Mechanics of materials at different length scales exhibit diverse characteristics that require proper regularity in the construction of numerical formulation. Strong and weak discontinuities, topological change in geometry, and singularities are a few examples that render difficulty in the construction of approximation functions with desirable regularity in solving mechanics problems at different length scales. In this talk, we first introduce meshfree approximation methods suitable for approximating smooth/non-smooth and local/non-local characteristics that could exist in multiple length scales with different dominant physics of a given problem. We show how the sub-scale local variables can be embedded in the upper-scale non-local variables under meshfree approximation, and how the sub-scale variables be coupled with the upper-scale variables in various multiscale computational paradigms. We also introduce the model order reduction in the scale of interest as well as a method for image based microstructure modeling to further facilitate the effectiveness of multiscale computation. Three model problems will be given to demonstrate the applicability of the proposed multiscale methods.
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