Attenuation Zones Research Articles

Periodic Solutions of Wave Propagation in a Strongly Nonlinear Monatomic Chain and Their Novel Stability and Bifurcation Analyses

Abstract A modified incremental harmonic balance (IHB) method is used to determine periodic solutions of wave propagation in discrete, strongly nonlinear, periodic structures, and solutions are found to be in a two-dimensional hyperplane. A novel method based on the Hill’s method is developed to analyze stability and bifurcations of periodic solutions. A simplified model of wave propagation in a strongly nonlinear monatomic chain is examined in detail. The study reveals the amplitude-dependent property of nonlinear wave propagation in the structure and relationships among the frequency, the amplitude, the propagation constant, and the nonlinear stiffness. Numerous bifurcations are identified for the strongly nonlinear chain. Attenuation zones for wave propagation that are determined using an analysis of results from the modified IHB method and directly using the modified IHB method are in excellent agreement. Two frequency formulae for weakly and strongly nonlinear monatomic chains are obtained by a fitting method for results from the modified IHB method, and the one for a weakly nonlinear monatomic chain is consistent with the result from a perturbation method in the literature.

Broadband surface wave manipulation by periodic barriers in unsaturated soil.

Periodic wave barriers have been widely used to manipulate elastic waves propagating in saturated and single-phase soil due to their attenuation zone properties. However, it is difficult to promote application of periodic barriers in unsaturated soils due to their complex constitutive relationship. In this study, manipulation of surface waves by periodic in-filled trench barriers in unsaturated soil has been studied based on the periodic theory. The dispersion relations of a periodic structure for surface waves in unsaturated soil are determined. The attenuation mechanism of evanescent surface waves is revealed. Next, the effects of several key parameters of unsaturated soil on the attenuation zones of the periodic in-filled trench barriers are comprehensively discussed. It is found that in a particular range for material parameter, the surface waves are attenuated over the entire frequency range due to the viscosity of fluid. Finally, a periodic in-filled trench barrier is designed according to a field test of ground vibration induced by a train, and its performances in mitigating surface waves propagating in unsaturated and saturated soils are conducted and compared by conducting analysis in time domain. This investigation provides a new insight for manipulating surface waves by periodic barriers. This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 1)'.

Surface wave mitigation by periodic wave barriers under a moving load: theoretical analysis, numerical simulation and experimental validation.

Periodic wave barriers (PWB) open a new window for vibration mitigation. However, the Doppler effect is rarely considered in most of the previous investigations on the control of ambient vibration induced by moving loads. This article reveals the significance of the speed and frequency of moving loads on surface waves, and improves the design method of PWB for ambient vibration reduction and isolation. First, the theoretical expression of the main frequency band of surface waves propagating in an elastic half-space caused by a moving load was obtained. Comparisons with the numerical results under three different types of traffic loads were also conducted and good agreement was found. Second, the theoretical expression and numerical results were verified by experimental studies. Some inherent properties of wave propagation caused by a moving load in an elastic half-space were also revealed. Third, two kinds of PWBs, i.e. periodic empty trench barrier and periodic pile barrier, were introduced to mitigate wave propagation. It has been confirmed that if the attenuation zones of PWB match the target frequency bands given by the theoretical expression, good vibration mitigation can be achieved. This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 1)'.

Nitrogen fate in riparian zones: Insights from experiments and analysis of sediment porosity and surface water-groundwater exchange

Riparian zones play a vital role in the river ecosystem. Solutes in vertical riparian zones are transported being by alternating hydraulic gradients between river water and groundwater, due to natural or human activities. This study investigates the impacts of porous sediments and alternating rate of surface water-groundwater on nitrogen removal in the riparian zone through experiments based on the field sampled. The experimental results, combined with dimensionless numbers (Péclet and Damköhler) and Partial Least Squares-Path Modeling, analyze the nitrogen fate responding to hydrodynamics changes. The results show that increased sediment porosity contributes to the ammonium removal, particularly when the oxygen content of river water is low, with the removal rate up to 72.57%. High ammonium content and dissolved organic carbon (DOC) in rural rivers lead to a constant low-oxygen condition (4 mg/L) during surface water-groundwater alternation, and promote denitrification. This threatens groundwater with ammonium pollution and causes accumulation at the top of vertical riparian zones during upwelling, potentially causing secondary river pollution. However, increasing the alternating rate hinders the nitrate denitrification and drastically changes in the redox environment of the riparian zone, despite contributing to ammonium removal. Rapid oxygen consumption during aerobic metabolism and nitrification in groundwater-surface water exchange created favorable conditions for denitrification. Floodplains sediment porosity is unfavorable for nitrification. This study improves understanding of coupled hydrologic and solute processes in vertical riparian zones, informing strategies for optimizing nitrogen attenuation and riparian zone construction.

Geometric Parameter Effects on Bandgap Characteristics of Periodic Pile Barriers in Passive Vibration Isolation

To investigate the impact of the geometric parameters of periodic pile barriers on bandgap characteristics in passive vibration isolation, a two-dimensional, three-component unit cell was developed using the finite element method (FEM). This study analyzed the bandgap properties of periodic pile barriers and validated the effectiveness of the FEM through model testing. The FEM was then methodically applied to evaluate the effects of pipe pile thickness, periodic constant, arrangement pattern, and cross-sectional shape on the bandgap characteristics, culminating in the proposition of a novel H-shaped cross-section for the piles. The results demonstrated that the FEM-calculated bandgap frequency range, featuring steel piles arranged in a square pattern, closely aligned with the attenuation zone in the model tests. The lower band frequency (LBF) was primarily influenced by the pipe pile’s inner radius, while the upper band frequency (UBF) was predominantly affected by its outer radius. As the periodic constant increased, the LBF, UBF, and the width of band gap (WBG) all decreased. Conversely, changing the arrangement pattern from square to hexagonal led to increases in UBF and WBG, while the LBF diminished. Notably, the WBG of the H-section steel piles, possessing the same cross-sectional area, was 1.31 times greater than that of the steel pipe piles, indicating an enhanced vibration isolation performance. Additionally, the impact of transverse and vertical characteristic dimensions of the H-shaped pile on the band gap distribution was assessed, revealing that the transverse characteristic dimensions exerted a more significant influence than the vertical dimensions.

Water-tank metabarrier for seismic Rayleigh wave attenuation.

An innovative concept of metabarrier is presented for seismic Rayleigh wave attenuation, which consists of a periodic array of cylindrical water tanks acting as resonant units above the soil surface. A pertinent theoretical framework is developed and implemented in COMSOL Multiphysics. The framework treats the dynamics of the water tank by a well-established three-dimensional linear, pressure-based model for fluid-structure interaction under earthquake excitation, accounting for the flexibility of the tank wall; furthermore, the soil is idealized as a homogeneous and isotropic medium. Floquet-Bloch dispersion analysis of the unit cell demonstrates the presence of relevant band gaps in the low-frequency range below 20 Hz and in the higher frequency range as well. The dispersion analysis is validated by comparison with the frequency-domain analysis of a soil domain with a finite array of water tanks. The band gaps are of interest to attenuate seismic Rayleigh waves and, more generally, Rayleigh waves caused by other ground vibration sources such as road or railway traffic. The water-tank resonant units are readily tunable by varying the water level, which allows changing opening frequencies/widths of the wave attenuation zones. This is a remarkable advantage over alternative seismic metamaterials that, in general, are not designed to be tunable.This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 2)'.

Heterogeneities of Short-Period S-Waves Attenuation Field in the Kuril-Kamchatka Region and their Relation to Large and Great Earthquakes

We study characteristics of short-period shear wave attenuation field in the lithosphere to pick out areas of possible preparation for large and great shallow earthquakes. We have processed more than 360 seismograms of events with source depths of 0–33 km recorded by station PET from two areas limited by coordinates of 45.0˚–50.5˚ N and 54.0˚–56.5˚ N, respectively (for brevity, we will call the areas as the southern and the northern ones). Besides, for the purposes of comparison we have analyzed seismograms by station KGB that recorded earthquakes from the area located between 52˚ и 54˚ N. We used the method based on analysis of Sn and Pn maximum amplitude ratio. We find that attenuation in the lithosphere of the northern area is generally much higher than in the southern one. At the same time attenuation in both areas is weaker than in the region of north-eastern Japan. Relatively lower attenuation corresponds to rupture zones of the great earthquakes of 1952 (Mw = 9.0) and 1963 (Mw = 8.6) occurred in the southern area more than 50 years ago. Higher attenuation is observed in the rupture zones of the recent events dated 1997 (Mw = 7.8), 2006 (Mw = 8.3) and 2018 (Mw = 7.3). The obtained data are in agreement with earlier conclusions stating that typical large subduction type earthquakes occur in the areas characterized by higher fluid content in the uppermost mantle; and large and great earthquakes are followed by deep fluids ascent during a few decades, which leads to attenuation decrease in the uppermost mantle. We also pick out the high attenuation zones where no large and great earthquakes (Mw ≥7.8) have occurred for quite a long time. We suggest that active processes of preparation for large earthquakes can be observed in these zones (first of all in the area of the Avacha Bay and to the east of it).

Propagation attenuation of elastic waves in multi-row infinitely periodic pile barriers: A closed-form analytical solution

Periodically arranged media exhibit attenuation zones for elastic waves, and the related periodic structures play a crucial role in the design of seismic isolation and vibration mitigation. From the perspective of wave theory, this paper presents a closed-form analytical solution for the propagation attenuation of elastic waves through infinitely periodic, multi-row pile barriers. The innovative approach begins by deriving the extended Graf’s addition theorem to transform wave potentials across arbitrary coordinate systems, overcoming the constraints of traditional formulas that limit pile centers to linear configurations. Subsequently, it completes the wave function expansion by utilizing the phase differences in the frequency domain of wave fields around different periodic elements, thus skillfully integrating the effects of incident and scattered wave fields. For the first time, this study delivers exact expressions for the scattering of various types of plane waves (SH, SV, and P) by multi-row pile barriers with an infinite periodic array, addressing the complexities of large pile numbers in previous theoretical studies. The proposed solution increases computational efficiency by tens of times and markedly reduces the majority of resource usage, improving the analysis of complex vibration isolation systems involving numerous piles. Building on the accuracy verification and convergence analysis, several numerical examples are conducted to explore the effects of pile row, pile spacing, and pile arrangement on the propagation attenuation. The findings elucidate the mechanisms driving vibration reduction design using periodic wave barrier, furthering its application in engineering structures.

Mechanisms of multi-bandgap inertial amplification applied in metamaterial sandwich plates

The design concept of integrating locally resonant metamaterials with sandwich plates has demonstrated promising prospects in the development of lightweight, load-bearing structures endowed with excellent capabilities for noise and vibration attenuation. However, achieving low-frequency vibration attenuation in the locally resonant metamaterial sandwich plates remains a challenging task that frequently requires the inclusion of additional centralized mass or heavy local resonators. This study proposes a novel multi-bandgap metamaterial sandwich plate with the lever-type inertial amplification mechanism (LIA-MMSP) for achieving the low-frequency vibration attenuation. Compared with the metamaterial sandwich plates incorporating multi-frequency local resonators (LR-MMSP) with equivalent additional mass, the LIA-MMSP exhibits the ability to achieve lower-frequency multiple bandgaps. The theoretical dynamic model is employed to elucidate the underlying mechanism behind the generation of multiple bandgaps at lower frequencies in the LIA-MMSP. The vibration attenuation performances of the LIA-MMSP are analyzed through both the finite element method and experiment study. The effect of various parameters on the vibration transmission characteristics of the LIA-MMSP is studied. The results show that the boundary frequencies of the LIA-MMSP are precisely one of the lever ratios of the LR-MMSP. By altering the lever ratio within the LIA-MMSP, precise fine-tuning and optimization of the low-frequency multiple bandgaps are achievable. When the attached mass is constrained, increasing the lever ratio enables the achievement of lower bandgaps. In addition, as the eigenfrequency of the primary lever-type IA resonator fp and secondary lever-type IA resonator fs decrease, both the first attenuation zone (AZ1) and the second attenuation zone (AZ2) of the LIA-MMSP shift towards lower frequencies. However, as fp decreases, the width of AZ1 expands, and the minimum accelerations within the AZs decrease even further. Moreover, a normalized comparison provides validation of the exceptional performance of the proposed LIA-MMSP in terms of lightweight design, as well as its ability to achieve low-frequency broadband vibration attenuation.

Bulk wave manipulation by periodic in-filled barriers in unsaturated soil

Band gaps or attenuation zones existing in periodic structures can filter out elastic waves propagating in both single-phase soils and saturated soils in specific frequency regions, which opens a window for designing new wave barriers to attenuate ambient vibration. However, vibration mitigation in unsaturated soil remains a significant challenge due to soil’s constitutive relation strongly relying on saturation, especially by periodic wave barriers. Based on local resonance mechanism, the performance of a periodic in-filled barrier for isolating low-frequency ambient vibration is studied. The periodic in-filled barrier is filled with a viscous material. Investigations found that the attenuation zones of the periodic wave barrier are robust to saturation and the viscosity of the in-filled material can broaden the frequency range for vibration reduction. More interestingly, periodic wave barriers in unsaturated soils can produce attenuation zones for P1 waves, which is impossible in saturated soils. Therefore, this investigation provides a promising insight for the application of periodic wave barriers to unsaturated soils.

Study on vibration attenuation of three-component local resonant pile with low-frequency attenuation zone

The paper explores the increasing concern over environmental vibrations resulting from machine operation and rail transit. Notably, low-frequency vibrations generate surface waves with characteristics like significant energy carrying capacity, long-distance propagation, and slow attenuation. The primary challenge in ground vibration attenuation lies in addressing low-frequency vibrations. The study employs the finite element method to investigate the vibration attenuation of a three-component local resonance pile in three-dimensional space. The analysis delves into the impact of the unit cell structure’s geometry size on bandgaps, and the influences of the period number on the Attenuation Zones (AZs) are also discussed. Furthermore, a comparison is made regarding the vibration attenuation performance of a two-component local resonance pile. The findings reveal that the appropriate design of a three-component local resonance pile can generate low-frequency complete bandgap. The concrete radius, rubber layer thickness, and buried depth of the pile significantly affect Rayleigh wave bandgap. Frequency domain analysis indicates that an increase in period number expands AZs into the bandgap range. Time domain analysis demonstrates a noticeable decrease in ground acceleration amplitude within the bandgap range. The research results presented in this paper offer a novel approach to address the low-frequency vibration attenuation.

Metamaterial sandwich plates with two-degree of freedom inertial amplified resonators for broadband low-frequency vibration attenuation

Sandwich plates are extensively utilized across various fields, encompassing building engineering, mechanical engineering, and aerospace engineering, owing to their exceptional stiffness-to-weight ratio. However, effectively attenuating the low-frequency and broadband vibrations of these plates poses a significant challenge. This paper proposes a new type of metamaterial sandwich plate that incorporates two-degree of freedom inertial amplified resonators (IA-MSPDF2), to attain two low-frequency band gaps (BGs) and achieve broadband vibration attenuation. The dispersion relation of the IA-MSPDF2 is calculated based on the Bloch-Floquet theorem, and the generation mechanism of two low-frequency BGs is analyzed through eigenmodes. Both numerical and experimental studies are conducted to substantiate the advantages associated with the presence of two BGs in the IA-MSPDF2 design. The results show that the enhanced coupling effect between the primary and secondary resonators of the IA-MSPDF2 leads to the band associated with the local resonance that shifts to lower frequencies, resulting in a Bragg scattering BG that arises above the locally resonant BG. Compared to the metamaterial sandwich plate with one-degree of freedom inertial amplified resonators (IA-MSPDF1) of equal mass, the IA-MSPDF2 exhibits an increased relative bandwidth of BG by 15%. Increasing the damping of the inertial amplified resonator causes two attenuation zones to widen and merge into a wider attenuation zone. The proposed IA-MSPDF2 can robustly and effectively attenuate the low-frequency and broadband vibration with a small mass cost, contributing to the further exploration and utilization of metamaterial sandwich plates in engineering applications.

Апроксимація дистрибутивної функції відбивної здатності поверхні поліном третього степеня

In the article a bidirectional reflectance distribution function based on a polynomial of the third degree is developed. The main disadvantages of Schlick, Phong, Blinn reflectance models are analyzed. The approximation of Blinn-Phong model by a cubic polynomial is proposed to improve the productivity of three-dimensional image formation. Formulas for the coefficients of the approximation cubic polynomial are calculated. The disadvantages of using the proposed cubic polynomial for reproducing the glare’s attenuation zone are considered. A function for high-precision reproduction of this zone is proposed, formulas of its coefficients are calculated. A combined function, combining a cubic polynomial for reproducing the glare’s epicenter zone and a function for reproducing the attenuation zone, is proposed. The plot of the combined function is built. It is shown that the combined function reproduces the glare’s epicenter and attenuation zones with small relative and absolute errors. The developed reflectance model provides a highly-productive formation of three-dimensional scenes in computer graphics systems. The proposed distribution function of surface reflectivity can be used in computer graphics systems.

Partitioning of Per‐ and Polyfluoroalkyl Substances to Weathered Light Non‐Aqueous Phase Liquid

AbstractPer‐ and polyfluoroalkyl substances (PFAS) and weathered petroleum hydrocarbons can be co‐contaminants at sites where aqueous film‐forming foam (AFFF) was used to extinguish high‐temperature petroleum fires. Here, we report on the partitioning of six PFAS between water and a weathered diesel and gasoline range hydrocarbon light non‐aqueous phase liquid (LNAPL) collected from the subsurface of a decommissioned refinery. Results showed that both the structural features of PFAS and the characteristics of LNAPL exhibited significant impacts on the magnitude of PFAS partitioned to the LNAPL. The LNAPL–water partition coefficients (Kl) of PFOA and PFOS to the weathered LNAPL measured in this work were nearly or more than 10 times greater than Kl values calculated by others for partitioning to single‐component, unweathered NAPLs, indicating that laboratory studies evaluating LNAPL–water partitioning studies using single component NAPLs can largely underestimate the Kl expected to be encountered at contaminated AFFF sites. Interactions with LNAPL can be important controllers of PFAS fate and transport, and the Kl values estimated in this work suggest that LNAPL could be a significant transport sink for PFAS in the field. Results from this work suggest that PFAS partitioning to weathered LNAPL is an important process to be considered when designing remedial strategies, particularly those incorporating monitored natural attenuation or natural source zone depletion.

A phononic crystal suspension for vibration isolation of acoustic loads in underwater gliders

Currently, it is a common practice to apply underwater gliders with acoustic loads for hydroacoustic detection, but the self-noise of the vehicle inevitably interferes with signal acquisition. In this study, a novel suspension is designed based on phononic crystals to realize vibration isolation of the acoustic loads in underwater gliders. The vibration properties of finite-period phononic crystals with different layouts of unit cells are studied by analyzing the attenuation zones, and the three-period phononic crystal with prolonged metal ends is determined to be the optimal structure of the suspension. Moreover, the effect of physical parameters on the underwater attenuation zones of the proposed suspension is further investigated through acoustic-structure coupling, resulting in the formation of a design scheme for the suspension. A vibration test is performed in an anechoic pool, which reveals that the phononic crystal suspension has a stable vibration isolation effect at 120–5000 Hz and achieves a maximum vibration isolation of about 7 dB compared with the original structure. The results in the present work provide a theoretical reference for applying phononic crystals for vibration isolation in underwater vehicles.

Broadband surface wave attenuation in porous soil by elastic metasurfaces

To mitigate surface waves in ambient vibration, a novel broadband elastic metasurface is proposed by periodically burying the layered in-filled pipes below the ground surface. The present metasurface is a simple structure and can be easily implemented. In contrast to the existing studies of metasurfaces in single-phase elastic soil, the coupling of elastic metasurfaces with saturated porous soil is investigated within the framework of Biot's theory. Note that Biot's poroelasticity can capture the complexity of soils in the presence of fluids. Both dispersion and transmission analysis are comprehensively conducted. The present investigation shows that the elastic metasurfaces are suitable for a wide range of fluid-saturated porous soil, and a significant surface wave attenuation is observed within the attenuation zone of elastic metasurfaces. In addition, the burying depth is the key point of the design of elastic metasurfaces. Specifically, the surface wave attenuation zone (SWAZ) will vanish when the depth is larger than three times the periodic constant. Besides, the SWAZ will be widened by stacking the layers of elastic metasurfaces. Three layers is enough for the metasurface to attenuate more than 80 % surface wave energy. Due to the similar dynamic characteristics of elastic metasurface in the fluid-saturated porous half-space compared to dry half-space, single-phased soil can be used to simplify the analysis of saturated soil with ideal fluids. To widen the scope of application in elastic metasurfaces, the feasibility of ambient vibration mitigation by elastic metasurfaces is experimentally and numerically verified. The experimental results confirm an obvious wave attenuation within the SWAZ, and the average insertion loss (IL︸) is 9.8 dB. Last but not least, the performance of elastic metasurfaces to mitigate ground-borne vibration induced by trains is numerically studied in time domain. Elastic metasurfaces outperform traditional trenches within the main frequency range, and IL︸ is increased by 3.9–7 dB. Thanks to the excellent wave attenuation performance and subwavelength thickness, these novel elastic metasurfaces are expected to open new avenues in ambient vibration mitigation in the costal area of saturated soil.

A novel buried periodic in-filled pipe barrier for Rayleigh wave attenuation: Numerical simulation, experiment and applications

Surface wave is a major concern in ambient vibration mitigation. To this end, a novel shallow buried periodic in-filled pipe barrier is proposed. Complex dispersion analysis is conduced to evaluate the decay of surface waves and the influence of material damping. Through a combination of numerical simulations and lab-scale experiments, the effectiveness and robustness of surface wave attenuation are verified. Result shows that the proposed periodic in-filled barrier exhibits excellent performance in low-frequency surface wave attenuation. Both the energy dissipation induced by material damping and local resonance of in-filled pipes contribute to surface wave attenuation. The influence of viscosity also enhances wave attenuation and it should be taken into full account in practice. In addition, these novel wave barriers are suitable for passive isolation as well as active isolation. To further widen the attenuation zones, the gradient layouts of in-filled pipes could be a good alternative considering the limited barrier width. Besides, the feasibility of periodic in-filled barriers to train-induced vibration mitigation is preliminary verified by analyzing the measured acceleration record in the time-domain. Thanks to the subwavelength size and shallow depth, these novel wave barriers are convenient for construction and cost-effective, thus showing great potential in ambient vibration mitigation.

Double-gradient seismic metamaterials with zero-frequency bandgap characteristic in a layered soil medium

Seismic metamaterials (SMs) are an artificially composite material designed within the sub-wavelength range. To attenuate low-frequency seismic surface waves, a metamaterial structural unit composed of a lead core and an auxetic foam coating layer was proposed; the higher impedance ratio results in the creation of a zero-frequency bandgap under layered soil conditions. To broaden the attenuation zone (AZ), double-gradient seismic metamaterials (DGSMs) were designed. The frequency domain analysis indicates that the attenuation area covering the range of 0–10 Hz is 89.8%. Through displacement-field analysis, the attenuation mechanism of DGSM on specific frequency surface waves can be fully explained. Finally, Taft seismic wave excitation is input for time history analysis, and the acceleration amplitude within 0–2 Hz decreases by 58.6%, which verifies the effectiveness of DGSMs in attenuation of low-frequency seismic waves.

NEW SURFACE REFLECTANCE MODEL WITH THE COMBINATION OF TWO CUBIC FUNCTIONS USAGE

In the article the model of light reflection based on the combination of two cubic bidirectional reflectance distribution functions is developed. The main components of color and the main requirements for reproducing the object`s glares are analyzed. The usage characteristics of Cook-Torrance, Bagher, Oren-Nayar, coupled Shirley reflection models are described. The advantages and disadvantages of the highly productive Blinn-Phong model are considered. The necessity of approximating the Blinn-Phong model by a function of low degree is justified. The characteristics of the cubic polynomial approximation of the Blinn-Phong model are determined. It was established that the main drawback of this approximation is a significant deviation of the function from the reference function in the glare’s attenuation zone. The combined function that combines two cubic functions is proposed. The first cubic function reproduces the glare’s epicenter, and the second replaces the specified function in the attenuation zone. A system of equations for calculating the coefficients of the second function was created. The formula for the connection point of two cubic functions is obtained. A graph of the developed combined model based on cubic functions is obtained. For the combined and original cubic functions a comparison of the maximum relative errors in the glare’s epicenter zone, the maximum absolute errors, and the relative errors at the inflection point was made. A three-dimensional plot of the absolute error of the combined cubic model from the Blinn-Phong model depending on the shininess and the angle value is built. Visualization results based on the combined and the original cubic functions are compared. It is confirmed that the proposed reflection model increases the realism of glare formation in the attenuation zone. The resulting combined reflection model provides a highly accurate approximation of the Blinn-Phong model and is highly efficient because the third power function is used.

Feasibility of vibration mitigation in unsaturated soil by periodic pile barriers

Periodic wave barriers have been widely used to mitigate ambient vibration. However, almost all the studies on the performance of periodic wave barriers are conducted under the assumption that soil is a single-phase or saturated medium. In fact, soil often consists of three phases: solid, liquid and gas, and it is more reasonable to be considered as an unsaturated soil. Existing research demonstrates that the dynamic properties of unsaturated soil have been significantly changed by gas phase, or the dynamic performance of unsaturated soil is completely different from that of single-phase soil and saturated soil. Especially, the dynamic performances of unsaturated soil cannot be approximated by the two extremes (saturated soil and single-phase soil). In this paper, the feasibility of vibration mitigation in unsaturated soil by periodic pile barriers is numerically validated. Firstly, based on the periodic theory, the complex band structures of P-SV waves in the periodic pile-unsaturated soil system are calculated. The influence of the saturation on the complex band structures is also investigated. Secondly, a numerical model for analyzing ambient vibration isolation is established. The responses of the numerical model are carried out in both frequency and time domains. The results indicate that periodic wave barriers in unsaturated soil can be designed with a wide attenuation zone covering the dominant frequencies of P1 waves, which is just a dream in saturated soil. Furthermore, S waves can also be greatly shielded in the target band. The present study removes the obstacles for the vibration mitigation in unsaturated soil by periodic pile barriers.