Session: VIB-04-01

Paper Number: 147606

147606 - The Atomistic Green's Function Method for Acoustic and Elastic Wave-Scattering Problems 

36th Conference on mechanical vibrations and sound

The Atomistic Green's Function method for acoustic and elastic wave-scattering problems

Hossein Khodavirdi, Zhun-Yong Ong, Ankit Srivastava

 

Abstract

Conventional methods for solving scattering problems are typically complex, with the complexity obscuring the interplay between the degrees of freedom in the scatterer and those in its environment. In this talk, we present the application of the Atomistic Green's Function (AGF) method [1,2], which originates from the non-equilibrium Green’s function method used to analyze ballistic phonon transport, to study acoustic and elastic wave scattering problems. 

We illustrate this method here by employing the AGF method to solve the scattering problem for a 1D slab, 1D phononic crystal or a defect in a 2D waveguide, and show how we can bypass the implementation of artificial truncating boundaries, such as perfectly matched layers or DtN maps. Atomistic Green’s Function method by itself is a novel method when applied to classical wave problems, and in addition to bypassing the need for non-reflecting boundary condition in solving scattering problems, it calculates Green’s function and effective Hamiltonian of the scatterer, tasks that are generally challenging. The calculation of Green’s functions as a means to solve scattering problems has been treated in earlier works via other approaches such as introducing interface response theory, while they follow complicated and relatively longer paths. Understanding the Green’s function as a powerful tool not only facilitates solving scattering problems but also opens doors to studying the dynamics of the system. AGF, in particular, offers an open system perspective in solving scattering problems. It views the problem through the lens of the interaction between a finite-dimensional space with a discrete spectrum and an infinite-dimensional environment with a continuum spectrum. The dynamics of this interaction are encapsulated within the effective Hamiltonian and Green’s function. This additional capability sets it apart from other methods typically used for solving scattering problems, which may not inherently provide insights into the system's dynamics. 

In addressing scattering problems, the AGF method converts the challenge of dealing with infinite-domain scattering into a more tractable finite problem focused solely on the scatterer. In one-dimensional scenarios, the method utilizes nearest-neighbor interactions and the sparsity of coupling dynamical matrices, resulting from employing lumped-mass technique, to derive the surface Green’s function analytically. This surface Green’s function captures the influence of the environment's degrees of freedom surrounding the scatterer. For two-dimensional scattering scenarios, the AGF method employs an iterative approach known as the decimation technique [3]. This technique efficiently computes the surface Green’s function/matrix, significantly streamlining the computation process for determining the self-energy matrix. This leads to explicit calculations of the effective Hamiltonian of the scatterer—a finite-dimensional operator/matrix encapsulating the dynamics of the scatterer's finite region interacting with its environment during the scattering phenomenon.

It is worth noting that this effective Hamiltonian furnishes us with the crucial scattering results, pivotal for comprehending scattering behavior. Additionally, the AGF method, along with its extensions, enables us to uncover the dispersion relation within the environment and provides mode-resolved contributions to the overall scattering process.

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References

[1].  Sadasivam, Sridhar, et al. "The atomistic Green's function method for interfacial phonon transport." Annual Review of Heat Transfer 17 (2014).

[2]. Ong, Zhun-Yong. "Atomistic S-matrix method for numerical simulation of phonon reflection, transmission, and boundary scattering." Physical Review B 98.19 (2018): 195301.

 

Presenting Author: Hossein Khodavirdi Illinois Institute of Technology

Presenting Author Biography: Khodavirdi is a PhD candidate of Mechanical Engineering at Illinois Institute of Technology. He received
his B.Sc. in mechanical engineering in 2019 from Iran university of science and technology, Tehran, Iran
and started his graduate studies at Illinois Tech in 2020. Khodavirdi has experience and publications in
acoustics steering techniques, Causality in elastodynamics, phononic crystals, and solving scattering
problems from the perspective of Open systems.

Authors:

Hossein Khodavirdi Illinois Institute of Technology
Zhun Yong Ong Institute of High-Performance Computing (IHPC)
Ankit Srivastava Illinois Institute of Technology