Multiscale modeling

7.5.1 Multiscale modeling

A wide range of techniques have been developed for doing CMS (Table 7.3), including (i) first principles or ab initio methods, (ii) molecular statics and dynamics (MS and MD), (iii) Monte Carlo (MC) methods, (iv) phase-field kinetic methods, (v) dislocation dynamics (DD), in addition to the more traditional finite element method (FEM) for continuum simulations. These methods fall into

a ‘multiscale’ perspective, meaning that each method best handles problems within a certain length scale, from nano to micro (first principles, MS, MD), then micro to meso (DD, phase field), and meso to macro (FEA), with the anticipation that the outcome of one scale is fed into the next one up as input, so that one can hope to calculate macroscopic behavior starting from quantum mechanics. Thus,

ab initio methods are for solving the Schrödinger equation directly, and because of the computational efforts involved, are applicable to nano-sized clusters of atoms of the order of 10 3 . MS and MD rely on established ‘interatomic’ potentials as input to compute, in the case of MD, the trajectories of the atoms by integrating Newton’s second law or, in the case of MS, the equilibrium positions of the atoms at zero temperature. MC is a class of methods for computing integrals by random number generation, and is particularly useful in computing thermodynamic functions which are ensemble averages in integral forms. DD is for calculating group dislocation behavior and works by treating the dislocations as ‘elastic’ curves which interact with one another through some simplified laws. Phase- field methods are based on kinetic laws such as the Langevin equation and are used to predict phase evolution, often taking into account fluctuations. Finally, FEM is for solving continuum problems governed by known constitutive laws, by dividing the domain in question into ‘finite elements’, so that the problem is converted into one of inverting algebraic matrices, which can be effectively handled by high-speed computers. Thus, in moving up each scale along the multiscale route described above, one loses rigor and accuracy, in exchange for the capability to handle larger systems.

In reality, although tremendous progress concerning the ‘multiscale’ approach has been made, parallel to the rapid growth of the power of computers during the past few decades, successful examples of accurate prediction of macroscopic, engineering behavior of materials remain extremely rare to date. The problem is twofold. First, while the capabilities at both the nano/micro and the macro ends are better established, those at the meso scale are still not accurate or robust enough.

Thus, while FEM procedures for macroplasticity are well established, and the same is true for

ab initio and MS/MD methods for accurate computation of the stability of phases, properties of clusters of atoms, monolithic defects such as interfaces, dislocations and so on, it remains very diffi- cult to link up these two ends, especially for real materials with usually complicated microstructures.

The corresponding methods such as DD or phase-field models are usually rather ad hoc, with a sig- nificant amount of unknowns hidden in the actual implementation. Secondly, while the spatial scales can be bridged as described above, problems with long temporal scales are much more difficult to

442 Physical Metallurgy and Advanced Materials Table 7.3 Multiscale approach of CMS.

Length scale

Size/complexity involved

Ab initio, molecular statics (MS), molecular dynamics (MD), Monte Carlo (MC)

Semi-continuum, bridging

Dislocation dynamics (DD), phase-field

kinetic models, etc. Macro

micro and macro scales

Continuum

Finite element method (FEM), etc.

(b) Figure 7.41 MD simulations involving millions of atoms. (a) Nanoindentation in aluminum thin

(a)

film (by J. Li). (b) Crack propagation in an fcc crystal (by F. Abraham). To reveal the generated dislocations, only atoms with high energies are shown.

simulate. Current supercomputers can at best handle MD problems with a real time scale of only up to ∼1 ms, and this drops rapidly to the nanosecond regime as the size of the problem increases beyond

∼10 6 atoms. Figure 7.41 shows two examples of large-scale MD simulations involving ∼10 6 atoms, concerning the evolution of dislocation structures during nanoindentation (Figure 7.41a) and crack propagation (Figure 7.41b) in fcc prototypic crystals. Although impressive to look at, both examples

involve time scales in the nanosecond regime and so their relevance to real experimental conditions is unclear. Many important material integrity problems such as creep, corrosion, time-dependent fracture, etc. involve time scales of months and years and, at present, they are simply out of the reach of the multiscale CMS approach.