Vibration mitigation performance and layout parameter optimization of honeycomb WIB under low-frequency vibrations induced by vibratory rollers

Abstract

To enhance the vibration mitigation performance of honeycomb wave impeding blocks (HWIB) against low-frequency environmental vibrations induced by vibratory rollers, this study systematically investigates the influence of material selection and layout parameters on HWIB vibration reduction effect through bandgap theory, numerical simulation, and backpropagation neural network-genetic algorithm. Key findings include: 1) the HWIB with rubber core-foam shell composite structure achieves 91% reduction in root mean square (RMS) acceleration at monitoring points; 2) three effective vibration isolation bandgaps (0-6 Hz, 11-14 Hz, and 19-21 Hz) are identified along the M-K direction; 3) filling depth and filling thickness are identified as critical layout parameters; 4) backpropagation (BP) neural network-genetic algorithm(GA) optimization further reduces RMS acceleration and acceleration power spectral density (PSD) amplitude by 9.8% and 26.7% respectively, demonstrating significant optimization effectiveness. The research provides technical support for downsizing vibration isolation trenches, improving low-frequency vibration control, and promoting HWIB applications in construction sites.

Keywords

subgrade engineering vibratory rollers HWIB damping material layout parameters backpropagation neural network-genetic algorithm bandgap theory

1. Introduction

The low-frequency environmental vibrations induced by vibratory roller construction adversely affect adjacent buildings and residents’ quality of life. Research indicates that roller-induced vibration energy primarily concentrates in the 0-35 Hz frequency range, with dominant frequencies distributed between 9-14 Hz.¹ Conventional vibration isolation trenches demonstrate limited effectiveness in controlling such low-frequency vibrations. While vibration isolation trenches (open trenches and filled trenches) attenuate energy through vibration wave reflection and scattering,² their performance is constrained by material properties and geometric dimensions: open trenches achieve less than 30% vibration reduction below 10 Hz,³ and filled trenches require depths equivalent to 1.5 times the Rayleigh wavelength to meet specifications,⁴ resulting in skyrocketing engineering costs and poor adaptability to confined construction sites. Consequently, developing efficient and compact low-frequency vibration isolation technologies has become an urgent demand in subgrade engineering.

In the broader context of structural vibration control, continuous efforts have been devoted to developing innovative vibration mitigation strategies to meet increasingly stringent engineering demands. For example, Zhang⁵ proposed an active rotary inertia driver system for flutter vibration control of bridges, demonstrating the growing demand for advanced vibration control technologies capable of addressing complex dynamic problems in practical engineering applications.

Wave Impeding Blocks (WIBs) offer innovative solutions for low-frequency vibration mitigation through wave impedance mismatch mechanisms that disrupt vibration transmission paths. Richard⁶ first proposed thin-walled circular borehole wave barriers, yet Woods⁷ identified practical limitations in which single-row pile barriers required pile spacing of less than 1/4 wavelength. Subsequent improvements include discontinuous barriers by Yang et al.,⁸ double-wall structures by Van Hoorickx et al.,⁹ and trench-WIB composite barriers by Zhou et al.,¹⁰ though their low-frequency performance remains constrained by material-structural interactions.

More recently, periodic energy-dissipating structures have attracted increasing attention for vibration mitigation due to their ability to manipulate wave propagation through bandgap mechanisms. Deng et al.¹¹ demonstrated that periodic additive acoustic black-hole arrays can achieve effective vibration damping by inducing local energy concentration and dissipation, highlighting the strong potential of periodic configurations for low-frequency vibration control. This line of research conceptually aligns with the periodic design philosophy of honeycomb wave impeding blocks. Honeycomb WIB (HWIB) exhibit unique advantages through periodic bandgap characteristics: Takemiya et al.^12–16 verified through numerical simulations and field tests that HWIB exhibit superior vibration mitigation effectiveness against low-frequency vibrations (3-5 Hz) compared to conventional pile rows; Gao et al.¹⁷ further developed the perforated wave barrier (DXWIB) system, extending isolation bandwidth through optimized burial depth and thickness. Material innovations including foam,¹⁸ sand-rubber mixtures,¹⁹ and rubber-concrete composite structures²⁰ significantly enhance mitigation performance. However, existing research predominantly focuses on the material selection of HWIBs or the structural form of HWIBs, lacking systematic exploration of layout parameters. Consequently, practical applications largely rely on empirical design, making it difficult to fully exploit HWIBs’ performance potential.

In parallel, considerable efforts have been devoted to the vibration analysis of complex structural systems subjected to low-frequency dynamic excitation. Talebitooti et al. systematically investigated the dynamic response and vibration characteristics of curved and layered structures using combined analytical and numerical approaches. Their early studies established high-fidelity vibration models to capture modal characteristics, frequency-dependent response, and wave transmission mechanisms under external excitations, highlighting the strong influence of structural geometry and boundary conditions on low-frequency vibration behavior.^21,22 More recently, this research framework was further extended to complex cellular and metamaterial-inspired structures, where the coupling between material properties, structural configuration, and wave propagation characteristics was shown to play a decisive role in vibration attenuation performance.²³ These studies demonstrate that effective low-frequency vibration mitigation cannot rely solely on empirical design or single-parameter tuning, but instead requires systematic vibration modeling and multi-parameter coupling analysis.

Several recent studies emphasize both the challenge of low-frequency isolation and the need for multi-parameter optimization under dynamic loading. Lu et al.²⁴ investigate a self-powered nonlinear vibration isolation system with a two-stage configuration designed to achieve broadband low-frequency vibration suppression through nonlinear dynamic mechanisms. Momen et al.²⁵ carried out multi-objective optimization of piezoelectric cantilevers under tire rotational excitation and demonstrated the complexity of optimizing coupled electromechanical parameters for target spectra. These works collectively underline that low-frequency mitigation often requires consideration of nonlinear dynamics, and that employing multi-objective and multi-parameter optimization in the design is crucial for achieving practical performance and accuracy.

In summary, despite the demonstrated effectiveness of periodic vibration mitigation concepts and HWIB-based barriers, two key research gaps remain. The first is the matching between bandgap characteristics and the vibration spectrum of vibratory rollers; the second is the strong coupling effects among layout parameters (filling depth, filling thickness, unit cell thickness, etc.), where traditional methods like orthogonal experimental design tend to converge to local optima.

This study makes three main contributions. Firstly, based on three-dimensional modeling, rubber and foam are selected from commonly used engineering materials as the core and shell materials of the HWIB. Secondly, the bandgap range of HWIB is specifically designed for the vibration spectrum of vibratory rollers, and its effectiveness in controlling low-frequency vibrations is verified through vibratory drum-HWIB-subgrade coupled model. Thirdly, a BP-GA optimization algorithm for layout parameters of the HWIB is presented.

This study, based on bandgap theory, numerical simulation, and BP neural network–genetic algorithm, systematically investigates the influence of HWIB material selection and layout parameters on the vibration mitigation effect against low-frequency environmental vibrations induced by vibratory rollers. First, a three-dimensional numerical model of HWIBs with different material combinations is established to select the optimal combination of core and shell materials. Second, according to HWIB vibration isolation bandgap theory, the bandgap characteristics of the selected scheme are calculated using COMSOL Multiphysics. Third, a vibratory drum-HWIB-subgrade coupled model is constructed, and in combination with a vibration mitigation effectiveness evaluation system, the mitigation performance of the selected scheme is comprehensively assessed. Finally, on the basis of orthogonal experiments, a BP neural network–genetic algorithm model is constructed to perform optimization of the HWIB layout parameters under the selected scheme.

2. Theoretical framework and methodology

2.1. HWIB vibration isolation bandgap theory

Assuming that the honeycomb periodic structure is an ideal elastic medium, satisfying the assumptions of perfect elasticity, homogeneity, isotropy, continuity, and small deformation, and under the condition of no initial external forces, the elastic wave equation is established as follows²⁶:

ρ ü = (λ + μ) \nabla (\nabla \cdot u) + μ \nabla^{2} u

(1)

where

ρ

is the density of the medium,

ü

is the second derivative of displacement with respect to time,

λ

and

μ

are the Lamé constants, and

u

is the displacement vector.

The unit cell of the honeycomb periodic structure represents the smallest periodic unit. In the plane, the entire honeycomb periodic structure can be generated by translating the unit cell along the non-coplanar vectors $a_{1}$ , $a_{2}$ , $a_{3}$ . The direct lattice vector $R$ intuitively characterizes the periodicity of the structure (Figure 1)²⁷:

R = n_{1} a_{1} + n_{2} a_{2} + n_{3} a_{3}

(2)

where

n_{i} (i = 1, 2, 3)

is an integer.

Because the honeycomb periodic structure exhibits spatial periodicity, the physical quantities at any point are correspondingly periodic²⁷:

F (r + R) = F (r)

(3)

where

r

is the position vector,

R

is the direct lattice vector, and

F (r)

is a periodic function. After Fourier transformation,

F (r)

can be expressed as²⁷:

F (r) = \sum_{G} f (G) e^{i G r}

(4)

where

G

is the reciprocal lattice vector,

G \cdot R = 2 n π

, n is an integer.

The vectors in the Fourier reciprocal space are expressed as²⁷:

b_{1} = 2 π \frac{a_{2} \times a_{3}}{a_{1} \cdot (a_{2} \times a_{3})}

(5)

b_{2} = 2 π \frac{a_{3} \times a_{1}}{a_{1} \cdot (a_{2} \times a_{3})}

(6)

b_{3} = 2 π \frac{a_{1} \times a_{2}}{a_{1} \cdot (a_{2} \times a_{3})}

(7)

Accordingly, the expression of the reciprocal lattice vector is obtained as (Figure 2)²⁷:

G = h_{1} b_{1} + h_{2} b_{2} + h_{3} b_{3}

(8)

where

h_{i} (i = 1, 2, 3)

is an integer.

As shown in Figure 3, a rhombus with an acute angle of 60° and side length a₀ is used to extract a portion of the periodic structure containing a complete unit cell for studying its eigenfrequencies and modes²⁷:

a_{1} = a_{0} a_{x}

(9)

a_{2} = a_{0} (\frac{1}{2} a_{x} + \frac{\sqrt{3}}{2} a_{y})

(10)

where

a_{x}

and

a_{y}

are the unit vectors along the x- and y-directions, respectively.

According to the calculation principle of reciprocal lattice vectors, the following expression can be obtained²⁷:

b_{1} = k_{0} (\frac{\sqrt{3}}{2} a_{x} - \frac{1}{2} a_{y})

(11)

b_{2} = k_{0} (a_{y})

(12)

where

k_{0} = \frac{4 π}{\sqrt{3} a_{0}}

According to Bloch’s theorem, the wave equation of the honeycomb periodic structure can be reformulated as:

u (r, t) = e^{i (k \cdot r - ω t)} u_{k} (r)

(13)

where

k

is the wave vector in Fourier reciprocal space;

ω

is the angular frequency (also referred to as the eigenvalue),

u = (u_{x}, u_{y}, u_{z})

is the displacement vector, and

t

is the time parameter.

According to the Bloch–Floquet theorem, the periodic boundary condition can be expressed as:

u (r + a) = {e^{i}}^{(k \cdot a)} u (r)

(14)

where

a

is the structural periodic constant.

The calculation of bandgaps for the honeycomb periodic structure using the finite element method is essentially equivalent to solving the unit cell eigenvalue equation derived from the simplified elastic wave equation in Eq. (13).²⁸

(K - ω^{2} M) U = 0

(15)

where

ω

is the eigenvalue,

K

the stiffness matrix,

M

the mass matrix, and

U

the displacement matrix.

In the honeycomb periodic structure, the wave vector $k$ can be given by the first irreducible Brillouin zone (Figure 4):

k = {\begin{cases} 0 \to 1; K \to Γ \\ 1 \to 2; Γ \to M \\ 2 \to 3; M \to K \end{cases}

(16)

By applying the principles of vector algebra, the boundary vectors of the first irreducible Brillouin zone can be expressed as²⁷:

K = \frac{2}{3} b_{1} + \frac{1}{3} b_{2} = (2 / 3, 1 / 3)

(17)

Γ = 0 b_{1} + 0 b_{2} = (0, 0)

(18)

M = \frac{1}{2} b_{1} + \frac{1}{2} b_{2} = (1 / 2, 1 / 2)

(19)

Thus, the piecewise vector expression of the wave vector $k$ is given by:

k = α k_{b 1} + β k_{b 2}

(20)

α = {\begin{cases} \frac{2}{3} (1 - k); 0 < k < 1 \\ 0.5 (k - 1); 1 < k < 2 \\ \frac{1}{6} (k + 1); 2 < k < 3 \end{cases}

(21)

β = {\begin{cases} \frac{1}{3} (1 - k); 0 < k < 1 \\ 0.5 (k - 1); 1 < k < 2 \\ - \frac{1}{6} (k - 5); 2 < k < 3 \end{cases}

(22)

the periodic boundary conditions are applied to the boundaries of the rhombic truncation of honeycomb structure. Subsequently, the wave vector

k

is parametrically scanned along the boundary

K - Γ - M - K

of the first irreducible Brillouin zone, from which the dispersion curves and bandgaps can be obtained.

2.2. Vibratory drum-HWIB-subgrade coupled model

2.2.1. Construction of numerical model

A vibratory drum-HWIB-subgrade coupled numerical model was established in ABAQUS finite element software, with global dimensions of 30 m (length) × 30 m (width) × 20 m (height). The subgrade has a thickness of 0.3 m and the composite subgrade has a thickness of 2.0 m, both with a width of 11 m, as shown in Figure 5. The vibratory drum, modeled as a rigid body with a diameter of 1.2 m, operates along a 20-meter compaction path at 3.6 km/h (1.0 m/s), with an angular velocity boundary condition applied at its centroid to simulate rolling motion. The HWIB consists of a rubber-filled core and foam shell, with material parameters detailed in Table 1. The subgrade uses field-prepared Group A fill, while the composite subgrade employs field-prepared Group B or C fill as per design requirements, and their mechanical properties are listed in Table 2.

Table 1.

Numerical simulation parameters of filling materials.

Material	Elastic modulusE/MPa	Poisson’s ratio $μ$	Density $ρ / (k g / m^{3})$	Damping ratio $ξ$	Filling type
Fine sand	75	0.35	1950	0.15	Core
Foam	2	0.25	30	0.12	Shell/Core
Rubber	2.85	0.4	1100	0.30	Shell/Core
Concrete	19085	0.2	2400	0.01	Shell
CFG	15000	0.2	2300	0.05	Shell

Table 2.

Finite element numerical model parameters.

Structural layer	Elastic modulusE/MPa	Poisson’s ratio $μ$	Densit $ρ / (k g / m^{3})$	Damping ratio $ξ$	Shear wave velocity $v_{s} / (m / s)$
Subgrade	105	0.30	1900	0.07	145.8
Composite subgrade	100	0.30	1850	0.07	144.2
Natural soil foundation	40	0.25	1850	0.08	135.2
Vibratory drum	212000	0.288	7860	-	-

The vibratory drum-subgrade contact used hard contact in a normal direction; tangential behavior adopted penalty friction (μ=0.2) with separation allowed. The HWIB-soil interface applied hard contact for normal interaction; tangential constraint is defined as rough friction (no slip). Both subgrade and HWIB were meshed with C3D8R elements (8-node reduced-integration hexahedral), uniform size 0.5 m, total elements ≈120,000. Infinite element boundary with CIN3D8 elements and a 5 m-thick transition layer eliminated wave reflection. The explicit dynamic solver ran with time step 1×10^-3 s for 20 s.²⁹

2.2.2. Experimental validation of numerical model

The test section is located in Xiaobi through Qingzhen East to Baiyun Southwest Loop Railway Guiyang Hub Contact Line Jinyang Station D1K62+516.82 ∼ D1K62+750 interval, both sides of the distribution of dense no frame foundation single brick wall structure of the private house group, of which the foot of the slope of the pile D1K62+516.82 fill roadbed from the private house from the vertical distance of only 7.8 m. The original site of the soil foundation mechanical properties of the weak and the existence of humus layer, using 2 m depth replacement treatment, typical cross-section design diagram as shown in Figure 6. The subgrade is filled in layers, each layer is compacted to a thickness of 30 cm, and the construction machinery is SR22M vibratory roller, with parameters as shown in Table 3.

Table 3.

SR22M vibratory roller parameters.

Item	Parameter
Mass	22000 kg
Front drum mass distribution	10700 kg
Vibratory drum width	2.14 m
Vibratory drum diameter	1.55 m
Wheelbase	3.14 m
Excitation frequency	28 Hz -32 Hz
Eccentric force	230 kN -395 kN

The monitoring section was set up at D1K62+516.82 pile number, and the 30 cm aggregate layer was paved after the compaction of the replacement fill reached the standard, and one monitoring point was set up at the distance of 2.5 m, 5.0 m and 7.5 m along the foot of the slope, and the spatial distribution is shown in Figure 7. L20-S vibrometer was used together with the TG-3 three-component acceleration transducer (frequency range of 0.1-500 Hz, resolution of 0.1 mg), and 100 ksps parallel acquisition and signal processing (including filtering, frequency domain conversion and integral operation) were realized by the Blasting Vibration Analysis System. The Blasting Vibration Analysis system realizes 100 ksps parallel acquisition and signal processing (including filtering, frequency and time domain conversion and integration operation). The spatial relationship between the monitoring points and the residential houses and the arrangement of the instruments are shown in Figure 8.

Figure 9 compares the acceleration time-history curves between numerical simulations and field tests under vibratory roller excitation at vertical offsets of 2.5 m, 5.0 m, and 7.5 m. The waveform distributions exhibit strong consistency. The relative errors between numerical and experimental results increase with vertical offset: 2.5 m (4.47%), 5.0 m (6.23%), and 7.5 m (9.51%), all below 10%, thereby validating model reliability.

2.3. Vibration mitigation effectiveness evaluation system

2.3.1. Time-domain and frequency-domain evaluation metrics

The root mean square (RMS) acceleration quantifies the overall intensity of vibration energy in the time domain. By comparing the RMS acceleration values without and with HWIB, the vibration mitigation effectiveness can be directly evaluated. The formula is³⁰:

A_{r m s} = \sqrt{\frac{1}{T} \int_{0}^{T} {[A_{i} (t)]}^{2} d t}

(23)

where

A_{r m s}

is the RMS acceleration,

T

is the time duration,

A_{i} (t)

is the instantaneous acceleration value at a given time.

The acceleration power spectral density (PSD) characterizes the distribution of vibration energy in the frequency domain. By analyzing the acceleration PSD curves in attenuation stage and filtering stage, the suppression capability of HWIB across different frequency ranges is assessed. The calculation adopts the filtered white-noise method³¹:

G_{q} (f) = G_{q} (n_{0}) n_{0}^{2} \frac{u}{{(f + f_{0})}^{2}}

(24)

where

G_{q} (n_{0})

is the road surface roughness power spectral density at the reference spatial frequency

n_{0}

, with units of m³,

n_{0}

is 0.1 m^-1,

f_{0}

is the lower limit of cut-off spatial frequency for rough road surface, which is 0.01 m^-1,

u

is the vehicle speed, with units of m/s.

2.3.2. Chaotic system analysis of environmental vibrations

In chaos theory, the Lyapunov exponent³² is employed to describe the dynamical characteristics of the system: The sign and magnitude of the Lyapunov exponent $λ_{i}$ along a certain direction describe the average rate of divergence ( $λ_{i} > 0$ ) or convergence ( $λ_{i} < 0$ ) of adjacent trajectories in phase space in that direction, while the maximum Lyapunov exponent $λ_{1}$ determines whether ( $λ_{1} \leq 0$ ) adjacent trajectories can converge to form a stable orbit or a stable point, or ( $λ_{1} > 0$ ) they cannot.

Let the chaotic time series $x (1), x (2), \dots, x (t), \dots$ be given. Using the C–C method,³³ the embedding dimension $m$ and the time delay $τ$ are obtained, and the reconstructed phase space is expressed as³⁴:

X (t) = {x (t), x (t + τ), \dots, x [t + (m - 1) τ]}, t = 1, 2, \dots, M

(25)

The maximum Lyapunov exponent, calculated via the Wolf algorithm to track the divergence rate of adjacent trajectories, is expressed as³⁴:

λ_{1} = \frac{1}{t_{M} - t_{0}} \sum_{i = 0}^{M} \ln L_{i}^{'} / L_{i}

(26)

where

t_{0}

is the initial time,

t_{i}

is the current time,

t_{M}

is the terminal time,

M = N - (m + 1) τ

N

is the endpoint of the time series,

L_{i}^{'} = | X (t_{i}) - X (t_{i - 1}) |

L_{i} = | X (t_{i}) - X_{i} (t_{i}) |

λ_{i} = \ln L_{i}^{'} / L_{i}

and

X_{i} (t_{i})

is a point in the neighborhood of

X (t_{i})

at time

t_{i}

with radius

ε

Using the Poincaré map to study the periodic motions of closed orbits in vibratory roller-induced environmental vibrations, the Poincaré map replaces the original Nth-order continuous flow with an (N-1)th-order discrete system. In the multidimensional space $\sum_{T i m e s e r i e s} = {(\ddot{x} (t), \dot{x} (t), x (t), t, Φ_{φ}) \in R^{4} \times S}$ , the phase plane $Φ_{φ}$ ³⁴:

Φ_{φ} = {(\ddot{x} (t), \dot{x} (t), x (t)) | x (t) = (\max (x (t)) + \min (x (t))) / 2}

(27)

by selecting

Φ_{φ}

as the Poincaré section and projecting the multidimensional space

\sum_{T i m e s e r i e s} = {\ddot{x} (t), \dot{x} (t), x (t), t, Φ_{φ}}

onto

{\ddot{x} (t), \dot{x} (t)}

via the stroboscopic method, the Poincaré fixed-point mapping diagram can be obtained.

2.4. Layout parameter optimization methods

2.4.1. Orthogonal experimental design

An orthogonal experiment was performed on five HWIB layout parameters: filling ratio, filling depth, unit cell thickness, filling thickness, and unit cell side length, to study their vibration reduction effects on environmental vibrations induced by vibratory roller subgrade construction. The physical definitions of these parameters are shown in Figure 10, and the orthogonal experimental design is listed in Table 4.

Table 4.

Orthogonal experimental design.

No.	Filling ratio	Filling depth/m	Unit cell thick-ness/m	Filling thick-ness/m	Unit cell side length/m	No.	Filling ratio	Filling depth/m	Unit cell thick-ness/m	Filling thick-ness/m	Unit cell side length/m
1	0	8	0.16	5.6	0.84	14	0.75	6	0.10	5.6	0.72
2	0.5	10	0.12	5.6	0.96	15	0	2	0.10	3.5	0.60
3	1	10	0.16	4.9	0.60	16	0	6	0.14	4.9	0.96
4	1	8	0.14	4.2	0.72	17	0.5	8	0.10	4.9	1.08
5	0.75	8	0.12	6.3	0.60	18	1	4	0.10	6.3	0.96
6	0.25	4	0.14	5.6	0.60	19	1	6	0.12	3.5	0.84
7	0.25	2	0.12	4.9	0.72	20	0.5	2	0.14	6.3	0.84
8	0	4	0.12	4.2	1.08	21	0.75	2	0.16	4.2	0.96
9	0.25	8	0.18	3.5	0.96	22	1	2	0.18	5.6	1.08
10	0.5	4	0.16	3.5	0.72	23	0.75	4	0.18	4.9	0.84
11	0.25	10	0.10	4.2	0.84	24	0.5	6	0.18	4.2	0.60
12	0.25	6	0.16	6.3	1.08	25	0	10	0.18	6.3	0.72
13	0.75	10	0.14	3.5	1.08	-	-	-	-	-	-

The vibration reduction efficiency of HWIB was assessed using the amplitude attenuation coefficient A_R proposed by Woods⁷:

A_{R} = \frac{A_{1}}{A_{2}}

(28)

in which

A_{1}

is the ground vibration RMS acceleration with HWIB,

A_{2}

is the ground vibration RMS acceleration without HWIB

2.4.2. BP-GA algorithm model

A BP neural network^35,36 was employed, where the five layout parameters (filling ratio, filling depth, unit cell thickness, filling thickness, unit cell side length) served as input values X of the input layer, and the corresponding RMS acceleration served as target output Y of the output layer. The hidden layer contained 11 neurons (determined by trial-and-error to minimize fitting error), with ReLU activation for the hidden layer and a linear function for the output layer. After iterative training, a functional relationship between the five parameters and RMS acceleration was established.

To ensure the generalization capability of the surrogate model, the orthogonal-experiment dataset was randomly divided into training and validation subsets following a 5-fold cross-validation strategy, in which 76% of the samples in each fold were used for training and 24% for validation. During the training process, the training and validation mean squared error (MSE) curves were monitored as learning curves, and early stopping was applied when the validation loss failed to decrease for several consecutive iterations. The training and validation curves showed consistent convergence trends without divergence, indicating that overfitting did not occur. With the surrogate model reliably established, it can now be integrated into an optimization framework using a genetic algorithm to efficiently explore the layout parameter optimization of HWIB.

In vibro-acoustic and acoustic optimization of composite structures, genetic algorithm (GA)-based optimization frameworks have been widely adopted due to their robustness in dealing with highly nonlinear relationships between structural design variables and vibration or sound transmission responses. In particular, GA and its multi-objective extensions (e.g., NSGA-II) have been successfully applied to optimize the vibro-acoustic performance of stiffened composite shells, porous-core sandwich structures and laminated composite cylindrical shells lined with porous cores involving multiple geometric and material parameters, where gradient-based optimization methods become inefficient or impractical.^37–39

To further reduce the computational burden associated with repeated high-fidelity numerical evaluations, GA is frequently coupled with surrogate models, such as neural networks, to rapidly predict fitness functions related to vibration or acoustic attenuation. This surrogate-assisted GA strategy has proven effective in the optimization of periodic and metamaterial-based structures for broadband vibration and noise suppression, while maintaining sufficient accuracy through subsequent numerical verification.⁴⁰ Accordingly, following these established optimization paradigms, a GA-based framework integrated with a BP neural network predictor and FEM re-evaluation is adopted in the present study to optimize the HWIB layout for enhanced vibration damping performance.

The variation range of the HWIB parameter is shown in Table 5. In this study, the optimization objective is to maximize the fitness function defined by Equation (29), which is predicted by a trained BP neural network based on combinations of input HWIB parameters. The genetic algorithm implemented in MATLAB first generates an initial population of parameters. Subsequently, candidate parameter sets from each iteration are fed into the BP neural network—with its parameters held constant—to compute the corresponding fitness values. Selection, crossover, and mutation operations are then performed according to these fitness values, and this process continues iteratively until the stopping criterion is reached. Through this integrated optimization framework, an optimal HWIB layout scheme that enhances vibration damping performance can be efficiently identified. To avoid potential model bias, the GA-identified optimal configurations were subsequently re-evaluated by the high-fidelity FEM model; if the discrepancy between BP-predicted and FEM-calculated RMS acceleration exceeded a predefined threshold, the corresponding samples were added back into the training set and the BP network was retrained, ensuring the reliability of the BP-GA optimization framework. The optimization process based on genetic algorithm is shown in Figure 11. The selected values for associated factors can be found in Table 6.

F (i) = \frac{1}{A_{r m s}}

(29)

Table 5.

The allowable range of structural parameters.

Parameter	Range
Filling ratio	[0,1]
Filling depth (m)	[2,10]
Unit cell thickness (m)	[0.10,0.18]
Filling thickness (m)	[3.5,6.3]
Unit cell side length (m)	[0.6,1.08]

Table 6.

Tuned values of the GA parameters.

GA parameter	Value
Population size	100
crossover probability	0.8
mutation probability	0.05
Number of variables	5
Stopping criterion	150 iterations or fitness change rate < 1%

3. Results and analysis

3.1. HWIB material selection

3.1.1. Material wave impedance

The transmission coefficient¹⁵ is mathematically expressed as:

γ_{12} = \frac{4 α_{12}}{\sqrt{{(1 + α_{12})}^{4} + {(1 - α_{12})}^{4} - 2 {(1 - α_{12}^{2})}^{2} \cos \frac{2 ω h}{V_{2}}}}

(30)

where

α_{12}

is the impedance ratio,

ω

is the vibration wave frequency,

h

is the HWIB Width,

V_{2}

is the HWIB shear wave velocity.

The impedance ratio between two materials is defined as:

α_{12} = \frac{ρ_{2} V_{2}}{ρ_{1} V_{1}}

(31)

Where

ρ_{i}

is density,

V_{i}

is shear wave velocity, natural soil foundation is denoted by subscript 1, and HWIB is denoted by subscript 2.

With a shear wave velocity of 163 m/s, vibration frequency range of 5 Hz-35 Hz, HWIB width of 3.5 m, and impedance ratio between HWIB and the natural soil foundation ranging from 0-10, the variation of transmission coefficient with impedance ratio is plotted in Figure 12. The results show that (1) under a constant vibration frequency, the transmission coefficient first increases and then decreases with increasing impedance ratio, reaching its maximum (full transmission) when the impedance ratio equals 1, and (2) under a constant impedance ratio, the transmission coefficient decreases as the vibration frequency increases. Therefore, a significant difference in wave impedance between HWIB and the natural soil foundation leads to improved vibration mitigation effectiveness.

3.1.2. Comparative analysis of core material performance

The environmental vibration propagation induced by vibratory rollers is a three-dimensional problem involving moving elliptical wavefronts. Therefore, considering only the one-dimensional transmission coefficient theory for material selection is insufficient. To determine the optimal core material for HWIB, the vibration mitigation performance of four materials (rubber, foam, water, and fine sand) was compared, with detailed parameters listed in Tables 1 and 7. In the tables, all materials were modeled using a linear elastic constitutive model, except for water, which was described by an equation of state.

Table 7.

Parameters of the equation of state for water.

Material	C₀/(m/s)	s	Gama₀	Density $ρ$ /( $k g \cdot m^{- 3}$ )	Filling type
Water	1483	0	0	1000	Core

It is noted that although materials such as rubber, foam, and soil may exhibit nonlinear and strain-rate-dependent behavior under large deformation or high loading conditions, the use of linear elastic constitutive models in the present study is justified for the following reasons. First, the numerical model was carefully calibrated and validated against field measurements under vibratory-roller excitation. Time-history comparisons at multiple monitoring points (vertical offsets of 2.5 m, 5.0 m, and 7.5 m) show good agreement between numerical predictions and field data, with relative errors below 10% (see Figure 9 and Section 2.2.2). This validation demonstrates that the linearized modelling used here captures the dominant vibration response under the tested vibratory-roller excitation and amplitude range.

Second, for the low-frequency environmental vibrations studied (dominant frequencies 0–35 Hz), the literature indicates that soils and many elastomeric materials behave approximately linearly (or can be represented by linear viscoelastic models) within the small-strain range, and that linear/linear-viscoelastic approximations are commonly adopted as a justified engineering simplification in similar vibration studies. Representative references on small-strain behavior and modelling practice are Castellón et al. (review on small-strain soil constitutive modelling)⁴¹ and Theland et al. (assessment of small-strain characteristics for vibration studies).⁴² These sources conclude that, when strains remain in the small-strain domain, linearized models can give reliable first-order predictions.

Finally, adopting linear elastic material models permitted us to perform a comprehensive parametric study (multiple material combinations, bandgap computations, Vibratory drum-HWIB-subgrade coupled model and global BP–GA optimization) within feasible computational time. Nonlinear models would substantially increase model complexity and computational cost. Similar modeling trade-offs between accuracy and computational efficiency have been discussed in the literature (e.g., Majidi et al., 2022⁴³; Yosef et al., 2024⁴⁴).

Through three-dimensional modeling analysis, as shown in Figure 13, the following results were drawn:

(1) Ranking of RMS acceleration (Figure 13(a)): rubber (0.0246 m/s²) < foam (0.0254 m/s²) < water (0.0266 m/s²) < fine sand (0.0301 m/s²). Rubber core exhibited the lowest RMS acceleration, showing an 18.3% reduction compared to fine sand, which confirms its superior capability in dissipating vibrational energy.

(2) Frequency-domain characteristics analysis (Figure 13(b)): Rubber exhibits remarkable vibration mitigation performance in the high-frequency range, while also providing good isolation effects in the mid- and low-frequency ranges. Foam materials demonstrate moderate vibration mitigation performance across all frequency ranges, making them a relatively balanced isolation material. In contrast, fine sand shows relatively limited isolation effects across all frequencies. Water can effectively attenuate high-frequency vibrations, but its suppression of mid- and low-frequency vibrations is weak.

(3) Mechanistic analysis: The significant wave impedance contrast between rubber ( $Z = ρ V_{s} = 1100 \times 163 = 1.793 \times 10^{5} k g / (m^{2} \cdot s)$ ) and the natural soil foundation ( $Z = ρ V_{s} = 1850 \times 135.2 = 2.501 \times 10^{5} k g / (m^{2} \cdot s)$ ) enhances vibrational wave reflection. Additionally, rubber’s high damping ratio (0.30) further suppresses vibration propagation through energy dissipation via internal friction.

3.1.3. Comparative analysis of shell material performance

Shell materials must balance structural stiffness and deformation energy absorption capacity. By comparing the vibration mitigation performance of foam, rubber, concrete, and CFG (Cement-Fly Ash Gravel), as shown in Table 1 and Figure 14, the following results are drawn:

(1) Ranking of RMS acceleration (Figure 14(a)): foam (0.0246 m/s²) < rubber (0.0268 m/s²) < CFG (0.0292 m/s²) < concrete (0.0316 m/s²). Foam exhibits the lowest RMS acceleration, showing a 22.2% reduction compared to concrete.

(2) Frequency-domain characteristics analysis (Figure 14(b)): Both foam and rubber materials exhibit certain vibration mitigation performance in the mid- and low-frequency ranges, with foam demonstrating superior isolation compared to rubber. The vibration mitigation performance of concrete and CFG materials are similar, with their dominant frequency distributions ranging from 10 Hz to 12 Hz, and their overall isolation effects are relatively limited.

(3) Mechanistic analysis: Foam’s low elastic modulus (E=2 MPa) enables significant elastic deformation under vibrational loads, thereby absorbing vibrational energy through structural deformation; concurrently, its low density (ρ=30 kg/m³) reduces the overall structural mass, minimizing inertial force-induced secondary disturbances to the subgrade.

3.1.4. Optimal material combination scheme

The vibration mitigation performance of HWIB has higher requirements on structural integrity. Numerical simulating of the vibration reduction effect of the HWIB with rubber core-foam shell composite structure under the action of vibratory drum indicates that the maximum local deformation of the core material is 2.1 mm and that of the shell material is 1.1 mm, as illustrated in Figures 15 and 16, both of which satisfy the structural integrity requirements of the HWIB.

From a synergistic perspective, the rubber core dissipates energy through high damping ratio, while the foam shell absorbs energy via elastic deformation, jointly reducing vibration transmissibility. From an engineering applicability standpoint, both rubber and foam are lightweight materials, enabling convenient construction and cost control, making them suitable for narrow spaces. By integrating the performance of the core-shell materials and HWIB’s structural integrity requirements, the rubber core–foam shell material combination has been identified as the optimal scheme.

3.2. Analysis of HWIB vibration reduction effect

3.2.1. Bandgap matching and vibration suppression

The Bloch periodic boundary conditions were set in COMSOL Multiphysics, as shown in Figure 17. The blue region represents foam material and the gray region represents rubber material, both governed by linear elastic constitutive models. Subsequently, the wave vector $k$ was scanned along the boundary of the first irreducible Brillouin zone to obtain the dispersion curves, as shown in Figure 18. The results are as follows:

(1) The straight-arm HWIB exhibits three distinct bandgaps in the M-K direction, bandgap 1 (0-6 Hz); bandgap 2 (11-14 Hz); bandgap 3 (19-21 Hz). The results demonstrate that the HWIB bandgap distribution matches well with the vibration spectrum of vibratory rollers (primarily concentrated in the 0-35 Hz frequency range, with dominant frequencies in 9-14 Hz), which excels in low-frequency suppression. This partly addresses the limitations of conventional vibration reduction trenches in low-frequency applications.

(2) By arranging the HWIB as shown in Figure 19 to force vibration waves to propagate as much as possible along the M-K direction (parallel to the HWIB outer contour), the utilization of bandgap vibration isolation effect is maximized.

It should be noted that the bandgaps identified in Figure 18 are all in the M-K direction, not the complete bandgaps. In a finite HWIB structure embedded in soil and subjected to moving-source excitation, incident vibration waves arrive from multiple directions. Consequently, only part of the incident energy can directly couple into the Bloch-wave bandgap mechanisms associated with the M–K direction. In addition to these directional bandgap effects, vibration attenuation in such a three-dimensional finite soil–structure coupled system is further governed by impedance mismatch and wave reflection at HWIB–soil interfaces, intrinsic material damping of the rubber and foam, engineering damping introduced by the core–shell configuration, and wave diffraction and scattering caused by the finite size and boundaries of the HWIB array. As a result, the effective attenuation bandwidth observed in practice can extend beyond the narrow frequency intervals predicted solely by idealized Bloch-wave analysis. Similar distinctions between sharp stop bands in ideal locally resonant metamaterials and broadened attenuation behavior in finite coupled systems have been reported in the literature, for example by Ebrahimi-Nejad and Kheybari, who demonstrated pronounced but relatively narrow locally resonant stop bands in composite metamaterial configurations.⁴⁵

(3) Time- and frequency-domain analyses were conducted to further evaluate the vibration mitigation performance of the HWIB. As shown in Figures 20–22, the acceleration amplitude increased with the growth of road surface roughness. The RMS acceleration values before vibration reduction were 0.136 m/s², 0.288 m/s², and 0.605 m/s², respectively, whereas after mitigation with HWIB, the RMS acceleration values decreased to 0.012 m/s², 0.025 m/s², and 0.052 m/s², corresponding to vibration reduction rates of 91.2%, 91.3%, and 91.4%.

As illustrated in Figures 20–22, the acceleration PSD curves after vibration mitigation exhibit two successive stages. In attenuation stage I, the acceleration PSD amplitude was significantly attenuated. Although the frequency range corresponding to this stage varied with different classes of road surface roughness, a pronounced reduction in acceleration PSD amplitude was consistently observed within the range of 4-35 Hz. In filtering stage II, within the frequency range of 50–128 Hz, the acceleration PSD amplitude did not exhibit substantial attenuation, but the fluctuation amplitude of the acceleration PSD curve decreased, tending toward stability.

3.2.2. Chaotic system suppression effect

(1) Figure 23(a)–(c) show the relationship between the Lyapunov exponent and rolling distance for road surface roughness of class B, C, and D, respectively. Regardless of whether HWIB is employed for vibration reduction, the maximum Lyapunov exponent remains greater than zero, indicating that the environmental vibration induced by the vibratory roller is unstable and exhibits distinct chaotic characteristics. After undergoing the startup stage of instability (0 m < rolling distance ≤ 2.5 m), the Lyapunov exponent enters a relatively stable attenuation stage (rolling distance > 2.5 m). With HWIB applied, the Lyapunov exponent during the attenuation stage consistently remains lower than that in the case without HWIB. Specifically, for Class B roughness, the maximum Lyapunov exponent decreases from 0.3511 to 0.2917 (a reduction of 16.9%); for lass C, it decreases from 0.2636 to 0.2321 (a reduction of 11.9%); and for class D, it decreases from 0.2023 to 0.1804 (a reduction of 10.8%). These results indicate that, after HWIB vibration reduction, the divergence of adjacent trajectories in the phase space is alleviated, and the motion trajectories become more stable.

(2) It is generally stipulated that the Poincaré fixed points are defined as the set of points left by trajectories crossing the Poincaré section from above to below. Figures 24–26 present the phase portraits and Poincaré fixed-point mappings of the environmental vibrations induced by the vibratory roller. As the road surface roughness class increases, the distribution region of the Poincaré fixed points becomes more extensive, indicating that greater roughness leads to more disordered motion of the closed-orbit periodic system. As shown in Figures 24 and 25, and 26(c), after vibration reduction with the HWIB, the trajectory distribution region of the closed-orbit periodic system converges, and the motion period is reduced. With the increase in road surface roughness class, the motion period decreases by 43%, 60%, and 24%, respectively. These results demonstrate that employing the HWIB effectively suppresses the chaotic disorder of the closed-orbit periodic system motion induced by the vibratory roller.

3.3. Layout parameter optimization

3.3.1. Parameter sensitivity analysis

Based on the orthogonal experimental results, as shown in Table 8, the influence of layout parameters on the amplitude attenuation coefficient was quantified through range analysis, as shown in Figure 27, and the parameter significance was further verified via Analysis of Variance (ANOVA) in Table 9. The following results are drawn.

(1) Range analysis results: Filling depth (amplitude attenuation coefficient range 0.5115) > Filling thickness (amplitude attenuation coefficient range 0.0562) > Unit cell side length (amplitude attenuation coefficient range 0.0269) > Unit cell thickness (amplitude attenuation coefficient range 0.0261) > Filling ratio (amplitude attenuation coefficient range 0.0224).

(2) ANOVA verification: Filling depth (P-value 1.07✕10^-6) > Filling thickness (P-value 0.01) > Unit cell thickness (P-value 0.079) > Unit cell side length (P-value 0.084) > Filling ratio (P-value 0.168).

(3) Filling depth and filling thickness are identified as critical layout parameters (P-value less than or equal to 0.01);

(4) The orthogonal experiment yields an optimal configuration of filling ratio = 1, filling depth = 10 m, unit cell thickness = 0.1 m, filling thickness = 4.9 m, and unit cell side length = 0.84 m.

Table 8.

Amplitude attenuation coefficients for different schemes.

No.	$A_{R M S}$ / $m . s^{- 2}$	$A_{R}$	No.	$A_{R M S}$ / $m . s^{- 2}$	$A_{R}$
1	0.0354963	0.122439	14	0.0525349	0.181876
2	0.0251876	0.045502	15	0.1873701	0.649553
3	0.0248702	0.085493	16	0.0475698	0.164701
4	0.0344717	0.118738	17	0.0329190	0.113442
5	0.0336831	0.116246	18	0.0935552	0.324297
6	0.1077454	0.37366	19	0.0588973	0.204033
7	0.1605301	0.556442	20	0.1740990	0.603564
8	0.0944781	0.327676	21	0.1677894	0.58168
9	0.045598	0.157448	22	0.1758695	0.609761
10	0.1067283	0.370225	23	0.0865527	0.30018
11	0.0316050	0.108687	24	0.0529034	0.183162
12	0.0622385	0.215807	25	0.0244802	0.083979
13	0.0347944	0.119873	-	-	-

Table 9.

ANOVA for HWIB amplitude attenuation coefficients.

	Type III sum of squares	Degree of freedom	Mean square	F-value	P-value(significance)
Filling ratio	0.001	4	0.000	2.848	0.168
Filling depth	0.870	4	0.217	1675.555	1.07✕10^-6
Unit cell thickness	0.002	4	0.001	4.789	0.079
Filling thickness	0.008	4	0.002	15.682	0.010
Unit cell side length	0.002	4	0.001	4.613	0.084

3.3.2. Optimization effectiveness of BP-GA algorithm

Based on the optimization results of the BP-GA algorithm model, as shown in Figures 28–31, the following results are drawn:

(1) BP neural network training performance: The mean squared error (MSE) of the training set converged to 1.2×10⁻⁴, and the prediction error of the test set was < 8%, indicating that the model accurately reflects the nonlinear relationship between layout parameters and vibration mitigation performance. Through weight analysis, filling depth (weight 0.41) and filling thickness (weight 0.33) exhibit the strongest influence on the output, which aligns with the conclusions from the orthogonal experiments (Figure 28).

(2) Global optimization via genetic algorithm: To overcome the problem of local optima, the genetic algorithm was executed 100 times, and the optimal combination was obtained through statistical analysis. Figure 29 presents the result of one such optimization, where it can be observed that the algorithm iterated for 130 generations. The fitness value converged rapidly within the first 10 generations, followed by a gradual convergence until the error in the fitness value satisfied the predefined threshold, thereby terminating the optimization process. The optimal parameter combination was identified as filling ratio = 0.7, filling depth = 10 m, unit cell thickness = 0.1 m, filling thickness = 6.3 m, and unit cell side length = 0.6 m (Figure 30).

(3) The optimization results demonstrate that after orthogonal experiment adjustments, the RMS acceleration decreased by 1.2% (from 0.0246 to 0.0243 m/s²) and the acceleration PSD amplitude decreased by 6.7% (from 0.0015 to 0.0014 (m/s²)²/Hz), whereas the BP-GA algorithm optimization achieved a more significant improvement: a 9.8% reduction in RMS acceleration (from 0.0246 to 0.0222 m/s²) and a 26.7% decrease in acceleration PSD amplitude (from 0.0015 to 0.0011 (m/s²)²/Hz), thereby confirming the superiority of the BP-GA algorithm in overcoming local optima through global search(Figure 31).

4. Conclusion

This study systematically investigates the vibration reduction effects of honeycomb wave impeding blocks (HWIB) on low-frequency environmental vibrations induced by vibratory rollers and their layout parameter optimization methods through theoretical analysis, numerical simulations, and BP-GA algorithm. The main conclusions are as follows.

(1) HWIB demonstrates significant advantages in low-frequency vibration isolation. When rubber-foam composite materials were used, the RMS acceleration at monitoring points decreased by approximately 91%. Moreover, the acceleration PSD amplitude exhibited a significant reduction in the frequency range of 4-35 Hz, while the fluctuation amplitude of the acceleration PSD curve decreased in the frequency range of 50-128 Hz.

(2) The vibration mitigation performance of HWIB stems from the synergistic interaction between periodic bandgap characteristics and material mechanical properties. The straight-arm HWIB exhibits three bandgaps (0-6 Hz, 11-14 Hz, and 19-21 Hz) in the M-K direction, which match well with the vibration spectrum of vibratory rollers, enabling effective suppression of low-frequency vibration. And the mitigation effect is further enhanced by leveraging material properties such as low elastic modulus and impedance contrast. This partly overcomes the limitations of conventional vibration reduction trenches in low-frequency ranges and reduces reliance on trench depth. Further analysis reveals that the use of HWIB suppresses the chaotic disorder of the environmental vibration induced by the vibratory roller, thereby significantly improving the stability of motion trajectories.

(3) In the optimization of layout parameters, filling depth and filling thickness were identified as critical control factors, with amplitude attenuation coefficient ranges of 0.5115 and 0.0562, respectively. Through global optimization using the BP-GA algorithm, the optimal parameter combination was determined: filling ratio of 0.7, filling depth of 10 m, unit cell thickness of 0.1 m, filling thickness of 6.3 m, and unit cell side length of 0.6 m. Compared with the unoptimized and orthogonal test optimization schemes, the BP-GA optimized scheme reduced the RMS acceleration by 9.8% and 8.6%, and the acceleration PSD amplitude by 26.7% and 21.4%, thereby confirming the superiority of the BP-GA algorithm.

Footnotes

ORCID iD

Junjie Yu

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research is supported by the Chongqing Municipal Natural Science Foundation (CSTB2023NSCQ-MSX0180), the Open Fund of Key Laboratory of Architectural Acoustic Environment of Anhui Province (AAE2022KFO2), Sichuan Provincial Transportation Technology Project (2024-A-01), and Sichuan Science and Technology Program (2025ZNSFSC0408).

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data Availability Statement

Data will be made available on request.*

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