former by the use of average numerical fluxes and the latter by the introduction of appropriate quadratic terms in the weak formulation. The semi‐discrete system of ordinary differential equations is integrated in time using a conventional second‐order central‐difference explicit scheme. A stability criterion for the time integration algorithm, accounting for the influence of the DG discretization stability, is derived for the equivalent linearized system. This approach naturally lends itself to efficient parallel implementation. The resulting DG computational framework is implemented in three dimensions via specialized interface elements. The versatility, robustness and scalability of the overall computational approach are all demonstrated in problems involving stress‐wave propagation and large plastic deformations. Copyright © 2007 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.     

Stability of an explicit multi-time step integration algorithm for linear structural dynamics equations

A proof of stability is developed for an explicit multi-time step integration method of the second order differential equations which result from a semidiscretization of the equations of structural dynamics. The proof is applicable to an algorithm that partitions the mesh into subdomains according to nodal groups which are updated with different time steps. The stability of the algorithm is demonstrated by showing that the eigenvalues of the amplification matrices lie within the unit circle and that a pseudo-energy remains constant. Bounds on the stable time steps for the nodal partitions are developed in terms of element frequencies.     

Phase‐field material point method for brittle fracture

The material point method for the analysis of deformable bodies is revisited and originally upgraded to simulate crack propagation in brittle media. In this setting, phase‐field modelling is introduced to resolve the crack path geometry. Following a particle in cell approach, the coupled continuum/phase‐field governing equations are defined at a set of material points and interpolated at the nodal points of an Eulerian, ie, non‐evolving, mesh. The accuracy of the simulated crack path is thus decoupled from the quality of the underlying finite element mesh and relieved from corresponding mesh‐distortion errors. A staggered incremental procedure is implemented for the solution of the discrete coupled governing equations of the phase‐field brittle fracture problem. The proposed method is verified through a series of benchmark tests while comparisons are made between the proposed scheme, the corresponding finite element implementation, and experimental results.     

A method for dynamic crack and shear band propagation with phantom nodes

A new method for modelling of arbitrary dynamic crack and shear band propagation is presented. We show that by a rearrangement of the extended finite element basis and the nodal degrees of freedom, the discontinuity can be described by superposed elements and phantom nodes. Cracks are treated by adding phantom nodes and superposing elements on the original mesh. Shear bands are treated by adding phantom degrees of freedom. The proposed method simplifies the treatment of element‐by‐element crack and shear band propagation in explicit methods. A quadrature method for 4‐node quadrilaterals is proposed based on a single quadrature point and hourglass control. The proposed method provides consistent history variables because it does not use a subdomain integration scheme for the discontinuous integrand. Numerical examples for dynamic crack and shear band propagation are provided to demonstrate the effectiveness and robustness of the proposed method. Copyright © 2006 John Wiley & Sons, Ltd.     

An implicit multi-time step integration method for structural dynamics problems

A multi-time step integration algorithm is developed based on the trapezoidal rule time integration method for finite element equations of motion. This algorithm uses nodal groups to partition the mesh into subdomains that are updated with different time steps. A␣stability analysis of the method shows that the scheme retains the unconditionally stable behavior of the trapezoidal rule and conserves the same pseudo energy as the parent algorithm. Several numerical examples are used to verify the stability of the method and to investigate the accuracy of the scheme.     

Three-dimensional molecular dynamics study of aperture shape effect on nanojet ejection

Three-dimensional molecular dynamics (MD) simulations of nanojet ejection with different aperture shapes are reported. The simulations use the Lennard-Jones 12-6 (LJ) potential to describe the intermolecular interaction. Using non-equilibrium MD, argon nanojet ejection is simulated under vacuum conditions. According to the analysis, different aperture shapes influence the ejection processes. The ejection speeds were 23.7 and 63.2?m/s respectively in the simulation. The speed of spurting atoms in type A nanojet was slower than the other types and it became more obvious when the process time increased. The variations in velocity, density, pressure, and temperature were found with the aid of MD. The liquid temperatures were set at 50, 100, 150, and 200?K, respectively, to examine nanojet break-up characteristics. The liquid temperature inside the nanojet was found to be a factor that induce break-up. A higher temperature led to faster nanojet break-up.     

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.     

Proper orthogonal decomposition‐based model order reduction via radial basis functions for molecular dynamics systems

Model order reduction for molecular dynamics (MD) systems exhibits intrinsic complexities because of the highly nonlinear and nonlocal multi‐atomic interactions in high dimensions. In the present work, we introduce a proper orthogonal decomposition‐based method in conjunction with the radial basis function (RBF) approximation of the nonlinear and nonlocal potential energies and inter‐atomic forces for MD systems. This approach avoids coordinate transformation between the physical and reduced‐order coordinates, and allows the potentials and inter‐atomic forces to be calculated directly in the reduced‐order space. The RBF‐approximated potential energies and inter‐atomic forces in the reduced‐order space are discretized on the basis of the Smolyak sparse grid algorithm to further enhance the effectiveness of the proposed method. The good approximation properties of RBFs in interpolating scattered data make them ideal candidates for the reduced‐order approximation of MD inter‐atomic force calculations. The proposed approach is validated by performing the reduced‐order simulations of DNA molecules under various external loadings. Copyright © 2013 John Wiley & Sons, Ltd.     

A GPU‐based preconditioned Newton‐Krylov solver for flexible multibody dynamics

This paper describes an approach to numerically approximate the time evolution of multibody systems with flexible (compliant) components. Its salient attribute is that at each time step, both the formulation of the system equations of motion and their numerical solution are carried out using parallel computing on graphics processing unit cards. The equations of motion are obtained using the absolute nodal coordinate formulation, yet any other multibody dynamics formalism would fit equally well the overall solution strategy outlined herein. The implicit numerical integration method adopted relies on a Newton–Krylov methodology and a parallel direct sparse solver to precondition the underlying linear system. The proposed approach, implemented in a software infrastructure available under an open‐source BSD‐3 license, leads to improvements in overall simulation times of up to one order of magnitude when compared with matrix‐free parallel solution approaches that do not use preconditioning. Copyright © 2015 John Wiley & Sons, Ltd.     

High‐order finite volume schemes on unstructured grids using moving least‐squares reconstruction. Application to shallow water dynamics

This paper introduces the use of moving least‐squares (MLS) approximations for the development of high‐order finite volume discretizations on unstructured grids. The field variables and their successive derivatives can be accurately reconstructed using this mesh‐free technique in a general nodal arrangement. The methodology proposed is used in the construction of two numerical schemes for the shallow water equations on unstructured grids: a centred Lax–Wendroff method with added shock‐capturing dissipation, and a Godunov‐type upwind scheme, with linear and quadratic reconstructions. This class of mesh‐free techniques provides a robust and general approximation framework which represents an interesting alternative to the existing procedures, allowing, in addition, an accurate computation of the viscous fluxes. Copyright © 2005 John Wiley & Sons, Ltd.     

Coupled meshfree and fractal finite element method for mixed mode two‐dimensional crack problems

This paper presents a coupling technique for integrating the element‐free Galerkin method (EFGM) with the fractal finite element method (FFEM) for analyzing homogeneous, isotropic, and two‐dimensional linear‐elastic cracked structures subjected to mixed‐mode (modes I and II) loading conditions. FFEM is adopted for discretization of the domain close to the crack tip and EFGM is adopted in the rest of the domain. In the transition region interface elements are employed. The shape functions within interface elements which comprise both the EFG and the finite element (FE) shape functions, satisfies the consistency condition thus ensuring convergence of the proposed coupled EFGM–FFEM. The proposed method combines the best features of EFGM and FFEM, in the sense that no special enriched basis functions or no structured mesh with special FEs are necessary and no post‐processing (employing any path independent integrals) is needed to determine fracture parameters, such as stress‐intensity factors (SIFs) and T‐stress. The numerical results show that SIFs and T‐stress obtained using the proposed method are in excellent agreement with the reference solutions for the structural and crack geometries considered in the present study. Also, a parametric study is carried out to examine the effects of the integration order, the similarity ratio, the number of transformation terms, and the crack length to width ratio on the quality of the numerical solutions. A numerical example on mixed‐mode condition is presented to simulate crack propagation. As in the proposed coupled EFGM–FFEM at each increment during the crack propagation, the FFEM mesh (around the crack tip) is shifted as it is to the new updated position of the crack tip (such that FFEM mesh center coincides with the crack tip) and few meshless nodes are sprinkled in the location where the FFEM mesh was lying previously, crack‐propagation analysis can be dramatically simplified. Copyright © 2010 John Wiley & Sons, Ltd.     

Estimating the critical time‐step in explicit dynamics using the Lanczos method

The goal of our paper is to demonstrate the cost‐effective use of the Lanczos method for estimating the critical time step in an explicit, transient dynamics code. The Lanczos method can provide a significantly larger estimate for the critical time‐step than an element‐based method (the typical scheme). However, the Lanczos method represents a more expensive method for calculating a critical time‐step than element‐based methods. Our paper shows how the additional cost of the Lanczos method can be amortized over a number of time steps and lead to an overall decrease in run‐time for an explicit, transient dynamics code. We present an adaptive hybrid scheme that synthesizes the Lanczos‐based and element‐based estimates and allows us to run near the critical time‐step estimate provided by the Lanczos method. Copyright © 2006 John Wiley & Sons, Ltd.