An introduction and tutorial on multiple-scale analysis in solids

Concurrent multiple-scale methods can be defined as those which combine information available from distinct length and time scales into a single coherent, coupled simulation. These methods have recently become both popular and necessary for the following reasons. One is the recent discovery of new, nanoscale materials, and the corresponding boom in nanotechnology research. Another factor is that experiments have conclusively shown the connection between microscale physics and macroscale deformation. Finally, the concept of linking disparate length and time scales has become feasible recently due to the ongoing explosion in computational power. We present a detailed introduction to the available technologies in the field of multiple-scale analysis. In particular, our review centers on methods which aim to couple molecular-level simulations (such as molecular dynamics) to continuum level simulations (such as finite element and meshfree methods). Using this definition, we first review existing multiple-scale technology, and explain the pertinent issues in creating an efficient yet accurate multiple-scale method. Following the review, we highlight a new multiple-scale method, the bridging scale, and compare it to existing multiplescale methods. Next, we show example problems in which the bridging scale is applied to fully non-linear problems. Concluding remarks address the research needs for multiple-scale methods in general, the bridging scale method in particular, and potential applications for the bridging scale. 2004 Elsevier B.V. All rights reserved.