Boundary Conditions For Wave Propagation Research Articles

Automated hyperparameter optimization for simulating boundary conditions for acoustic and elastic wave propagation using deep learning

We have developed a deep learning framework to simulate the effect of boundary conditions for wave propagation in anisotropic media. To overcome the challenges associated with the stability of conventional implementation of boundary conditions for strongly anisotropic media, we develop an efficient algorithm using deep neural networks. We incorporate the hyperparameter optimization (HPO) workflow in the deep learning framework to automate the network training process. Hyperparameter selection is a crucial step in model building and has a direct impact on the performance of machine learning models. We implement three different HPO techniques, namely random search, Hyperband, and Bayesian optimization, for simulating boundary conditions and compare the strengths and drawbacks of these techniques. We train the network using a few shot locations and time slices enabling the network to learn how to remove boundary reflections and simulate wave propagation for unbounded media. The automated deep learning framework with HPO improves the performance of deep learning models by achieving the optimal minima and significantly improves the efficiency of the workflow. The benefit of this approach is its simple implementation and significant reduction of reflections at the boundaries, especially in the case of tilted transverse isotropic media. We validate our approach by comparing wave propagation at the boundaries using our algorithm with the output obtained using the unbounded media simulated by padding the model. Tests on different models with acoustic and elastic wave propagation verify the effectiveness of our approach.

A stepwise artificial boundary condition for wave propagation in elasto-plastic media

Wave propagation in elasto-plastic media is often encountered in earthquake geotechnical engineering. However, dynamic boundary conditions in the numerical simulation of wave propagation problems are often developed based on linear elastic assumptions. In this study, the method of characteristics is utilized to propose a novel stepwise artificial boundary condition (S-ABC) for wave propagation in elasto-plastic media. The numerical implementation of S-ABC within a finite element procedure in time domain is developed. Typical numerical examples are chosen to verify the efficiency of this ABC, including: 1) a semi-infinite soil column model subjected to impulse load, where the influence of unloading wave front in elasto-plastic media is discussed; 2) an infinite and a finite soil column model under cyclic seismic wave input, where the proposed S-ABC is compared with the traditional viscoelastic artificial boundary condition (V-ABC). Numerical results demonstrate the significantly higher accuracy and robustness of the proposed S-ABC for nonlinear and elasto-plastic media compared to traditional V-ABC. The influence of unloading wave front on artificial boundary condition for elasto-plastic media is also revealed, which has not been reported previously.

Impact of Internal Faraday Shields on RF Driven Hydrogen Discharges

At RF plasma reactors operated at high power, internal Faraday shields are required to shield dielectric vessel or windows from erosion due to isotropic heat and particle fluxes. By utilizing a flexible and diagnostically well-equipped laboratory setup, crucial effects that accompany the application of internal Faraday shields at low-pressure hydrogen (and deuterium) RF discharges are identified and quantified in this contribution. Both an inductively coupled plasma (ICP) utilizing a helical coil and a low-field helicon discharge applying a Nagoya-type III antenna at magnetic fields of up to 12 mT are investigated. Discharges are driven at 4 MHz and in the pressure range between 0.3 and 10 Pa while the impact of the Faraday shields on both the RF power transfer efficiency and spectroscopically determined bulk plasma parameters (electron density and temperature, atomic density) is investigated. Three main effects are identified and discussed: (i) due to the Faraday shield, the measured RF power transfer efficiency is globally reduced. This is mainly caused by increased power losses due to induced eddy currents within the electrostatic shield, as accompanying numerical simulations by a self-consistent fluid model demonstrate. (ii) The Faraday shield reduces the atomic hydrogen density in the plasma by one order of magnitude, as the recombination rate of atoms on the metallic (copper) surfaces of the shield is considerably higher compared to the dielectric quartz walls. (iii) The Faraday shield suppresses the transition of the low-field helicon setup to a wave heated regime at the present conditions. This is attributed to a change of boundary conditions for wave propagation, as the plasma is in direct contact with the conductive surfaces of the Faraday shield rather than being operated in a laterally fully dielectric vessel.

Local Transparent Boundary Conditions for Wave Propagation in Fractal Trees (II). Error and Complexity Analysis

This work is dedicated to a refined error analysis of the high-order transparent boundary conditions introduced in the companion work [8] for the weighted wave equation on a fractal tree. The construction of such boundary conditions relies on truncating the meromorphic series that represents the symbol of the Dirichlet-to-Neumann operator. The error induced by the truncation depends on the behaviour of the eigenvalues and the eigenfunctions of the weighted Laplacian on a self-similar metric tree. In this work we quantify this error by computing asymptotics of the eigenvalues and bounds for Neumann traces of the eigenfunctions. We prove the sharpness of the obtained bounds for a class of self-similar trees.

Fractional Buffer Layers: Absorbing Boundary Conditions for Wave Propagation

Fractional Buffer Layers: Absorbing Boundary Conditions for Wave Propagation

Local Transparent Boundary Conditions for Wave Propagation in Fractal Trees (I). Method and Numerical Implementation

This work is dedicated to the construction and analysis of high-order transparent boundary conditions for the weighted wave equation on a fractal tree, which models sound propagation inside human l...

Transparent boundary conditions for wave propagation in fractal trees: convolution quadrature approach

In this work we propose high-order transparent boundary conditions for the weighted wave equation on a fractal tree, with an application to the modeling of sound propagation in a human lung. This article follows the recent work (Joly et al. in Netw Heterog Media 14(2):205–264, 2019), dedicated to the mathematical analysis of the corresponding problem and the construction of low-order absorbing boundary conditions. The method proposed in this article consists in constructing the exact (transparent) boundary conditions for the semi-discretized problem, in the spirit of the convolution quadrature method developed by Ch. Lubich. We analyze the stability and convergence of the method, and propose an efficient algorithm for its implementation. The exposition is concluded with numerical experiments.

Practical Absorbing Boundary Conditions for Wave Propagation on Arbitrary Domain

Practical Absorbing Boundary Conditions for Wave Propagation on Arbitrary Domain

Numerical simulation and absorbing boundary conditions for wave propagation in a semi-infinite media with a linear isotropic hardening plastic model

The theory of plastic wave propagation finds its applications in many geotechnical earthquake engineering problems. In this paper, a linear isotropic hardening plastic material is chosen to investigate the plastic wave propagation in a semi-infinite media. The dynamic response of a semi-infinite bar subjected to a triangular load is first derived. The numerical modelling scheme is then proposed and verified by the existing and derived analytical solutions. Novel artificial boundary conditions (ABCs), which are designed for transient dynamic analysis of plastic dilatational wave propagation problem in the media with a linear hardening plastic property, are developed and incorporated into the numerical scheme. Numerical examples illustrate the accuracy of the ABCs in reflecting the complex phenomenon of wave interaction, and reveal good absorption of outgoing stress waves. The study outcome provides reference for developing suitable ABCs for media with complicated nonlinear mechanical properties, and will facilitate the numerical technique in solving soil-structure seismic interaction problems.

Modified Finite Difference Schemes on Uniform Grids for Simulations of the Helmholtz Equation at Any Wave Number

We construct modified forward, backward, and central finite difference schemes, specifically for the Helmholtz equation, by using the Bloch wave property. All of these modified finite difference approximations provide exact solutions at the nodes of the uniform grid for the second derivative present in the Helmholtz equation and the first derivative in the radiation boundary conditions for wave propagation. The most important feature of the modified schemes is that they work for large as well as low wave numbers, without the common requirement of a very fine mesh size. The superiority of the modified finite difference schemes is illustrated with the help of numerical examples by making a comparison with standard finite difference schemes.

Absorbing boundary conditions for wave propagation in viscoelastic media

We derive absorbing boundary conditions for viscoelastic waves. Error estimates are given for such absorbing boundary conditions. We treat the problem in the space-frequency domain formulation after taking the Fourier transformation in time. Well-posedness is shown for the elliptic problems thus created, and Fourier invertibility is also proved. We also present some results of numerical simulations.

An absorbing boundary condition for wave propagation in saturated poroelastic media - Part I: formulation and efficiency evaluation : G. Degrande & G. De Roeck, Soil Dynamics & Earthquake Engineering, 12(7), 1993, pp 411–421

An absorbing boundary condition for wave propagation in saturated poroelastic media - Part I: formulation and efficiency evaluation : G. Degrande & G. De Roeck, Soil Dynamics & Earthquake Engineering, 12(7), 1993, pp 411–421

An absorbing boundary condition for wave propagation in saturated poroelastic media - part II: finite element formulation : G. Degrande & G. De Roeck, Soil Dynamics & Earthquake Engineering, 12(7), 1993, pp 423–432

An absorbing boundary condition for wave propagation in saturated poroelastic media - part II: finite element formulation : G. Degrande & G. De Roeck, Soil Dynamics & Earthquake Engineering, 12(7), 1993, pp 423–432

On the accuracy of absorbing boundary conditions for wave propagation

In practical calculations it is often essential to truncate the computational domain by introducing artificial boundaries. The manner in which boundary conditions are approximated has an important influence on the accuracy of the results. In this paper we present a time domain technique to quantify the spurious reflection of waves from an artificial boundary, obtained by using a nonreflecting boundary condition based on rational approximations.

An absorbing boundary condition for wave propagation in saturated poroelastic media — Part II: Finite element formulation

In a finite element formulation for dynamic soil-structure interaction, an absorbing boundary condition is needed to model wave propagation towards infinity. When the soil is saturated, its dynamic behaviour can be modelled by means of Biot's poroelastic theory. In Part I (Degrande, G. & De Roeck, G., Soil Dynamics & Earthquake Eng., 1993, 12(7), 411-21), a local absorbing boundary condition for wave propagation in saturated poroelastic media has been developed. In the present paper, this boundary condition is implemented in an irreducible finite element formulation for a compressible pore fluid. Spurious reflections for oblique incident waves on the absorbing boundary contribute to the solution errors. Therefore, a spectral element method, based on classical analytical solution techniques, is used to assess the accuracy of the finite element formulation.

An absorbing boundary condition for wave propagation in saturated poroelastic media — Part I: Formulation and efficiency evaluation

In a finite element formulation for dynamic soil-structure interaction, an absorbing boundary condition is needed to model wave propagation in an unbounded problem domain. When the soil is saturated, its dynamic behaviour can be modelled by means of Biot's poroelastic theory. Based on an analytical study of wave propagation in a saturated poroelastic medium, a frequency-dependent local absorbing boundary condition can be obtained at the expense of spurious reflections for oblique incident waves. Reflection curves are presented for all types of incident wave and varying dimensionless frequencies. Alternatively, the effective energy ratio allows the definition of a measure for the overall efficiency of the absorbing boundary condition. Based on these analytical investigations, good wave-absorbing capabilities of the absorbing boundary condition have been revealed. Encouraged by these results, the absorbing boundary condition is implemented in an irreducible finite element formulation for a compressible pore fluid, as discussed in Part II (Degrande, G. & De Roeck, G., Soil Dynamics & Earthquake Eng., 1993, 12(7), 423-32).

On absorbing boundary conditions for wave propagation

This paper is concerned with the development of methods for constructing stable artificial boundary conditions for wavelike equations in a general and automatic way. The one-dimensional problem of a semi-infinite, inhomogeneous, elastic bar is studied here as a prototype situation. For this problem a family of efficient artificial boundary conditions is obtained using geometrical optics in the Laplace transform domain for generating outgoing solutions, together with a stability criterion based on energy integrals to insure that the resulting artificial boundaries are dissipative. Numerical examples illustrate the efficacy of this approach. The paper also includes some remarks about the extension of the proposed method to a more general two-dimensional situation.