Multiscale Methods

Abstract

This comprehensive chapter is dedicated to the discussions on a number of multiscale methods. It begins by discussing the concepts of the homogenization methods and covers both the mathematical and computational homogenization techniques and presents the state-of-the-art concepts of the enriched multiscale homogenization. Then, the atomistic and molecular modellings are discussed in detail by covering the concepts of statistical mechanics and the governing equations of motion along with a review on major available potential functions. The important subject of the sequential multiscale method is discussed by providing the details of a practical simulation, which spans the extremely wide range of MD simulations, nanoscale analysis, macroscale computations, micro scale investigation and macro scale study of concrete specimen with additive carbon nanotube (CNT) fibres. Then, the concurrent multiscale schemes are examined. Discussion begins by the quasicontinuum (QC) approach and the fundamentals of the bridging domain (BDM) and the bridging scale (BSM) methods, and continues to address the concepts of the recently developed dis-ordered multiscale method (DCMM), which is expected to perform well for amorphous and polymeric material structures and biomolecule applications. Finally, the variable node multiscale method (VNMM) and its extension to the enriched multiscale method (EMM) are presented.