Multiscale modeling or multiscale mathematics is the field of solving problems that 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).

An example of such problems involve the Navier–Stokes equations for incompressible fluid flow.

${\displaystyle {\begin{array}{lcl}\rho _{0}(\partial _{t}\mathbf {u} +(\mathbf {u} \cdot \nabla )\mathbf {u} )=\nabla \cdot \tau ,\\\nabla \cdot \mathbf {u} =0.\end{array}}}$

In a wide variety of applications, the stress tensor ${\displaystyle \tau }$ is given as a linear function of the gradient ${\displaystyle \nabla u}$. Such a choice for ${\displaystyle \tau }$ has been proven to be sufficient for describing the dynamics of a broad range of fluids. However, its use for more complex fluids such as polymers is dubious. In such a case, it may be necessary to use multiscale modeling to accurately model the system such that the stress tensor can be extracted without requiring the computational cost of a full microscale simulation.

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

The recent surge of multiscale modeling from the smallest scale (atoms) to full system level (e.g., autos) related to solid mechanics that has now grown into an international multidisciplinary activity was birthed from an unlikely source. Since the US Department of Energy (DOE) national labs started to reduce nuclear underground tests in the mid-1980s, with the last one in 1992, the idea of simulation-based design and analysis concepts were birthed. Multiscale modeling was a key in garnering more precise and accurate predictive tools. In essence, the number of large-scale systems level tests that were previously used to validate a design was reduced to nothing, thus warranting the increase in simulation results of the complex systems for design verification and validation purposes.

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.

The advent of parallel computing also contributed to the development of multiscale modeling. Since more degrees of freedom could be resolved by parallel computing environments, more accurate and precise algorithmic formulations could be admitted. This thought also drove the political leaders to encourage the simulation-based design concepts.