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This paper aims to elucidate the clear visibility of attenuating seismic waves (SWs) with forest trees as natural metamaterials known as forest metamaterials (FMs) arranged in a periodic pattern around the protected area. In analyzing the changeability of the FM models, five distinct cases of “metawall” configurations were considered. Numerical simulations were conducted to study the characteristics of bandgaps (BGs) and vibration modes for each model. The finite element method (FEM) was used to illustrate the generation of BGs in low frequency ranges. The commercial finite element code COMSOL Multiphysics 5.4a was adopted to carry out the numerical analysis, utilizing the sound cone method and the strain energy method. Wide BGs were generated for the Bragg scattering BGs and local resonance BGs owing to the gradual variations in tree height and the addition of a vertical load in the form of mass to simulate the tree foliage. The results were promising and confirmed the applicability of FEM based on the parametric design language ANSYS 17.2 software to apply the boundary conditions of the proposed models at frequencies below 100 Hz. The effects of the mechanical properties of the six layers of soil and the geometric parameters of FMs were studied intensively. Unit cell layouts and an engineered configuration for arranging FMs based on periodic theory to achieve significant results in controlling ground vibrations, which are valuable for protecting a large number of structures or an entire city, are recommended. Prior to construction, protecting a region and exerting control over FM characteristics are advantageous. The results exhibited the effect of the ‘trees’ upper portion (e.g., leaves, crown, and lateral bulky branches) and the gradual change in tree height on the width and position of BGs, which refers to the attenuation mechanism. Low frequency ranges of less than 100 Hz were particularly well suited for attenuating SWs with FMs. However, an engineering method for a safe city construction should be proposed on the basis of the arrangement of urban trees to allow for the shielding of SWs in specific frequency ranges.

Seismic waves (SWs) are one of the physical phenomena that affect people and their property, and because the occurrence of this phenomenon is difficult to prevent, it must be minimized (

In natural disasters, most damage is caused by elastic surface waves (

Natural available metamaterials of forest trees can be used at the subwavelength scale of low-frequency BGs (

The increasing interest in natural metamaterials opens new paths to protect threatened areas of EW propagation. A qualitative investigation has been carried out to study the effects of various geometrical characteristics, including green belt configurations, resonator height, radius, lateral side branches, multilayers of soil, water content, and mechanical properties (i.e., elastic modulus, density, and Poisson’s ratio), on the elimination and control of BG characteristics (

Accordingly, in this paper, 3D simulations of FMs, including their upper parts, are numerically highlighted and modeled in an engineering way. The aim is to identify the role of urban trees in the attenuation of EWs, as well as the effect of different heights of periodic structures upward and downward on creating wide BGs in the frequencies of interest less than 100 Hz. Research methods have shifted to metamaterials, rearranging periodic structures in soil in a specific geometric way. FMs’ extensive range of applications toward urban areas includes the capacity to generate BS and LR BGs. The increasing interest in metamaterials is opening up new paths for future study toward defending major cities and buildings that are in danger from the propagation of EWs. Within the context of the significant interest shown in research activities, an opportunity has arisen to investigate the benefits offered by urban trees in the formation of environmental systems that are free from the effects of noise pollution and secure against the dangers posed by earthquakes and other forms of natural calamity.

This paper introduces a metawall composed of a forest, in which each unit has a unique configuration in terms of height, width, lattice, and distance from the source of EW energy. The proposed model investigates the role of the upper part of forest trees in attenuating SWs. It demonstrates the effectiveness of a 3D simulation of metawall with various configurations to assess the effect of foliage to create BGs on dispersion curves of wave propagation. Furthermore, this study examines how the width and position of BGs are influenced by multiple layers of soil, each with its own mechanical properties. The innovation of this work centers around the use of forest trees as natural metamaterials, termed FMs, for attenuating SWs. This paper introduces an innovative concept: using forest trees, arranged in a periodic pattern around a protected area, as a method for seismic wave suppression. It explores how different configurations of these forest metamaterials, including variations in tree height and the addition of mass to simulate foliage, can create band gaps in low-frequency ranges. This approach is novel in its integration of natural elements into engineering solutions for seismic wave attenuation, providing a sustainable and ecologically friendly method to protect urban areas from seismic activity. The main objective of this study is to determine the optimal distribution pattern of forest trees around designated conservation areas for maximum protection and effectiveness.

This work advances numerically in simulating SWs to address specific problems. It focuses on the effectiveness of FMs under SW excitation at low frequencies less than 100 Hz. The metawall of FMs represents that resonators interact with SWs in substrate soil; thus, behavior toward attenuating SWs in mechanism is called BG in the field of metamaterials, in which FMM is used as a natural material and the low frequency range is extended to 100 Hz. Through swapping waves via the first irreducible Brillouin zone, the features of BG are realized to attenuate SW propagation.

This work is limited to the analysis of SWs by using a 3D model. The velocities of primary and secondary waves differ depending on how particles are distributed along the longitudinal and transverse axes. To avoid complexity, when the band structures of the FM resonator are arranged in a periodic pattern, EWs are attenuated. The lattice is considered a function of the wavelength of EW in the soil substrate. Herein, it equals to 50% of λ_{ω}, where λ_{ω} is the wavelength of EW in the first BG of BS. The commercial code of COMSOL Multiphysics 5.4a software is valuable to solving the model through the solid mechanics study module. COMSOL Multiphysics 5.4a is also adequate to calculate the corresponding eigenvalues of the unit cell frequency modes of the first irreducible Brillouin zone and swap the periodic arrangement in ΓXMΓ as shown in _{ω}, as demonstrated in

Seismic waves and periodic FM barriers,

A periodic structure has a similar effect to the atomic periodic potential field on electrons. On the basis of this similarity, this study directly refers to the lattice in physics to describe the periodicity of structures. The concept of this phenomenon extends periodic structures to nanoscale. A periodic structure is represented as “metawall,” which can be simplified into a cantilever bending wall on soil as locally vertical resonators, FMs. The effects of tree branches and crowns are considered when establishing the numerical model. The influence of roots on scattering is negligible, on the basis that the roots act as a connecting tool between the soil and the trees. Different configurations of a unit cell are demonstrated to represent the unit cell of urban trees in real images. The mechanical properties of unit cell configurations include the following:

Mechanical properties for the proposed FM (

Model |
^{3}) |
E (MPa) | υ |
---|---|---|---|

Foliage | 200 | 23.8 | 0.4 |

FMM | 1,000 | 2000 | 0.3 |

Soil | 2,000 | 200 | 0.35 |

An inhomogeneous mechanism is used by replacing an equivalent “metawall” filled with an isotropic medium of soil (

The mathematically governing equation for the propagation of waves in a medium is considered the same as that in a linear elastic homogeneous medium (

The proposed FMs should satisfy periodic boundary conditions based on Bloch theory as:_{x}, r_{y}, r_{z}) refers to the components of location vector matching on the FM boundary, u(r) is the displacement vector of the “metawall” nodes, and a is the lattice constant. Then, through substituting Eq.

To verify the reliability of the calculation methods in this paper with those applied in the electromagnetic field and postprocessing methods, two previous research models are selected; both are carried out on a nanoscale and are similar to the model studied in this work.

Validation of the dispersion curve through case verification with (

In this section, the focus is on analyzing wave propagation and mode shapes in the context of EWs interacting with forest metamaterials. By employing numerical simulations, the study examines the combined effects of bulk and surface waves, which are fundamental to EW formation due to subsurface complexities. This section delves into the dispersion characteristics of these waves, highlighting their interactions with the metamaterials. EWs’ BG characteristics are predicted via numerical simulations. Various bulk and surface waves combine to form EWs as a result of the subsurface formation’s complexity. Seismic surveys can produce EWs when the shear wave velocity of one layer is substantially lower than the velocity of the layer above it. The mode shape analysis is consistent with periodicity, as shown in

Various model shapes

Bulk and surface wave modes are included in the dispersion curves of the 3D model. SCM is used to differentiate between pure wave modes because of the peculiarities of EW propagation. The surface wave’s dispersion curve is shown under the sound line, and the BGs visible in the blue hatched area are its dispersion gaps. Seismic surface wave BGs and bulk wave modes with extremely high frequencies can be found outside the sound cone. Moreover, the dispersion properties of elastic waves (EWs) and their interactions with forest metamaterials are explored. The mode shapes relevant to the analysis capture both surface and bulk wave dynamics. Emphasis is placed on modeling the upper sections of forest trees, representing tree branches and biomass as vertical loads to simulate foliage effects. This approach aligns the simulation outcomes with real-world forest configurations, ensuring a realistic representation of wave interactions within varied forest densities.

Bulk waves travel down into deeper soil layers, while surface waves travel along the soil surface (_{s} is the shear wave velocity of the upper soil layer, which can be calculated as:

The distribution of energy in Eq. _{1} is the sedimentary soil, and h_{s} is the total height of soil layers; strain energy density is integrated to achieve elastic strain energy (

According to the results of this calculation, wave mode coupling occurs at the resonance attachment point between the longitudinal patterns of FMs and the vertical motion of incoming waves. This coupling causes SWs to move in phase, and the periodic arrangement of trees acts constructively to provide a large BG. The upper section of FM depicts supplementary branches and/or foliage that appear to improve the coupling strength even further. It is demonstrated a significant degree of vibration reduction within the BG frequency by establishing the boundary conditions of the 3D model. The vibration modes in _{1}) and (P_{2}). Owing to the bending of FMs within SCM, not all FM resonances can produce LR BGs. The vibration mode only differs in the direction of vibration and has the same vibration deformation. Several straight bands can be seen in the dispersion curves of

To absorb propagating EWs, a symmetric load of boundary condition is added to the left and right sides of the model, which is perpendicular to the

With regard to physical properties, such as acceleration and displacement in the z-axis, maximum values are known as amplitudes. Over a certain range, ARF exhibits some degree of freedom. FRF, also known as the transfer function, is defined as the ratio of the complex output amplitude to the complex input amplitude. The frequency response function (FRF) represents the output per unit sinusoidal input at a specific frequency for a sinusoidal input. Measuring the frequency response typically entails stimulating the system with an input signal, determining the resulting output signal, calculating the frequency spectra of the two signals, and comparing the spectra to isolate the system’s effect. In this case, the sources are turned on sequentially. It is also found that when the first source excites the FM, FRF can be calculated by dividing the FM response signal by the model without the FM signal. The average value of FRF is also used to describe the efficiency of the attenuation mechanism, and it can be calculated using the following formula:

Mesh size and viscoelastic boundary conditions for 3D metawall with periodic FMM.

The analysis of the dynamic responses of “metawall” models is conducted using the ANSYS17.2 finite element software. In this study, the source of the excitation wave is set up at (2a) away from the closest side surface, and the observation (receiver) line is located at (2a) away from the other side. Viscoelastic boundaries are mainly interpreted as artificial boundaries. A damper for absorbing the energy of the wave propagating toward the boundary is utilized, in which the elastic modulus of the spring and the damping constant of the damper are determined by chosen material parameters of the periodic FMs. A comprehensive mesh convergence study was conducted, employing a user-controlled adaptive mesh for the unit cell. This study ensured that the maximum element size was maintained at less than one-tenth of the wavelength of an incident wave, adhering to rigorous accuracy standards. To scientifically quantify mesh convergence, it is incorporated specific metrics, including the percentage of allowable error. The results showed convergence with a maximum element size of 0.2 m and a finer mesh, which was determined to be adequate for our simulations. Through the ANSYS 17.2 parametric design language, when the viscoelastic boundary conditions are applied, the periodic FMs are set perpendicular to the x-axis and the bottom with the z-axis. A frequency-domain analysis is used instead of a time-domain analysis in COMSOL Multiphysics 5.4a because of the enormous computing time required. The x-direction end face of the wall receives a predetermined displacement. The other end face is subjected to a traction-free boundary condition, and the wall’s lateral faces are subjected to periodic boundary conditions.

Numerical computation of the incident SW values is performed using extensions of the eigenmode method via COMSOL Multiphysics 5.4a. The 3D model, with the mechanical properties presented in _{m}) is 41.5 Hz. The corresponding wavelength

This wavelength is almost twice the lattice constant, or _{t} = 11.5 m, and the radius of the bottom of the trunk is r = 0.23 m. The soil depth should be sufficiently large compared to the tree spacing (_{S2} = 25.4 m, for ground height and, h _{S1} = 4.6 m, for the guiding layer. For the starting frequency represented by P_{1}, the tree and soil exhibit some degree of deformation. The bands are not fully opened, and only BGs in the “ГX” region are determined. The best approach to explain the wave reflections induced by FMs is creating a model and calculating the finite size of “metawall”.

In this study, the FEM is employed to simulate the interaction between seismic waves and forest metamaterials. The FEM allows for detailed analysis of the complex geometries and material properties inherent in the proposed model. It is utilized a discretized model of the physical system, subdividing the forest metamaterials into smaller, manageable finite elements. The results from FEM simulations are instrumental in understanding and optimizing the design and arrangement of forest metamaterials for effective seismic wave attenuation. The simulation modeling is realized via ANSYS 17.2. The viscoelastic boundary elements are accurately modeled to a sufficient level. Boundary finite elements are added to the surfaces parallel to the x direction and the bottom face. To simplify the computation model, periodic boundary conditions are also added to the surface that is orthogonal to the

3D band structure of FMMs:

In the past few years, numerical simulations have been extensively used to determine the effectiveness of periodic structures for the attenuation of EWs and earthquakes in general and to study the factors affecting the attenuation mechanism in particular. Designing periodic structures based on the results of modular analysis is usually a fast, economical, and high-precision method. Therefore, numerical simulations are conducted to examine a group of models of natural metamaterials, arrange them as periodic structures in soil, know the effectiveness of 3D models of these resonators for mitigating the propagation of SWs, and prove the most effective models for attenuating SWs and protecting cities from the dangers of ground vibrations. The mechanism of the attenuation for SW hazard by periodic FMs is applied in this section in consideration of the effects of several factors, including multilayered soil with different properties, “tree–soil” unit cell with various configurations and lattice shapes, FMs’ upper part, bulky side branches, and foliage with crown, and metawall with gradually changing trunk height. The numerical simulation of these factors is performed on the basis of the finite element analysis of the 3D model in five cases.

The simulations in the model of this case are repeated as a transitional step, using soil parameters, via COMSOL Multiphysics 5.4a. Medium dense, dry, uniform sand is chosen as the type of soil because it exhibits the most linear behavior in reality. The assigned soil is still considered as a linear elastic material. On the basis of the assumption that studying and modeling soil layers with inhomogeneity and different consolidated soil characteristics, as well as the universal properties of many layers, are difficult, there may be considerable attenuation in ground vibrations. This condition validates the study’s findings and concludes that resonant trees reduce vibrations better than single homogeneous layers of sedimentary soil, confirming the effectiveness of FMs in attenuating SWs in other stratified ground conditions. To make this study universal, layered soils are considered in the model of

FMs on multilayers of soil strata:

Properties of layers for simulating a multilayer model (

Soil type |
^{3}) |
E (MPa) | Depth (m) | Height (m) | υ |
---|---|---|---|---|---|

Fill | 1830 | 37 | 0–2 | 2 | 0.4 |

Clayey silt | 1939 | 44 | 2–4 | 2 | 0.4 |

Sandy silt | 1888 | 153 | 4–9.0 | 5 | 0.4 |

Silt | 1898 | 141 | 9.0–15.5 | 6.5 | 0.45 |

Sandy silt | 1837 | 90 | 15.5–25.5 | 10 | 0.45 |

Silty clay | 1847 | 155 | 25.5–30 | 4.5 | 0.45 |

The dispersion curves in the direction of ΓX are presented in _{t} = 11.5 m, the radius of the bottom of the trunk is r = 0.3 m, and the dimension of the foliage volume is 1.5 m × 1.5 m × 1 m. The properties of the foliage and the forest metamaterials (FMs) are consistent with those outlined in

The resonant modes of the infinitely periodic resonator act in a way that helps produce low-frequency BGs when periodicity conditions are applied. When the soil medium is replaced with layered soil and the geometric characteristics of this instance are maintained, the resonance of LR depends on the resistance misalliance and coupling of SW modes between the FMs and soil. The increase in the width of BG is attributed to the interaction with the resonant FMs. Because the velocity patterns of SWs change in each layer of a homogeneous medium, surface impulses travel at amplitudes and phase velocities. The ground effect results in deductive and eradicative intervention, causing a rise and minimizing the BG width and having an effective influence on the attenuation mechanism. For acoustically rigid and stiff surfaces, the changes in the magnitude of elastic modulus and in relative bulk density led to a decay in the attenuation ratio, thus improving the effectiveness of periodic barriers. For spongy soil surfaces, such as clay, sandy, forestay sand, and gray sandy clay, ground actions can result in relevant low frequencies. At high frequencies, soft ground can absorb SWs, so the resonance’s amplitude is changed. The attenuation mechanism is determined clearly in the vibration modes of these models, as shown in

Owing to the unstable, irregular movement of trees’ leaves caused by wind and the disparity in the shapes and sizes of foliage and secondary branches, they differ from one species of tree to another (

The LR BGs are highlighted under the sound line. The exact ranges are 7.32–13.71 Hz, 21.18–26.32 Hz, and 37.17–46.62 Hz. In the actual process using the COMSOL Multiphysics 5.4a software, the vibration modes plotted in

The modes of vibration for unit cells indicate noticeable changes in vibration modes at points from P_{1} to P_{8} after damping the surface propagation of EW through the upper part of FMs. In the case of a foliage model, the increase in dimensions of foliage BG width declines. This case presents a detailed investigation into the effect of tree foliage and branches on ground vibration attenuation, particularly at low frequencies below 20 Hz. It demonstrates that these natural elements significantly contribute to the creation of Rayleigh wave band gaps, which are instrumental in reducing ground-borne vibrations. This finding emphasizes the potential of incorporating natural structures into seismic metamaterial design, offering ecological and effective solutions for vibration control. The inclusion of foliage, in particular, introduces vital band gaps at lower frequencies, enhancing the overall efficiency of vibration mitigation strategies. That is, the coupling of FMs and soil deteriorates with a negative effect, which means that the foliage can induce further noise when the incident wave interacts with the structure, especially in wind.

In addition to the considerable influence of layered soil and treetops, another interesting feature appears, which may significantly affect the propagation of directed waves through forestation of FMs by increasing tree heights. In this case, FMs are arranged in a special periodic model, with an increase in FMs’ height from 4 times to 8 times the lattice constant. The FM parameters are as follows: the lattice is a = 2 m, the trunk radius at bottom is r = 0.3 m, the trunk radius at top is 0.5 r, and height changes from 8 m beside the wave sources to 15 m beside the recording point. The soil contains two layers with 4 and 26 m heights from up to down with the same parameters provided on

As the height of FM increases, it interacts with the inducing surface wave, generating extremely wide BGs in the low frequency range of interest below 100 Hz. To determine the variant restrictions of the BGs in the 3D models of FMs, the dispersion curves based on SCM and EDM methods typically establish the BGs’ characteristics. The gradual change in FMs’ height effects results in deductive and eradicative intervention, causing a rise and minimizing the SW attenuation. The sound line divides the dispersion relations of FMs for medium heights into LR and BS BGs. The unit cell motion shows that the vibration is related to FM oscillations. The FMs show a large sinusoidal distortion in LR BGs in the ranges of 24.15–43.86 Hz and 47.18–51.62 Hz and a strong surface distortion, almost as strong as the ground movement of the surface wave below the sound line. On the contrary, such distortion is noticeably attenuated beyond the sound line to generate some of the BS BGs of periodic FMs in many ranges, such as the ranges of 56.21–61.46 and 64.4–79.12 Hz. The analysis finding in the case of gradually increasing FM height results in extremely wide BGs for LR and BS BGs. Although BGs do not appear at low frequencies below 20 Hz, the BGs produced by BS correspond to surface waves converting into shear waves in the deep layers of soil.

This case includes a numerical analysis of a model with varying heights of FMs and considers the cutoff frequency for the propagating wave with decreasing FM height from h_{t} = 15 m toward shorter trees, h_{t} = 8 m. The other parameters used in this case are the same as those presented in case (III). The analysis in this case is suitable to illustrate surface wave propagation along FMs with decreasing stem height, which becomes increasingly adapted to being converted into shear wave in deeper soil layers. The calculation methods here are limited to explaining the effect of gradually reducing FM height. The results show that when a tree reaches a certain height, coupled with the resonant frequency of the vector, it is converted into a common wave, not adapted to pass the cutoff frequency and thus simply reflected back to the lower layers of soil. This strongly asymmetric behavior of surface love waves was shown in a previous study by Maurel et al. (

In this case, the main concern is understanding how the attenuation effectiveness changes as surface waves move in the ΓXMΓ direction. As shown in

This work concluded that FMs acted as a harvesting device for SWs through wide BGs, as evidenced by the hybridization phenomenon in numerical simulation. To this end, surface wave attenuation using FMs has been studied intensively on the basis of periodicity theory. The effects of various parameters, such as layered soil, foliage, and gradually changing tree height, have been comprehensively discussed. Conclusions are summarized as follows:

The surface waves at low frequencies below 100 Hz are refracted and deflected in the unit cell and converted into shear waves, as proven through numerical simulations of different configurations of FMs in this work.

The changes in BGs’ characteristics due to foliage are discernible. The results of additional characteristics are revealed in the case of the response of foliage in the low-frequency range of SWs, in which the first BGs start with small frequencies less than 10 Hz and show strong attenuation of the propagation of SWs. However, their effect becomes negative during wind waves, which causes further noise.

According to the findings of this study, the wave propagation in layered soil results in refracted seismic surface waves at frequencies less than 50 Hz. Thus, the layered soil is believed to be the optimal damper median for surface waves because the waves are reflected multiple times as they travel in the various layers of soil.

The gradual change in FMs’ height results in the rising the starting point of BGs, but it also generates wide BGs for LR and BS BGs.

The local amplification of ground vibration control by FMs can be directly applied to the improvement of urban ecosystem functions and has extremely important application potential in urban ecological construction.

The raw data supporting the conclusion of this article will be made available by the authors, without undue reservation.

QA-S: Conceptualization, Methodology, Writing–review and editing, Writing–original draft, Investigation. JH: Conceptualization, Methodology, Data curation, Writing–review and editing, Supervision, Project administration. MA: Writing–review and editing, Formal Analysis, Resources, Supervision, Project administration, Validation, Funding acquisition. SM: Formal Analysis, Resources, Writing–review and editing, Validation. AA: Conceptualization, Methodology, Formal Analysis, Writing–review and editing, Validation, MA-H: Writing–review and editing, Resources. YG: Writing–review and editing, Funding acquisition, Formal Analysis, Resources. HA: Formal Analysis, Writing–review and editing, Software, Visualization, Validation.

The author(s) declare that financial support was received for the research, authorship, and/or publication of this article. This study is supported via funding from Prince Sattam bin Abdulaziz University project number (PSAU/2023/R/1444).

Authors gratefully acknowledge the Beijing Municipal Education Commission for their financial support through the Innovative Transdisciplinary Program “Ecological Restoration Engineering” and Peiyang Future Scholar Scholarship. The authors also appreciate the financial support given by the Islamic University of Madinah, Saudi Arabia.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

SW, Seismic wave; FM, Forest metamaterial; FEM, Finite element method; BG, Bandgap; EW, Elastic wave; LR, Local resonance; 3D, Three-dimension model; IBZ, Irreducible Brillouin Zone; SCM, Sound cone method; EDM, Energy density method; ARF, Amplitude reduction factor; FRF, Frequency response function.