Introduction

It is by now widely recognized that materials are inherently of multiscale, hierarchical character.^48,54 Material behavior should not be considered as monolithic properties that manifest only at phenomenological levels, as historically taught. Rather, important properties and material responses can arise at a myriad of length scales and the

phenomenological behavior of the continuum follows from their atomic and microscopic structures. The significance of multiscale analysis naturally follows this understanding and it brings the hope that new concepts and methods can be developed based on low scale structures, behaviors and physics. In fact, in the past 20 years one sees a wide scope of interests and intensive research activities in developing multiscale methods. To name a few, quasicontinuum (QC),⁵⁵ CADD^56-58 are examples for the direct coupling (DC) methods between atoms and FE nodes of continuum in the concurrent multiscale analysis. Specifically, QC uses the Cauchy-Born rule to transfer atomistic energy to the strain energy density of FE analysis and CADD, short for coupled atomistic discrete dislocation dynamics, combines atomic scale analysis with discrete dislocation analysis in a continuum to perform multiscale modeling. Here, approaches that relate atoms and finite element nodes in a one to one manner, or through a form of interpolation, will be referred to as DC methods. According to this definition, most of the existing multiscale methods belong to the DC methods. On the other hand GP^18,19,48 and ESCM³² belong to the non-DC methods category. The embedded statistical coupling method (ESCM) employs statistical averaging over selected time intervals and volume in atomistic subdomains at the MD/FE interface to determine nodal displacement for the continuum FE domain. The other non-DC method, called the generalized particle dynamics method (GP), is proposed to use constant material neighbor link cells (NLC) at the interface region to mutually transfer information from the bottom-scale up or from the top-scale down to quantitatively link variables at different scales. Readers interested in the details of classification, historic development, and applications of multiscale analysis are referred to “Multiscale Analysis of Deformation and Failure of Materials”.¹⁹

Looking back on the developments of multiscale analysis in the past two decades, two basic issues for extending engineering applications of concurrent multiscale simulations can be addressed. The first one is how to quantify the accuracy of atomistically-based multiscale simulation and the second one is how to enlarge the model dimension to the minimum size necessary to make the model realistic.

The first issue is obvious since experimental accuracy verification of the multiscale analysis at atomistic/nano scale is difficult even when using high-resolution experimental methods. The most popular approach so far is to compare the results of

multiscale analysis with a fully atomistic simulation method such as molecular dynamics (MD). Examples can be found for dislocation at notches⁴⁸ and dislocations passing through scale boundaries.⁵⁰ Among those efforts, there are two notable verification studies. Curtin and Miller³⁶ used a one-dimensional (1D) spring model to compare the scale transition regions of the various methods such as QC/CLS,⁵⁹ QC-GFC⁶⁰/FEAt⁶¹ and CADD with the fully atomistic model. Their results confirm that besides the QC-GFC and FEAt methods most multiscale models do not truly meet the requirement for a seamless transition at the interface between atomic and FE domains. This discontinuity is caused by the intrinsic incompatibility of constitutive behavior between material models being coupled together at the two sides of the scale interface which causes non-physical phenomena at the interface, including the so-called ghost forces.⁶⁰

Specifically, the behavior of the continuum on one side of the scale interface is local but that of atoms on the other side is non-local in nature. Here, non-local constitutive behavior indicates that the force at any atom depends not only on those atoms closest to it, but also on the atoms nearby, which are not direct neighbors with the atom at hand but within its neighborhood through interatomic forces; local behavior indicates that the force (stress) of a material point depends only on the deformation gradient (strain) at the same point of the continuum. Due to this local behavior, nodes in the continuum region cannot feel interactions from other nodes nearby as their atomic counterparts in the atomistic region. Based on the mechanism of this incompatibility, a dead ghost force correction method of QC, (i.e., QC-GFC) was developed⁶⁰ to use the ghost force as a dead force to recalculate the result. This correction method allows this approach to exhibit a seamless transition at the scale interface, as mentioned in the previous paragraph. It is therefore necessary to recalculate the ghost forces, in some cases, say, after each re-meshing, since the ghost forces may change.

The other verification performed is the benchmark computation of 14 models carried out by Miller and Tadmor.⁵³ The fully atomistic simulation is the benchmark against which one compares the multiscale models. Here, the corresponding atomistic solution is considered as the exact solution. The test used a common framework to examine the accuracy and efficiency of these methods using a test problem of single crystal aluminum containing a dipole of Lomer dislocations. While this test is significant,

the quantitative criterion for accuracy comparison used the global error which covers displacements of all atoms in the simulation system. The shortcomings of the method for accuracy verification are twofold: First, it is difficult to judge how accurately this method describes the local behavior of the material, say, at the interface which makes it difficult to guide improvement of the interface design. Second, it can also be hard to make a judgment for how accurately it describes the continuum behavior for engineering applications. In fact, the benchmark test indicated itself that slight error in global energy estimation can lead to profound effects on the resulting dislocation motion and, in turn, the continuum behavior.

The second issue related to a model size requirement needs some explanation even though it is a common problem that appears frequently during model design. This issue can be addressed from the following four aspects. Firstly, accuracy verification for low scales is important to find the deformation mechanisms, such as crack nucleation in fatigue, dislocation patterns in fatigue and creep, the inherent inhomogeneity of plastic deformation, the statistical nature of brittle failure and the effects of size, geometry and stress state on yield properties.⁶² To make this finding meaningful, however, one must link these low-scale dynamics to material behavior and be characterized in the continuum scale for applications. This requires a relatively large continuum model size. Otherwise the approach of predicting material behavior by implicitly averaging atomistic/microscopic dynamics may not be valid. Secondly, if the model size is small the boundary conditions (BC) may likely affect the local fields of forces and displacements which are near the regions of interest. In turn, it will change the behavior of highly important domains (e.g. interfaces, crack-tips and flaws). In most practical cases, BC cannot be perfectly maintained and will have oscillatory and random perturbations. Various BCs can be accepted if they are sufficiently far away from the interested regions following the concept of Saint Venant's principle.^63,51 Otherwise, the obtained low-scale phenomena observed can be qualitatively different which can cause the instability of atomic motions, microstructural evolutions and unexpected material failure. Thirdly, some mathematical solutions for the continuum require the medium to be infinitely or sufficiently large to obtain results close to the analytical solution such as the crack-tip solution of linear elastic fracture mechanics (LEFM). In this case, model size

must not be small for a reasonable result. The fourth aspect is that for microsystems and nanotechnology, the model size should be equal/larger than micrometers or at least being sub-micrometers so the problem of micro- or nano-sensors/activators can be more accurately simulated. For nanotechnology, this is true for some designs since nanotubes, nanofibers, etc. need to be assembled and embedded in a matrix which has a certain size requirement. Thus, investigating the model size effects and choosing a minimum model size necessary for the accuracy requirement is essential.

This work is our first effort to address the issue of model size effects on accuracy.

It is obvious that the premise for investigating this issue is to have new methodology for developing sufficiently large size models such that the effects can be investigated systematically by varying the model size. Thus, this paper focuses on the introduction of the new methodology within the framework of the GP and GP-FEA method, where the GP-FEA is a further development of the GP method. This new method can make the model size as large as needed. The use of this new methodology to investigate the model size effects will be exemplified in future works. To address the first issue, accuracy verification is conducted by a comparison of the GP-FEA simulation result with a classical solution of continuum theory. This classical problem is a two-dimensional elasticity solution of a specimen with a central hole under tensile load.⁶⁴ This solution from elasticity theory is accurate but using it to verify the accuracy of atomistically-based simulation faces new challenges. Firstly, the explicit expression needed to be modified so the direct comparison can be available. Secondly, the data process for the atomistic scale needs to be done correctly which should involve a group of atoms near a continuum point not a single atom in that position. The third challenge concerns how to estimate the error by the comparison. All these issues will be addressed through the example of a 2D hole specimen.

The Chapter is organized as follows: Section B will introduce the proposed GP- FEA method in detail. Section C will extend the elasticity solution to an explicit displacement expression around the hole of a 2-D plate specimen under tension and then make a comparison with the multiscale simulation. The paper is ended with summary and conclusions in Section D.