Implementation Aspects of the Bridging Scale Method and Application to Intersonic Crack Propagation

D.E. Farrell, H.S. Park and W.K. Liu
International Journal for Numerical Methods in Engineering 2007; 71:583-605.

Abstract

The major purpose of this work is to investigate the performance of the Bridging Scale Method (BSM), a multiscale simulation framework for the dynamic, concurrent coupling of atomistics to continua, in capturing shear-dominant failure. The shear-dominant failure process considered in this work is intersonic crack propagation along a weak plane in an elastic material, similar to the seminal molecular dynamics (MD) simulations by Abraham and Gao (Physical Review Letters 2000; 84(14):3113-3116). We show that the BSM simulations accurately capture the essential physics of the intersonic crack propagation, including the formation of a daughter crack and the sudden acceleration of the crack to a velocity exceeding the material shear wave speed. It is also demonstrated that the non-reflecting boundary condition can adequately dissipate the strongly localized wave formed by the Mach cone after the crack accelerates beyond the material shear wave speed. Finally, we provide the algorithm for our implementation of the BSM, as well as the code used to determine the damping kernels via a newly adopted technique which is less expensive than previous methods.

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A Phonon Heat Bath Approach for the Atomimstic and Multiscale Modeling of Solids

E.G. Karpov, H.S. Park and W.K. Liu
International Journal for Numerical Methods in Engineering 2007; 70:351-378.

Abstract

We present a novel approach to numerical modeling of the crystaline solid as a heat bath. The approach allows bringing together a small and large crystalline domain, and model accurately the resulting interface using harmonic assumptions for the larger domain, which is excluded from the explicit model and viewed only as a hypothetic heat bath. Such an interface is non-reflective for the elastic waves, as well as providing a thermostatting for the small domain. The small domain can be modeled with a standard molecular dynamics approach, and its interior may accommodate arbitrary nonlinearities. The formulation involves a normal decomposition for the random thermal motion term R(t) in the generalized Langevin equation for solid-solid interfaces. Heat bath temperature serves as a parameter for the distribution of the normal mode amplitudes found from the Gibbs canonical distribution for the phonon gas. Spectral properties of the normal modes (polarization vectors and normal frequencies) are derived from lattice stiffness properties. Approach results in a physically motivated random force term R(t) derived consistently to represent the correlated thermal motion of lattice atoms. We describe the method in detail, and demonstrate applications to one and two-dimensional lattice structures.

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Bridging Scale Methods for Nanomechanics and Materials

W.K. Liu, H.S. Park, D. Qian, E.G. Karpov, H. Kadowaki and G.J. Wagner
Computer Methods in Applied Mechanics and Engineering 2006; 195:1407-1421.
(Invited paper: Special Issue in Honor of the 60th Birthday of Prof. T.J.R. Hughes).

Abstract

Inspired by the pioneering work of Professor T.J.R. Hughes on the variational multiscale method, this document summarizes recent developments in multiple-scale modeling using a newly developed technique called the bridging scale. The bridging scale consists of a two-scale decomposition in which the coarse scale is simulated using continuum methods, while the fine scale is simulated using atomistic approaches. The bridging scale offers unique advantages in that the coarse and fine scales evolve on separate time scales, while the high frequency waves emitted from the fine scale are eliminated using lattice impedance techniques. Recent advances in extending the bridging scale to quantum mechanical/continuum coupling are briefly described. The method capabilities are demonstrated via quasistatic nanotube bending, dynamic crack propagation and dynamic shear banding.

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Non-reflecting Boundary Conditions for Atomistic, Continuum and Coupled Atomistic/Continuum Simulations

H.S. Park, E.G. Karpov and W.K. Liu
International Journal for Numerical Methods in Engineering 2005; 64:237-259.

Abstract

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 neighbor atomic interactions; it is also demonstrated that the identical procedure can be used for finite element and meshfree 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).

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Multiscale Boundary Conditions in Crystalline Solids: Theory and Application to Nanoindentation

E.G. Karpov, H. Yu, H.S. Park, W.K. Liu, J. Wang and D. Qian
International Journal of Solids and Structures 2006; 43:6359-6379

Abstract

This paper presents a systematic approach to treating interfaces between the localized (fine grain) and peripheral (coarse grain) domains in atomic scale simulations of crystalline solids. Based on Fourier analysis of regular lattices structures, this approach allows elimination of the unnecessary atomic degrees of freedom over the coarse grain, without involving an explicit continuum model for the latter. The mathematical formulation involves compact convolution operators that relate displacements of the interface atoms and the adjacent atoms on the coarse grain. These operators are defined by geometry of the lattice structure, and interatomic potentials. Application and performance are illustrated on quasistatic nanoindentation simulations with a crystalline gold substrate. Complete atomistic resolution on the coarse grain is alternatively employed to give the benchmark solutions. The results are found to match well for the multiscale and the full atomistic simulations.

This paper is available in PDF form .

Three-dimensional Bridging Scale Analysis of Dynamic Fracture

H.S. Park, E.G. Karpov, P.A. Klein and W.K. Liu
Journal of Computational Physics 2005; 207:588-609

Abstract

This paper presents a three-dimensional generalization of the bridging scale concurrent method, a finite temperature multiple scale method that couples molecular dynamics (MD) to finite elements (FE). The generalizations include the numerical calculation of the boundary condition acting upon the reduced MD region, as such boundary conditions are analytically intractable for realistic three-dimensional crystal structures. The formulation retains key advantages emphasized in previous papers, particularly the compact size of the resulting time history kernel matrix. The coupled FE and reduced MD equations of motion are used to analyze dynamic fracture in a three-dimensional FCC lattice, where interesting physical phenomena such as crack branching are seen. The multiple scale results are further compared to benchmark MD simulations for verification purposes.

This paper is available in PDF form .

The Bridging Scale for Two-dimensional Atomistic/Continuum Coupling

H.S. Park, E.G. Karpov, W.K. Liu and P.A. Klein
Philosophical Magazine 2005; 85 (1):79-113

Abstract

In this paper, we present all necessary generalizations to extend the bridging scale, a finite temperature multiple scale method which couples molecular dynamics (MD) and finite element (FE) simulations, to two dimensions. The crucial development is a numerical treatment of the boundary condition acting upon the reduced atomistic system, as such boundary conditions are analytically intractable beyond simple one-dimension systems. The approach presented in this paper offers distinct advantages compared to previous works, specifically the compact size of the resulting time history kernel, and the fact that the time history kernel can be calculated using an automated numerical procedure for arbitrary multi-dimensional lattice structures and interatomic potentials. We demonstrate the truly two-way nature of the coupled FE and reduced MD equations of motion via two example problems, wave propagation and dynamic crack propagation. Finally, we compare both problems to benchmark full MD simulations to validate the accuracy and efficiency of the proposed method.

This paper is available in PDF form .

A Temperature Equation for Coupled Atomistic/Continuum Simulations

H.S. Park, E.G. Karpov and W.K. Liu
Computer Methods in Applied Mechanics and Engineering 2004; 193:1713-1732.
(Invited paper: Special Issue on Multiple Scale Methods for Nanoscale Mechanics and Materials).

Abstract

We present a simple method for calculating a continuum temperature field directly from a molecular dynamics (MD) simulation. Using the idea of a projection matrix previously developed for use in the bridging scale, we derive a continuum temperature equation which only requires information that is readily available from MD simulations, namely the MD velocity, atomic masses and Boltzmann constant. As a result, the equation is valid for usage in any coupled finite element (FE)/MD simulation. In order to solve the temperature equation in the continuum where an MD solution is generally unavailable, a method is utilized in which the MD velocities are found at arbitrary coarse scale points by means of an evolution function. The evolution function is derived in closed form for a 1D lattice, and effectively describes the temporal and spatial evolution of the atomic lattice dynamics. It provides an accurate atomistic description of the kinetic energy dissipation in simulations, and its behavior depends solely on the atomic lattice geometry and the form of the MD potential. After validating the accuracy of the evolution function to calculate the MD variables in the coarse scale, two 1D examples are shown, and the temperature equation is shown to give good agreement to MD simulations.

This paper is available in PDF form .

An Introduction and Tutorial on Multiple Scale Analysis in Solids

H.S. Park and W.K. Liu
Computer Methods in Applied Mechanics and Engineering 2004; 193:1733-1772.
(Invited paper: Special Issue on Multiple Scale Methods for Nanoscale Mechanics and Materials).

Abstract

Concurrent multiple scale methods can be defined as those which combine information available from distinct length and time scales into a single coherent, coupled simulation. These methods have recently become both popular and necessary for the following reasons. One is the recent discovery of new, nanoscale materials, and the corresponding boom in nanotechnology research. Another factor is that experiments have conclusively shown the connection between microscale physics and macroscale deformation. Finally, the concept of linking disparate length and time scales has become feasible recently due to the ongoing explosion in computational power. We present a detailed introduction to the available technologies in the field of multiple scale analysis. In particular, our review centers on methods which aim to couple molecular level simulations (such as molecular dynamics) to continuum level simulations (such as finite element and meshfree methods). Using this definition, we first review existing multiple scale technology, and explain the pertinent issues in creating an efficient yet accurate multiple scale method. Following the review, we highlight a new multiple scale method, the bridging scale, and compare it to existing multiple scale methods. Next, we show example problems in which the bridging scale is applied to fully nonlinear problems. Concluding remarks address the research needs for multiple scale methods in general, the bridging scale method in particular, and potential applications for the bridging scale.

This paper is available in PDF form .

An Introduction to Computational Nanomechanics and Materials

W.K. Liu, E.G. Karpov, S. Zhang and H.S. Park
Computer Methods in Applied Mechanics and Engineering 2004; 193:1529-1578.
(Invited paper: Special Issue on Multiple Scale Methods for Nanoscale Mechanics and Materials).

Abstract

Many arenas of research are rapidly advancing due to a combined effort between engineering and science. In some cases, fields of research that were stagnating under the exclusive domain of one discipline have been imbued with new discoveries through collaboration with practitioners from the second discipline. In computational mechanics, we are particularly concerned about the technological engineering interest by combining engineering technology and basic sciences through modeling and simulations. These goals have become particularly relevant due to the emergence of the field of nanotechnology, and the related burst of interest in nanoscale research. In this introductory article, we first briefly review the essential tools used by nanoscale researchers. These simulation methods include the broad areas of quantum mechanics, molecular dynamics and multiple-scale approaches, based on coupling the atomistic and continuum models. Upon completing this review, we shall conclusively demonstrate that the atomistic simulation tools themselves are not sufficient for many of the interesting and fundamental problems that arise in computational mechanics, and that these deficiencies lead to the thrust of multiple-scale methods. We summarize the strengths and limitations of currently available multiple-scale techniques, where the emphasis is made on the latest perspective approaches, such as the bridging scale method, multi-scale boundary conditions, and multi-scale fluidics. Example problems, in which multiple-scale simulation methods yield equivalent results to full atomistic simulations at fractions of the computational cost, are shown. We conclude by discussing future research directions and needs in multiple-scale analysis, and also discuss the ramifications of the integration of current nanoscale research into education.

This paper is available in PDF form .