Multiscale mathematics

In engineering, mathematics, physics, chemistry, bioinformatics, computational biology, meteorology and computer science, multiscale modeling or multiscale mathematics is the field of solving problems which have important features at multiple scales of time and/or space. Important problems include multiscale modeling of fluids, solids, polymers, proteins,nucleic acids as well as various physical and chemical phenomena (like adsorption, chemical reactions, diffusion).

Horstemeyer 2009, 2012 presented a historical review of the different disciplines (solid mechanics, numerical methods, mathematics, physics, and materials science) for solid materials related to multiscale materials modeling.

Essentially, the idea of filling the space of system level “tests” was then proposed to be filled by simulation results. After the Comprehensive Test Ban Treaty of 1996 in which many countries pledged to discontinue all systems level nuclear testing, programs like the Advanced Strategic Computing Initiative (ASCI) were birthed within the Department of Energy (DOE) and managed by the national labs within the US. Within ASCI, the basic recognized premise was to provide more accurate and precise simulation-based design and analysis tools. Because of the requirements for greater complexity in the simulations, parallel computing and multiscale modeling became the major challenges that needed to be addressed. With this perspective, the idea of experiments shifted from the large scale complex tests to multiscale experiments that provided material models with validation at different length scales. If the modeling and simulations were physically based and less empirical, then a predictive capability could be realized for other conditions. As such, various multiscale modeling methodologies were independently being created at the DOE national labs: Los Alamos National Lab (LANL), Lawrence Livermore National Laboratory (LLNL), Sandia National Laboratories (SNL), and Oak Ridge National Laboratory (ORNL). In addition, personnel from these national labs encouraged, funded, and managed academic research related to multiscale modeling. Hence, the creation of different methodologies and computational algorithms for parallel environments gave rise to different emphases regarding multiscale modeling and the associated multiscale experiments.

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