Refined Modeling of Heterogeneous Medium for Ground-Penetrating Radar Simulation

Abstract

Ground-penetrating radar (GPR) has been widely used for subsurface detection and testing. Numerical simulations of GPR signal are commonly performed to aid the interpretation of subsurface structures and targets in complex environments. To enhance the accuracy of GPR simulations on heterogeneous medium, this paper proposes a hybrid modeling method that combines the discrete element method with a component fusion strategy (DEM–CFS). Taking the asphalt pavement as an example, three 3D stochastic models with distinctly different porosities are constructed by the DEM–CFS method. Firstly, the DEM is utilized to establish the spatial distribution of random coarse aggregates. Then, the component fusion strategy is employed to integrate other components into the coarse aggregate skeleton. Finally, the GPR response of the constructed asphalt models is simulated using the finite-difference time-domain method. The proposed modeling method is validated through both numerical and laboratory experiments and demonstrates high precision. The results indicate that the proposed modeling method has high accuracy in predicting the dielectric constant of heterogeneous media, as generated models are closely aligned with real-world conditions.

1. Introduction

Within various non-destructive testing methods, ground-penetrating radar (GPR) is recognized as one of the most effective and efficient tools, offering the advantages of high flexibility and fine resolution [1,2]. Therefore, GPR has been widely applied for monitoring and assessing civil infrastructures. In pavement engineering, GPR has been utilized for measuring pavement layers’ density and thickness, locating hidden defects, detecting water infiltration [3,4,5,6], etc. GPR delineates subterranean structures by receiving and analyzing electromagnetic (EM) waves reflected and diffracted inside the pavement layers [7]. As the most commonly used pavement material, asphalt mixtures are a kind of multiphase heterogeneous material composed of aggregates, asphalt, voids, and other phases. As the dielectric constant of each phase is different in the mixture, undesired EM scattering inevitably occurs in the subsurface background media, making the GPR image hardly resemble the complex subsurface geometry. In such cases, accurate interpretation of GPR data is challenging. To address this challenge, numerical simulations are commonly implemented to validate the subsurface structure interpretated from the GPR data [8,9,10,11]. These simulations not only aid in understanding EM wave interactions with underground media and the influence of measurement parameters on GPR performance, but also facilitate the assessment and comparison of various processing and imaging methods across diverse detection scenarios. This approach is more time- and cost-efficient compared to constructing physical tests.

There are various numerical techniques used for EM wave simulations, such as the finite-difference time-domain (FDTD) method [12], the finite element method (FEM) [13], the pseudo-spectral time-domain (PSTD) method [14], and the spectral element method (SEM) [15]. Among these methods, the FDTD method is widely preferred for GPR simulation, due to its advantage of easy implementation. While the success of a GPR simulation and inversion mainly relies on the construction of a dielectric model representing the realistic subsurface media and structures, existing medium modeling methods include equivalent homogeneous medium and random heterogeneous approaches. The equivalent homogeneous approach assumes the layer to be uniform, where an average dielectric constant is derived from the dielectric properties and volume fractions of each phase. This calculation can be performed on various equivalent homogeneous models, such as the complex refractive index model (CRIM) [16], Rayleigh model [17], Sihvola model [18], and Bruggeman–Hanai–Sen (BHS) model [19]. By treating the medium as uniform with this equivalent dielectric constant, the modeling process is significantly simplified, greatly reducing the modeling efforts. However, such a simplified model omits the EM scattering phenomena in heterogeneous background media, often leading to noticeable discrepancies between real GPR responses and simulation results [20,21]. Therefore, many efforts have been devoted to establishing heterogeneous dielectric models for GPR simulations, considering a multiphase, discrete, and randomly distributed nature of subsurface medium. For example, Jiang et al. developed an ellipsoidal autocorrelation function for generating random heterogeneous models [22], Li et al. developed a collision detection algorithm for generating multiple bonded irregular polygon particles [23], and Benedetto et al. introduced a random sequential absorption (RSA) paradigm for water seepage simulation in hot-mix asphalt [24], and subsequently extended it for GPR simulations on railway ballast [25]. In addition to the mathematical modeling methods above, Yan et al. developed an ellipsoidal discrete element method (DEM) to simulate a dynamic compaction process of granular materials [26], which was later extended by Liu et al. to investigate EM wave scattering in densely packed discrete random media [27]. Benefiting from scanning techniques, Kerem et al. developed an image recognition method for generating complex buried objects for GPR numerical models [28]. Later, Fan et al. proposed a sample image identification and model generation method to construct asphalt models [29]. With the advancements in machine learning techniques, Dai et al. introduced a deep-learning-based GPR forward solver for predicting B-scans of subsurface objects buried in heterogeneous soil [30]. Similarly, Zheng and Wang developed a physics-informed neural network (PINN) solver for GPR wave propagation, effectively avoiding numerical dispersion and simplifying the process for handling complex irregular models without the need for specific meshing [31]. However, these methods suffer from two limitations: (1) the regular geometries of course aggregates exhibit notable deviations from actual situations, and (2) the continuous distribution of fine aggregates/elements is simply modelled with one or several types of dielectric materials.

To address the afore-mentioned limitations, this paper proposes an innovative heterogeneous medium modeling method, which integrates the DEM with a component fusion strategy (CFS). This method (termed DEM–CFS) enhances the fidelity and precision of heterogeneous models by customizing the aggregates’ shapes and optimizing the continuous distribution of fine elements while balancing computational efficiency. The rest of this paper is organized as follows. The proposed modeling and EM simulation methods are introduced in Section 2, and validated in Section 3 through numerical and laboratory experiments. Conclusions are summarized in Section 4.

2. Methods

2.1. DEM–CFS for Heterogeneous Model Generation

In this study, we take dry asphalt pavement as an example to examine the DEM–CFS for modeling heterogeneous media. The asphalt medium comprises aggregates, asphalt binder, and air voids, which occupy variable volume fractions for different asphalt mixture designs. Meanwhile, porosity is a key parameter that controls the asphalt’s bearing capacity and permeability [32]. For a target porosity (φ_a), the volume fractions of asphalt binder and aggregates can be calculated by [33]

$${\mathsf{\phi }}_{\mathrm{b}}={\displaystyle \frac{{\mathit{P}}_{\mathrm{b}}{\mathit{G}}_{\mathrm{mm}}(1-{\mathsf{\phi }}_{\mathrm{a}})}{{\mathit{G}}_{\mathrm{b}}}}$$

(1)

$${\mathsf{\phi }}_{\mathrm{s}}=1-{\mathsf{\phi }}_{\mathrm{a}}-{\mathsf{\phi }}_{\mathrm{b}}$$

(2)

where G_mm, G_b, and P_b refer to the maximum specific density of the asphalt mixture, the specific density of the asphalt binder, and the asphalt content of the asphalt mixtures by weight, respectively.

In this paper, the generation of a dielectric model of a heterogeneous asphalt mixture is performed in two steps. Firstly, the DEM is employed to construct a 3D skeleton of coarse aggregates. Then, the CFS is utilized to integrate the remaining components into the skeleton structure, resulting in a multiphase heterogeneous asphalt model. The flowchart of the proposed DEM–CFS is illustrated in Figure 1.

To accurately replicate the real construction process of asphalt mixtures, it is crucial to consider mechanical modeling stages in the DEM model. Specifically, two stages, packing and pressing, are implemented to ensure that the model reflects the actual construction process of the pavement materials. These two stages are depicted in Figure 2. It is worth mentioning that various convex “clump” templates with different angularity and texture indices are incorporated to mimic the irregular nature of crushed stones to a certain extent. This irregularity, compared to simplified basic shapes, benefits the simulation of scatterings at the edges and corners and enhances the reliability of simulated GPR responses [34]. Moreover, the proportions of aggregates of various sizes are arranged in accordance with the gradation curve, thereby establishing an authentic graded structure. It is important to note that the generation of fine aggregates entails significant computational efforts. Furthermore, these fine aggregates may be misclassified as air voids in the subsequent modeling processes, leading to a higher air void content than the target porosity. Thus, the DEM is used only for simulating coarse aggregates, which are the crushed stones with a size larger than 2.36 mm in this study.

Since only coarse aggregates are considered in the DEM model, it is necessary to adjust the volume proportions of these aggregates. Specifically, the proportions of fine aggregates are excluded, and each coarse aggregate bin’s proportion is scaled up accordingly, ensuring that the total volume proportion of the coarse aggregate equals 100%. Accordingly, the total mass allocated in all clumps generated in the DEM model can be calculated as

$${\mathit{M}}_{\mathrm{DEM}}={\mathit{V}}_{\mathrm{am}}{\mathsf{\phi }}_{\mathrm{s}}{\mathit{G}}_{\mathrm{se}}{\mathit{P}}_{\mathrm{ca}}$$

(3)

where V_am represents the volume of the entire model, G_se denotes the specific density of the aggregates, and P_ca is the weight percentage of coarse aggregates.

Except for fine aggregates, the DEM does not perform substantive modeling of asphalt binder materials, due to its focus on discrete particle interactions rather than continuous material properties. However, accurately simulating asphalt behavior is crucial for creating a precise model of asphalt mixtures. Thus, to capture the asphalt’s viscoelastic properties and the interaction dynamics between the binder and aggregates accurately, the Hertz–Mindlin model (with Johnson–Kendall–Roberts cohesion) is applied as the material constitutive relation. This model is implemented by setting an additional cohesive energy on the particle surfaces to simulate the adhesive forces between the asphalt binder and the aggregates. This cohesive model is able to represent the contact state of asphalt binder acting as a liquid bridge within a particle-based framework [35]. During the DEM simulation, the gravitational acceleration is configured at 9.8 N/kg, and other mechanical parameters are detailed in

Table 1. Particle characteristics in the DEM simulation [35,36].

Parameter	Description	Value	Unit
E	Young’s modulus	55	GPa
ν	Poisson’s ratio	0.35	-
ρ	Density	2.60–3.20	kg/m3
Re	Restitution coefficient	0.3	-
μs	Static friction coefficient	0.7	-
μr	Rolling friction coefficient	0.01	-
γ	Surface energy density	20	J/m2

.

In this study, the FDTD method is selected for GPR wavefield propagation due to its ease of implementation and ability to handle complex boundary conditions. To adapt the DEM particle units for FDTD simulation, the 3D skeleton of coarse aggregate generated by the DEM is discretized into hexahedral cells utilizing a voxelization algorithm. To align with the FDTD grid configuration and avoid numerical dispersion, the size of each voxel must be at least ten times smaller than the smallest wavelength of the EM waves, which is expressed by [37]

$$\mathsf{\Delta }\mathit{l}={\displaystyle \frac{{\mathsf{\lambda }}_{\min }}{10}}={\displaystyle \frac{\mathit{c}}{10{\mathit{f}}_{\max }\sqrt{{\mathsf{\epsilon }}_{\mathrm{r}}}}}$$

(4)

where λ_min represents the smallest wavelength, c denotes the velocity of EM waves in free space, f_max specifies the max frequency of the EM waves, which is at the level of −40 dB relative to the peak at the center frequency, and ε_r is the dielectric constant of the background media. Given that the center frequency of GPR used for pavement inspection is typically around 2 GHz [38], the f_max is likely be around 4 GHz for a ricker wavelet. Assuming the dielectric constant of asphalt pavement to be 6, the grid size employed for the following FDTD simulation should be less than 3 mm. Accordingly, we conducted three distinct discretization operations with varying voxel sizes, i.e., 3 mm, 1 mm, and 0.5 mm, respectively. The reconstructed cross-sections of the models at these resolutions are shown in Figure 3. The computational resources required for these three models are 0.7 MB, 18 MB, and 142 MB in memory, and 3 s, 10 s, and 5 min in computation time, respectively. The discretization results indicate that increasing mesh density reduces the staircase approximation error in aggregate edge regions, but significantly increases computation cost. To achieve an optimal balance between model accuracy and computational efficiency, a grid size of 1 mm is adopted for model discretization in this study.

After the discretization, the geometric model is divided into coarse aggregate and free space voxels. Then, the free space voxels are replaced with fine aggregates, asphalt binder, and additional edge elements. Here, the additional edge elements are specifically introduced to compensate for any geometric discretization errors that may have occurred during the voxelization process. The volume of these edge elements can be calculated by

$${\mathit{V}}_{\mathrm{ee}}={\mathit{V}}_{\mathrm{am}}{\mathsf{\phi }}_{\mathrm{s}}{\mathit{P}}_{\mathrm{ca}}-{\mathit{K}}_{\mathrm{ca}}{\mathsf{\delta }}_{\mathrm{v}}$$

(5)

where K_ca represents the number of coarse aggregate voxels, and δ_v denotes the volume of a voxel, which is 1 mm³ in this study.

To increase the modeling precision, the fine aggregates are classified into two categories based on the size of aggregates: one group for aggregates smaller than one grid size, and another group for aggregates larger than one grid size. These larger aggregates, with size ranging from 1 mm to 2.36 mm, are integrated into the discretized model using multiple 3D voxel clusters. Each cluster consists of four interconnected voxels, arranged in various spatial configurations, taking an example depicted in Figure 4.

Meanwhile, we introduce a CFS for both particles smaller than 1 mm and the edge elements, which cannot effectively occupy a single grid space. This strategy introduces two types of composite voxels, i.e., the aggregate–asphalt and the asphalt–void voxels. The former represents a voxel composed of fine aggregates and asphalt binder. Similarly, the latter represents a mix of asphalt binder and air voids. The deliberate exclusion of an aggregate–void voxel is to simulate asphalt’s coating effect on aggregates in real mixtures, reducing direct aggregate–air interactions. To achieve a realistic representation of the pavement mixture, all integrated voxels are strategically positioned adjacent to the coarse aggregate voxels. Furthermore, to quantitatively capture the intrinsic variability inherent in these two-phase materials, the aggregate–asphalt and asphalt–void materials are assigned into ‘l’ and ‘m’ groups, respectively. These groups are systematically sorted by their levels of dielectric constant, with the sequence ε_aa,1 < … < ε_aa,l for the aggregate–asphalt material, and ε_av,1 < … < ε_av,m for the asphalt–void material. Additionally, it is assumed that the voxel counts corresponding to the various levels of dielectric constants follow a normal distribution, with (n_aa,1, …, n_aa,l) ~ G (μ₁, s₁) for the aggregate–asphalt material, and (n_av,1, …, n_av,m) ~ G (μ₂, s₂) for asphalt–void material. Here, μ₁ and μ₂ are the mean values, and s₁ and s₂ are the standard deviations. For a specified target volume fraction of asphalt (φ_b) and aggregates (φ_s), the voxel counts n_aa,i (i = 1, …, l) and n_av,j (j = 1, …, m) are, respectively, derived by

$${\mathit{n}}_{\mathrm{aa},i}={\displaystyle \frac{{\mathsf{\phi }}_{\mathrm{s}}{\mathit{P}}_{\mathrm{fa},1}{\mathit{V}}_{\mathrm{am}}+{\mathit{V}}_{\mathrm{ee}}}{{\mathsf{\delta }}_{\mathrm{v}}}}\cdot {\displaystyle \frac{1}{{\displaystyle \sum _i\left[(1-{\varphi }_{\mathrm{i}})\cdot {\mathit{p}}_{\mathrm{aa},i}/{\mathit{p}}_{\mathrm{aa},1}\right]}}}\cdot {\displaystyle \frac{{\mathit{p}}_{\mathrm{aa},i}}{{\mathit{p}}_{\mathrm{aa},1}}},\left(\mathit{i}=1, \dots ,\mathit{l}\right)$$

(6)

$${\mathit{n}}_{\mathrm{av},\mathit{j}}={\displaystyle \frac{{\mathsf{\phi }}_{\mathrm{b}}{\mathit{V}}_{\mathrm{am}}-{\displaystyle \sum _i{\mathit{n}}_{\mathrm{aa},i}{\varphi }_{\mathit{i}}}}{{\mathsf{\delta }}_{\mathit{v}}}}\cdot {\displaystyle \frac{1}{{\displaystyle \sum _j\left({\varphi }_{\mathit{j}}{\mathit{p}}_{\mathrm{av},j}/{\mathit{p}}_{\mathrm{av},1}\right)}}}\cdot {\displaystyle \frac{{\mathit{p}}_{\mathrm{av},j}}{{\mathit{p}}_{\mathrm{av},1}}},\left(\mathit{i}=1, \dots ,\mathit{l}; \mathit{j}=1, \dots ,\mathit{m}\right)$$

(7)

where P_fa,1 represents the weight percentage of fine aggregates smaller than the grid size; V_am denotes the total volume of the model; δ_v is the volume of a voxel; p_aa,i and p_av,j are the probabilities of the i-th and j-th values given to the normal distribution; and ϕ_i and ϕ_j signify the asphalt content in the corresponding voxel, achieved using the two-phase Rayleigh model [17], as detailed in Equations (8) and (9), respectively.

$${\varphi }_{\mathit{i}}={\displaystyle \frac{3{\mathsf{\epsilon }}_{\mathrm{b}}({\mathsf{\epsilon }}_{\mathrm{aa},i}-{\mathsf{\epsilon }}_{\mathrm{s}})}{{\mathsf{\epsilon }}_{\mathrm{aa},i}{\mathsf{\epsilon }}_{\mathrm{b}}-{\mathsf{\epsilon }}_{\mathrm{aa},i}{\mathsf{\epsilon }}_{\mathrm{s}}+2{\mathsf{\epsilon }}_{\mathrm{b}}^2-2{\mathsf{\epsilon }}_{\mathrm{b}}{\mathsf{\epsilon }}_{\mathrm{s}}}},\left(\mathit{i}=1, \dots ,\mathit{l}\right)$$

(8)

$${\varphi }_{\mathit{j}}={\displaystyle \frac{3{\mathsf{\epsilon }}_{\mathrm{b}}({\mathsf{\epsilon }}_{\mathrm{av},j}-{\mathsf{\epsilon }}_{\mathrm{a}})}{{\mathsf{\epsilon }}_{\mathrm{av},j}{\mathsf{\epsilon }}_{\mathrm{b}}-{\mathsf{\epsilon }}_{\mathrm{av},j}{\mathsf{\epsilon }}_{\mathrm{a}}+2{\mathsf{\epsilon }}_{\mathrm{b}}^2-2{\mathsf{\epsilon }}_{\mathrm{b}}{\mathsf{\epsilon }}_{\mathrm{a}}}},\left(\mathit{j}=1, \dots ,\mathit{m}\right)$$

(9)

Here, ε_s, ε_b, and ε_a represent the standard dielectric constants of course aggregate, asphalt binder, and air voids, respectively, as determined by field tests or empirical data. In this study, ε_s, ε_b, and ε_a are assigned as 10.5, 3.0, and 1.0, respectively [39,40], and the conductivity of both aggregate and asphalt is 10⁻⁵ S/m. Accordingly, the upper and lower limits of the dielectric constant for the aggregate–asphalt material are defined as 3.0 and 10.5, corresponding to ϕ_i values of 100% and 0, respectively. These limits are divided into thirty-one groups (l = 31), with the dielectric constant in each group satisfying

$${\mathsf{\epsilon }}_{\mathrm{aa},\mathrm{i}}=\left(\mathit{i}- 1\right)\left(10.5 - 3.0\right)/ 30+3.0,\left(\mathit{i}=1, \dots ,31\right).$$

(10)

Similarly, the dielectric constant range for the asphalt–void material is [1.0, 3.0], corresponding to ϕ_j values from 0 (voids) to 100% (pure asphalt). This range is divided into nine groups (m = 9), with the values in each group satisfying

$${\mathsf{\epsilon }}_{\mathrm{av},j}=\left(\mathit{j}- 1\right)\left(3.0 - 1.0\right)/ 8+1.0,\left(j=1, \dots ,9\right).$$

(11)

Figure 5 illustrates an example of the voxel count distribution of the aggregate–asphalt material. It is clear that its dielectric constant is distributed in a normal fashion. Then, the statistical methodology is extended to other aggregate voxels, which represent particles larger than 1 mm, to accommodate the normal distribution of their dielectric constant. This expansion is realized by incorporating a variability of ±s_s to ε_s, ensuring that ε_s ± s_s adheres to a normal distribution. In this study, s_s equals 1.7.

The process of constructing a three-phase asphalt pavement model using the DEM–CFS algorithm is summarized in

Table 2. The process of the DEM–CFS algorithm.

Step	Description
1	Generate a physics-based coarse aggregate model using the discrete element method (DEM), and then discretize it into Yee grids using a voxelization algorithm.
2	Calculate the average radius for fine aggregates larger than the grid, and incorporate them into the discretized model by replacing the interstitial spaces with similarly sized clusters.
3	Fuse the sub-grid components into aggregate–asphalt and asphalt–void materials. Define their dielectric constant distribution, calculate the voxel counts for each dielectric constant using Equations (6) and (7), and position these voxels adjacent to the coarse aggregate voxels.
4	Upon the completion of the component fusion strategy (CFS), coarse aggregate regions are segmented using the watershed algorithm, with a variable dielectric constant assigned to each region to ensure that the overall distribution follows the expected normal curve.

.

2.2. Workflow of the RSA Method

In this study, the RSA method is also applied to generate asphalt pavement models for comparison, as both RSA and our method can utilize the same input parameters, thereby enhancing the comparability of the results. A notable challenge with the RSA method is its tendency to prematurely reach the “jamming limit” [41], where no additional particles can be placed without overlapping with the existing ones. This prevents achieving the required number of particles, particularly in a 3D modeling scenario. Furthermore, the RSA method requires an overlap detection operation each time a new particle is generated. Its computation efficiency decreases sharply as the number of particles increases, significantly increasing computational costs in 3D modeling. Therefore, the RSA method is employed to generate 2D pavement models in this work. The process of constructing a three-phase heterogeneous model using the RSA algorithm is outlined as follows [40].

(1)

Define the particle size of the coarse aggregate at each grade as the median of the upper and lower sieve sizes.

(2)

Calculate the quantity of coarse aggregates according to the proportion of the corresponding gradation and the specifically defined particle size.

(3)

Place non-overlapping circular particles of varying sizes randomly within the modeling domain, in line with predefined statistical quantities.

(4)

The circles representing coarse aggregates are discretized, and the interstitial spaces are filled with fine aggregates, represented as unit squares.

(5)

Upon the completion of aggregate placement, the areas not occupied by aggregates are designated as asphalt binder. Air voids are finally introduced by randomly removing squares identified as asphalt binder to achieve a pre-defined porosity.

The flowchart of the RSA algorithm for generating three-phase asphalt mixture is illustrated in Figure 6.

2.3. Methods for Dielectric Constant Estimation

The theoretical dielectric constant of the generated asphalt mixture model (ε_am) is derived using the modified Böttcher model with a polarization coefficient of 1.28 [42], given by

$${\displaystyle \frac{{\mathsf{\epsilon }}_{\mathrm{am}}-{\mathsf{\epsilon }}_{\mathrm{a}}}{2.28{\mathsf{\epsilon }}_{\mathrm{am}}+0.72{\mathsf{\epsilon }}_{\mathrm{a}}}}={\displaystyle \frac{{\mathit{V}}_{\mathrm{s}}({\mathsf{\epsilon }}_{\mathrm{s}}-{\mathsf{\epsilon }}_{\mathrm{a}})}{1.28{\mathsf{\epsilon }}_{\mathrm{am}}+0.72{\mathsf{\epsilon }}_{\mathrm{a}}+{\mathsf{\epsilon }}_{\mathrm{s}}}}+{\displaystyle \frac{{\mathit{V}}_{\mathrm{b}}({\mathsf{\epsilon }}_{\mathrm{b}}-{\mathsf{\epsilon }}_{\mathrm{a}})}{1.28{\mathsf{\epsilon }}_{\mathrm{am}}+0.72{\mathsf{\epsilon }}_{\mathrm{a}}+{\mathsf{\epsilon }}_{\mathrm{b}}}}.$$

(12)

The dielectric constant of the asphalt mixture in the numerical experiment, as well as the specimen in the laboratory experiment, is estimated from the two-way travel time (TWTT) of EM waves propagating through it, as given by [43]

$${\mathsf{\epsilon }}_{\mathrm{am}}={\left(0.5\mathit{c}\cdot \mathit{h}/\mathit{t}\right)}^2$$

(13)

where h, t, and c denote the known thickness of the specimen, the time delay of the GPR signals reflected from the top and bottom of the specimen, and the speed of light in vacuum, respectively. The TWTT of the reflection signals is determined from the maximum amplitude picked from the envelope.

3. Numerical Simulation and Lab Experiment Verification

3.1. Numerical Experiments

In this section, we applied the DEM–CFS and RSA methods to establish dielectric models for three distinct types of asphalt mixtures, designated as Type 1, Type 2, and Type 3, respectively. The aggregate gradation for each mixture type is detailed in

Table 3. Aggregate gradation for each mixture type.

Sieve Size (mm)	Passing Ratio (%)
	Type 1-Top	Type 1-Bottom	Type 2	Type 3
26.5	100.0	100.0	100.0	100.0
19	100.0	97.5	100.0	100.0
16	100.0	90.0	100.0	100.0
13.2	87.0	80.0	100.0	87.0
9.5	63.7	64.0	97.0	63.7
4.75	22.0	22.0	29.0	22.0
2.36	16.5	15.0	19.0	16.5
1.18	14.0	13.0	16.0	14.0
0.075	4.0	4.0	7.0	4.0

. Notably, Type 1 is specifically designed as a double-layer structure with a 40 mm A-top and a 60 mm A-bottom.

Table 4. Asphalt mixture information for the specimens.

Mixture Type	φa	Gmm	Gb	Pb	Gse	${\mathit{M}}_{\mathbf{DEM}}$	$\overline{{\mathsf{\epsilon }}_{\mathbf{aa},\mathit{i}}}$	$\overline{{\mathsf{\epsilon }}_{\mathbf{av},\mathit{j}}}$
Type 1-top	20.0%	2.74	1.042	4.0%	2.93	6.3 kg	6.5	1.2
Type 1-bottom	25.0%	2.73		4.2%		9.0 kg	6.5	1.1
Type 2	10.5%	2.66		6.0%		16.2 kg	8.5	2.1
Type 3	6.0%	2.69		5.0%		16.4 kg	8.7	2.2

presents the target porosity for each mixture type, as well as other specific design parameters.

The 2D RSA models have dimensions of 0.3 m × 0.1 m, while the DEM–CFS models are 0.3 m × 0.3 m × 0.1 m. Figure 7 illustrates one of the RSA models and one slice of the corresponding DEM–CFS model. It is evident that the proposed DEM–CFS can better model the continuous variation of the dielectric constant of the mixture component than the RSA method. Furthermore, the double-layer structure presented in Figure 7a is well bonded, with no delamination (air layer) between the two layers. This indicates that the DEM–CFS model accurately resembles actual pavement condition.

Figure 8 depicts a 3D model for the GPR simulation. A perfect electric conductor (PEC) layer is set under the asphalt pavement to enhance the bottom reflection. The spacing between the transmitting (Tx) and receiving (Rx) antennas is 6 cm, and their height from the pavement surface is 5 mm. A Ricker wavelet with a central frequency of 2 GHz is employed for the source excitation. A recursive-integration-based complex-frequency-shifted perfectly matched layer (CFS-RIPML) [44] is employed as the absorption boundary condition.

For each numerical model, 16 GPR traces were simulated with a uniform trace interval of 1 cm. To accurately locate the pavement top reflection, the air coupling wave, which was separately simulated in a homogeneous air model, was removed. Figure 9 shows an example of the simulated GPR traces on the numerical models generated by the two methods. The time delay between the top and bottom reflections is further used to calculate the pavement dielectric constant, and the results are given in Table 6.

3.2. Laboratory Experiments

To examine the accuracy of the simulation results, we conducted laboratory experiments on nine asphalt mixture specimens. Specifically, three specimens were prepared for each of three asphalt pavement types given in Table 4. The parameters and dimensions were the same as those used in the numerical models. A static GPR measurement was carried out at the center of each specimen, under which a metal plate was placed, as shown in Figure 10. A commercial GPR system with a 2 GHz antenna was used for the measurement. Data processing, including DC removal, bandpass filtering, and time-zero correction, were applied to the raw data. Figure 11 compares the processed GPR A-scans recorded on the three types of asphalt specimen. Similarly, the time delay between the top and bottom reflections is further used to calculate the pavement permittivity.

Since the pressure during compaction is concentrated in the central area of the asphalt specimen, the pavement thickness at the central region of the specimen tends to be slightly lower than the designed height. Thus, a vernier caliper and a laser distance meter were utilized to measure the pavement thicknesses at several fixed points in the central area and estimate the average thickness. The measurement results of each specimen are summarized in

Table 5. Summary of measured thickness for each specimen.

Mixture Type	Type 1			Type 2			Type 3
	A	B	C	A	B	C	A	B	C
Thickness (mm)	102.29	102.52	100.48	97.30	90.70	92.90	96.87	98.12	96.99

.

3.3. Results of Relative Permittivity

Table 6. Dielectric constant of different asphalt pavement estimated by Equation (13) in the laboratory and the numerical experiments.

Mixture Type	Theoretical Value	Lab Test		DEM–CFS Model		RSA Model
		Value	mRE	Value	mRE	Value	mRE
Type 1	6.33	6.39 ± 0.16	1.93%	6.47 ± 0.2	3.38%	8.02 ± 0.1	26.63%
Type 2	7.55	7.56 ± 0.11	1.07%	7.53 ± 0.1	0.65%	8.25 ± 0.1	9.27%
Type 3	8.34	8.21 ± 0.39	4.40%	8.29 ± 0.1	1.25%	8.60 ± 0.1	3.10%

lists the dielectric constant of the three types of pavements estimated in the numerical and laboratory experiments, along with the theoretical values. To evaluate the accuracy of the laboratory measurements and precision of the numerical models, the mean relative error (mRE) between the theoretical and estimated dielectric constant is calculated.

The results in Table 6 prove a strong agreement between the laboratory test results and theoretical values. This means that the dielectric parameters of the coarse aggregates and the asphalt are assigned with a good accuracy. Additionally, the proposed DEM–CFS models exhibit superior performance over the RSA models, as evidenced by the much lower mREs of dielectric constant. Especially in the scenario of high porosity, the result of the RSA model significantly deviates from the expected value. The primary reason for this deviation is the lack of sufficient continuity in material variations in the RSA models, leading to significant step changes at material interfaces. These discontinuities cause numerical errors in the FDTD method, and the increased number of interfaces under high-porosity conditions further amplifies these cumulative errors.

3.4. Sensitivity Analysis

During the numerical experiments, we observed that the DEM–CFS models’ results are sensitive to changes in the mean values μ₁ and μ₂. Deviations from the optimal parameters can lead to significant errors in estimated dielectric constant. These two parameters are closely related to porosity (φ_a). Specifically, increased porosity necessitates a corresponding decrease in μ₁ and μ₂. However, if μ₁ and μ₂ are reduced to a large extent, it may result in an unreasonable distribution of the asphalt phase, leading to significant errors in the simulation results. Consequently, this section quantifies the relationship between μ₁, μ₂, and φ_a.

To conduct this analysis, the FDTD method is selected due to its efficiency in large-scale simulations and superior performance in handling complex boundary conditions, making it an ideal choice for the sensitivity analysis. The sensitivity analysis is conducted on eight asphalt models, with the porosity of each model ranging from 6% to 20%. In these models, G_s, P_b, and G_b are set to 3.0, 5.0%, and 1.042, respectively. Theoretically, the distribution of coarse aggregates in all models should remain consistent to avoid potential impacts on the analysis results. However, when all models use a single coarse aggregate distribution and the mixture’s porosity reaches 14%, the volumetric fraction of the fine aggregates becomes anomalously low. Therefore, the eight models are divided into two groups: a low-porosity group (6%–12%) using a uniform aggregate model based on the gradation curve of Type 3, and a high-porosity group (14%–20%) utilizing the gradation information of Type 1-top.

The setup of the FDTD model is identical to that described in Section 3.1. By repeatedly adjusting μ₁ and μ₂ for each model, the mRE between the simulated outcomes and the theoretical values is controlled within 2%. The final parameter settings are shown in

Table 7. Modeling information and simulation results in the parameter sensitivity analysis.

Mixture Type	Modeling Information			Theoretical Value	Simulation Results
	φa	μ1	μ2		Value	mRE
Mix 1	6%	9.5	2.6	8.18	8.18 ± 0.1	0.83%
Mix 2	8%	9.0	2.3	7.92	7.99 ± 0.1	0.96%
Mix 3	10%	8.5	2.1	7.67	7.74 ± 0.1	0.92%
Mix 4	12%	8.0	1.7	7.41	7.42 ± 0.1	0.53%
Mix 5	14%	7.0	1.4	7.16	7.14 ± 0.1	0.43%
Mix 6	16%	6.7	1.3	6.92	6.93 ± 0.1	0.26%
Mix 7	18%	6.3	1.2	6.67	6.74 ± 0.1	1.05%
Mix 8	20%	6.0	1.2	6.43	6.41 ± 0.1	0.31%

. Using the least squares algorithm to analyze eight sets of data points, the best fit line for the relationship among μ₁, μ₂, and φ_a was determined as follows:

$${\displaystyle \frac{{\phi }_{\mathrm{a}}-13.0}{0.96}}={\displaystyle \frac{{\mu }_{\mathrm{ab}}-7.63}{-0.25}}={\displaystyle \frac{{\mu }_{\mathrm{bv}}-1.76}{-0.1}}.$$

(14)

By using this formula, we can efficiently and accurately determine the optimal values of μ₁ and μ₂ that correspond to a given porosity. This can remove the need for iterative tuning, and drastically enhance the computation efficiency.

4. Discussion and Conclusions

In this paper, we propose a hybrid modeling method, DEM–CFS, for precise GPR simulations on heterogeneous medium. Its accuracy has been validated through both numerical and laboratory experiments on three types of asphalt pavement mixture, ranging from dense-graded to open-graded. The mRE between the estimated dielectric constant of the DEM–CFS models and the theoretical values exhibits significantly smaller errors than those of the RSA models, particularly under high-porosity conditions (φ_a = 23%), where the mRE is nearly one order of magnitude lower—specifically, 3.38% vs. 26.63%. Additionally, our numerical experiments revealed a strong dependency between porosity and optimal parameter settings. Using these insights, we optimize the CFS parameter selection process for CFS, significantly enhancing its applicability and eliminating the need for iterative tuning.

The improvement of our method relies on the employment of the DEM to address the challenge of irregular aggregate shapes and the spatial packing inefficiency, also called “jamming limit”, in 3D modeling. Moreover, we meticulously integrate the fine aggregates and asphalt binder through the CFS. This approach reflects the small-scale elements’ non-uniformities and statistical characteristics for capturing the macroscopic continuous distribution of the dielectric constant of the mixture. These advancements showcase the method’s versatility in constructing a wide range of complex models that incorporate particulate elements as their fundamental structure, such as sand, ballast, and large-grain soil–rock mixtures, paving the way for GPR simulations on various heterogeneous media. However, the DEM–CFS method is computationally intensive, especially for materials like sand or soil, where the numerous fine particles significantly increase computational demands. Consequently, using DEM–CFS for standard sand models may not be ideal unless the research involves specific physical simulations, such as studying water flow seepage through soil. Moreover, the method requires substantial prior information, potentially necessitating extensive calibration experiments. Future research should focus on optimizing its computational efficiency, expanding its application to other types of heterogeneous media, and developing more automated techniques for parameter extraction and model calibration.