It is shown in this Chapter that the auto-duality feature of the GP method can be used to help mitigate the unrealistic wave propagation dynamics caused by higher scale GP representations. This illustrates the need for atoms in locations where the deformation gradient is large and that higher scale particles may be used in areas of small deformations. There are two dynamic problems that occur within GP scales higher than the atomistic scale.
GP Dynamic Problem 1. The wave speed in higher scales is proportional to the scale ratio. This is due to the farther reaching influence of higher scales for the same
reason why higher scales naturally have a proportionally greater surface effect. It causes the gradient of the strain field to be smaller than it would be for a pure atomistic representation while keeping the acceleration consistent. This mismatch between acceleration and strain gradient causes a perceived increase in the wave speed; it changes the wave equation.
In order to address this problem a more accurate way to calculate the strain gradient must be made. The current inverse mapping method is a very fast and convenient method that is appropriate for quasi-static problems due to its assumption based on the Cauchy Born rule. However, when the strain gradient is non-zero it causes a deviation in the wave equation. If there was a way to interpolate the local/atomistic strain gradient at a given particle location then this problem could be mitigated. Put another way, the non-locality of GP high scale domains increases proportionally with the scale ratio. To maintain the atomistic strain gradient, the same degree of non-locality as on the atomistic scale is needed.
GP Dynamic Problem 2. The temperature of high scales is not well defined;
using only the inverse mapping method to calculate temperature does not make sense due to the multiscale nature of velocity distributions that compose thermal energy. Since a particle's velocity is the average of the atoms' that it is composed of, there is no guarantee that the kinetic/thermal energy of those atoms is consistent with the particle that represents them.
From this perspective the temperature of a high scale domain could be represented by a sum of two terms, the particles' kinetic temperature plus the internal kinetic energy of the implicit atoms the particles represent. The larger the particle scale the greater the internal thermal contribution of the implicit atoms (lost DOF) to the domain temperature.
The heat from the lost DOF due to lumping may be calculated in the same way as is used in the MPM multiscale method.104
This brief dynamical study of the GP method for use in dynamic applications is instructive. It clearly illustrates the needs still wanting in the GP method and suggests certain possible solutions to these tough problems. Current work in this field will help to guide the development of the GP method into a more advanced future incarnation.
PARTICLE-BASED MULTISCALE ANALYSIS PROGRAM (PMAP) STRUCTURE
A. Introduction
Many theoretical models seek to investigate the effects of material defects such as pores, dislocations, microcracks, etc. on the actual applications of processing, loading and service conditions. These applications tend to be continuum level fields such as strain, fatigue, electric field, temperature, pressure, etc. To have these material defects evolve naturally in these fields there must be sufficient freedom such that boundary and model size effects do not modify the natural defect evolution. This multiscale capability is the main goal of the GP and GP-FEA methods that differentiate it from traditional Molecular Dynamics (MD). The features of the GP and GP-FEA methods allow them to contend for this goal opening the gateway to pursue other applications such as materials strengthening and toughening by tailoring the design of the nano and microstructure or as a tool for designing nanotechnological devices and microsystems for different functions.
In order to model these multiscale characteristics of materials and to realize multiscale modeling theories, certain numerical methods must be utilized. Computer programs effectuate these methods; quantitatively tracking the motions of atoms and particles as they move through space and time. Most simulation programs are designed for one type of length and time scale, such as molecular dynamic and coarse-graining programs like LAMMPS105, DLPOLY106,107, GROMACS108, NAMD109, the discrete dislocation dynamics method which can be simulated by software such as, ParaDis110, microMegas111, TRIDIS112 and finite element analysis software like ABAQUS79, and ANSYS113. There are coarse graining methods that can reduce an atomic structure into representative particles or beads, MARTINI23 is an example and can be run within MD software like NAMD109 and GROMACS108. They are able to handle two different model scales hierarchically, the atomistic and one level up, a coarse-grained representation within a single program framework. These are mainly designed for soft materials in biological applications their accuracy is based on the atomistic structure thus they are a
Class-I multiscale technique. However it is advantageous for all hierarchical scale simulation methods to be able to run within a single framework where one scale's information can be used in higher scale simulations and results can be directly compared in a simple and easy way.
This is the philosophy that caused the development of the OCTA114 and VOTCA115 the Versatile Object-oriented Toolkit. The former is an integrated simulation system which utilizes four different meso-scale simulation engines: COGNAC116 (COarse Grained molecular dynamics program by NAgoya Cooperation), PASTA117 (Polymer rheology Analyzer with Slip-link model of enTAnglement), SUSHI118 (Simulation Utilities for Soft and Hard Interfaces), and MUFFIN119 (MultiFarious FIeld simulator for Non-equilibrium system).
Concurrent multiscale frameworks also exist and commonly come in a single framework. Perhaps the most well-known is the Quasi-Continuum Method (QC).120 Their code is available publicly and is able to set up the material lattice, grains and the FE mesh, eliminate the ghost force and run the simulation without the need of any third party software. It is currently limited to 2D, crystals with BCC or FCC and the original code does not allow for finite temperature calculation although there has been work to extend it,121-123 QC is not MD but can do Molecular Mechanics (MM) calculations i.e. energy minimization. There is also LibMultiScale124 which is a parallel framework for coupled multiscale methods. This framework provides an API which makes it possible to program coupled simulations of pre-existing codes. MD codes such as Stamp from CEA and LAMMPS105 have been coupled to a unique FEM code libMesh125. It currently is based on the Bridging Domain Method (BDM) of T. Belytschko and S. Xiao.100 Which uses a Lagrange multiplier method or augmented Lagrangian method for enforcing the kinematic constraints in the overlapping subdomain where the total Hamiltonian is a linear combination of the molecular and continuum Hamiltonians which can handle the spurious wave reflection problem in the overlapping domain.102,103 They have also developed an explicit algorithm and a multi-time step method for BDM. Most simulation codes used in literature are in-house private codes that are designed for specific applications or to prove a concept. Never the less these private codes are extremely important for the advancement of multiscale simulation techniques; one cannot advance
computational simulation methods without writing source code for the computers to execute. Without computer code brilliant ideas will flounder and die; an intimate knowledge of the physical mechanisms, equations, and algorithms are essential for correct simulations.
The program used in this work is a particle-based multiscale analysis program (PMAP) with the capacity of simulating molecular dynamics with numerous interfaces coupling higher-scale particle dynamics concurrently to the atomistic domain via the Generalized Particle Dynamics Method (GP). The model size may be further extended by coupling the high-scale particle domains to finite element meshes. These coupled scales are designed to provide the atomistic domain with realistic boundary conditions so that real world applications may be investigated. This illustrates PMAP as a complete framework for concurrent multiscale analysis, including MD, various particle scales and ultimately an FEM continuum. In this chapter the structure of PMAP will be discussed first as a brief overview, then in more detail about its three parts, initialization, equilibration, and loading. Explaining the structure and process flow of this multiscale framework is important for those who wish to develop their own multiscale simulation code; it is best to begin with the general idea of how the code works before delving into the details of particular subroutines and functions. These details are discussed in the Appendix for instructive purposes.