Numerical Analysis of the Dispersive Behaviour of Buried Elastic Periodic Structures

Abstract

Train-induced vibrations negatively impact residents in nearby buildings and are increasingly recognized as a public health concern. To address this issue, both effective mitigation measures and simplified design procedures are essential. This study investigates the mitigation pattern induced by an array of stiff inclusions employing a modal dispersive analysis. However, applying this type of analysis to a half-space medium presents challenges. To overcome this limitation, a wave-scattering methodology is proposed. This approach enables the computation of the mitigation pattern in a specific direction and at a particular location. It also highlights the conditioning energy content, thereby identifying the key frequency target for attenuation.

1. Introduction

The growing demand for a more efficient and reliable rail transportation network requires higher train operating speeds [1,2]. Technology progress has allowed for the development of ever-faster trains, such as the TGV in France, with an operating speed of 320 km/h, and several others in China travelling at 380 km/h [3], among others. However, higher operating speeds introduce significant drawbacks, as they increase the demands on the infrastructure and the magnitude of ground-borne vibrations. Train-induced vibrations negatively impact residents of nearby buildings and are increasingly being recognized as a public health concern [1]. Consequently, several strategies have been studied over the years [4] in order to reduce the negative effects of these vibrations and, ultimately, the discomfort of residents.

Usually, mitigation measures are grouped according to their location within the track–ground–building system and are typically applied to: (i) the vibration source (track) [5,6]; (ii) the receiver (building) [7,8]; and (iii) the propagation path (ground) [9,10,11]. A new type of mitigation measure applied to the propagation path, known as a seismic metamaterial, has recently attracted increasing attention [12].

The study of metamaterials, which began with the attenuation of electromagnetic waves, have expanded over the last decade to the fields of ultrasounds, thermal fluctuation, acoustics, and elastic waves [13,14,15]. From an engineering perspective, a metamaterial is a structure with particular characteristics, in this case concerning attenuation, that are not found in natural or ordinary materials. In elastodynamics, a solution of periodically embedded inclusions acts as a seismic metamaterial, as it can interrupt the energy propagation within a specific frequency range that a single inclusion cannot attenuate [16]. This filtering effect, known as the Bragg effect, occurs due to the solution’s periodicity and leads to the development of a band-gap (a frequency interval without energy content) [17,18].

Over the last decade, several authors have investigated the band-gaps provided by seismic metamaterials formed by embedded soil inclusions as an effective vibration mitigation measure [19,20,21]. Contrary to what was seen in previous works, Brule et al. [22] designed a seismic metamaterial using a grid of holes in the soil to filter frequencies near 50 Hz. The first experimental evidence to prove the Bragg effect in a seismic metamaterial is attributed to Brule et al. [23]. In a full-scale test, the authors applied in depth a 50 Hz excitation and demonstrated the effectiveness of the grid of holes in absorbing vibrations. The diversity of seismic metamaterials in the literature led Brule et al. [14] to propose four distinct groups: (i) seismic soil-metamaterials [19,20,23]; (ii) buried mass-resonators [24,25]; (iii) above-surface resonators [12,26]; and (iv) auxetic materials [27]. Despite recent advances, the application of seismic metamaterials to attenuate vibrations induced by rail traffic remains an open topic requiring further investigation [28]. Noteworthy is the work of Castanheira-Pinto [16], who assessed the mechanical behavior of a solution composed of periodic embedded inclusions parallel to a railway. Following this concept, Albino et al. [29] conducted an exhaustive parametric study of a periodic arrangement of vertical inclusions, analyzing variables such as penetration depth and soil stratigraphy. Recently, Li et al. [3] evaluated the effectiveness of a seismic metamaterial solution using vertical inclusions to mitigate vibrations from passing trains.

The mechanical behavior of a metamaterial is first evaluated using dispersion theory. This allows determining the frequencies with energy content and, consequently, the band-gaps where waves cannot propagate through the phononic crystal. However, conclusions drawn exclusively from a dispersion diagram are limited, as the theory identifies all possible propagation modes, but cannot distinguish those that are most significant for a given incident direction. To overcome this, numerical modelling of realistic scenarios is required, but these models are often computationally demanding. In this context, and in an attempt to combine the efficiency of dispersion analysis with the accuracy of detailed numerical modelling, this article proposes an advanced dispersion analysis methodology. A step-by-step approach is used to study a mitigation measure consisting of an embedded inclusion parallel to the track. The analysis begins with the 2D behavior in a full-space medium, proceeds to the 3D behavior in a full-space scenario, and concludes with the 3D behavior in a half-space scenario.

Taking the above into consideration, the novelty of this article lies in the proposal of a new methodology to compute the dispersive behavior of continuous media under realistic scenarios, thereby enabling the study of mitigation measures such as metastructures. By applying the proposed methodology, it becomes possible to estimate the metastructure’s mechanical behavior, specifically to identify the frequency bands where attenuation is expected. It should also be highlighted that the proposed methodology can be used to optimize the metastructure’s design since it offers a lower computational cost compared to conventional finite element analysis.

2. Description of the Numerical Model

2.1. Numerical Model: 2.5D FEM-PML

Train-induced vibrations are generally not associated with high strain fields in the soil. Thus, it is appropriate to consider the wave propagation within the linear elastic domain. The fundamental equation governing wave propagation in the space–frequency domain is then expressed by:

$$(\lambda +2\mu )\nabla (\nabla \cdot u(x,y,z,\omega ))-\mu \cdot \nabla \cdot \nabla \cdot u(x,y,z,\omega )+{\omega }^2\cdot \rho \cdot u(x,y,z,\omega )=0$$

(1)

where $u(x,y,z,\omega )$ is the displacement vector, $\lambda$ and $\mu$ are Lamé’s constants, $\rho$ is the mass density of the soil, and $\omega$ is the angular frequency.

Railway tracks are often considered longitudinally invariant structures, as shown in Figure 1a. This allows for the application of transformed techniques to compute the 3D vibration field, eliminating the need to model a large global domain. Indeed, performing a Fourier transform on the longitudinal coordinate converts the spatial domain into a wavenumber domain ($k_1$), which then allows the computation of the 3D vibration field by discretizing merely a cross-section of the structure.

The 3D vibration field is then obtained by combining solutions achieved for different wavenumbers through an inverse Fourier transform:

$$u^{3D}(x,y,z,\omega )={\displaystyle \frac{1}{2\pi }}{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}{u}^{2.5D}(k_1,y,z,\omega )}{e}^{-i(x-x_0)k_1}d{k}_1$$

(2)

where $u^{2.5D}$ and $u^{3D}$ represent the displacement field in the wavenumber–frequency domain and in the space–frequency domain, respectively. Integral convergence, in most cases, is achieved considering wave numbers between −20 rad/m and 20 rad/m. A special reference should be made to the wavenumber $k_1=0$ rad/m, as it represents a 2D plane–strain analysis.

The fundamental wave propagation equation is frequently solved using numerical methodologies, among which the finite element method (FEM) is widely adopted due to its ability to deal with complex cross-section geometries. Implementing the 2.5D concept within a FEM formulation is straightforward, with the global equilibrium equation expressed by:

$$(K(k_1,\omega )-{\omega }^{2}M(k_1,\omega ))\cdot u^{2.5D}(k_1,\omega )=F_n(k_1,\omega )$$

(3)

where $F_n$ represents the applied external nodal forces, and $K$ and $M$ the global stiffness and mass matrix, respectively, which can be computed using Equation (4).

While the finite element method is quite versatile for solving complex geometries, its application to the elastodynamic modeling of continuous media presents a clear limitation: it is unable to consider the infinite nature of the medium. To address this and satisfy the Sommerfeld radiation condition, it is necessary to employ a technique to absorb the energy at the model’s boundaries [30].

Perfectly matched layer (PML) stands out as a powerful technique for dealing with domain truncation. Their implementation involves introducing new variables to the well-known FE equations to determine the element’s mass and stiffness matrices. These variables are known as stretch functions, expressed in Equation (4) by ${\lambda }_y$ and ${\lambda }_z$, which are associated with attenuation in the y- or z-direction. For finite elements without attenuation characteristics, the stretch functions assume a unitary value, which simplifies Equations (4) and (5) to the well-known finite element method equations.

$$K(k_1,\omega )={\displaystyle \underset{z}{\int }{\displaystyle \underset{y}{\int }{B}^T(k_1,{\lambda }_y,{\lambda }_z)\cdot D\cdot B(k_1,{\lambda }_y,{\lambda }_z)\cdot {\lambda }_y(k_1,\omega ,y)\cdot {\lambda }_z(k_1,\omega ,z)dydz}}$$

(4)

$$M(k_1,\omega )={\displaystyle \underset{z}{\int }{\displaystyle \underset{y}{\int }{N}^T(k_1,{\lambda }_y,{\lambda }_z)\cdot \rho \cdot N(k_1,{\lambda }_y,{\lambda }_z)\cdot {\lambda }_y(k_1,\omega ,y)\cdot {\lambda }_z(k_1,\omega ,z)dydz}}$$

(5)

where $D$ refers to the constitutive matrix and N to the shape function matrix.

With the aim of testing the reliability and accuracy of the implemented model, a simple theoretical example was simulated, and the results were compared with the explicit analytical solution proposed by Tadeu and Kausel [31]. Figure 2 shows the main characteristics of the problem, as well as the geometry of the adopted 2.5D finite element mesh. For the numerical analysis, the symmetry and anti-symmetry conditions of the problem were exploited by applying the boundary conditions to the 2.5D finite element mesh, as indicated in Figure 2b.

The infinite medium is excited by a vertical load that is harmonic along the longitudinal coordinate x-axis, with a wavenumber k₁. Figure 3 shows the vertical displacements along the y-axis for dimensionless wavenumbers k₁, defined as k₁ = k₁Cs/ω, at frequencies of 10 Hz and 75 Hz, respectively.

Analyzing the results presented in Figure 3, it can be concluded that the agreement between the numerical and theoretical solutions is excellent, regardless of the wavenumber or frequency.

Additional details on the numerical model and its validation are available in the authors’ previous publications [32,33].

2.2. Dispersion Analysis

By definition, a phononic crystal is a periodic arrangement designed to disrupt wave propagation over a frequency range linked to the periodicity of its section. According to the Floquet–Bloch theory, such periodic structures can be analyzed using exclusively a unit cell, although periodic conditions must be applied along the section’s borders, as shown in Figure 4a [12].

The imposition of these periodic boundary conditions in the aforementioned 2.5D finite element model enables the computation of the dispersion diagram for a selected section. This involves calculating the eigenvalues of Equation (6) for a specific wave vector (a combination of $k_y$ and $k_z$).

$$(K-{\omega }^{2}M)\cdot u^{2.5D}=0$$

(6)

To account for all propagating vibration modes, the wave vector must discretize the highlighted borders of the first irreducible Brillouin zone, expressed in Figure 4b [34]. A detailed explanation of the dispersion diagram computation is provided in the next section.

3. Full-Space Analysis of a 2D Array of Inclusions

3.1. Description of the Case Study

Dispersion diagram analysis offers significant advantages in elastodynamic problems by fully characterizing the mechanical behavior. As previously mentioned, dispersion diagrams define the frequency content for each propagation direction. Consequently, the mechanical behavior observed in a real scenario results from combining the contributions from all propagation directions.

Although elastodynamic problems, particularly ground-borne vibrations, are clearly three-dimensional, it is advantageous to start with the simplified case: a 2D full-space medium. Accordingly, a full-space medium containing an infinite phononic crystal in both directions, as illustrated in Figure 5a, is assumed. Performing a dispersion analysis using Floquet–Bloch theory merely requires the modelling of a unit cell (Figure 5b), given that periodic boundary conditions can effectively replicates its periodicity in both directions. This approach is valuable not only for providing insight into the mechanical behavior, but also for its reduced computational effort due to the considerably smaller required mesh.

3.2. Homogeneous Scenario

The dispersion diagram is determined by calculating the eigenvalues (eigenfrequencies) of Equation (6) for every combination of $k_y$ and $k_z$. The results of this analysis for a scenario without inclusions are presented in the diagram in Figure 6.

The diagram is typically divided into three distinct areas, ΓX, XM, and MΓ, each with distinct wave propagation characteristics. To clarify these characteristics, it is helpful to examine the $k_y$ and $k_z$ values for each extreme direction: Direction Γ represents the consideration of 0 for both wavenumbers, corresponding only to plane waves propagating in the y- and z-directions. Direction X refers to the wavenumber pair of $k_y={\displaystyle \frac{\pi }{a}}$ rad/m, where $a$ is the section thickness expressed in Figure 3a and $k_z=0$. As observed, region ΓX combines the propagation of plane waves in z-direction with different wave configurations in the y-direction, up to the maximum of $k_y={\displaystyle \frac{\pi }{a}}$. Since direction M is achieved by applying $k_y=k_z={\displaystyle \frac{\pi }{a}}$, it can be observed that the XM region comprises the inclination of the resulting wave. Finally, MΓ describes the propagation of waves with a 45° inclination for several wave configurations. By combining the results for each region as explained, the full mechanical behavior for the section can be understood.

Since the results shown in Figure 4 are related to a homogeneous section, there is energy content across the entire frequency range considered and for all directions.

When the strain field in elastodynamic problems is perturbed by a dynamic loading, several seismic waves are generated. It is well established that each seismic wave exhibits a specific configuration, leading to different propagation patterns. Consequently, the analysis of a dispersion diagram in elastodynamic media, as employed in this paper, is crucial for evaluating both the natural frequencies and the vibration mode shapes. Thus, the second and third vibration modes for the pair $k_y=k_z=0$ rad/m are illustrated in Figure 7.

Figure 7 illustrates the dependency between the modal configuration and its originating propagating wave. As shown, the second vibration mode represents a rigid body movement and can only be excited by a P-wave. In contrast, S-waves will induce the third vibration mode for the condition of $k_y=k_z=0$.

3.3. Soft Inclusion Scenario

This paper aims to present a methodology for assessing the mechanical behavior of elastodynamic problems and to investigates the impact of adopting a periodic solution to mitigate ground-borne vibrations. For that purpose, and recognizing that assuming a steel inclusion in the soil comprises a double-effect situation—increasing both stiffness and mass—the individual impact of each was initially assessed. Consequently, two scenarios were modelled, considering the inclusions with a 50% increase in stiffness and density, respectively, related to the surrounding soil. The results were directly compared to the dispersion diagram for the homogeneous case, expressed in Figure 6. The adoption of an inclusion with increased stiffness yielded the dispersion diagram presented in Figure 8a, demonstrating an upward shift in the frequency content compared to the homogeneous case.

This phenomenon is easily understood by reducing the problem to a single degree of freedom, where the natural frequency can be achieved by computing the square root of the stiffness divided by the system’s mass. As such, increasing the overall stiffness will inevitably increase the natural frequencies of the section. However, the increase in vibration frequencies was not homogeneous across all vibration modes, with some showing a higher increase than others. Since some modes explore the bending behavior of the section, local variations in properties are emphasized. Conversely, modes characterized by a rigid body movement, such as the first two, are less sensitive to local properties, experiencing a lower increase in vibration frequencies. An interesting point to highlight is the morphology of the third vibration mode for direction Γ, presented in Figure 8b. Adopting a stiffer material resulted in less distortion in the center of the mesh, where the inclusion is inserted, thereby enhancing the vibration mode’s sensitivity to the stiffness contrast between the inclusion and the surrounding soil.

A similar analysis was conducted for the scenario with increased inclusion density. The dispersion diagram for the first four modes is presented in Figure 9a, with the modes for the homogeneous case superimposed. Increasing the system’s mass produces the opposite result to that identified previously: a reduction in vibration frequencies. Although this outcome contrasts with the previous findings, the justification relies in the same premise, verifying that the increase in mass causes a decrease in the natural frequency. As before, adopting a heavier material resulted in a less distorted mesh (Figure 9b), enhancing the vibration mode’s sensitivity to the local details of the mesh.

Having explored the isolated effects of increased mass and stiffness, an additional model was built that combines both increases. Figure 10 presents the dispersion diagram obtained by adopting a stiffer and heavier inclusion, alongside the dispersion diagram for the homogeneous case.

Four points, indicating the so-called “pseudo-gaps”, are marked in both figures. These represent frequencies at which negligible energy content (or an attenuated response) is expected and their determination is based on the relation between frequency and wave velocity:

$$f={\displaystyle \frac{c}{\lambda }}$$

(7)

where $f$ represents the frequency, $c$ the wave velocity, and $\lambda$ the wavelength.

The prediction of pseudo-gap frequencies, as seen in Figure 10, is associated with a certain propagating direction. For direction X (representing the case of a $k_y={\displaystyle \frac{\pi }{a}}$ rad/m and $k_z=0$), it can be computed by replacing the associated wavelength $\lambda ={\displaystyle \frac{2\pi }{\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$a$}\right.}}=2a$ in Equation (7). Therefore, the pseudo-gap frequency for direction M ($k_y=k_z={\displaystyle \frac{\pi }{a}}$) is obtained by replacing the wavelength in Equation (7) by $\lambda =\sqrt{2}a$. Contrary to what is observed in acoustics, where only a single wave type propagates, elastodynamic media involve the development of several wave types. Consequently, the equation used to determine pseudo-gaps for direction X must be applied to each of the propagated waves, P and S, as expressed below:

$$\begin{gathered} f_P={\displaystyle \frac{c_P}{2a}}=132.1\mathrm{Hz} \\ {f}_S={\displaystyle \frac{c_S}{2a}}=66.7\mathrm{Hz} \end{gathered}$$

(8)

And for direction M:

$$\begin{gathered} f_P={\displaystyle \frac{c_P}{\sqrt{2}a}}=187.4\mathrm{Hz} \\ {f}_S={\displaystyle \frac{c_S}{\sqrt{2}a}}=94.6\mathrm{Hz} \end{gathered}$$

(9)

where $c_P$ and $c_S$ represent the velocity of waves P and S, respectively.

Comparing the results between the reinforced and the homogeneous scenarios (Figure 8), a dissociation of the vibration modes for the pseudo-gap frequencies is observed, evidencing the filtering effect inherent to phononic crystals. Adopting stiffer materials leads to a more pronounced dissociation effect, resulting, as further presented, in a real bang-gap (frequency band without any energy content regardless of direction). While the inclusion promotes heterogeneity in the analyzed section, it is important to emphasize that the morphology of the vibration modes observed in the homogeneous scenario remains unchanged with the adoption of stiffer inclusions. Instead, the frequency at which the mode occurs shifts due to the different material properties. To clarify this concept, the second vibration mode for X direction of the dispersion diagram was determined for both the homogeneous soil and the reinforced one, as shown in Figure 11. As observed, the vibration mode’s morphology remains unchanged, despite occurring at a different frequency.

It is also important to mention that pseudo-gaps are intrinsically related to wave typology, which justifies the development of a shear vibration mode for the P-wave pseudo-gap. This fact illustrates the increased complexity of conducting elastodynamic modal analyses compared to those in acoustics, as the former involves multiple waves with distinct propagation morphologies that condition the response.

3.4. Steel Inclusion Scenario

As previously stated, the article’s main objective is to present a low-computational-cost methodology for evaluating the mitigation behavior promoted by a periodic solution. For this purpose, a conceptual example was created, adopting a steel inclusion as a mitigation measure.

Figure 12 presents the properties of the materials used and the corresponding dispersion diagram. This example is similar to the previous one, but it considers a stiffer inclusion, leading to a greater expression of pseudo-gaps and the development of an effective band-gap. As can be seen, there is a frequency band without energy content, which means a complete attenuation of the response in a real application.

To assess the effectiveness of modal analysis in properly identifying the behavior of a periodic set of inclusions, a wave tube example was constructed. This wave tube included three steel inclusions acting as a mitigation measure.

Considering that the dispersion diagram illustrates the propagation of both P- and S-waves, the wave tube was subdivided into two variants to simplify the discussion of the results: one where the excitation comprised exclusively P-waves and the another consisting only of S-waves. A schematic illustration of the wave tube, along with the finite element mesh adopted for the study, can be seen in Figure 13.

Two calculations were performed for each incident wave: one for a set of three steel inclusions and the another for a homogeneous scenario (reference). The results obtained for P-wave excitation are shown in Figure 14a, while the previously discussed dispersion diagram for the steel inclusion is shown in Figure 14b. From the perspective of modal analysis (Figure 14b), an attenuated response was expected for a frequency range between 110 Hz and 130 Hz (band-gap).

However, according to the wave tube results (Figure 14a), the adoption of steel inclusions develops a wider attenuation (in the range of 80–130 Hz) than that expected from the modal analysis. The wave tube results registered an attenuated response even for a frequency range where the third vibration mode is expected by the modal analyses. This behavior stems from the fact that the system’s third vibration mode is related to the propagation of S-waves, as expressed in Figure 15, and is, therefore, not excited by a P-wave.

A similar analysis was developed where the applied loading corresponded to S-waves. The resulting displacement is expressed in Figure 16a. One aspect that immediately stands out is the existence of energy content for the frequency band related to the third vibration mode, unlike the previous result. Since this vibration mode is activated exclusively by the propagation of S-waves, the result is easily understood. In the frequency range considered in the analysis, the vibration modes depend mainly on P-wave propagation, with the energy content registered for an S-wave being less pronounced and practically attenuated (as seen in Figure 16a). However, it is important to emphasize that the analysis was developed under plane–strain conditions, which may not adequately represent the mechanical behavior for a three-dimensional analysis.

4. Full-Space Analysis of a 3D Array of Inclusions

Considering a more realistic scenario, this section aims to study the three-dimensional behavior of the previous periodic arrangement of steel inclusions. As the present study was performed using a 2.5D FEM model, the imposition of three-dimensional space is easily reached considering $k_1$ different than zero. To better understand this fact, Figure 17 provides a schematic illustration of the wave front for two cases. As can be seen in Figure 17a, for a plane–strain condition, i.e., for $k_1=0$, the incident wave simultaneously impact every point of the modelled section, exhibiting the exact same response for any considered cross-section. Conversely, Figure 17b illustrates the wave front configuration for $k_1\ne 0$ rad/m, where an inclination is observed when compared to the 2D case. In this context, the incident wave strikes different points of the modelled section at varying times, leaving to a three-dimensional behavior. Bearing this in mind, the dispersion diagram for the steel inclusion case, previously assessed under plane–strain conditions ($k_1=0$ rad/m), was now determined for $k_1=0.02$ rad/m.

To differentiate the three-dimensional vibrational behavior, the modes from the two-dimensional analysis (Figure 12b) are superimposed on the 3D dispersion diagram in Figure 18. The comparison reveals additional vibration modes not identified in the 2D case, which are associated with energy propagation along the longitudinal direction. Furthermore, a notable reduction in the band-gap is observed in the 3D analysis compared to the 2D result.

To better understand the new vibration mode shapes, Figure 19 presents three snapshots of the third vibration mode, which develops around 11 Hz for Γ direction. This mode is characterized by a longitudinal vibration of the inclusion, a phenomenon that is only active within a three-dimensional vibration field. Consequently, because the motion is exclusively longitudinal, the resulting vertical and horizontal displacements are negligible. This outcome demonstrates an inherent complexity of modal analysis in continuous media, where the presence of modal energy is not a sufficient condition to predict the response magnitude across all displacement components.

Considering that $k_1$ represents the front angle of the waves relative to the modeled section, it is relevant to evaluate the system response pattern as higher $k_1$ values are considered, which reflects the incidence of more inclined waves. To this end, the dispersion diagram for a $k_1=0.20$ rad/m is presented in Figure 20. As observed, the response pattern at this higher $k_1$ is clearly different from the previous case, with a distinct shift in the energy content. Notably, the band-gap is entirely suppressed, indicating that, for larger incidence angles, the interaction among inclusions is completely lost. This phenomenon has been previously reported by the authors [16], who noted that for oblique wave incidence, the attenuation mechanism is not provided by a group effect but by a guided phenomenon throughout the inclusion.

5. Half-Space Analysis of a 3D Array of Inclusions

Until this point, the analyses have assumed exclusively P- and S-wave propagation, since the scenario consisted of a full-space medium. However, such configuration does not fully exploit the potential advantages of periodic inclusions as a mitigation solution. From an engineering perspective, surface traffic scenarios are more relevant, as they are particularly suitable for adopting inclusions arrays aligned to the track. In the presence of a free-surface, additional wave types are excited, making the wavefield significantly more complex. To address this, this section adopts the half-space scenario presented in [16], providing a more realistic context for modal analysis. Although the referenced work involves a completely different type of analysis, it will be used in this section for result validation, as the wave scattering analysis proposed in this section is expected to lead to the same conclusions as those obtained from a fully numerical simulation.

Figure 21 shows the geometric properties that allow a modal analysis to be performed on just a slice of the section.

As previously mentioned, the material properties adopted for the present study are identical to those specified in [16] and are provided in

Table 1. Material properties adopted for the study.

Material	Density (kg/m3)	Young’s Modulus (MPa)	ν (-)	ζ (-)	Cs (m/s)
Soil	1700	116	0.33	0.001	160
Inclusion	2700	4416	0.2	0.001	825.5

where ν and ζ refer to Poisson’s ratio and hysteretic damping coefficient, respectively.

.

As can be easily seen, half-space media only require the prescription of periodic boundary conditions in one direction, since the other, in this case the top border, is free. At the top and bottom borders, free-surface (null stress) and null-displacement boundary conditions are imposed, respectively. In this context, to compute the dispersion diagram in a half-space scenario, it is only necessary to consider the variation in $k_y$_, which can be discretized in the ΓX region. Thus, Figure 22 illustrates the resulting dispersion diagram for a plane–strain analysis ($k_1=0$ rad/m) considering the stiffer inclusions from [16], as expressed in Table 1. As shown, the result is completely different from the full-space medium, as it is not possible to identify any gap in the frequency range, unlike in the previous case (Figure 18). The complexity of this response pattern arises from two distinct factors: first, the modelled section is substantially larger than the previous ones, leading to a greater number of vibration modes; second, the surface induces not only propagation modes associated with P- and S-waves, but also Rayleigh waves.

As already mentioned, the modal analysis in elastodynamic media presents significant challenges. The presence of energy in a vibration mode does not necessarily imply that this vibration mode governs the response in a specific direction. Consequently, extracting direct and meaningful information from the dispersion diagram, particularly regarding the attenuation behavior, is generally not feasible, except in very simplified cases. To address this limitation, while maintaining the computational efficiency associated to small FE meshes, the authors propose an alternative procedure. This approach involves computing the nodal displacements in response to a unit point load, as illustrated in Figure 23a. In this context, the dispersion diagram is determined by solving the systems of equations for $k_y$-frequency pairs. Finally, the directional nodal dispersion diagram is achieved by plotting in a color scale the displacements obtained for every $k_y$-frequency pair.

To emphasize the difference between the proposed analyses, the cross-section presented in Figure 21a was also solved for a homogenous case and the dispersion wave previously described by a modal analyses, with the results superimposed and expressed in Figure 24. Since modal analysis does not inherently fulfill the radiation condition of Sommerfeld, two scenarios were considered: one including an absorption layer at the bottom of the mesh and one without it. It is important to clarify that the color dispersion diagram represents the vertical displacement at a specific receiver point, whereas the modal analyses determine all energetic modes, regardless of the excited direction.

Although the dominant energetic mode remains unchanged between the scenarios with and without a bottom absorbing layer, the influence of wave reflections at the bottom of the section is clearly noticeable. Fictitious energetic modes can arise when the Sommerfeld radiation condition is not met, potentially leading to distorted conclusions. Nevertheless, a significant advantage of employing wave scattering analyses over a modal approach is the ability to determine the energy content for the direction of interest, while simultaneously preserving the modal information of the analysis.

Another noteworthy aspect is the complexity observed in the response pattern of a continuous medium. As shown in Figure 24, the first vibration mode exhibits high energy in the vertical direction up to a certain frequency. This conclusion is supported by the superposition of the first vibration mode predicted by modal analysis (dashed lines) with the energetic mode identified through wave scattering analysis. However, for frequencies above 60 Hz, no single predominant mode of vibration is apparent. Instead, the response appears as a combination of several modes. Given that the present case refers to a homogeneous scenario, no attenuated frequency is expected, as every frequency has a certain energetic value of $k_y$. This detailed assessment of the vertical response would have been impossible to achieve using exclusively the results provided by a modal analysis.

Following a similar strategy, the numerical dispersion diagram was determined for the previously presented scenario with three stiffer inclusions. It is important to note that the properties assumed for the study are precisely the same as those assumed in the previous authors’ article [16]. Figure 25 presents the numerical dispersion diagrams for both the homogeneous case, previously shown in Figure 24, and for the inclusion case obtained under plane–strain conditions.

The effectiveness of the mitigation measure can be assessed through a direct comparison of the obtained dispersion diagrams. The adoption of stiffer inclusions not only decreases the energy magnitude of the vibration modes, but also significantly inhibits the energy propagation within specific frequency bands. These inhibited bands are the so-called band-gaps, and, in the present example, a clear band-gap is observed between 60 Hz and 70 Hz. A reduction in the response spectrum between 70 Hz and 100 Hz is also evident. Considering that the article that served as the basis for the study developed in this section addresses the attenuation effect in a realistic scenario, caused by the adoption of an array of inclusions distributed horizontally (Figure 21a), it is possible to compare the results obtained in this paper with those shown in [16].

In [16], the insertion loss for the section depicted in Figure 21a was presented. This was computed for a set of dimensionless wave numbers using the following equation:

$$K1(-)={\displaystyle \frac{c_S}{\omega }}{k}_1$$

(10)

where $K1$ is the dimensionless wave number, and $\omega$ (rad/s) is the frequency applied.

The insertion loss of the vertical displacement presented in the article is expressed in Figure 26. In this figure, the red colors refer to attenuated areas and the blue ones occur when the mitigation solution’s effect was not observed. To establish a comparison between the dispersion diagram (Figure 25b) and the insertion loss obtained for a real scenario (Figure 26), the non-dimensionlization of the wave number, as established in the present analysis, is required. As previously mentioned, the dispersion diagrams from Figure 25 were computed for a plane–strain condition, i.e., $k_1=0$. Using Equation (10) to convert it into the dimensionless wavenumber, it is possible to conclude that it corresponds to the first horizontal line plotted in Figure 26.

Based on these considerations, it can be observed that, for the 2D case ($k_1=0$), the insertion loss diagram predicts a significant attenuation of the response between 60 Hz and 70 Hz, which corroborates the information obtained from the dispersion diagram (Figure 25b). Attenuation is also present in the 70 Hz–90 Hz range, although with less expression than in the previous interval, while a considerable attenuation is observed again for frequencies above 90 Hz. These findings are consistent with the energy distribution depicted in Figure 25b, where a considerable decrease in the energy content occurs between 80 Hz and 90 Hz, followed by a pronounced drop beyond 90 Hz.

As presented in Figure 26, two additional curves, corresponding to $k_1=0.45$ rad/m and $k_1=0.8$ rad/m, are superimposed on the insertion loss diagram. The corresponding numerical dispersion diagrams are presented in Figure 27. The most evident feature in this dispersion diagram is the upward shift of the first vibration mode, which is activated for higher frequencies with the increase in the longitudinal wavenumber ($k_1$). This behavior is easily justified with the insertion loss diagram presented in Figure 26. This diagram shows that both wavenumber curves ($k_1=0.45$ rad/m and $k_1=0.8$ rad/m) in the low-frequency range have a dimensionless wavenumber ($K1$) that falls either within the evanescent region or the zone that triggered the wave guidance phenomenon along the inclusion, as explained in [16]. Consequently, only frequencies above 25 Hz lead to $K1$ values for a $k_1=0.8$ rad/m outside the wave-guided region, and, therefore, express energy content, as proved in Figure 27b.

Another noteworthy aspect is the band-gap previously identified between 60 Hz and 70 Hz. This band-gap experiences a slight upwards shift, indicating that the attenuation effect is triggered for slightly higher frequencies. Although this variation is practically residual, it is corroborated by the insertion loss diagram obtained for a realistic scenario.

A final point concerns the computational cost of the two analyses described above. As shown, the dispersive behavior of the metamaterial in this section was obtained by two entirely different methods. First, a fully resolved finite-element simulation was performed; second, the dispersive analysis proposed in this article. The computational resources used for these analyses consisted of a system equipped with a 12th Gen Intel (R) Core (TM) i7-1255U processor and 16 GB of RAM. Under these conditions, the full finite-element simulation requires a mesh of 21,725 nodes and took approximately 9000 s to compute (for frequencies between 1 and 100 Hz, and 31 wavenumbers). By contrast, the wave-scattering method introduced in this paper exploits the metamaterial’s periodicity, reducing the mesh to just 2082 nodes and cutting the processing time to 780 s for the same range of frequencies and wavenumbers. This significant reduction in computational time is a clear advantage of the approach presented in this work.

6. Conclusions

The mitigation behavior induced by an array of stiff inclusion was assess by a modal dispersive analysis. Initially, a 2D full-space modal analysis was performed. This revealed that introducing steel inclusions into the adopted medium lead to a complete attenuation, known as band-gap, for the frequency range from 100 Hz to 120 Hz. However, a wave tube example showed a wider band-gap than the one expressed in the modal analysis for the P-wave application. This discrepancy illustrates the inherent complexity of wave propagation in elastodynamic media, as the generation of a specific wave typology is intrinsically linked to the configuration of the external excitation. Consequently, energetic modes induced by S-waves propagation will not be activated by P-wave excitation.

The subsequent step involved the computation of the mitigation behavior of stiffer inclusions in a 3D full-space scenario using modal dispersive analysis. In this case, new vibration modes appeared, distinct from those identified in the 2D analysis, specifically consisting of longitudinal vibration modes of the inclusion. A loss of efficiency in the mitigation pattern was also observed, evidenced by a reduction in band-gap width, when inclined waves were considered.

Finally, a 3D half-space analysis was developed, which revealed a complex behavior. Given that the free surface induces the appearance of new vibration modes related to the propagation of surface waves, the resulting dispersion diagram is uninterpretable. To overcome this limitation, the authors proposed an approach to compute the dispersion diagram through a wave dispersion analysis. This method allows for the identification of which energy content is conditioning the response in a given direction and at a specific point. To validate the accuracy of this methodology in computing the mitigation pattern induced by an array of inclusions, a comparison was established with a result previously publish by the authors in [16].

The proposed wave-scattering methodology accurately reproduces the attenuation patterns reported in [16], which thereby provide a rigorous validation of the proposed numerical model. Unlike conventional dispersive modal analyses, which cannot resolve vibrational energy modes along a specific direction, the proposed wave-scattering method directly computes the corresponding energy distribution. This enables the determination of attenuation capabilities for a given direction. The results offer promising insights that may support future efforts in designing and optimizing metamaterial geometries and constitutive properties, for which further parametric investigations are recommended.