Special Section on Multiscale Materials Modeling

Abstract

Materials modeling is intrinsically multiscale and has motivated much of the development of multiscale methods for both analysis and simulation. Multiscale features of materials can originate from a variety of sources, including multiple physics, atomistic effects, defects, multiscale continuum structures, and multiscale behavior of the PDEs or other elements of mathematical models. The following special section on multiscale materials modeling was motivated by the Fourth International Conference on Multiscale Materials Modeling (MMM-2008), which was held in Tallahassee, Florida, October 27–31, 2008. Anter El-Azab, Max Gunzberger, and Sidney Yip served as guest editors for this special section. Molecular dynamics using effective potentials is an appropriate model for the mechanical properties of fluids and solids at the atomistic scale. The most basic multiscale issue is the connection between molecular dynamics and continuum mechanics, and analysis of this connection is one of the most important goals of mathematical materials science. Besides its basic scientific importance, a practical outcome of this analysis can be a molecular-based determination of the constitutive equations for the material. In “Self-Consistent Multiscale Modeling in the Presence of Inhomogeneous Fields” by Ruichang Xiong, Rebecca L. Empting, Ian C. Morris, and David J. Keffer, computational results from molecular dynamics for an isothermal fluid are averaged to directly evaluate the equation of state for fluid pressure using a novel thermostat to keep the system at a fixed temperature. An alternative approach to relating atomistic and continuum mechanics is taken by Pablo Seleson, Michael L. Parks, Max Gunzberger, and Richard B. Lehoucq in their article “Peridynamics as an Upscaling of Molecular Dynamics.” Peridynamics is a continuum model for solid mechanics involving nonlocal interactions, as in molecular dynamics. This article shows that peridynamics can be interpreted as an upscaling of molecular dynamics. A principal computational difficulty with molecular dynamics is the large number of atoms required for realistic systems and the resulting large computational complexity. For materials with defects, molecular dynamics may be necessary near the defects, but far from the defects a coarse grained description may be adequate and much more computationally efficient. “Multiscale Analysis across Atoms/Continuum by a Generalized Particle Dynamics Method” by Jinghong Fan develops a new multiscale computational approach for particle/continuum mechanics, in which the spatial domain is divided into different regions. In each region the material is represented as a system of particles, but the size and spacing of the particles are different in different regions. Large molecules, such as peptides and proteins, can arrange themselves in various spatial configurations or conformations. The dynamics of these conformations are difficult to directly represent and simulate, because of the large size of the molecules. In their paper “Mean Field Approximation in Conformation Dynamics,” Gero Friesecke, Oliver Junge, and Péter Koltai develop a mean field description of conformation dynamics, based on molecular dynamic simulations over short times and an assumption of statistical independence of the eigenfunctions of the related transfer operators. At macroscopic scales, the most natural theory is a continuum description based on partial differential equations (PDEs). This can already require a multiscale description due to spontaneous formation of multiscale phenomena such as fluid turbulence or mechanical shear bands, or inherent multiscale structures as composite materials or porous media. These phenomena lead to multiscale analysis of the PDEs for the continuum theory. The Hamilton–Jacobi equation is a canonical equation for transport of material or other quantities. In “Homogenization of Metric Hamilton–Jacobi Equations,” Adam M. Oberman, Ryo Takei, and Alexander Vladimirsky derive a new approach to homogenization for static Hamilton–Jacobi equations. They relate the Hamilton–Jacobi solution to a distance function for a corresponding metric and show that the homogenized equation also induces a metric. This provides a simplified description of the homogenization. For materials with microscopic heterogeneity, a stochastic description of the heterogeneities is a useful mathematical model. In their article “A Bounded Random Matrix Approach for Stochastic Upscaling,” Sonjoy Das and Roger Ghanem develop a maximum entropy approach to probabilistic construction of the matrices of coefficients for the constitutive laws, taking advantage of the energy bounds on the constitutive coefficients. Bone is a particularly complex and important multiscale material. The article “Micromechamical Modeling of the Anisotropy of Elastic Biological Composites” by Jamila Rahmoun, Fahmi Chaari, Eric Markiewicz, and Pascal Drazetic derives a mathematical model for the effective elastic parameters of trabecular bone, based on micromechanical modeling and using the multiscale geometry obtained from x-ray microtomography. In summary, this collection of seven papers provides a sampling of some of the most important issues in multiscale modeling and simulation for materials. It shows the range of issues and results in this field of great current interest.