Perfectly Matched Layer (Pml) for Transient Wave Propagation in a Moving Frame of Reference

Madsen, Stine Skov; Krenk, Steen; Hededal, Ole

doi:10.7712/120113.4819.c1228

Cited by 3 publications

(6 citation statements)

References 8 publications

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“…Here, the convolution PML (C‐PML) proposed in the work of Matzen for time‐domain analyses is employed. In the PML region, the coordinate variables x i , i = 1,2 of the problem are transformed into the stretched coordinate variables

{\overset{true˜}{x}}_{i}

, in each direction, given as

{\overset{true˜}{x}}_{i} = \int_{0}^{x_{i}} s_{i} ((),, x_{i}, ω) d x_{i}, i = 1, 2,

where ω is the angular frequency and s i are the stretched coordinate functions, proposed by Kuzuoglu and Mittra, ie,

s_{i} ((), x_{i} ω) = κ_{ι} ((),, x_{i}, ω) + \frac{β_{i} ((), x_{i})}{α_{i} ((), x_{i}) + i ω} .

…”

Section: Extension Of the Second Coupling Case To 2d Problemsmentioning

confidence: 99%

“…Since this is applicable to the case considered here, these values are adopted. The function β i ( x i ) is defined in the works of Matzen and Madsen et al and the references therein as follows:

β_{i} ((), x_{i}) = β_{i}^{\max} {(\frac{x_{i} - x_{i}^{0}}{2 d_{i}})}^{3},

where

β_{i}^{\max} = - ((), 4 c_{p} \log_{10} R_{0}) false/ ((), 2 d_{i})

, R 0 = 10 −8 is the theoretical reflection coefficient at normal incidence,

x_{i}^{0}

is the location of the PML interface, and d i is the PML thickness in each direction.…”

Section: Extension Of the Second Coupling Case To 2d Problemsmentioning

confidence: 99%

“…58 Here, the convolution PML (C-PML) proposed in the work of Matzen 59 for time-domain analyses is employed. In the PML region, the coordinate variables x i , i = 1,2 of the problem are transformed into the stretched coordinate variablesx i , in each direction, given as 58,60…”

Section: Partial Trapping Of Pulses In Pd-fe Modelsmentioning

confidence: 99%

“…Since this is applicable to the case considered here, these values are adopted. The function i (x i ) is defined in the works of Matzen 59 and Madsen et al 60 and the references therein as follows:…”

Section: Partial Trapping Of Pulses In Pd-fe Modelsmentioning

confidence: 99%

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Wave reflection and cut‐off frequencies in coupled FE‐peridynamic grids

Giannakeas

Papathanasiou

Bahai

2019

Numerical Meth Engineering

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Summary Reflections are typically observed when pulses propagate across interfaces. Accordingly, spurious reflections might occur at the interfaces between different models used to simulate the same medium. Examples of such coupled models include classical continuum descriptions with molecular dynamics or peridynamic (PD) grids. In this work, three different coupling approaches are implemented to couple bond‐based PDs with finite element (FE) solvers for solid mechanics. It is observed that incorporation of an overlapping zone, over which the coupling between FE and PD occurs, can lead to minimization of the reflected energy compared to a standard force coupling at the FE domain/PD grid interface. However, coupling with other existing methodologies, like the addition of ghost particles, achieves comparable accuracy at lower computational cost. Furthermore, the prudent selection of the discretization parameters is of pivotal importance as they control the high frequency cut‐off limit. Mismatch between the cut‐off frequencies of the different descriptions can lead to unrealistic results.

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{\overset{true˜}{x}}_{i}

, in each direction, given as

{\overset{true˜}{x}}_{i} = \int_{0}^{x_{i}} s_{i} ((),, x_{i}, ω) d x_{i}, i = 1, 2,

where ω is the angular frequency and s i are the stretched coordinate functions, proposed by Kuzuoglu and Mittra, ie,

s_{i} ((), x_{i} ω) = κ_{ι} ((),, x_{i}, ω) + \frac{β_{i} ((), x_{i})}{α_{i} ((), x_{i}) + i ω} .

…”

Section: Extension Of the Second Coupling Case To 2d Problemsmentioning

confidence: 99%

β_{i} ((), x_{i}) = β_{i}^{\max} {(\frac{x_{i} - x_{i}^{0}}{2 d_{i}})}^{3},

where

β_{i}^{\max} = - ((), 4 c_{p} \log_{10} R_{0}) false/ ((), 2 d_{i})

, R 0 = 10 −8 is the theoretical reflection coefficient at normal incidence,

x_{i}^{0}

is the location of the PML interface, and d i is the PML thickness in each direction.…”

Section: Extension Of the Second Coupling Case To 2d Problemsmentioning

confidence: 99%

Section: Partial Trapping Of Pulses In Pd-fe Modelsmentioning

confidence: 99%

Section: Partial Trapping Of Pulses In Pd-fe Modelsmentioning

confidence: 99%

See 2 more Smart Citations

Wave reflection and cut‐off frequencies in coupled FE‐peridynamic grids

Giannakeas

Papathanasiou

Bahai

2019

Numerical Meth Engineering

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“…The PML is considered to be the best absorbing boundary condition available and is widely used for various science and engineering applications such as soil-structure interaction, seismic wave propagation, non-destructive evaluation of engineering Corresponding author, Assistant Professor, E-mail: jwkang@hongik.ac.kr structures, etc. (Xu et al 2013, Mahmoud and Luo 2009, Mahmoud et al 2010, Madsen et al 2013, Sagiyama et al 2013.…”

Section: Introductionmentioning

confidence: 99%

Hybrid perfectly-matched-layers for transient simulation of scalar elastic waves

Pakravan

Kang

Newtson

et al. 2014

Structural Engineering and Mechanics

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This paper presents a new formulation for forward scalar wave simulations in semi-infinite media. Perfectly-Matched-Layers (PMLs) are used as a wave absorbing boundary layer to surround a finite computational domain truncated from the semi-infinite domain. In this work, a hybrid formulation was developed for the simulation of scalar wave motion in two-dimensional PML-truncated domains. In this formulation, displacements and stresses are considered as unknowns in the PML domain, while only displacements are considered to be unknowns in the interior domain. This formulation reduces computational cost compared to fully-mixed formulations. To obtain governing wave equations in the PML region, complex coordinate stretching transformation was introduced to equilibrium, constitutive, and compatibility equations in the frequency domain. Then, equations were converted back to the time-domain using the inverse Fourier transform. The resulting equations are mixed (contain both displacements and stresses), and are coupled with the displacement-only equation in the regular domain. The Newmark method was used for the time integration of the semi-discrete equations.

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Probability Methods for Stability Design of Open Pit Rock Slopes: An Overview

Abdulai

Sharifzadeh

2021

Geosciences

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The rock slope stability analysis can be performed using deterministic and probabilistic approaches. The deterministic analysis based on the safety concept factor uses fixed representative values for each input parameter involved without considering the variability and uncertainty of the rock mass properties. Probabilistic analysis with the calculation of probability of failure instead of the factor of safety against failure is emerging in practice. Such analyses offer a more rational approach to quantify risk by incorporating uncertainty in the input variables and evaluating the probability of the failure of a system. In rock slope engineering, uncertainty and variability involve a large scatter of geo-structural data and varied geomechanical test results. There has been extensive reliability analysis of rock slope stability in the literature, and different methods of reliability are being employed for assessment of the probability of failure and the reliability of a slope. Probabilistic approaches include Monte Carlo simulation (MCS), the point estimate method (PEM), the response surface method (RSM), first- and second-order reliability methods (FORMs and SORMs), and the first-order second-moment method (FOSM). Although these methods may be complicated, they provide a more complete definition of risk. Probabilistic slope stability analysis is an option in most commercial software; however, the use of this method is not common in practice. This paper provides an overview of the literature on some of the main probabilistic reliability-based methods available for the design of the rock slope in open pit mining. To demonstrate its applicability, the paper investigates the stability of a rock slope in an open pit mine in the Goldfields region, Western Australia. Two different approaches were adopted: deterministic stability analysis using two-dimensional limit equilibrium and finite element shear strength reduction methods using SLIDE and RS2 software, respectively, and probabilistic analysis by applying the MCS and RSM methods in the limit equilibrium method. In this example, the slope stability analysis was performed using the Spencer method with Cuckoo search optimization to locate the critical slip surface. The results obtained were compared and commented on.

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Perfectly Matched Layer (Pml) for Transient Wave Propagation in a Moving Frame of Reference

Abstract: Abstract. In relation to the development of a Rolling Wheel Deflectometer (RWD), which is a non

Cited by 3 publications

References 8 publications

Wave reflection and cut‐off frequencies in coupled FE‐peridynamic grids

Wave reflection and cut‐off frequencies in coupled FE‐peridynamic grids

Hybrid perfectly-matched-layers for transient simulation of scalar elastic waves

Probability Methods for Stability Design of Open Pit Rock Slopes: An Overview

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