This is a general strategy ofdecomposing functions or more generally signals into components atdifferent scales. A well-known example is the wavelet representation(Daubechies, 1992). This is a strategy for choosing thenumerical grid or mesh adaptively based on multi-scale analysis what is known about thecurrent approximation to the numerical solution. Usually one finds alocal error indicator from the available numerical solution based onwhich one modifies the mesh in order to find a better numericalsolution.

Alphanumerical scales

For this reason, direct applications of the first principle are limited to rather simple systems without much happening at the macroscale. At LANL, LLNL, and ORNL, the multiscale modeling efforts were driven from the materials science and physics communities with a bottom-up approach. Each had different programs that tried to unify computational efforts, materials science information, and applied mechanics algorithms with different levels of success.

Mesoscopic and multiscale modelling in materials

Forexample, if the macroscale model is the gas dynamics equation, then anequation of state is needed. When performing molecular dynamicssimulation using empirical potentials, one assumes a functional formof the empirical potential, the parameters in the potential areprecomputed using quantum mechanics. 5, which depicts the macroscale strains that correspond to the minimum potential energy for each load step, superimposed on the resolved design space. 5b–d, it is clearly demonstrated that only a small proportion of the permissible macroscale strain space has been resolved in order to derive the displacement field illustrated in Fig. The proportion of the resolved strain space amounts to 1.45% of the total strain space.

Multiple-Scale Analysis

The geometrical parameter bounds are selected to facilitate access during the optimization procedure to significant variations in both Poisson’s ratio and axial stiffness. Future studies should seek to experimentally validate the multiscale geometries presented within this paper. In addition, it would be interesting to explore higher-order homogenization schemes for the multiscale structural analysis, with the aim to exploit instabilities such as buckling at both scales to realize larger deformations. Multiscale topology optimization frameworks permit the microscale description of a structure to be spatially optimized to fulfill functional objectives at the macroscale. This permits hierarchical structures with superior mechanical (Imediegwu et al. 2019; Murphy et al. 2021b), thermal (Imediegwu et al. 2021), and dynamic (Nightingale et al. 2021) performance to be generated relative to conventional topology optimization-based approaches.