Dynamic compound wavelet matrix method for multiphysics and multiscale problems

Abstract

The paper presents the dynamic compound wavelet method (dCWM) for modeling the time evolution of multiscale and/or multiphysics systems via an "active" coupling of different simulation methods applied at their characteristic spatial and temporal scales. Key to this "predictive" approach is the dynamic updating of information from the different methods in order to adaptively and accurately capture the temporal behavior of the modeled system with higher efficiency than the (nondynamic) "corrective" compound wavelet matrix method (CWM), upon which the proposed method is based. The system is simulated by a sequence of temporal increments where the CWM solution on each increment is used as the initial conditions for the next. The numerous advantages of the dCWM method such as increased accuracy and computational efficiency in addition to a less-constrained and a significantly better exploration of phase space are demonstrated through an application to a multiscale and multiphysics reaction-diffusion process in a one-dimensional system modeled using stochastic and deterministic methods addressing microscopic and macroscopic scales, respectively.

Linking simulations and experiments for the multiscale tracking of thermally induced martensitic phase transformation in NiTi SMA

“…The inherent capability of the wavelets to provide scale specific resolution makes it an alternative choice to apply in the field of localized feature extraction from irregular/ non-stationary data. Since the theory and motivation for wavelet analysis are already widely covered in the literature, we do not repeat those details here, but simply provide a brief overview and refer the reader to the following major references on the subject [5][6][7][8][9]14,22]. In this study an important predictive feature is added to the compound wavelet matrix (CWM) method.…”

“…n N > , the maximum number of scales that can be coupled is n . However, in the CWM method instead of adopting all of the n fine simulation scales, a cutoff scale ( ) Then, at all scales above the cutoff scale ( ) cut n , the wavelet coefficients are those from the fine simulation which are repeated periodically [6], a process that is valid when the process is stationary or quasi stationary. However, when there are long range correlations in the process (such as the martensitic phase transformation, where fluctuations present in the fine simulation are correlated colored noise) repetitions should be restrained over small scales.…”

“…In general, fine scale simulations are inherently stochastic, thereby an appropriate quantification of uncertainty associated with material properties requires a large number of simulations, and each simulation is typically by itself computationally demanding. In concurrent multiscaling methods [4][5][6][7], simulations are performed simultaneously in different spatial regions using different sub methods. More accurate fine scale simulations are performed in regions where such accuracy is needed, and are connected to regions where coarse scale simulations are performed.…”

“…Wavelet based methods form a viable concurrent multiscale modeling toolbox [1][2][3][6][7][8][9]. They substantially reduce the computational overhead, without compromising accuracy, required to transfer information from one physical scale to another.…”

Review of wavelet methods for the solution of reaction–diffusion problems in science and engineering

“…Mishra et al [72] introduced the wavelet-based spatial scaling of coupled reaction-diffusion fields. Muralidharan et al [73] discussed the dynamic compound wavelet matrix method for multiphysics/multiscale problems. Frantziskonis et al [74] established the time-parallel multiscale/multiphysics framework.…”