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As a highly efficient absorbing boundary condition, Perfectly Matched Layer (PML) has been widely used in Finite Difference Time Domain (FDTD) simulation of Ground Penetrating Radar (GPR) based on the first order electromagnetic wave equation. However, the PML boundary condition is difficult to apply in GPR Finite Element Time Domain (FETD) simulation based on the second order electromagnetic wave equation. This paper developed a non-split perfectly matched layer (NPML) boundary condition for GPR FETD simulation based on the second order electromagnetic wave equation. Taking two-dimensional TM wave equation as an example, the second order frequency domain equation of GPR was derived according to the definition of complex extending coordinate transformation. Then it transformed into time domain by means of auxiliary differential equation method, and its FETD equation is derived based on Galerkin method. On this basis, a GPR FETD forward program based on NPML boundary condition is developed. The merits of NPML boundary condition are certified by compared with wave field snapshots, signal and reflection errors of homogeneous medium model with split and non-split PML boundary conditions. The comparison demonstrated that the NPML algorithm can reduce memory occupation and improve calculation efficiency. Furthermore, numerical simulation of a complex model verifies the good absorption effects of the NPML boundary condition in complex structures.

Numerical modeling of ground penetrating radar (GPR) is an important means to study high-frequency electromagnetic wave detection, which plays a critical role in theoretical research of electromagnetic wave propagation in underground structures and guiding for processing and interpretation of actual data [

When using a computer to carry out GPR numerical simulation, due to limited memory space, mesh space is always truncated somewhere to form a finite region. Then strong non-physical electromagnetic reflection interference waves will be generated at the truncations of the mesh space. Therefore, the truncation boundaries must be properly treated to eliminate or weaken such spurious boundary reflections. To this end, predecessors have done a lot of research work and developed many kind of boundary conditions, such as Sarma boundary condition [

It should be noted that the aforementioned PMLs and improved boundary conditions mostly are proposed for electromagnetic field modeling based on FDTD method. FDTD is a numerical calculation method based on first-order electromagnetic wave equations while FETD is a method based on second-order equations. Therefore, the PML boundary conditions widely used in FDTD cannot be directly applied to FETD. To solve this problem, Komatitsch and Tromp [

Based on the researches of Komatitsch, Tromp, Basu, Matzen, Liu, etc., and considering the FETD’s advantages of easiness to deal with irregular meshes and free surface boundary conditions as well as flexible representation of irregular boundaries and complex geometries, we propose an efficient non-split PML boundary condition by using complex stretching coordinate transformation and auxiliary variables. Furthermore, we give the computation format of FETD in time domain by using Galerkin approximation technique and develop the FETD forward modeling algorithm of second-order GPR wave equation based on the non-split PML boundary condition. The numerical examples show that, compared with the split PML boundary condition, the non-split PML boundary condition has made up the deficiency of splitting in electromagnetic wave field, ensuring the calculation accuracy and improving the calculation efficiency of full-wave-fields numerical modeling.

According to the electromagnetic wave theory, the propagation of GPR electromagnetic wave in underground medium satisfies the Maxwell’s equations [

μ ε ∂ 2 E z ∂ t 2 + μ σ ∂ E z ∂ t = ∂ 2 E z ∂ x 2 + ∂ 2 E z ∂ y 2 . (1)

where, E z is the electric field strength (V/m) in the z direction; H x , H y respectively are the magnetic field strength (A/m) in the x, y direction; t is the time (s), μ , σ , ε respectively are the permeability (H/m), electrical conductivity (S/m) and dielectric constant (F/m) of the medium.

According to the PML boundary condition theory [

p ˜ = p − i ω ∫ 0 p d p ( s ) d s , ( p = x , p = y ) . (2)

where, p ˜ is the complex coordinate, i = − 1 is the imaginary unit, d p ( p ) is the boundary attenuation coefficient, which is a real function attenuating with the coordinate p, and ω is the angular frequency.

From Equation (2), we can respectively obtain the first and second order partial derivatives relation for p and p ˜ as follows:

∂ ∂ p ˜ = i ω i ω + d p ∂ ∂ p , ∂ 2 ∂ p ˜ 2 = ( i ω i ω + d p ) 2 ∂ 2 ∂ p 2 − ( i ω ) 2 d ′ p i ω + d p ∂ ∂ p . (3)

Transforming Equation (1) to the frequency domain, we can obtain the following equation:

∂ 2 E ˜ z ∂ x 2 + ∂ 2 E ˜ z ∂ y 2 = ( − ω 2 μ ε + i ω μ σ ) E ˜ z . (4)

where E ˜ z is a Fourier transform of E z with respect to time. Substituting Equation (3) into Equation (4), we can derive the second-order electromagnetic wave equation in the complex stretching coordinate system as follows:

i ω i ω + d x ∂ ∂ x ( i ω i ω + d x ∂ E ˜ z ∂ x ) + i ω i ω + d y ∂ ∂ y ( i ω i ω + d y ∂ E ˜ z ∂ y ) = ( − ω 2 μ ε + i ω μ σ ) E ˜ z (5)

Deriving Equation (5) we can obtain:

( i ω i ω + d x ) 2 ∂ 2 E ˜ z ∂ x 2 + ω 2 d ′ x ( i ω + d x ) 3 ∂ E ˜ z ∂ x + ( i ω i ω + d y ) 2 ∂ 2 E ˜ z ∂ y 2 + ω 2 d ′ y ( i ω + d y ) 3 ∂ E ˜ z ∂ y = ( − ω 2 μ ε + i ω μ σ ) E ˜ z (6)

In the second-order electromagnetic wave Equation (6), the electromagnetic field is split into four terms under the traditional split PML boundary condition:

E ˜ z = E ˜ z , 1 + E ˜ z , 2 + E ˜ z , 3 + E ˜ z , 4 , (7)

The split results corresponding to Equation (6) can be expressed as:

{ ( − ω 2 μ ε + i ω μ σ ) E ˜ z , 1 = ( i ω i ω + d x ) 2 ∂ 2 E ˜ z ∂ x 2 , ( − ω 2 μ ε + i ω μ σ ) E ˜ z , 2 = ω 2 d ′ ( x ) ( i ω + d x ) 3 ∂ E ˜ z ∂ x , ( − ω 2 μ ε + i ω μ σ ) E ˜ z , 3 = ( i ω i ω + d y ) 2 ∂ 2 E ˜ z ∂ y 2 , ( − ω 2 μ ε + i ω μ σ ) E ˜ z , 4 + ω 2 d ′ y ( i ω + d y ) 3 ∂ E ˜ z ∂ y . (8)

Performing an inverse Fourier transform on both side of Equation (8) with respect to ω . Then the time domain wave equation satisfied in the PML region is as follows:

{ ( μ ε + μ σ ∫ 0 t ) ( ∂ t + d x ) 2 E z , 1 = ∂ 2 E z ∂ x 2 ( μ ε + μ σ ∫ 0 t ) ( ∂ t + d x ) 3 E z , 2 = d ′ x ∂ E z ∂ x ( μ ε + μ σ ∫ 0 t ) ( ∂ t + d x ) 2 E z , 3 = ∂ 2 E z ∂ y 2 ( μ ε + μ σ ∫ 0 t ) ( ∂ t + d x ) 3 E z , 4 = d ′ y ∂ E z ∂ y (9)

Equation (9) is the second-order GPR wave equation based on the split PML boundary condition. Where, the second and fourth equations need to calculate the third derivative of the electric field E ˜ z with respect to time and first integral, which will take more computation time. To avoid the calculation of the third derivative of E ˜ z with respect to time, the intermediate variables P ˜ x , P ˜ y are introduced. Letting ( ∂ t + d x ) E ˜ z , 2 = P ˜ x , ( ∂ t + d y ) E ˜ z , 4 = P ˜ y , the second and fourth equations in Equation (9) can be rewritten as:

{ ( μ ε + μ σ ∫ 0 t ) ( ∂ t + d x ) 2 P x = d ′ x ∂ E z ∂ x ( μ ε + μ σ ∫ 0 t ) ( ∂ t + d y ) 2 P y = d ′ y ∂ E z ∂ y (10)

To reduce the terms of split in electromagnetic field, the first two terms and the last two terms of Equation (6) respectively are combined into one term [

{ ( μ ε + μ σ / i ω ) ( i ω + d x ) 2 E ˜ 1 , z = ∂ 2 E ˜ z ∂ x 2 − d ′ x P ˜ x ( μ ε + μ σ / i ω ) ( i ω + d y ) 2 E ˜ 2 , z = ∂ 2 E ˜ z ∂ y 2 − d ′ y P ˜ y ( i ω + d ( x ) ) P ˜ x = ∂ E ˜ z ∂ x ( i ω + d ( y ) ) P ˜ y = ∂ E ˜ z ∂ y (11)

According to the Fourier transform theory, the time domain signal U ( t ) and its frequency signal U ˜ ( ω ) satisfied that U ( t ) = i ω ⋅ U ˜ ( ω ) , applied it into Equation (11), we can obtain Equation (12) as follows:

{ ( μ ε + μ σ ∫ 0 t ) ( ∂ t + d x ) 2 E 1 , z = ∂ 2 E z ∂ x 2 − d ′ x P x ( μ ε + μ σ ∫ 0 t ) ( ∂ t + d y ) 2 E 2 , z = ∂ 2 E z ∂ y 2 − d ′ y P y ( ∂ t + d x ) P x = ∂ E z ∂ x ( ∂ t + d y ) P y = ∂ E z ∂ y (12)

The comparison between Equation (9) and Equation (12) shows that the improved split PML boundary condition has the advantages over the traditional split PML boundary condition: 1) avoiding the calculation of third derivative with respect to time by splitting the displacement into two terms in the absorbing layer; 2) reducing calculation amount and improving calculation speed since the electromagnetic field is only split into two terms.

To avoid electromagnetic field splitting in Equation (5), the variables β x , β y are introduced [

β x = 1 + d x / i ω , β y = 1 + d y / i ω . (13)

The following identities can be obtained by deriving Equation (13):

{ ( i ω ) 2 β x β y = ( i ω ) 2 + i ω ( d x + d y ) + d x d y β y / β x = 1 + ( d y − d x ) / ( i ω + d x ) β x / β y = 1 + ( d x − d y ) / ( i ω + d y ) (14)

Multiplying both sides of Equation (5) by β x β y and substituting Equation (14) into it, we can obtain:

( ( i ω ) 2 + i ω ( d x + d y ) + d x d y ) μ ε E ˜ z + ( i ω + ( d x + d y ) + d x d y i ω ) μ σ E ˜ z = ∂ 2 E ˜ z ∂ x 2 + ∂ ∂ x ( d y − d x i ω + d x ∂ E ˜ z ∂ x ) + ∂ 2 E ˜ z ∂ y 2 + ∂ ∂ y ( d x − d y i ω + d y ∂ E ˜ z ∂ y ) (15)

Introducing the intermediate variables P x , P y , Q , defined as:

P ˜ x = d y − d x i ω + d x ∂ E ˜ z ∂ x , P y = d x − d y i ω + d y ∂ E ˜ z ∂ y , Q = d x d y i ω E ˜ z . (16)

Substituting Equation (16) into Equation (15) and deriving it, we can obtain:

[ ( i ω ) 2 + i ω ( d x + d y ) + d x d y ] μ ε E ˜ z + [ i ω + ( d x + d y ) ] μ σ E ˜ z + μ σ Q ˜ = ∂ 2 E ˜ z ∂ x 2 + ∂ P ˜ x ∂ x + ∂ 2 E ˜ z ∂ y 2 + ∂ P ˜ y ∂ y (17)

Performing an inverse Fourier transform on Equation (16) and Equation (17) to time domain, we can obtain:

{ μ ε ∂ 2 E z ∂ t 2 + [ ( d x + d y ) μ ε + μ σ ] ∂ E z ∂ t + [ d x d y μ ε + ( d x + d y ) μ σ ] E z + μ σ Q = ∂ 2 E z ∂ x 2 + ∂ P x ∂ x + ∂ 2 E z ∂ y 2 + ∂ P y ∂ y , ∂ P x ∂ t + d x P x = ( d y − d x ) ∂ E z ∂ x , ∂ P y ∂ t + d y P y = ( d x − d y ) ∂ E z ∂ y , ∂ Q ∂ t = d x d y E z . (18)

Equation (18) is the second-order GPR wave equation based on the non-split PML boundary condition. Comparison among Equations (9), (12) and (18) shows that the non-split PML boundary condition neither needs split the electromagnetic field, nor needs the calculation of third derivative with respect to time, which means less calculation amount and higher calculation speed.

According to the principle of the Galerkin method [

{ ∫ Ω ∂ 2 E z ∂ t 2 φ d Ω + ∫ Ω [ ( d x + d y ) + σ ε ] ∂ E z ∂ t φ d Ω + ∫ Ω [ d x d y + ( d x + d y ) σ ε ] E z φ d Ω + ∫ Ω σ ε Q φ d Ω − ∫ Γ 1 μ ε ( ∂ E z ∂ x n x + ∂ E z ∂ y n y ) φ d Γ + ∫ Ω 1 μ ε ( ∂ E z ∂ x ∂ φ ∂ x + ∂ E z ∂ y ∂ φ ∂ y ) d Ω = − ∫ Ω 1 μ ε ∂ P x ∂ x d Ω − ∫ Ω 1 μ ε ∂ P y ∂ y d Ω ∫ Ω ∂ P x ∂ t φ x d Ω + ∫ Ω d x P x d Ω = ∫ Ω ( d y − d x ) ∂ E z ∂ x d Ω ∫ Ω ∂ P y ∂ t φ y d Ω + ∫ Ω d y P y d Ω = ∫ Ω ( d x − d y ) ∂ E z ∂ y d Ω ∫ Ω ∂ Q ∂ t φ d Ω = ∫ Ω d x d y E z d Ω (19)

where, Γ , Ω respectively are the calculation region and its boundary. n = ( n x , n y ) T is the outer normal vector of the boundary. φ = ( φ x , φ y ) T is the virtual displacement vector. Dirichlet boundary condition with a displacement of 0 is used for the PML boundary and free boundary condition is used for the upper boundary. The finite element equation corresponding to Equation (19) can be derived with the Galerkin finite element method:

{ M E ¨ z + M 1 E ˙ z + M 2 E z + M 3 Q = K E z + K x P x + K y P y M P ˙ x + C x P x = C x x E z M P ˙ y + C y P y = C y y E z M Q ˙ = C x y E z (20)

Equation (20) is the FETD GPR wave equation based on the non-split PML boundary condition. Where, the coefficient matrixes are expressed as follows:

{ M = ∫ Ω N N T d Ω , M 1 = ∫ Ω ( d x + d y + σ / ε ) N N T d Ω , M 2 = ∫ Ω ( d x d y + ( d x + d y ) σ / ε ) N N T d Ω , M 3 = ( σ / ε ) ∫ Ω N N T d Ω , K = 1 μ ε ∫ Ω ( ∂ N ∂ x N T ∂ x + ∂ N ∂ y N T ∂ y ) d Ω , K x = 1 μ ε ∫ Ω ( ∂ N ∂ x N T ∂ x ) d Ω , K y = 1 μ ε ∫ Ω ( ∂ N ∂ y N T ∂ y ) d Ω , C x = ∫ Ω d x N N T d Ω , C y = ∫ Ω d y N N T d Ω , C x x = ∫ Ω ( d y − d x ) ∂ N ∂ x N T d Ω , C y y = ∫ Ω ( d x − d y ) ∂ N ∂ y N T d Ω , C x y = ∫ Ω d x d y N N T d Ω , (21)

where, N is the shape function of the calculation region, M is the mass matrix, C is the damping matrix, S is the load vector; U ¨ and U ˙ indicates the first and second derivative of the electric field E z with respect to time, respectively.

The general form of the equations in Equation (21) is:

M U ¨ + C U ˙ + K U = S (22)

To solve Equation (22), we use the Newmark difference algorithm which is a time integration algorithm with energy conservation. Using the algorithm, we can update the point ( U n + 1 , U ˙ n + 1 ) based on the point ( U n , U ˙ n ) . And the iterative formula of the Newmark time-stepping algorithm for Equation (22) is as follows:

{ U n + 1 = U n + Δ t U ˙ n + Δ t 2 [ ( 0.5 − β ) a n + β a n + 1 ] , U ˙ n + 1 = U ˙ n + Δ t [ ( 1 − γ ) a n + γ a n + 1 ] , a n + 1 = M − 1 ( S − C U ˙ n − K U n ) . (23)

The above matrix formed by finite elements is a large sparse matrix which requires a large memory space for storing all elements. In this paper, the stiffness matrix adopts compressed storage row (CSR) format and only needs to store its non-zero elements, which can greatly reduce the storage space needed [

To verify the advantages of the non-split PML boundary condition proposed in this paper, a homogeneous medium model of 2.2 m × 2.2 m is established with a relative dielectric constant of 5.0 and electrical resistivity of 0.001 Ω·m. For simplicity, the model space is discretized using structured 4-node quadrilateral meshes with a total of 220 × 220 meshes with a size of 0.01 m × 0.01 m. The PML absorbing boundary layer around the model is 0.2 m thick, occupying 20 meshes as a unit as shown in

The model is performed with FETD forward modeling under PML-free, split PML and non-split PML boundary conditions respectively, and the snapshots of wave fields at different times are obtained as shown in Figures 2-4.

8.5 ns, the reflections from the upper and left boundaries continue to spread to the middle of the model. Meanwhile, the wave fronts have propagated to the right and bottom boundaries and been reflected. The above non-physical reflected waves seriously interfere with the propagation of GPR waves in the model space.

electric field energy gradually propagates into the PML layers after 6 ns and attenuates rapidly to zero after 18 ns. It is clear that both of the PML boundary conditions have good absorption effects.

In order to better analyze the absorption effects of the two PML boundary conditions, we use the formula E r r o r d b = 20 × log ( | E S − E r e f | / | E r e f max | ) to calculate the reflection errors at the receiver under the two different boundary conditions, as shown in

For better analysis of the calculation efficiency of the split and non-split PML boundary conditions, we program the FETD GPR forward modeling under the two boundary conditions using the Matlab platform and test them on the same computer (Lenovo Think Centre M8300t). The statistics of CPU time consuming and memory occupation under the two boundary conditions is shown as

Homogeneous medium | Split PML boundary condition | Non-split PML boundary condition |
---|---|---|

CPU time consuming (ns) | 45.6 | 23.8 |

Memory occupation (MB) | 32.5 | 24.8 |

To verify the absorption effects of the non-split PML boundary condition in the FETD forward modeling of complex GPR model, a complex geo-electric model is established as shown in ^{−1} and a lower layer medium of soil with relative dielectric constant of 10.0 and electrical conductivity of 0.002 S∙m^{−1}. In the lower layer medium, three circular anomalies with a radius of 0.1m are buried at different depths. From left to right, their relative dielectric constants and center coordinates are 20, 15, 81 and (1.4 m, 1.0 m), (2.4 m, 1.1 m), (3.4 m, 1.2 m), respectively. The model is discretized by quadrilateral meshes with a total of 480 × 180 meshes and a distance of 0.01 m between each other. In addition, there are 20 PML boundary layers on the outer boundary of the calculation region. At last, a zero-phase Ricker wavelet with a center frequency of 500 MHz is given as an excitation source on the surface, and FETD method based on the non-split PML boundary condition is used for forward modeling with a time step of 0.02 ns and a window length of 40 ns.

the water pipe medium and the background medium. Based on wide angle method, the profile of GPR forward modeling is shown as

To better understand the absorption effects of non-split PML boundary condition on the strong reflections from model boundaries, the excitation source is positioned in the center of the surface and the wave field snapshots of different times are obtained through the FETD forward modeling, as shown in

wave fronts propagate to the three circular anomalies, from which the reflected waves spread and propagate; meanwhile, the reflected waves from the undulating interface all have entered the PML layers and have been completely absorbed. In

In this paper, based on the second order electromagnetic wave equation, an efficient non-split PML boundary condition is proposed by using the complex stretching coordinates and introducing the auxiliary functions. Furthermore, the FETD GPR wave equation under the non-split PML boundary condition is derived based on the Galerkin method. The forward modeling examples show that spurious reflections from the boundaries can be well absorbed under the non-split PML boundary condition. Compared with the split PML boundary condition, the non-split PML boundary condition can reduce memory occupation and improve calculation efficiency without loss of forward modeling accuracy. Therefore, based on the convenient and efficient non-split PML boundary condition and combined with the FETD method, which can represent complex geometric models easily by using irregular quadrilateral and triangular meshes, we can perform rapid and highly accurate FETD forward modeling of complex GPR models.

This research is financially supported by the National Natural Science Foundation of China (41574078, 41604102 and 41604039) and supported by the National Natural Science Foundation of Guangxi (2016GXNSFBA380082, 2016GXNSFBA380215, 2018GXNSFAA138059).

The authors declare no conflicts of interest regarding the publication of this paper.

Zhang, Z., Wang, H.H., Wang, M.L., Guo, X. and Guo, G.H. (2019) Non-Split PML Boundary Condition for Finite Element Time-Domain Modeling of Ground Penetrating Radar. Journal of Applied Mathematics and Physics, 7, 1077-1096. https://doi.org/10.4236/jamp.2019.75073