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Modeling and simulation across different length and time scales enables atomic-level understanding of materials properties and behavior which manifest at the meso- and macro-scales. Case studies of theoretical strength of crystals and defect nucleation and mobility illustrate the recent progress on studying unit processes. A future challenge lies in probing complex functional behavior of real materials.

In this chapter we review recent advances in numerical simulation of micro and nanoflows. For coarse-grained simulation of microfuidics, we present an overview of Lattice Boltzmann, Brownian dynamics, stochastic rotation dynamics, and smoothed particle hydrodynamics methods and discuss the dissipative particle dynamics method in detail as it shares many features with the other methods. In the area of nanoflows, we review recent advances in non-equilibrium molecular dynamics methods focusing on the development of self-consistent and grand canonical methods for electric-field mediated transport. We present examples showing the significance of quantum effects in nanoflows. Finally, we discuss multiscale modeling focusing on direct coupling of molecular dynamics with Navier-Stokes equations and hierarchical coupling of quantum, molecular dynamics and classical fluid equations.

This chapter reveals that sampling plant diversity at multiple spatial scales may allow for a deeper understanding of relationships between species distributions and composition shifts relative to changing environmental gradients. Like single-scale techniques, different multi-scale techniques have various strengths and weaknesses, and some may be better suited than others for particular study goals and objectives. No single design will be the “end all, be all” for plant diversity sampling. All designs have advantages and disadvantages that must be carefully evaluated. An experimental approach is almost always warranted. Several examples provide a glimpse of the profound potential of multi-scale sampling for plant diversity.

Multiphysics, multiscale models present significant challenges in terms of computing accurate solutions and for estimating the error in information computed from numerical solutions. In this chapter, we discuss error estimation for a widely used numerical approach for multiphysics, multiscale problems called multiscale operator decomposition. In this approach, a multiphysics model is decomposed into components involving simpler physics over a relatively limited range of scales, and the solution is sought through an iterative procedure involving numerical solutions of the individual components. After describing the ingredients of adjoint-based a posteriori analysis, we describe the extension to multiscale operator decomposition solution methods. While the particulars of the analysis vary considerably with the problem, there are several key ideas underlying a general approach to treat operator decomposition multiscale methods.

Building on a general abstraction of the steps and transformations of a multiscale analysis, this chapter considers an approach to the development of multiscale simulation in which interoperable components can be effectively combined to address a wide range of multiscale simulations. Key concerns in the development of these interoperable components are maximizing the ability to use existing single simulation tools and supporting adaptive simulation control methods. In addition to indicating specific tools that have been developed to support multiscale simulations, an example adaptive atomistic/continuum simulation procedure is demonstrated.

We study mixed multiscale finite element methods (MsFEM) on unstructured coarse grids. Unstructured grids are often used when highly heterogeneous reservoirs are discretized via irregular anisotropic fine grids. Our study is motivated by the development of coarse-scale models for coupled flow and transport equations in a multi-phase system. An unstructured coarse grid is often used to upscale the transport equation with hyperbolic nature in a highly heterogeneous reservoir. Solving the flow equation on the same coarse grid provides a general robust coarse-scale model for the multi-phase flow and transport at a low CPU cost. We present numerical results when both the flow and transport equations are solved on the coarse grid. Numerical examples involve highly channelized permeability as well as a 3-D reservoir model using an unstructured fine grid. In all examples, we show that our approach can provide an accurate approximation of the resolved solution at a much lower cost. We also study the convergence of the mixed multiscale finite element method on unstructured coarse grids.

A broad range of natural and man-made materials, such as the trabecular bone, aerogels have hierarchical microstructure. Performing efficient design of structures made from such materials requires the ability to integrate the governing equations of the respective physics on supports with complex geometry. The traditional approach is to devise constitutive equations which are either calibrated based on experiments or on micromechanics considerations. However, traditional homogenization cannot be used in most of these cases in which scale decoupling does not exist and the structure geometry lacks translational symmetry. Several efforts have been made recently to develop new formulations of mechanics that include information about the geometry in the governing equations. This new concept is based on the idea that the geometric complexity of the domain can be incorporated in the governing equations, rather than in the definition of the boundary conditions, as usual in classical continuum mechanics. In this chapter we review the progress made to date in this direction.

A model reduction approach aimed at systematically reducing computational complexity of multiple scale homogenization for nonlinear history-dependent problems is developed. The so-called N-scale model reduction (NMR) approach is based on reformulating a sequence of nonlinear unit cell problems at multiple scales in terms of inelastic deformation modes that a priori satisfy equilibrium equations at multiple scales and thus eliminating the need for costly solution of discretized nonlinear equilibrium. The N-scale model reduction approach is validated on tube crush tests.

This chapter presents a review of concurrent multiscale methods for coupling continuum, molecular, and quantum mechanics with a particular emphasis on the Bridging Domain Method. An expanded taxonomy of multiscale methods is presented which highlights the role and characteristics of concurrent multiscale methods. Several popular 0 K methods are summarized: master-slave coupling, ONIOM, the Bridging Domain Method, the Bridging Scale Method and the Quasicontinuum method. The topic of ghost forces in the Bridging Domain Method is studied and the stability properties of Lagrange multiplier coupling methods are reviewed. Several methods for coupling molecular dynamics with continua are described: master-slave and handshake methods, the Bridging Domain Method, and the Bridging Scale Method. The conservation properties of the Bridging Domain Method are demonstrated. Numerical examples are presented which focus on the reduction of spurious phonon reflections from the continuum/molecular interfaces in the Bridging Domain Method. Examples of the simulation of cracks and defects in graphene using a modified ONIOM method to couple continuum, molecular, and quantum subdomain are also given.

Coarse-grained molecular dynamics (CGMD) is a computer modeling technique that couples conventional molecular dynamics (MD) in some spatial regions of the simulation to a more coarse-grained description in others. This concurrent multiscale modeling approach allows a more efficient use of computer power as it focuses only on those degrees of freedom that are physically relevant. In the spirit of finite element modeling (FEM), the coarse-grained regions are modeled on a mesh with variable mesh size. CGMD is derived solely from the MD model, however, and has no continuum parameters. As a result, it provides a coupling that is smooth and provides control of errors that arise at the coupling between the atomistic and coarse-grained regions. In this chapter, we review the formulation of CGMD, describing how coarse graining, the systematic removal of irrelevant degrees of freedom, is accomplished for a finite temperature system. We then describe practical implementation of CGMD for large-scale simulations and some tests of validity. We conclude with an outlook on some of the directions future development may take.

We review some recent developments in the coupling of atomistic and continuum models based on the blending of the two models in an overlap, or bridge, subdomain. These coupling schemes resemble overlapping domain decomposition methods. However, their analysis and development is complicated by the non-local force model employed by the atomistic model. We present an abstract framework for atomistic-to-continuum (AtC) coupling methods and formulate precise mathematical notions of patch and consistency tests for the methods. The framework admits both force-based or energy-based coupling methods and allows us to identify four general classes of blending methods. We subject each class to patch and consistency tests and discuss important implementation issues such as: the enforcement of displacement continuity constraints in the bridge region; internal vs. external blending; the role of ghost forces, or forces that arise from coupling nonlocal and local models of force, and how they can be mitigated by blending methods.

Building on the complementary advantages of Renormalization Group (RG) and multigrid (MG) methods, Systematic Upscaling (SU) comprises rigorous procedures for deriving suitable variables and corresponding numerical equations (or statistical relations) that describe a given physical system at progressively larger scales, starting at some fine scale where the physical laws are known (in the form of a partial differential system, or a statistical-mechanics Hamiltonian, or Newton laws for moving particles, etc.). Unlike RG, the SU algorithms include repeated coarse-to-fine transitions, which are essential for (1) testing the adequacy of the set of coarse-level variables (thus providing a general tool for constructing that set); (2) accelerating the finer-level simulations; and, most importantly (3) confining those simulations to small representative subdomains. No substantial scale separation is assumed; as in MG, small scale ratio between successive levels is in fact important to ensure slowdown-free simulations at all scales. Detailed examples are given in terms of local-interaction systems at equilibrium, and extensions are briefly discussed to long-range interactions, dynamic systems, low temperatures, and more.

The chapter deals with multiple scales in both space and time. First, the state-of-the-art is presented. Then, we discuss a family of computational approaches using time-space homogenization. Emphasis is put on the time aspects.

Computational modelling of materials behaviour is becoming a reliable tool of scientific investigation, complementary to traditional theory and experimentation. The Multiscale Materials Modelling (MMM) approach reflects the realization that continuum and atomistic analysis methods are complementary. This chapter describes the most popular numerical techniques of each component that make up the MMM paradigm for modelling nano- and micro-systems: Quantum Mechanics (QM), Molecular Dynamics (MD), Monte Carlo (MC), Dislocation Dynamics (DD), Statistical Mechanics (SM), and Continuum Mechanics (CM).

Spatial structure is the result of locational decisions reflecting criteria operating at a variety of scales. Thus the CA land use model is augmented with a regionalized, spatial interaction based dynamic model of population and employment that captures spatial processes operating at intermediate scales. The model is structurally very similar to typical transportation models, but with output representing yearly migrations of activity rather than hourly flows. The regional activity totals generated by the model are translated into cell demands for the various land uses, so the CA is now constrained regionally rather than globally. This integrated multiscale model continues to be linked interactively to demographic, economic and other relevant models. For some applications a transportation model is included in order to produce a high-resolution LUTI (Land Use – Transportation Interaction) model. Applications to Dublin (including one for wastewater treatment infrastructure planning) The Netherlands, Puerto Rico, and New Zealand are discussed.

Chapter 9 is the core of the book, as it touches on the essence of composing: organizing sound. While other chapters examine different elements deployed in the game, this chapter addresses the game itself: the construction of mesostructure and macroform. The organization of a composition is intertwined with its materials and tools. The unique materials and tools of electronic music lead to new forms of organization. The studio-based practice of multiscale composition applies these tools directly to the organization of a work—in the presence of sound and aligned with human perception.

Hybrid simulation is particularly useful in population health, since healthcare systems are characterized by both dynamic and stochastic complexity and the use of one single simulation approach may result in an oversimplified model that fails to address the real problem. This chapter presents the foundational concepts of hybrid simulation modeling and describes how the various stages in developing a single-method model can be adapted for hybrid simulation. These are illustrated by two examples from population health: age-related macular degeneration and dementia. In both cases, hybrid simulation has enabled the model to reflect the complexity of the decisions facing population health planners, who have to consider individual patient variability and the uncertainty of health outcomes from a “whole-system” perspective. The chapter presents a set of guidelines for modelers, showing how an integrated, multiscale simulation modeling framework can be developed, validated, and exploited for population health problems. The integration of micro-level modeling with macro-level modeling approaches, grounded in foundational complex systems properties and theories, can capture aspects of health systems that a single-method approach cannot.