A novel spectral approach to multi-scale modeling

Giacomo Landi

doi:10.17918/etd-3491

In this work, we present a novel approach for predicting the elastic, thermo-elastic and plastic fields in three-dimensional (3-D) voxel-based microstructure datasets subjected to uniform periodic boundary conditions. Such localization relationships (linkages) lie at the core of all multi-scale modeling frameworks and can be efficiently formulated in a Discrete Fourier Transforms (DFT) - based knowledge system. This new formalism has its theoretical roots in the statistical continuum theories developed originally by Kroner [1]. However, in the approach described by Kroner, the terms in the series were established by selecting a reference medium and numerically evaluating a complex series of nested convolution integrals. This approach is largely hampered by the principal value problem, and exhibits high sensitivity to the properties of the selected reference medium. In the present work, the same series expressions have been recast into much more computationally efficient representations using DFTs. The spectral analysis transforms the complex integral relations into relatively simple algebraic expressions involving polynomials of structure parameters and morphology-independent influence coefficients. These coefficients need to be established only once for a given material system. The main advantage of the new DFT-based framework is that it allows easy calibration of Kroner's expansions to results from finite element methods, thereby overcoming all of the main obstacles associated with the principal value problem and the need to select a reference medium. This approach can be seen as an efficient procedure for data-mining the results from computationally expensive numerical models and establishing the underlying knowledge systems at a selected length scale in multi-scale modeling problems. The set of influence coefficients described above constitutes the underlying knowledge for a given deformation and can be easily stored and recalled as and when needed in a multi-scale modeling effort. In this work, the new mathematical formalism is first presented in a generalized framework, and its viability is then demonstrated in the study of the elastic, thermo-elastic and plastic responses of a selected class of two-phase and multi-phase material systems. Finally, once all of these linkages have been proven robust, ideas for further improvements of the model are addressed.

A novel spectral approach to multi-scale modeling

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A novel spectral approach to multi-scale modeling

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