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Stability and reflection/transmission analysis of a dynamic hybrid coupling scheme for elastic wave propagation
Siddhesh Raorane, Tadeusz Uhl, Pawel Packo, Michael Leamy

Last modified: 2020-07-21


In this paper the stability and reflection/transmission properties of a dynamic hybrid coupling scheme coupling two dynamic transient models for elastic wave propagation, namely, the Local Interaction Simulation Approach and Cellular Automata for Elastodynamics, are analyzed. The phenomena of stability and reflection/transmission always arise in numerical methods with the former being critical for all models, while the latter being significant when numerical models are coupled. Varying material properties, spatial discretization parameters and time steps in the coupled numerical models highly affect the stability and reflection/transmission properties, and thus it becomes important to study and analyze these phenomena as they determine the accuracy of the results to a great extent.

CAFE is based on cellular automata, while LISA is based on finite differences, and a hybrid approach coupling these two methods is appealing because of CAFE's capability to allow non-uniform meshes - used e.g. for modelling crack – that can be coupled to LISA's rectangular meshes while taking advantage of the built-in LISA features of being implemented with graphical processing units on the uniform portion of the domain. The aforementioned coupling scheme is capable of coupling different material properties, spatial discretization parameters and time steps in LISA and CAFE, and thus its stability and reflection/transmission properties need to be understood and analyzed.

The analysis of stability is done in the complex domain of β parameter with the stability given by two vectors in β space. Each vector represents individual model’s stability. This contrasts with a usual case of an explicit time integration scheme, where only a single vector represents the stability in the β space. Rotations of these two vectors in the β space are in opposite directions, and their critical combinations drive the stability of the coupled model. The stability was investigated for three model configurations, namely, no restrictions on model parameters case, equal Young’s moduli in LISA and CAFE case, and equal Young’s moduli and equal densities in LISA and CAFE case. The latter two configurations are of practical interest as they are likely to be common choices while coupling two domains like structure with localized defects.
For the reflection/transmission analysis, the impedance of the LISA domain and CAFE domain of the dynamic coupling scheme are determined. A mismatch between these impedances gives rise to reflections in the coupled model.  In this paper, equations to predict the amount of reflection/transmission in the coupling scheme are derived. These equations, capable of predicting reflections for any combination of material properties and/or spatial discretization parameters in the coupling scheme, are validated against results from numerical simulations.


numerical methods, multi-scale, elastic wave propagation, dynamic coupling, reflection, stability

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