Adaptive simulation of wave propagation problems including dislocation sources and random media

doi:10.1007/s11709-019-0536-4

Front. Struct. Civ. Eng.

2019, Vol. 13

Issue (5) : 1054-1081 https://doi.org/10.1007/s11709-019-0536-4

RESEARCH ARTICLE

Adaptive simulation of wave propagation problems including dislocation sources and random media

Hassan YOUSEFI(

), Jamshid FARJOODI, Iradj MAHMOUDZADEH KANI

School of Civil Engineering, College of Engineering, University of Tehran, Tehran, Iran

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Abstract

An adaptive Tikhonov regularization is integrated with an h-adaptive grid-based scheme for simulation of elastodynamic problems, involving seismic sources with discontinuous solutions and random media. The Tikhonov method is adapted by a newly-proposed detector based on the MINMOD limiters and the grids are adapted by the multiresolution analysis (MRA) via interpolation wavelets. Hence, both small and large magnitude physical waves are preserved by the adaptive estimations on non-uniform grids. Due to developing of non-dissipative spurious oscillations, numerical stability is guaranteed by the Tikhonov regularization acting as a post-processor on irregular grids. To preserve waves of small magnitudes, an adaptive regularization is utilized: using of smaller amount of smoothing for small magnitude waves. This adaptive smoothing guarantees also solution stability without over smoothing phenomenon in stochastic media. Proper distinguishing between noise and small physical waves are challenging due to existence of spurious oscillations in numerical simulations. This identification is performed in this study by the MINMOD limiter based algorithm. Finally, efficiency of the proposed concept is verified by: 1) three benchmarks of one-dimensional (1-D) wave propagation problems; 2) P-SV point sources and rupturing line-source including a bounded fault zone with stochastic material properties.

Keywords adaptive wavelet adaptive smoothing discontinuous solutions stochastic media spurious oscillations Tikhonov regularization minmod limiter

Corresponding Author(s): Hassan YOUSEFI

Just Accepted Date: 24 April 2019 Online First Date: 10 July 2019 Issue Date: 11 September 2019

Cite this article:

Hassan YOUSEFI,Jamshid FARJOODI,Iradj MAHMOUDZADEH KANI. Adaptive simulation of wave propagation problems including dislocation sources and random media[J]. Front. Struct. Civ. Eng., 2019, 13(5): 1054-1081.

URL:

https://academic.hep.com.cn/fsce/EN/10.1007/s11709-019-0536-4
https://academic.hep.com.cn/fsce/EN/Y2019/V13/I5/1054

Fig.1 Grid and smoothing adaptations for adaptive estimations. In illustrations (a), (c) and (e), solid and dashed lines represent estimated functions f(x) and variable smoothing parameters p(x), respectively. There, adapted grid points are also presented, where

? = 10 − 3

. For each estimated function f(x), corresponding estimation errors

| e (x) |

is presented in the same column, in the bellow row

Fig.2 Wavelet based graded tree. (a) Adapted points in different resolutions and corresponding graded tree; (b) redistribution of adapted points based on the graded tree. In Fig. (a), solid points, thick horizontal gray lines are adapted points and corresponding leaves. Black inclined lines show the tree and horizontal-vertical dashed gray lines illustrate boundaries of the spatio-resolution representation

Fig.3 Different approaches for detection of function variation. (a) The function y(x); (b) position of leaves in different resolution levels obtained by the graded tree; (c) smoothing parameter p obtained by the wavelet-based graded tree; (d) zones with significant variations detected by the MINMOD-based detector; (e) smoothing parameter p obtained by the by the MINMOD based approach

Fig.4 Effects of noise on variation detectors for different approaches. (a) The random noise added to function y(x); (b) position of leaves obtained by the graded tree; (c) smoothing parameter p based on the wavelet leaves; (d) zones of significant variations detected by the MINMOD-based approach; (e) smoothing parameter p obtained by the by the MINMOD-based approach

Fig.5 Effects of the MINMOD limiters on the estimations of the first derivative for noisy data. (a) estimation of dy/dxby the FD method; (b–d) Estimation of dy/dx with different MINMOD limiter definitions,

D j i

where j∈{I,II,III}

Fig.6 Comparison of results obtained by different numerical methods by the L-Curve at t = 0.2

Fig.7 Comparison of results obtained from different numerical methods at t = 0.2. In each figure, the black and gray lines correspond to numerical and exact solutions, respectively. (a) SS: p = 1; common finite difference; (b) SS: p = 0.99; (c) SS: p = 0.9; (d) SS: p = 0.7; (e) HOTG: α = γ = 0.5; (f) HOTG: α = 1.5, γ = 0.5; (g) CTG: α = γ = 0.5; (h) FE: 4th Runge-Kutta; (i) α-GDTI: α = α_max⁽²⁾?; (j) α-GDTI; γ = 0.6, α = α_max⁽²⁾?; (k) TDG; (l) FE−(GB−TI): p = 0.54

Fig.8 Effects of constant and adaptive smoothing in numerical simulations in the wave-propagation problem at t = 0.4. The problem contains waves of different magnitudes. Figures (a–b) and (c–d) correspond to the adaptive and constant smoothing, respectively. For the adaptation, assumed parameters are: p₀= 0.98, α_d= 0.0075,

Δ p N S = 0.01

= 0.01 and α_NS= 0.07

Fig.9 Numerical and theoretical scattering attenuation functions

Q s − 1

(a–c) and (d–f) belong to stochastic media with large and small correlation length, respectively. Dashed and solid lines represent theoretical and numerical attenuation functions

Fig.10 Fig. 10 Schematic shape of the computational domain; W₁ and W₂ denote the back-ground domain and the fault zone, respectively; “S” denotes the P-SV point source; and r₁– r₆are the receivers

Fig.11 Fig. 11 Snapshots of solutions u_x, u_z, and corresponding adapted grids at 0.076, 2.108, 3.203 and 5.0762 s. It is assumed that p is adaptive and for grid-adaptations the threshold value is ? = 0.25 ×10⁻⁸

Fig.12 Smoothing effects on solutions; smooth and non-smooth solutions recorded in receivers r₁ and r₄. The smoothing leads to stable results and adapted smoothing ends to more permanent displacements around the source (the r₁ receiver) due to less dissipation effects. Sufficiently far from the source, both adapted and constant smoothing lead to same results (the r₄ receiver)

Fig.13 Effects of the numerical dispersion on discontinuous solutions. (a) The solution with an adaptive smoothing; (b) the solution without the smoothing stage, this solution includes dispersion effects

Fig.14 The P-SV source modeling by considering the fault zone around the source. Multi-reflected waves in the fault zone leakage continuously to the back-ground domain

Fig.15 Snapshots of solutions u_x, u_z, and corresponding adapted grids at 0.473, 2.108, 3.203, and 5.0762 s. It is assumed that p is adaptive and the threshold value is ? = 0.25 × 10⁻⁸

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