IEEE TRANSACTIONS ON COMPONENTS, PACKAGING AND MANUFACTURING TECHNOLOGY, VOL. 5, NO. 12, DECEMBER 2015
1839
Efficient Noniterative Implicit Time-Stepping Scheme Based on E and B Fields for Sequential DG-FETD Systems Qingtao Sun, Luis Eduardo Tobon, Qiang Ren, Yunyun Hu, and Qing Huo Liu, Fellow, IEEE Abstract— The discontinuous Galerkin finite-element timedomain (DG-FETD) method with implicit time integration has an advantage in modeling electrically fine-scale electromagnetic problems. Based on domain decomposition methods, it avoids the direct inversion of a large system matrix as in the conventional FETD method; by employing implicit time integration, it obviates an extremely small time-step interval to maintain stability as in explicit schemes. Based on curl-conforming basis functions for the electric field intensity E field and divergence-conforming basis functions for the magnetic flux density B field, a new noniterative implicit time-stepping scheme is proposed to efficiently solve sequentially ordered systems for electrically fine-scale problems. Compared with the previous EH-based scheme, the new scheme introduces fewer unknowns and, thereby, results in a smaller matrix system. Based on the Crank–Nicholson algorithm for time integration, the matrix system is in a block tridiagonal form. Then, through separating the surface unknowns from the volume unknowns, a block lower-diagonal-upper (LDU) decomposition is implemented, reducing the computational complexity of the original system. The adaptivity of parallel computing in subdomain level during preprocessing further helps shorten the computation time. Numerical results confirm that the proposed LDU scheme presents improved efficiency in terms of memory and CPU time while retaining the same accuracy, compared with the previous implicit block-Thomas method. With respect to the explicit Runge–Kutta method and the standard FDTD, it also shows an advantage in CPU time. The proposed scheme will help improve the performance of DG-FETD in modeling electrically fine-scale problems. Index Terms— Basis functions, curl-conforming elements, discontinuous Galerkin finite-element time domain (DG-FETD), divergence-conforming elements, electrically fine-scale problems, implicit time stepping, lower-diagonal-upper (LDU) decomposition, sequential order.
I. I NTRODUCTION
T
HE FINITE-ELEMENT method is widely used for the numerical modeling of electromagnetic problems due
Manuscript received February 19, 2015; revised July 13, 2015; accepted October 25, 2015. Date of publication November 19, 2015; date of current version December 11, 2015. Recommended for publication by Associate Editor M. S. Tong upon evaluation of reviewers’ comments. (Corresponding author: Qing Huo Liu.) Q. Sun, Q. Ren, Y. Hu, and Q. H. Liu are with the Department of Electrical and Computer Engineering, Duke University, Durham, NC 27708 USA (e-mail:
[email protected];
[email protected];
[email protected];
[email protected]). L. E. Tobon is with the Department of Computer Science and Engineering, Pontificia Universidad Javeriana at Cali, Cali 56710, Colombia (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCPMT.2015.2496192
to its flexibility of discretizing complex structures. The discontinuous Galerkin finite-element time-domain (DG-FETD) method shows great potential in modeling large problems by splitting the original computational domain with a large number of unknowns into smaller subdomains with lower degrees of freedom (DoFs) [1]–[12]. Through spatial discretization, a large linear system is transformed into a combination of a set of smaller systems, reducing the difficulty and complexity of the problems [11], [13]. However, DG-FETD still encounters another problem of employing an optimal time-stepping scheme to present good adaptability for large linear systems for electrically fine-scale problems, where the discretization steps are much smaller than a wavelength. Explicit schemes, featuring more efficient time stepping compared with implicit ones, are limited by the Courant–Friedrichs–Lewy (CFL) condition [14]. For electrically fine-scale problems, explicit schemes would require an extremely small time-step increment to maintain stability, resulting in a huge number of time iterations to complete the simulation. In contrast, implicit time-stepping schemes show an advantage in unconditioned stability, thus allowing large time-step intervals. Their numerical performance can be affected as the system matrix becomes large by requiring higher memory with respect to explicit ones. For the first-order Maxwell’s curl equations, typical implicit schemes include the alternating-direction-implicit (ADI) method [15]–[17] and the Crank–Nicholson-based methods, both of which are unconditionally stable. However, ADI suffers from severe numerical dispersion and anisotropy as the time-step interval increases, which significantly affects the modeling accuracy [18], [19]. By contrast, the Crank–Nicholson-based methods can provide results with better accuracy [19]–[23]. Moreover, for a sequentially ordered system, the block-Thomas (BT) algorithm can be further employed to accelerate time stepping for the resulting block tridiagonal linear system [12], [24]. For the second-order wave equation, the Newmark-beta method can be employed to yield an unconditional stable scheme by choosing β ≥ 1/4 [25]. For the methods based on the first-order Maxwell’s curl equations, different combinations of field variables to be solved results in different schemes. Moreover, the incorrect choices of basis functions may result in spurious modes, which contaminate field variation in time domain. By employing curl-conforming and divergence-conforming basis functions [26]–[30], field variables E, H, B, and D can be spatially discretized, as the curl-conforming basis
2156-3950 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
1840
IEEE TRANSACTIONS ON COMPONENTS, PACKAGING AND MANUFACTURING TECHNOLOGY, VOL. 5, NO. 12, DECEMBER 2015
Fig. 1. Curl-conforming and divergence-conforming basis functions for the E and B fields.
are the unknowns. Then, the weak forms of (1) and (2) can be obtained through Galerkin’s method (i) (i) (i) ∂E (i) −1 (i) dV = p · ∇ × (i) B dV p ·μ (i) (i) ∂t (i) (i) (i) − (i) p · σe E + Ji d V (i) (i) + (i) × μ−1 Btot d S p · nˆ ∂(i)
functions are employed to discretize E and H, and the divergence-conforming basis functions are employed to discretize B and D. The EH scheme requires mixed-order curl-conforming basis functions for E and H (e.g., En Hn+1 or En+1 Hn ) to suppress spurious modes [11], [31]–[34]. This constraint leads to a linear system with large DoFs. By contrast, the EB scheme allows the same order discretization for E and B (e.g., En Bn ) [35], [36], and has smaller DoFs. Thus, in this paper, the EB scheme is employed to solve electrically fine-scale problems. Most previous DG-FETD methods are based on explicit schemes [1]–[5], [8], [10], [11], [37], [38]. However, for electrically fine-scale structures, an implicit or semi-implicit scheme is preferable, as the number of time steps can be greatly reduced [12], [39], [40]. This paper presents a new noniterative implicit method to accelerate time stepping for DG-FETD. The proposed scheme, according to the block tridiagonal property for linear systems with sequentially ordered subdomains [12], reorders the system matrix by separating the surface unknowns from the volume unknowns, and then implements a block lower-diagonal-upper (LDU) decomposition. Finally, the original linear system is transformed into a three-step LDU matrix solution, requiring lower computational complexity. The accuracy of the new scheme is validated through a short microstrip line case and a cavity case. Its efficiency is further demonstrated through a long microstrip line case and an interconnect package case against the previous implicit scheme—the BT method, the explicit Runge–Kutta method of fourth order (RK4), and the standard FDTD method, in terms of CPU time, memory usage, and some other evaluating parameters. II. G OVERNING E QUATIONS Maxwell’s curl equations are given as follows:
∂E = ∇ × μ−1 B − σe E − J ∂t ∂B = −∇ × E − σm μ−1 B − M ∂t
(1) (2)
where E and B represent the electric field intensity and the magnetic flux density, respectively; , μ, σe , and σm are the permittivity, permeability, electric conductivity, and magnetic conductivity of the medium, respectively; and J and M are the imposed electric and magnetic current densities, respectively. The curl-conforming basis functions and the divergenceconforming basis functions are chosen to discretize the E field and the B field (as depicted in Fig. 1), respectively, in the form of E = j e j j and B = j b j j , and e j and b j
(i) q ·
(i)
∂B(i) ∂t
dV = −
(3)
(i) ∇ × (i) · E d V − q
(i)
(i)
(i) q
−1 (i) · σm(i) μ(i) B(i) + Mi d V (i) − (i) × Etot d S q · nˆ ∂(i) (i) ≈− (i) · (∇ ×E )d V − (i) q q (i)
(i)
−1 · σm(i) μ(i) B(i) + Mi(i) d V (i) (i) × Etot )d S − q · (nˆ ∂(i) (i) + (i) × E(i) )d S q · (nˆ ∂(i)
(4)
where the superscript (i ) denotes the i th subdomain, (i) represents its volume, ∂(i) represents its boundaries, including shared surfaces with adjacent subdomains and other boundaries (such as perfect electric conductor and perfect magnetic conductor), nˆ (i) represents its outward normal unit vector, and (·)tot denotes the total field on the boundaries. Note that integration by part is performed once in (3) and twice in (4) to obtain the surface integrals. The reason for twice integration by part in (4) is that the curl of divergenceconforming basis functions gives a null subspace after the first integration by part. Furthermore, during the second integration by part, one of Etot is approximated as E(i) in the surface integral terms. The Riemann solver (upwind flux) can be adopted to evaluate the surface integrals [5], [6], [10]–[12], [34], [41], [42]. When all the subdomains are sequentially ordered, for the i th subdomain, only the (i −1)th and (i +1)th subdomains are adjacent to it; thus, the linear system can be finally expressed in a compact form as M(i)
i+1 ∂u(i) = L(i, j ) u( j ) + q(i) , i = 1, . . . N ∂t
(5)
j =i−1
where M(i) is the mass matrix; u(i) is the unknowns to be solved; L(i, j ) contains the stiffness matrices, damping matrices, and interface matrices, resulting from the Riemann solver; and q(i) is the source term. For brevity, the exact expressions for the matrices are not elaborated here, as similar formulation can be found in [12] and [34]. III. B LOCK LDU D ECOMPOSITION For N subdomains with a sequential order, based on the Crank–Nicholson algorithm, the whole linear system presents
SUN et al.: EFFICIENT NONITERATIVE IMPLICIT TIME-STEPPING SCHEME BASED ON E AND B FIELDS
a block tridiagonal form as follows: ⎤ ⎡ (1,1) A B(1,2) 0 ... 0 ⎥ ⎢ B(2,1) A(2,2) B(2,3) ... 0 ⎥ ⎢ ⎥ ⎢ . .. (3,2) (3,3) ⎥ ⎢ 0 B A . . . ⎥ ⎢ ⎥ ⎢ . . . . .. .. .. (N−1,N) ⎦ ⎣ .. B (N,N−1) (N,N) 0 0 ... B A ⎡ (1) ⎤ ⎡ (1) ⎤ u νn ⎢ n+1 ⎥ ⎢ ⎥ ⎢ u(2) ⎥ ⎢ (2) ⎥ ⎢ n+1 ⎥ ⎢ νn ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ (3) ⎥ (3) ⎥ ×⎢ ⎢ un+1 ⎥ = ⎢ νn ⎥ ⎢ . ⎥ ⎢ . ⎥ ⎢ . ⎥ ⎢ . ⎥ ⎣ . ⎦ ⎣ . ⎦ (N) un+1 νn(N)
(6)
where 1 A(i,i) = M(i) − tL(i,i) 2 1 (i, j ) B = − tL(i, j ) 2 νn(i) = [ −B(i,i−1)
C(i,i)
⎡
(i−1)
un
⎤
⎢ (i) ⎥ (i) ⎥ −B(i,i+1) ] ⎢ ⎣ un ⎦ +tqn+ 12 (i+1) un
1 C(i,i) = M(i) + tL(i,i) . 2 Directly inverting the matrix in (6) is difficult for a large problem. Previously, the BT method was introduced to accelerate the procedure, which solves the submatrix system for each subdomain instead [12]. However, as the number of DoFs increases in each subdomain, this method can still be costly in terms of memory and CPU time. We, hereby, propose a new block LDU scheme to further accelerate the solution procedure. To illustrate the scheme in detail, we simply apply it to a two-subdomain system (1,1) (1,2) (1) (1) B u ν A = . (7) B(2,1) A(2,2) u(2) ν (2) First, the system is reordered according to the type of the unknowns. In particular, the unknowns on the surface connecting the two subdomains (surface unknowns) are separated from the unknowns inside the subdomains (volume unknowns) as in ⎤ ⎡ 0 AVS+ (1,1) 0 AV (1,1) ⎢ 0 AV (2,2) 0 AVS− (2,2) ⎥ ⎥ ⎢ (1,1) ⎣ AS+ V (1,1) 0 AS+ BS+ S− (1,2) ⎦ 0 AS− V (2,2) BS− S+ (2,1) AS− (2,2) ⎡ (1) ⎤ ⎡ (1) ⎤ uV νV ⎢ (2) ⎥ ⎢ (2) ⎥ ⎢u ⎥ ⎢ν ⎥ ⎢ V ⎥ ⎢ V ⎥ ×⎢ ⎥=⎢ ⎥ (8) ⎢ u(1) ⎥ ⎢ ν (1) ⎥ ⎣ S+ ⎦ ⎣ S+ ⎦ uS(2) −
1841
unknowns connected to the (i + 1)th subdomain, while “−” denotes the surface unknowns connected to the (i −1)th subdomain. For the two-subdomain system, therefore, (1) and uS(1) uV + represent the volume unknowns inside the first subdomain and the surface unknowns connected to the second (2) (2) subdomain, respectively; similarly, uV and uS− represent the volume unknowns inside the second subdomain and the surface unknowns connected to the first subdomain, respectively. Note that the surface unknowns of the EB scheme in this paper is different from those in the EH scheme as proposed in [33] and [43]. For instance, the surface unknowns of the EB scheme in a tetrahedron element involve all the E unknowns on the surface and the B unknowns in the element, while the surface unknowns of the EH scheme are only related to all the E and H unknowns on the surface. This is the main difference of the LDU decomposition when applied to the EH scheme and the EB scheme. Finally, the transformed matrix in (8) is decomposed in a block LDU form as follows: ⎡ (1) ⎤ ⎡ (1) ⎤ uV νV ⎢ (2) ⎥ ⎢ (2) ⎥ ⎢u ⎥ ⎢ν ⎥ ⎢ V ⎥ ⎢ V ⎥ LDU ⎢ (9) ⎥=⎢ ⎥ ⎢ u(1) ⎥ ⎢ ν (1) ⎥ ⎣ S+ ⎦ ⎣ S+ ⎦ (2)
uS− where
⎡
I 0
0 I 0
(2)
νS− 0 0 I 0
⎤ 0 0⎥ ⎥ 0⎦ I
⎢ (10) L=⎢ ⎣ WS+ V (1,1) 0 WS− V (2,2) ⎤ ⎡ (1,1) DV 0 0 0 ⎥ ⎢ 0 DV (2,2) 0 0 ⎥ (11) D=⎢ (1,1) (1,2) ⎦ ⎣ 0 0 DS+ DS+ S− (2,1) (2,2) 0 0 DS− S+ DS− ⎡ ⎤ (1,1) 0 I 0 WVS+ (2,2) ⎥ ⎢0 I 0 W VS− ⎥. U=⎢ (12) ⎣0 0 ⎦ I 0 0 0 0 I From the expressions (10) to (12), we can clearly find that L and U are lower and upper triangular matrices, representing coupling matrices between surface unknowns and volume unknowns of each subdomain; note that L is not necessarily the transpose of U. Matrix D exhibits a block diagonal form, containing self-coupling matrices between volume unknowns of each subdomain and also matrices for surface unknowns. By doing this, the original linear equation (6) becomes lighter to solve than the direct inversion or the BT method, and the solution procedure for the transformed linear equation (9) involves forward and backward substitutions for the L and U block triangular matrices, and the BT method for the surface unknowns, as detailed below.
νS(2) −
where the subscript V denotes the volume unknowns and the subscript S denotes the surface unknowns, and the sign “+,” for the i th subdomain in general, denotes the surface
A. Preprocessing In order to solve the transformed linear equation (9), we first need to prepare some intermediate submatrices before time
1842
IEEE TRANSACTIONS ON COMPONENTS, PACKAGING AND MANUFACTURING TECHNOLOGY, VOL. 5, NO. 12, DECEMBER 2015
(i, j )
(i,i) (i,i) stepping, such as D(i,i) V , DS± S∓ , WS± V , and DS± , by (i,i)
DV
(i, j )
C. Solve for Dz = y
(i,i)
= AV
(i, j )
DS± S∓ = BS± S∓
(i,i) WS(i,i) = A(i,i) ± V DV S± V (i,i)
(i,i)
(i,i)
(i,i)
DS± = AS± − WS± V AVS± . From the formulations, WS(i,i) ± V can be obtained through the based on lower-upper prefactorization of the submatrix D(i,i) V (i,i) decomposition; then, with WS(i,i) , D can be obtained. Note ±V S± (i,i) that the submatrix WVS± in (12) does not necessarily present a transpose relationship with WS(i,i) ± V , and to explicitly obtain it, we need to solve the following equation: (i,i) (i,i) D(i,i) V WVS± = AVS± .
(13)
During practical implementation, however, we can avoid this through a simple transformation, which will be detailed in the following step. This can help reduce the computational load for preprocessing. In general, the preprocessing step is a procedure requiring heavy computational load, as it involves matrix inversion and multiplication. For the LDU scheme, fortunately, this step can be parallelized at the subdomain level to prepare the matrices before time stepping, thus significantly reducing the CPU time, especially for cases with many subdomains. This is a distinctive characteristic of the new scheme with respect to the BT scheme, which is a sequentially based algorithm at the subdomain level, that is, the BT scheme would require the information communication between different subdomains, and, thus, is not suitable for parallelization. To demonstrate this advantage, we implement a primary parallel experiment for the LDU scheme using the parallel computing toolbox in MATLAB. After preparing the intermediate submatrices, (9) can be solved in the following three steps.
After vector y is obtained, the equation related to the D matrix can be written as ⎡ (1) ⎤ ⎤ zV ⎡ (1,1) 0 0 0 DV ⎢ (2) ⎥ (2,2) zV ⎥ ⎥⎢ ⎢ 0 D 0 0 ⎥ V ⎥⎢ ⎢ ⎥ ⎢ ⎣ 0 0 DS+ (1,1) DS+ S− (1,2) ⎦ ⎢ z(1) ⎥ + ⎦ ⎣ S 0 0 DS− S+ (2,1) DS− (2,2) (2) zS− ⎡ (1) ⎤ yV ⎢ (2) ⎥ ⎢y ⎥ ⎢ V ⎥ =⎢ ⎥ (15) ⎢ y(1) ⎥ ⎣ S+ ⎦ yS(2) − where z is also an intermediate vector. , which is related In fact, for (15), we only need to solve z(i) S± to the surface unknowns of each subdomain, by employing the BT method with the form (1) (1) yS+ zS+ DS+ S− (1,2) DS+ (1,1) = (16) (2,1) (2,2) (2) (2) DS− S+ DS− zS− yS− and z(i) V can be treated implicitly during the next step. D. Solve for Uu = z After vector z is obtained, the equation related to the U matrix can be written as ⎡ (1) ⎤ ⎡ (1) ⎤ zV ⎡ ⎤ uV (1,1) 0 I 0 WVS+ ⎢ (2) ⎥ ⎢ (2) ⎥ ⎢u ⎥ ⎢z ⎥ ⎢0 I 0 WVS− (2,2) ⎥ V ⎥ ⎢ V ⎥ ⎢ ⎥⎢ ⎥=⎢ ⎥. (17) ⎣0 0 ⎦⎢ ⎢ u(1) ⎥ ⎢ z(1) ⎥ I 0 ⎣ S+ ⎦ ⎣ S+ ⎦ 0 0 0 I uS(2) z(2) − S− From (17), we can obtain the surface unknowns by uS(i)± = z(i) . S± Then, for the volume unknowns, we arrive at (i) (i,i) (i) = z(i) uS+ − WVS− (i,i) uS(i)− . uV V − WVS+
B. Solve for Ly = ν The first step is to solve for the equation related to the L matrix as ⎡
I ⎢ 0 ⎢ ⎣ WS+ V (1,1) 0
0 I 0
WS− V (2,2)
0 0 I 0
⎤
⎡
(1) yV
⎤
⎡
νV(1)
⎤
0 ⎢ ⎥ ⎢ ⎥ ⎢ y(2) ⎥ ⎢ ν (2) ⎥ 0⎥ ⎢ ⎢ ⎥ ⎥ V V ⎥⎢ ⎥=⎢ ⎥ (14) 0 ⎦ ⎢ y(1) ⎥ ⎢ ν (1) ⎥ ⎣ S+ ⎦ ⎣ S+ ⎦ I yS(2) νS(2) − −
where y is an intermediate vector. As this is a simple lower triangular system, it is easy to arrive at (i)
(i)
(i)
(i)
yV = νV
(i)
yS± = νS± − WS± V (i,i) νV .
(18)
Note that (18) is a general equation to obtain the volume unknowns of the i th subdomain. For the two-subdomain (1) (2) system, uS− and uS+ do not exist, and we can treat them as zero, so that we can have a universal equation for all the subdomains. By performing the left multiplication of D(i,i) V for (i,i) (i) (i) both the sides of (18), together with (13) and DV zV = yV from the previous step, we finally arrive at (i) (i) (i,i) (i) D(i,i) uS+ − AVS− (i,i) uS(i)− V uV = νV − AVS+
(19)
which can be solved based on the prefactorized submatrix D(i,i) V during preprocessing. Through the above transformation, we avoid the necessity of obtaining WVS± (i,i) in advance. From the aforementioned procedure, we can observe that the new LDU scheme solves equations involving volume and surface unknowns separately for each subdomain, rather than
SUN et al.: EFFICIENT NONITERATIVE IMPLICIT TIME-STEPPING SCHEME BASED ON E AND B FIELDS
1843
Fig. 2. Geometry of the short microstrip line. (a) Whole structure. (b) Cross section.
as a whole matrix, which reduces the dimension of the linear system and, thereby, the computational complexity, with respect to the previous implicit BT method. In particular, for a subdomain with DoFs of M = M S + MV (M S and MV represent the surface DoFs and the volume DoFs, respectively), the computational complexity of the BT method is O(M 2 ) [44], in general, while for LDU, it is O((max(MV , M S ))2 ). In addition, since the new scheme is based on the Crank–Nicholson algorithm and also a noniterative solver, it is unconditionally stable and has no convergence issue, thus facilitating the modeling of electrically fine-scale problems by allowing large time-step intervals and meanwhile maintaining high accuracy, compared with explicit methods and iterative methods. The accuracy and efficiency of the new scheme are demonstrated in Section IV.
Fig. 3. Results of the short microstrip line with the LDU scheme and FDTD. (a) Scattered voltages (note that the curve of LDU Port 1 exists under the squares). (b) Magnitude of scattering parameters S11 and S21 (the curve LDU S21 exists under the circles).
IV. N UMERICAL R ESULTS AND D ISCUSSION A. Case 1 (Short Microstrip Line) To demonstrate its accuracy, we first apply the new scheme to a short 50- microstrip line with an active lumped port at one end and a load at the other end, as depicted in Fig. 2(a) and (b). The first derivative of the Blackman–Harris window (BHW) pulse with a central frequency of 18.5 GHz is adopted as the excitation source function [45], [46]. The original model is split into two subdomains to take advantage of domain decomposition. The numerical results from DG-FETD are shown in Fig. 3(a) and (b) with FDTD as a reference. From Fig. 3(a) and (b), we can observe both the scattered voltages and the scattering parameters from the new scheme agree well with the reference, demonstrating its accuracy. B. Case 2 (Electric Dipole in a Cavity) To further demonstrate its accuracy and efficiency, we apply the new scheme to a cavity case with an electric dipole source
Fig. 4. Mesh of the cavity model with four subdomains at a sampling density of 32 PPWs for the first scenario.
in it, as shown in Fig. 4. The dipole source is polarized in (0, 0, 1), and has a BHW pulse with a central frequency of 2.78 GHz. Fig. 5(a) and (b) presents the calculated transient electric and magnetic fields at a receiver located at (−1.5, 0, 0) cm. To demonstrate the efficiency, we investigate the performance of the proposed scheme under two scenarios. In the first scenario, the cavity is divided into
1844
IEEE TRANSACTIONS ON COMPONENTS, PACKAGING AND MANUFACTURING TECHNOLOGY, VOL. 5, NO. 12, DECEMBER 2015
Fig. 5. Calculated time-varying fields at the receiver (3, 0, 0.2) cm by the two schemes when the electric dipole polarized along (0, 0, 1) is located at (−1.5, 0, 0) cm. (a) E z . (b) H y .
Fig. 7. Comparison of computational costs between the BT and LDU schemes for different DoFs. (a) Total DoFs versus memory usage. (b) Total DoFs versus CPU time.
Fig. 6.
Comparison of DoFs of the EH scheme and the EB scheme.
four subdomains, the sampling density [in terms of points per wavelength (PPWs)] is gradually increased from 20 to 33 with respect to the wavelength for the highest frequency 6 GHz (−40 dB below the peak), and the corresponding DoFs of each subdomain increase from 9000 to 40 000. In the second scenario, to analyze the correlation between the speedup and the ratio of MV /M S , we change the number of subdomains from three to seven. As the number of subdomains increases, the proportion of the surface unknowns becomes larger for each subdomain, thereby helping us understand how it will affect the performance of the proposed scheme. The comparisons of the new EB-based LDU scheme with other schemes are shown in Figs. 6, 7(a) and (b), and 8(a) and (b). Fig. 5(a) and (b) shows excellent agreement between the new scheme and the BT method, demonstrating the LDU
scheme has the same accuracy as BT, which can also be observed from the formulation. The advantage of employing the EB scheme over the EH scheme can be clearly found in Fig. 6; as for the same sampling density in terms of E, the EH scheme requires approximately three times more DoFs. This is because the EH scheme has to employ mixed-order basis functions to discretize E and H, while the EB scheme allows the same order discretization for E and B. Undoubtedly, the EB scheme can expect much smaller computational overheads involved in the matrix inversion than the EH scheme. From Fig. 7(a) and (b), the new scheme is demonstrated to consume lower memory usage than the BT method in the first scenario. As the DoF increases, the difference becomes larger. For instance, when the total DoFs reach 150k, the BT scheme could approximately consume twice the memory usage of the LDU scheme. In terms of CPU time, although the difference between the two methods is not as apparent as for memory usage, it shows an increasing trend with DoFs. The first scenario indicates that the new scheme would exhibit more impressive performance for cases with large DoFs with respect to the BT method. Fig. 8(a) and (b) demonstrates the performance of the new scheme in the second scenario. As the number of subdomains increases from 3 to 7, the ratio of MV /M S decreases
SUN et al.: EFFICIENT NONITERATIVE IMPLICIT TIME-STEPPING SCHEME BASED ON E AND B FIELDS
1845
Fig. 9. Geometry of the long microstrip line with holes (courtesy of Dr. H. Braunisch, Intel). (a) Geometry parameters. L = 16.054 mm, W = 0.02662 mm, Ws = 1.26 mm, T = 0.01563 mm, H1 = 0.02072 mm, and H2 = 0.01792 mm. (b) Structure view.
Fig. 8. Comparison of computational costs between the BT and LDU schemes for different ratios of MV /M S . (a) Different values of MV /M S versus memory usage. (b) Different values of MV /M S versus CPU time.
from 22 to 7. Theoretically, we should expect a larger difference between LDU and BT in terms of numerical performance. The computational complexity of LDU for this scenario is in the level of O(MV2 ), while the computational complexity of BT is O(M 2 ). Their difference should be proportional to (M/MV )2 = (1 + M S /MV )2 . Therefore, as the ratio of MV /M S decreases, (M/MV )2 becomes larger, indicating larger difference. The speedup effect should be more obvious for the LDU scheme. From Fig. 8(a) and (b), we can also confirm such a trend, demonstrating the effectiveness of the proposed scheme. C. Case 3 (Long Microstrip Line With Complicated Structures) Then, we apply the new scheme to a long 50- microstrip line model with more complicated structures provided by Intel Corporation (courtesy of Dr. H. Braunisch), which features many holes in it. The detailed geometry of the model is depicted in Fig. 9(a) and (b). The first derivative of the BHW pulse with a central frequency of 18.5 GHz is employed for the simulation. Note that the smallest wavelength for this model (corresponding to the highest frequency of 40 GHz) is approximately 250 times larger than the smallest dimension,
Fig. 10. Mesh of the long microstrip line (ten subdomains). DoFs for each subdomain are ∼100 000).
indicating an electrically fine-scale problem, and the original model is divided into ten subdomains to reduce the DoFs for each subdomain, as shown in Fig. 10. The average surface DoFs M S , volume DoFs MV , and total DoFs M for each subdomain are 5 × 103 , 9.5 × 104 , and 1 × 105 , respectively. The results are compared in Fig. 11(a) and (b) with both the FDTD and BT methods. The detailed computational overheads are listed in Table I. Fig. 11(a) and (b) shows satisfactory consistency between the LDU scheme and the references for both the scattered voltages and the scattering parameters, although a slight mismatch for the low frequency exists between LDU and FDTD. The computational costs shown in Table I further demonstrate the efficiency of the proposed scheme with respect to the references. In addition to the total CPU time and memory, some other parameters, such as the total number of unknowns, the time-step interval t, the CPU time for preprocessing (corresponding to the CPU time consumed in the preprocessing step), and the CPU time per time step,
1846
IEEE TRANSACTIONS ON COMPONENTS, PACKAGING AND MANUFACTURING TECHNOLOGY, VOL. 5, NO. 12, DECEMBER 2015
Fig. 12. Geometry of the interconnect package. (a) Whole structure. (b) Side view.
Fig. 11. Calculated scattered voltages and scattering parameters by the LDU, BT, and FDTD schemes. (a) Scattered voltages. (b) Scattering parameters. TABLE I C OMPUTATIONAL C OSTS FOR THE L ONG M ICROSTRIP L INE C ASE
are also investigated to comprehensively evaluate the performance of different methods. Compared with FDTD, the total number of unknowns is reduced by a factor of 10.7, while the total CPU time is approximately shortened to 10% of that of FDTD. Although, in terms of CPU time per time step, FDTD shows an advantage over the new scheme (approximately 2 times faster), it is balanced by the even larger number of time iterations (approximately 588 times) required to complete the simulation. In terms of memory, however, the new scheme requires a larger cost, as FDTD involves no matrix operations. Compared with the explicit RK4 method, similar to FDTD, the new scheme shows a significant improvement in t (400 times) and total CPU time (approximately 90 times), even though they have the same DoFs. For such an electrically
fine-scale case, the explicit schemes would require a much smaller time-step interval to meet the CFL stability condition, thus resulting in much more time iterations and longer CPU time. Note that, for the implicit BT and LDU schemes, the time step t is chosen as 1000 fs (25 points per period with respect to the highest frequency of 40 GHz) to retain similar accuracy as the explicit ones, although they are unconditionally stable. With respect to the BT method, the new scheme exhibits an overwhelming advantage as it reduces the total CPU time by 2.4 times and the memory by 3.1 times. The CPU time per time step is also reduced by a factor of 3.1. Another interesting point is the 2.2 times reduction in the CPU time for the preprocessing step, which consumes approximately half of the total CPU time for the implicit schemes. This is attributed to the application of parallel computing in the new scheme (parallel computing toolbox in MATLAB) to solve the matrices simultaneously for each subdomain. For the BT method, however, parallel computing in the level of subdomains does not apply, as it is a series algorithm and the solution of the whole linear system requires a sequential operation subdomain by subdomain. D. Case 4 (Interconnect Package) Finally, we apply the scheme to an interconnect package with complicated layered structures and four ports, which originates from Intel Corporation (courtesy of Dr. H. Braunisch). The detailed geometry of the model is depicted in Fig. 12(a) and (b). The central frequency for this simulation is 9.25 GHz, and the smallest wavelength (corresponding to the highest frequency of 20 GHz) is approximately 1000 times larger than the smallest dimension, indicating a typical electrically fine-scale problem. The mesh of DG-FETD is depicted in Fig. 13, as the original model is divided into five subdomains. The average surface DoFs M S , volume DoFs MV , and total DoFs M for each subdomain
SUN et al.: EFFICIENT NONITERATIVE IMPLICIT TIME-STEPPING SCHEME BASED ON E AND B FIELDS
Fig. 13.
1847
Mesh of the interconnect package. TABLE II C OMPARISON OF C OMPUTATIONAL C OSTS FOR THE I NTERCONNECT PACKAGE C ASE
are 1.2 × 104 , 1.08 × 105 , and 1.2 × 105 , respectively. The calculated scattered voltages and scattering parameters are compared in Figs. 14(a)–(d) and 15 with FDTD, where Port 1 in Fig. 12(a) functions as the excitation source and Ports 2–4 work as loads. The detailed computational overheads are listed in Table II. As shown in Figs. 14(a)–(d) and 15, both the scattered voltages and the scattering parameters from the LDU scheme agree well with the reference. From the computational costs, we can observe that the explicit FDTD method would require an extremely dense mesh with 252 million DoFs to capture the fine structures in the model, while, in contrast, the FETD method by employing a tetrahedron mesh can easily discretize the model with only 0.58 million DoFs. The dense mesh in FDTD further results in a time-step interval of 291 times smaller than LDU, consequently leading to a 13.6-time longer CPU time to complete the simulation. Note that, for the implicit BT and LDU schemes, based on the criterion of similar accuracy, the time step t is chosen as 2500 fs (20 points per period with respect to the highest frequency of 20 GHz). Another reason for the new scheme to consume less CPU time is that this model is a package with multiports. For FDTD, when the four ports shift their function as an excitation source and a load, it requires exactly four times the CPU time to obtain the results. By contrast, the LDU scheme, as a direct solver, allows the source alternation as the right-hand side term while maintaining the matrices from the preprocessing step. In particular, for LDU, we only implement preprocessing once and time stepping four times when different ports are chosen as an excitation source. This saves the CPU time for preprocessing, in this case, 3 × 3.4 hours as shown in Table II. For explicit RK4, although it is also a direct solver and has the same DoFs as LDU, the majority of the CPU time is consumed on time stepping instead of preprocessing. Compared with LDU, its total CPU time is approximately 23.4 times longer. Thus, for packages with many ports, the new scheme would
Fig. 14. Calculated scattered voltages by the LDU and FDTD schemes for the interconnect package. (a) Scattered voltage for Port 1. (b) Scattered voltage for Port 2. (c) Scattered voltage for Port 3. (d) Scattered voltage for Port 4.
show more advantages over FDTD and explicit RK4 in terms of CPU time. The computational cost for the BT scheme is also shown in Table II, which consumes longer CPU time and much larger memory than the proposed scheme.
1848
IEEE TRANSACTIONS ON COMPONENTS, PACKAGING AND MANUFACTURING TECHNOLOGY, VOL. 5, NO. 12, DECEMBER 2015
Fig. 15. Calculated scattering parameters by the LDU and FDTD schemes for the interconnect package.
V. C ONCLUSION Based on the EB scheme with curl-conforming and divergence-conforming basis functions, the block LDU decomposition is proposed for sequentially ordered DG-FETD systems as an efficient noniterative time-stepping scheme for modeling electrically fine-scale problems. Through reducing the dimension of the original linear system, the new scheme presents a significant memory efficiency compared with the previous BT method while maintaining the same accuracy. Furthermore, for electrically fine-scale problems with a large number of unknowns, it shows an impressive CPU-time advantage over the BT method, the explicit Runge–Kutta method, and the standard FDTD method, due to the lighter linear system, its adaptability for parallel computing in subdomain level during preprocessing, and the unconditional stability. It is expected that the new scheme will significantly improve the performance of the DG-FETD method in modeling electrically fine-scale problems for future applications. ACKNOWLEDGMENT The authors would like to thank H. Braunisch, Intel Corporation for providing the long microstrip line model and the interconnect package model. R EFERENCES [1] J. S. Hesthaven and T. Warburton, “High–order nodal discontinuous Galerkin methods for the Maxwell eigenvalue problem,” Philos. Trans. Roy. Soc. London A, Math., Phys. Eng. Sci., vol. 362, no. 1816, pp. 493–524, 2004. [2] T. Lu, P. Zhang, and W. Cai, “Discontinuous Galerkin methods for dispersive and lossy Maxwell’s equations and PML boundary conditions,” J. Comput. Phys., vol. 200, no. 2, pp. 549–580, Nov. 2004. [3] X. Ji, T. Lu, W. Cai, and P. Zhang, “Discontinuous Galerkin time domain (DGTD) methods for the study of 2-D waveguidecoupled microring resonators,” J. Lightw. Technol., vol. 23, no. 11, pp. 3864–3874, Nov. 2005. [4] B. Cockburn, F. Li, and C.-W. Shu, “Locally divergence-free discontinuous Galerkin methods for the Maxwell equations,” J. Comput. Phys., vol. 194, no. 2, pp. 588–610, Mar. 2004. [5] N. Canouet, L. Fezoui, and S. Piperno, “Discontinuous Galerkin timedomain solution of Maxwell’s equations on locally-refined nonconforming Cartesian grids,” COMPEL, Int. J. Comput. Math. Elect. Eng., vol. 24, no. 4, pp. 1381–1401, 2005.
[6] T. Xiao and Q. H. Liu, “Three-dimensional unstructured-grid discontinuous Galerkin method for Maxwell’s equations with well-posed perfectly matched layer,” Microw. Opt. Technol. Lett., vol. 46, no. 5, pp. 459–463, Sep. 2005. [7] P. Monk and G. R. Richter, “A discontinuous Galerkin method for linear symmetric hyperbolic systems in inhomogeneous media,” J. Sci. Comput., vol. 22, no. 1, pp. 443–477, Jun. 2005. [8] G. Cohen, X. Ferrieres, and S. Pernet, “A spatial high-order hexahedral discontinuous Galerkin method to solve Maxwell’s equations in time domain,” J. Comput. Phys., vol. 217, no. 2, pp. 340–363, Sep. 2006. [9] A. Buffa and I. Perugia, “Discontinuous Galerkin approximation of the Maxwell eigenproblem,” SIAM J. Numer. Anal., vol. 44, no. 5, pp. 2198–2226, 2006. [10] J.-H. Lee, J. Chen, and Q. H. Liu, “A 3-D discontinuous spectral element time-domain method for Maxwell’s equations,” IEEE Trans. Antennas Propag., vol. 57, no. 9, pp. 2666–2674, Sep. 2009. [11] J. Chen, Q. H. Liu, M. Chai, and J. A. Mix, “A nonspurious 3-D vector discontinuous Galerkin finite-element time-domain method,” IEEE Microw. Compon. Lett., vol. 20, no. 1, pp. 1–3, Jan. 2010. [12] J. Chen, L. E. Tobon, M. Chai, J. A. Mix, and Q. H. Liu, “Efficient implicit–explicit time stepping scheme with domain decomposition for multiscale modeling of layered structures,” IEEE Trans. Compon., Packag. Manuf. Technol., vol. 1, no. 9, pp. 1438–1446, Sep. 2011. [13] P. Li and L. J. Jiang, “A hybrid electromagnetics-circuit simulation method exploiting discontinuous Galerkin finite element time domain method,” IEEE Microw. Compon. Lett., vol. 23, no. 3, pp. 113–115, Mar. 2013. [14] R. Courant, K. Friedrichs, and H. Lewy, “On the partial difference equations of mathematical physics,” IBM J. Res. Develop., vol. 11, no. 2, pp. 215–234, Mar. 1967. [15] T. Namiki, “A new FDTD algorithm based on alternating-direction implicit method,” IEEE Trans. Microw. Theory Techn., vol. 47, no. 10, pp. 2003–2007, Oct. 1999. [16] F. Zheng, Z. Chen, and J. Zhang, “A finite-difference time-domain method without the Courant stability conditions,” IEEE Microw. Guided Wave Lett., vol. 9, no. 11, pp. 441–443, Nov. 1999. [17] M. Movahhedi, A. Abdipour, A. Nentchev, M. Dehghan, and S. Selberherr, “Alternating-direction implicit formulation of the finiteelement time-domain method,” IEEE Trans. Microw. Theory Techn., vol. 55, no. 6, pp. 1322–1331, Jun. 2007. [18] S. G. Garcia, T. W. Lee, and S. C. Hagness, “On the accuracy of the ADI-FDTD method,” IEEE Antennas Wireless Propag. Lett., vol. 1, no. 1, pp. 31–34, Feb. 2002. [19] G. Sun and C. W. Trueman, “Unconditionally stable Crank–Nicolson scheme for solving two-dimensional Maxwell’s equations,” Electron. Lett., vol. 39, no. 7, pp. 595–597, Apr. 2003. [20] G. Sun and C. W. Trueman, “Unconditionally-stable FDTD method based on Crank–Nicolson scheme for solving three-dimensional Maxwell equations,” Electron. Lett., vol. 40, no. 10, pp. 589–590, May 2004. [21] G. Sun and C. W. Trueman, “Efficient implementations of the Crank–Nicolson scheme for the finite-difference time-domain method,” IEEE Trans. Microw. Theory Techn., vol. 54, no. 5, pp. 2275–2284, May 2006. [22] Y. Yang, R. S. Chen, and E. K. N. Yung, “The unconditionally stable Crank Nicolson FDTD method for three-dimensional Maxwell’s equations,” Micro. Opt. Technol. Lett., vol. 48, no. 8, pp. 1619–1622, Aug. 2006. [23] R.-S. Chen, L. Du, Z. Ye, and Y. Yang, “An efficient algorithm for implementing the Crank–Nicolson scheme in the mixed finite-element time-domain method,” IEEE Trans. Antennas Propag., vol. 57, no. 10, pp. 3216–3222, Oct. 2009. [24] G. Meurant, “A review on the inverse of symmetric tridiagonal and block tridiagonal matrices,” SIAM J. Matrix Anal. Appl., vol. 13, no. 3, pp. 707–728, 1992. [25] S. D. Gedney and U. Navsariwala, “An unconditionally stable finite element time-domain solution of the vector wave equation,” IEEE Microw. Guided Wave Lett., vol. 5, no. 10, pp. 332–334, Oct. 1995. [26] A. Bossavit, “Whitney forms: A class of finite elements for threedimensional computations in electromagnetism,” IEE Proc. A, Phys. Sci., Meas. Instrum., Manage. Edu.-Rev., vol. 135, no. 8, pp. 493–500, Nov. 1988. [27] J.-F. Lee and Z. Sacks, “Whitney elements time domain (WETD) methods,” IEEE Trans. Magn., vol. 31, no. 3, pp. 1325–1329, May 1995.
SUN et al.: EFFICIENT NONITERATIVE IMPLICIT TIME-STEPPING SCHEME BASED ON E AND B FIELDS
[28] M. F. Wong, O. Picon, and V. Fouad Hanna, “A finite element method based on Whitney forms to solve Maxwell equations in the time domain,” IEEE Trans. Magn., vol. 31, no. 3, pp. 1618–1621, May 1995. [29] A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics. New York, NY, USA: IEEE Press, 1998. [30] J. C. Nedelec, “Mixed finite elements in R3 ,” Numer. Math., vol. 35, no. 3, pp. 315–341, Sep. 1980. [31] J. Chen and Q. H. Liu, “A non-spurious vector spectral element method for Maxwell’s equations,” Prog. Electromagn. Res., vol. 96, pp. 205–215, 2009. [32] L. E. Tobón, J. Chen, and Q. H. Liu, “Spurious solutions in mixed finite element method for Maxwell’s equations: Dispersion analysis and new basis functions,” J. Comput. Phys., vol. 230, no. 19, pp. 7300–7310, Aug. 2011. [33] L. E. Tobon and Q. H. Liu, “New efficient and naturally parallelizable time integration algorithm applied to sequential domains for DG-TD,” in Proc. IEEE Antennas Propag. (APSURSI), Jul. 2013, p. 50. [34] L. E. Tobón, Q. Ren, and Q. H. Liu, “A new efficient 3D discontinuous Galerkin time domain (DGTD) method for large and multiscale electromagnetic simulations,” J. Comput. Phys., vol. 283, pp. 374–387, Feb. 2015. [35] N. Marais and D. B. Davidson, “Numerical evaluation of high-order finite element time domain formulations in electromagnetics,” IEEE Trans. Antennas Propag., vol. 56, no. 12, pp. 3743–3751, Dec. 2008. [36] Q. Ren, L. E. Tobón, and Q. H. Liu, “A new 2D non-spurious discontinuous Galerkin finite element time domain (DG-FETD) method for Maxwell’s equations,” Prog. Electromagn. Res., vol. 143, pp. 385–404, 2013. [37] L. Fezoui, S. Lanteri, S. Lohrengel, and S. Piperno, “Convergence and stability of a discontinuous Galerkin time-domain method for the 3D heterogeneous Maxwell equations on unstructured meshes,” ESAIM, Math. Model. Numer. Anal., vol. 39, no. 6, pp. 1149–1176, 2005. [38] S. Gedney et al., “The discontinuous Galerkin finite-element timedomain method solution of Maxwell’s equations,” ACES J., vol. 24, no. 2, pp. 129–142, 2009. [39] A. Kanevsky, M. H. Carpenter, D. Gottlieb, and J. S. Hesthaven, “Application of implicit–explicit high order Runge–Kutta methods to discontinuous-Galerkin schemes,” J. Comput. Phys., vol. 225, no. 2, pp. 1753–1781, Aug. 2007. [40] V. Dolean, H. Fahs, L. Fezoui, and S. Lanteri, “Locally implicit discontinuous Galerkin method for time domain electromagnetics,” J. Comput. Phys., vol. 229, no. 2, pp. 512–526, Jan. 2010. [41] V. Shankar, A. H. Mohammadian, and W. F. Hall, “A time-domain, finite-volume treatment for the Maxwell equations,” Electromagnetics, vol. 10, nos. 1–2, pp. 127–145, 1990. [42] A. H. Mohammadian, V. Shankar, and W. F. Hall, “Computation of electromagnetic scattering and radiation using a time-domain finitevolume discretization procedure,” Comput. Phys. Commun., vol. 68, nos. 1–3, pp. 175–196, Nov. 1991. [43] L. E. Tobon, Q. Ren, Q. Sun, J. Chen, and Q. H. Liu, “New efficient implicit time integration method for DGTD applied to sequential multidomain and multiscale problems,” Prog. Electromagn. Res., vol. 151, pp. 1–8, 2015. [44] A. George, “Nested dissection of a regular finite element mesh,” SIAM J. Numer. Anal., vol. 10, no. 2, pp. 345–363, 1973. [45] Q. H. Liu, “Transient electromagnetic modeling with the generalized K -space (GkS) method,” Microw. Opt. Technol. Lett., vol. 7, no. 18, pp. 842–848, Dec. 1994. [46] Q. H. Liu, “The PSTD algorithm: A time-domain method requiring only two cells per wavelength,” Microw. Opt. Technol. Lett., vol. 15, no. 3, pp. 158–165, Jun. 1997.
Qingtao Sun received the B.S. and M.S. degrees in geodetection and information technology from the China University of Petroleum, Qingdao, China, in 2009 and 2012, respectively. He is currently pursuing the Ph.D. degree with the Department of Electrical and Computer Engineering, Duke University, Durham, NC, USA. His current research interests include forward modeling with finite difference, finite element and hybrid methods, and inverse problems in geophysical exploration.
1849
Luis Eduardo Tobon received the B.S. degree in electronics engineering and the M.S. degree in material science from the Universidad del Quindio, Armenia, Colombia, in 2003 and 2007, respectively, and the Ph.D. degree from Duke University, Durham, NC, USA, in 2013. He has been with the Department of Electrical Engineering and Computer Science, Pontificia Universidad Javeriana at Cali, Cali, Colombia, since 2007, where he is currently a Research Associate Professor. His current research interests include computational electromagnetics applied to multiscale problems, geophysical subsurface sensing, electronics packaging, microwaves and electronic devices, and wireless communication and propagation. Qiang Ren received the B.S. degree in electrical engineering from Beihang University, Beijing, China, in 2008, and the M.S. degree in electrical engineering from the Institute of Acoustics, Chinese Academy of Sciences, Beijing, in 2011. He is currently pursuing the Ph.D. degree with Duke University, Durham, NC, USA. He has been a Research Assistant with the Department of Electrical and Computer Engineering, Duke University, since 2011. His current research interests include finite element and spectral element methods, hybrid fast time domain solvers, wave propagation in anisotropic medium, beam forming, array optimization, and electromagnetics inversion. Yunyun Hu received the B.S. and M.S. degrees in geophysical well logging from the China University of Petroleum, Qingdao, China, in 2010 and 2013, respectively. She is currently pursuing the Ph.D. degree with the Department of Electrical and Computational Engineering, Duke University, Durham, NC, USA. Her current research interests include computational electromagnetics, reservoir modeling, subsurface sensing, and inverse problem.
Qing Huo Liu (S’88–M’89–SM’94–F’05) received the B.S. and M.S. degrees in physics from Xiamen University, Xiamen, China, in 1983 and 1986, respectively, and the Ph.D. degree in electrical engineering from the University of Illinois at Urbana–Champaign, Urbana, IL, USA, in 1989. He was with the Electromagnetics Laboratory, University of Illinois at Urbana–Champaign, as a Research Assistant from 1986 to 1988 and a Post-Doctoral Research Associate from 1989 to 1990. He was a Research Scientist and Program Leader with Schlumberger-Doll Research, Ridgefield, CT, USA, from 1990 to 1995. From 1996 to 1999, he was an Associate Professor with New Mexico State University, Las Cruces, NM, USA. Since 1999, he has been with Duke University, Durham, NC, USA, where he is currently a Professor of Electrical and Computer Engineering. He has authored over 300 refereed journal and 450 conference papers in conference proceedings. His current research interests include computational electromagnetics and acoustics, inverse problems, and their applications in geophysics, nanophotonics, and biomedical imaging. Dr. Liu is a fellow of the Acoustical Society of America and the Electromagnetics Academy. He was a recipient of the 1996 Presidential Early Career Award for Scientists and Engineers from the White House, the 1996 Early Career Research Award from the Environmental Protection Agency, and the 1997 CAREER Award from the National Science Foundation. He currently serves as inaugural Editor in Chief of the IEEE Journal on Multiscale and Multiphysics Computational Techniques, an Associate Editor of the IEEE T RANSACTIONS ON G EOSCIENCE AND R EMOTE S ENSING, and an Editor of the Journal of Computational Acoustics. He was the Guest Editor-in-Chief of the P ROCEEDINGS OF THE IEEE of the Special Issue on Large-Scale Electromagnetics Computation and Applications in 2013. He serves as an IEEE Antennas and Propagation Society Distinguished Lecturer from 2014 to 2016.
