Bridging Scale Method Research Articles

A damping boundary condition for atomistic-continuum coupling

The minimization of spurious wave reflection is a challenge in multiscale coupling due to the difference of spatial resolution between atomistic and continuum regions. In this study, a new damping condition is presented for eliminating spurious wave reflection at the interface between atomistic and continuum regions. This damping method starts by a coarse–fine decomposition of the atomic velocity based on the bridging scale method. The fine scale velocity of the atoms in the damping region is reduced by applying nonlinear damping coefficients. The effectiveness of this damping method is verified by one- and two- dimensional simulations.

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

ABSTRACTIn this paper, we present a regression hybrid method that calculates shape sensitivity coefficients 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 coefficients 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 coefficients 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 efficiency of the proposed method.

Multiscale hydro-mechanical analysis of unsaturated granular materials using bridging scale method

Purpose – The purpose of this paper is to extend the bridge scale method (BSM) developed for granular materials with only the solid phase to that taking into account the effects of wetting process in porous continuum. The granular material is modeled as partially saturated porous Cosserat continuum and discrete particle assembly in the coarse and fine scales, respectively. Design/methodology/approach – Based on the mass and momentum conservation laws for the three phases, i.e. the solid skeleton, the pore water and the pore air, the governing equations for the unsaturated porous Biot-Cosserat continuum model in the coarse scale are derived. In light of the passive air pressure assumption, a reduced finite element model for the model is proposed. According to the decoupling of the fine and coarse scale calculations in the BSM, the unsaturated porous Cosserat continuum model using the finite element method and the discrete element model using the discrete element method for granular media are combined. Find...

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.

Continuum-based sensitivity analysis for coupled atomistic and continuum simulations for 2-D applications using bridging scale decomposition

The concept of linking simulations at multiple length and time scales is found useful for studying local physical phenomena such as crack propagation. Many multi-scale methods, which couple molecular dynamics models with continuum models, have been proposed over the last decade. One of the most advanced methods developed recently is the bridging scale method, in which the total displacement is decomposed into orthogonal coarse and fine scales. This paper presents the continuum-based sensitivity analysis for two-dimensional coupled atomistic and continuum problems using the bridging scale method. A variational formulation for the bridging scale decomposition is developed based on the Hamilton's principle. The continuum-based variational formulation provides a uniform and generalized system of equations from which the differential equations can be obtained naturally. The sensitivity expressions for both direct differentiation method (DDM) and adjoint variable method (AVM) are derived in a continuum setting. Due to its efficiency for crack problems, the direct differentiation method is chosen to be implemented numerically and applied to two examples, including a crack propagation problem. Both material and sizing design variables are included to reveal the impact of design changes at the macroscopic level to the responses at the atomistic level. Also demonstrated is the feasibility of achieving the variation of the time history kernel computed using numerical procedures. The sensitivity coefficients calculated are shown to be accurate compared with overall finite difference method. The physical implications of the sensitivity results are also discussed, which accurately predict the behavior of the structural responses.

A bridging scale method for granular materials with discrete particle assembly – Cosserat continuum modeling

Based on the bridging scale method (BSM) initially presented in nano mechanics [1,2], a new version of the BSM that couples discrete particle assembly modeling using the discrete element method (DEM) and Cosserat continuum modeling using the finite element method (FEM) at both micro–macro scale levels respectively is proposed for multiscale analysis of granular materials. The DEM is only applied to the limited localized regions for accurately simulating discontinuous failure phenomena in microscopic scale, meantime, the FEM that costs much less computational time and storage space covers the whole domain, with permitting to adopt different time step sizes for the time integration schemes in coarse and fine scales respectively. As a consequence, both computational accuracy and efficiency of the proposed BSM are greatly enhanced. With a bridging scale displacement (including translations and rotations) decomposition and based on the virtual work principle applied to the FEM nodes of the Cosserat continuum and the particle centers of the discrete particle assembly respectively, two decoupling sets of equations of motion of the combined coarse–fine scale system are formulated. The interfacial condition between the coarse and fine scale regions in the quasi-static and dynamic loading cases are presented and discussed. The non-reflecting boundary condition and its implementation, which is capable of effectively eliminating spurious reflected waves at the interfaces between the coarse and fine scale regions, are described in detail. The numerical results for 2D example problems demonstrate the applicability and advantages of the present BSM, and the performances of the non-reflecting boundary condition implemented to simulate dynamical responses in geo-structures composed of granular materials.

Multiscale methods for mechanical science of complex materials: Bridging from quantum to stochastic multiresolution continuum

AbstractRecent development in the multiscale method based on the bridging scale concept is presented with an emphasis on complex material systems. The bridging scale method (BSM) was originally proposed by Wagner and Liu (J. Comput. Phys. 2003; 190:249–274) as an effective way of treating the interface in coupled atomistic/continuum simulation. Since its publication, the BSM has become a very useful paradigm that has been applied to solve a host of problems in mechanical sciences of complex material systems. In this paper, we present a review on the recent developments. We first describe the application of BSM to the coupled atomistic/continuum simulation of dynamic fracture. The important extensions within the framework of space–time method and multiscale non‐equilibrium molecular dynamics are then presented. We then focus on the multiresolution continuum theory that inherits the BSM concept in the analysis of heterogeneous material structures. Recent work of incorporating statistical factors into this model based on the concurrent nested homogenization of randomness of the material structures is highlighted. Finally, we present the use of the bridging scale concept in resolving the electron‐mechanical coupling mechanism. The robustness of the BSM is demonstrated through many benchmark problems and application examples. Copyright © 2010 John Wiley & Sons, Ltd.

MULTISCALE SIMULATION OF MICROCRACK BASED ON A NEW ADAPTIVE FINITE ELEMENT METHOD

In this paper, a new adaptive finite element (FE) framework based on the variational multiscale method is proposed and applied to simulate the dynamic behaviors of metal under loadings. First, the extended bridging scale method is used to couple molecular dynamics and FE. Then, macro damages evolvements of those micro defects are simulated by the adaptive FE method. Some auxiliary strategies, such as the conservative mesh remapping, failure mechanism and mesh splitting technique are also included in the adaptive FE computation. Efficiency of our method is validated by numerical experiments.

A mesoscopic bridging scale method for fluids and coupling dissipative particle dynamics with continuum finite element method

A multiscale procedure to couple a mesoscale discrete particle model and a macroscale continuum model of incompressible fluid flow is proposed in this study. We call this procedure the mesoscopic bridging scale (MBS) method since it is developed on the basis of the bridging scale method for coupling molecular dynamics and finite element models [G.J. Wagner, W.K. Liu, Coupling of atomistic and continuum simulations using a bridging scale decomposition, J. Comput. Phys. 190 (2003) 249–274]. We derive the governing equations of the MBS method and show that the differential equations of motion of the mesoscale discrete particle model and finite element (FE) model are only coupled through the force terms. Based on this coupling, we express the finite element equations which rely on the Navier–Stokes and continuity equations, in a way that the internal nodal FE forces are evaluated using viscous stresses from the mesoscale model. The dissipative particle dynamics (DPD) method for the discrete particle mesoscale model is employed. The entire fluid domain is divided into a local domain and a global domain. Fluid flow in the local domain is modeled with both DPD and FE method, while fluid flow in the global domain is modeled by the FE method only.The MBS method is suitable for modeling complex (colloidal) fluid flows, where continuum methods are sufficiently accurate only in the large fluid domain, while small, local regions of particular interest require detailed modeling by mesoscopic discrete particles. Solved examples – simple Poiseuille and driven cavity flows illustrate the applicability of the proposed MBS method.

Algorithms for bridging scale method parameters

Multiple scale simulation methods have been the focus of much attention in recent years. The Bridging Scale method (BSM) of Liu and co-workers was developed as a framework for the dynamic concurrent coupling of atomistics and continua. A key feature of the BSM is a non-reflecting atomistic interfacial condition which relies on a time history integral. In this work, we present a practical review of the relevant theory as well as a new algorithm used in the determination of the kernel function of the time history integral. We also provide a detailed description of the random displacement terms from the interfacial condition, which are required for finite temperature BSM simulations.

Implementation aspects of the bridging scale method and application to intersonic crack propagation

AbstractThe 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 (Phys. Rev. Lett. 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. Copyright © 2007 John Wiley & Sons, Ltd.

A mathematical framework of the bridging scale method

AbstractIn this paper, we present a mathematical framework of the bridging scale method (BSM), recently proposed by Liu et al. Under certain conditions, it had been designed for accurately and efficiently simulating complex dynamics with different spatial scales. From a clear and consistent derivation, we identify two error sources in this method. First, we use a linear finite element interpolation, and derive the coarse grid equations directly from Newton's second law. Numerical error in this length scale exists mainly due to inadequate approximation for the effects of the fine scale fluctuations. An modified linear element (MLE) scheme is developed to improve the accuracy. Secondly, we derive an exact multiscale interfacial condition to treat the interfaces between the molecular dynamics region ΩD and the complementary domain ΩC, using a time history kernel technique. The interfacial condition proposed in the original BSM may be regarded as a leading order approximation to the exact one (with respect to the coarsening ratio). This approximation is responsible for minor reflections across the interfaces, with a dependency on the choice of ΩD. We further illustrate the framework and analysis with linear and non‐linear lattices in one‐dimensional space. Copyright © 2005 John Wiley & Sons, Ltd.

Perfectly matched multiscale simulations

A multiscale method is proposed. It combines the so-called bridging scale method and the perfectly matched layer method to form a robust and versatile multiscale algorithm. The method can efficiently eliminate the spurious reflections/diffractions from the artificial atomistic/continuum interface by matching the impedance at the interface of the molecular dynamic region and the perfectly matched layer. Moreover, it is shown in this paper that the method can capture anharmonic interaction among nonuniformly distributed atoms in a local region.

An introduction and tutorial on multiple-scale analysis in solids

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 non-linear 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.

An introduction to computational nanomechanics and materials

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.