Multiscale modeling is utilized to investigate the domain of the product at multiple levels of scales to increase the speed of calculations rather than simulating all the details at once. In the multiscale modeling techniques, homogenization methods are utilized to pass the information at finer scales to the macroscopic or human scale level. However, the homogenization methods are usually application dependent or different homogenization methods are used at different levels. In addition, homogenized properties are often determined by complex, explicit equations that links the fine and coarse scale domains. As a result, the multiscale modeling techniques possess increased computational complexity especially when the uncertainty effects at finer scales are considered.
To address these issues, a multi-level upscaling framework is formulated that can enable a top-down decomposition of the system or product at macroscale level into multiple sub-levels (e.g. mesoscale level, microscale level, etc.) without penetrating the underlying equations. For this purpose, an improved simulation-based stochastic upscaling method, in which the complexity of the fine scale model under uncertainty is replaced by the homogenized coarse scale parameters by matching the probabilistic performance between fine and coarse scale models, is proposed and integrated into the bottom-up multiscale modeling framework. Polynomial Chaos Expansion (PCE) is employed in the upscaling procedure along with an efficient objective function called exponential loss function to handle the computational burden caused by the input uncertainties. An effective hierarchical validation approach is proposed to ensure that the coarse scale models corresponding to each level can produce accurate simulations at multiple levels. Additively manufactured cellular structures are used as application examples to demonstrate the efficacy of the proposed multiscale modeling process. The proposed multi-level upscaling framework is integrated into the design optimization process of cellular structures to identify the optimal microstructure type and geometric properties for each element in the structure under the uncertainties induced by the manufacturing process parameters. The cellular microstructure is replaced by a homogeneous medium using the proposed upscaling method to eliminate the computational burden of simulating the structure with detailed cellular elements.