ADI-FDTD方法在乳腺癌检测正向计算中的应用

传统的时域有限差分(Finite-Difference Time-Domain, FDTD)算法受到稳定性条件的制约, 时间步长受限于空间网格的尺寸.医学应用讲究即时性, 为提高成像的速度, 文中采用无条件稳定的交替隐式时域有限差分(Alternating-Direction Implicit Finite-Difference Time-Domain, ADI-FDTD)算法替代传统的FDTD算法进行正向计算, 通过实验得出采用ADI-FDTD算法在保证精度的前提下, 计算时间可缩短为FDTD算法的四分之一, 为乳腺癌微波即时成像提供了可能.     

A 3-D precise integration time-domain method without the restraints of the courant-friedrich-levy stability condition for the numerical solution of Maxwell's equations

In this paper, a new three-dimensional time-domain method for solving vector Maxwell's equations, called the precise-integration time-domain (PITD) algorithm, is proposed in order to eliminate the Courant-Friedrich-Levy (CFL) condition restraint. The new algorithm is based on the precise-integration technique. It is shown that this method is quite stable even when the CFL condition is not satisfied. Although the memory requirement of the PITD method is much larger than that of the finite-difference time-domain (FDTD) method, this new algorithm is very appealing since the time step used in the simulation is no longer restricted by stability. As a result, computation speed can be improved. Therefore, if the minimum cell size in the computational domain is required to be much smaller than the wavelength, this new algorithm is more efficient than the FDTD scheme. Theoretical proof of the unconditional stability is shown and numerical results are presented to demonstrate the effectiveness and efficiency of the method. It is found that the accuracy of the PITD is independent of the time-step size.     

ADI-FDTD在二维散射问题中的应用

利用交替隐式时域有限差分(ADI—FDTD)这一新方法计算二维电磁散射问题。研究了ADI—FDTD方法的入射波设置、连接边界条件、PML吸收边界和近远场变换等关键技术。与传统FDTD方法相比，ADI—FDTD的时间步长不受时间步长和空间步长的稳定性条件(CFL约束条件)限制。在该方法中，可选取较大的时间步长进而提高计算效率。最后还给出了金属和介质柱散射截面的数值算例，证实了ADI—FDTD方法处理散射问题的有效性和实用性。     

无时间约束FDTD方法在三维散射中的应用

本文阐述了一种无时间约束条件的FDTD方法(ADI-FDTD)在三维目标电磁散射中的应用.由于散射问题的复杂性,文中分别推导了ADI-FDTD原始方程在连接边界条件、吸收边界条件和近远场外推等关键处的修正方程,并提出了ADI-FDTD方法中的时间步长上限.通过算例表明该方法与传统FDTD方法相比,时间步长可突破传统时间－空间约束条件,它的选取能远大于原有时间步长,对同一散射问题,总计算时间步可以相应大幅度减少,进而提高FDTD方法在计算散射问题中的效率.最后,数值计算显示了该方法的计算精度,并通过图表给出与传统FDTD计算时间的比较.     

UPML媒质中无条件稳定的二维ADI-FDTD方法

对单轴各向异性PML（UPML）媒质中二维TM波的交变方向隐式时域有限差分方向（ADI-FDTD），通过计算实例表明，ADI-FDTD方法在UMPL媒质中是无条件稳定的，其时间步长不受CFL稳定性条件的限制，并且当计算区域内具有精细差分网格时，其计算效率明显优于传统的时域有限差分方向（FDTD）。     

基于FDTD算法的新型亚网格技术

提出了一种基于空间滤波（Spatial Filtering-Finite-Difference Time-Domain,SF-FDTD）算法的亚网格技术,使得FDTD算法的Courant-Friedrich-Levy（CFL）稳定性条件可通过空间频域滤波操作得以提高,从而获得高稳定度FDTD算法.进一步将SF-FDTD算法应用到亚网格技术中,可使亚网格区域时间步长的选取取与粗网格一致,从而极大地提高了计算效率.数值计算结果表明,在求解带有精细结构的电磁问题上,所提算法具有较高的准确性和有效性.     

一种非条件稳定的隐式时域有限差分法

介绍一种基于交替方向隐式(ADI)技术的时域有限差分法(FDTD).该方法是非条件稳定的,时间步长不再受到Courant稳定条件的限制,而是由数值色散误差来确定.与传统的FDTD相比,ADI-FDTD增大了时间步长,从而缩短了总的计算时间,特别是当空间网格远小于波长时,优点更加突出.首次把完全匹配层(PML)边界条件应用到ADI-FDTD计算中,采用幂指数形式的时间步进算法,推导了相应的迭代公式.进行了实例计算,并与传统FDTD的结果对比,验证了ADI-FDTD的有效性与优越性.     

Theoretical Proof of Unconditional Stability of the 3-D ADI-FDTD Method

In order to eliminate Courant-Friedrich-Levy(CFL) condition restraint and improvecomputational efficiency,a new finite-difference time-domain(FDTD)method based on the alternating-direction implicit(ADI) technique is introduced recently.In this paper,a theoretical proof of the stabilityof the three-dimensional(3-D)ADI-FDTD method is presented.It is shown that the 3-D ADI-FDTDmethod is unconditionally stable and free from the CFL condition restraint.     

Unconditionally Stable Bilinear Transformation FDTD Algorithm for Modeling Double-Negative Meta-material Electromagnetic Problems

Accurate and unconditionally stable finite difference time domain (FDTD) algorithm is presented for modeling electromagnetic wave propagation in double-negative (DNG) meta-material domains. The proposed algorithm is based on incorporating the Bilinear transformation technique into the FDTD implementations of Maxwell’s equations. The stability of the proposed approach is studied by combining the von Neumann method with the Routh-Huwitz criterion and it has been observed that the proposed algorithm is free from the Courant-Friedrichs-Lewy (CFL) stability limit of the conventional FDTD scheme. Furthermore, the proposed algorithm is incorporated with the split-step FDTD scheme to model two-dimensional problems. Numerical examples carried out in one and two dimensional domains are included to show the validity of the proposed algorithm.     

3-D ADI-FDTD method-unconditionally stable time-domain algorithmfor solving full vector Maxwell's equations

We previously introduced the alternating direction implicit finite-difference time domain (ADI-FDTD) method for a two-dimensional TE wave. We analytically and numerically verified that the algorithm of the method is unconditionally stable and free from the Courant-Friedrich-Levy condition restraint. In this paper, we extend this approach to a full three-dimensional (3-D) wave. Numerical formulations of the 3-D ADI-FDTD method are presented and simulation results are compared to those using the conventional 3-D finite-difference time-domain (FDTD) method. We numerically verify that the 3-D ADI-FDTD method is also unconditionally stable and it is more efficient than the conventional 3-D FDTD method in terms of the central processing unit time if the size of the local minimum cell in the computational domain is much smaller than the other cells and the wavelength     

A higher order FDTD method in Integral formulation

We present a fourth-order (4, 4) finite-difference time-domain (FDTD)-like algorithm based on the integral form of Maxwell's equations. The algorithm, which is called the integro-difference time-domain (IDTD) method, achieves its fourth-order accuracy in space and time by taking into account the spatial and temporal variations of electromagnetic fields within each computational cell. In the algorithm, the electromagnetic fields within each cell are represented by space and time integrals (or integral averages) of the fields, i.e., the electric and magnetic fluxes (D,B) are represented by the surface-integral average, and the electric and magnetic fields (E,H) by the line and time integral average. In order to relate the integral average fields in the staggered update equations, we have obtained constitutive relations for these fields. It is shown that the IDTD update equations combined with the constitutive relations are fourth-order accurate both in space and time. The fourth-order correction terms are represented by the modified coefficients in the update equations; the numerical structure remains the same as the conventional second-order update equations and more importantly does not require the storage of field variables at the previous time steps to obtain the fourth-order accuracy in time. Furthermore, the Courant-Friedrichs-Lewy (CFL) stability criteria of this fourth-order algorithm turns out to be identical to the stability limits of conventional second-order FDTD scheme based on differential formulation.     

Conformal method to eliminate the ADI-FDTD staircasing errors

The alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method is an unconditionally stable method and allows the time step to be increased beyond the Courant-Friedrich-Levy (CFL) stability condition. This method is potentially very useful for modeling electrically small but complex features often encountered in applications. As the regular FDTD method, however, the spatial discretization in the ADI-FDTD method is only first-order accurate for discontinuous media; several researchers have shown that the errors can be very high when the regular ADI-FDTD method is applied to such discontinuous media. On the other hand, the conformal FDTD method has recently emerged as an efficient FDTD method with higher order accuracy. In this work, a second-order accurate ADI-FDTD method using the conformal approximation of spatial derivatives is proposed. This new scheme, called the ADI-CFDTD method, retains the second-order accuracy in both temporal and spatial discretizations even for discontinuous media with metallic structures, and is unconditionally stable. 2D and 3D examples demonstrate the efficacy of this method and its application in EMC problems.     

A modified FDTD (2, 4) scheme for modeling electrically largestructures with high-phase accuracy

A new fourth-order finite-difference time-domain (FDTD) scheme has been developed that exhibits extremely low-phase errors at low-grid resolutions compared to the conventional FDTD scheme. Moreover, this new scheme is capable of combining with the standard Yee (1966) scheme to produce a stable hybrid algorithm. The problem of wave propagation through a building is simulated using this new hybrid algorithm to demonstrate the large savings in computing resources it could afford. With this new development, the FDTD method can now be used to successfully model structures that are thousands of wavelengths large, using the present day computer technology     

用于电磁波多尺度问题求解的高效FDTD-PITD混合算法北大核心CSCD

针对电磁波多尺度问题的高效仿真需求,提出了基于亚网格技术的时域有限差分(FDTD)方法与时域精细积分(PITD)方法的混合数值算法。该混合算法的基本思想是采用局部亚网格技术分别对精细结构区域以及其他区域进行剖分,并应用FDTD方法和PITD方法分别对粗网格区域与细网格区域进行求解,同时构建信息交互策略交换细网格区域与粗网格区域的计算信息。一方面该方法减少了电磁波多尺度问题的网格剖分数目,显著降低了内存需求;另一方面由于应用于细网格区域的PITD方法不受Courant-Friedrich-Levy(CFL)数值稳定性条件的限制,该混合方法能够采用较大的时间步长进行仿真,减少了迭代步数以及CPU执行时间。数值计算结果验证了混合算法的稳定性、可行性以及高效性。     

Analyzing electromagnetic structures with curved boundaries onCartesian FDTD meshes

In this paper, a new finite-difference time-domain (FDTD) algorithm is investigated to analyze electromagnetic structures with curved boundaries using a Cartesian coordinate system. The new algorithm is based on a nonorthogonal FDTD method. However, only those cells near the curved boundaries are calculated by nonorthogonal FDTD formulas; most of the grid is orthogonal and can be determined by traditional FDTD formulas. Therefore, this new algorithm is more efficient than general nonorthogonal FDTD schemes in terms of computer resources such as memory and central processing unit (CPU) time. Simulation results are presented and compared to those using other methods     

二维无条件稳定时域有限差分方法

基于半隐式的Crank-Nicolson差分格式给出了一种无条件稳定时城有限差分方法。和传统FDTD法中采用的显式差分格式不同，对Maxwell方程组采用半隐式差分格式，在时间和空间上仍然是二阶精确的。但时间步长不再受稳定性条件的限制，只需考虑数值色散误差对其取值的制约。利用分裂场完全匹配层吸收边界截断计算空间，为保证PML空间的无条件稳定性，其方程也采用半隐式差分格式。数值结果表明相同条件下US-FDTD方法与传统FDTD方法的计算精度是相同的，而且在增大时间步长时US-FDTD方法是稳定的和收敛的。可以预见US-FDTD方法在模拟具有电小结构问题时具有实际意义。     

A locally conformed finite-difference time-domain algorithm ofmodeling arbitrary shape planar metal strips

A general algorithm for modeling arbitrary shape planar metal strips by the finite-difference-time-domain (FDTD) method is presented. With this method, fields in the entire computation domain are computed by the regular FDTD algorithm except near metal strips, where special techniques proposed herein are applied. Unlike the case for globally conformed finite-difference algorithms, the computation efficiency of the regular FDTD method is maintained while high space-resolution is obtained by this locally conformed finite-difference method. Numerical tests have verified that a higher computation accuracy is achieved by this scheme than by the conventional staircase approximation. The modeling of electrical characteristics of two crossed strip lines is provided as an example     

Analysis of microstrip discontinuities by FDTD method combined with SOC technique

In this paper, the application of the finite-difference time-domain (FDTD) algorithm combined with the short-open calibration SOC technique to three-dimensional microstrip discontinuity is presented. This SOC technique is directly accommodated in the FDTD algorithm. It is used to remove the unwanted parasitic errors brought by the approximation of the impressed voltage sources and the feed lines. The FDTD is formulated in such a way that the port voltages and currents are explicitly represented through relevant network matrices. This new method is also used to analyze finite periodic structures. The scattering parameters of the whole periodic structure can be approximately obtained through analyzing only one cell of it. The results for microstrip discontinuities and finite periodic structures are compared with the conventional FDTD method.     

Three-Dimensionally Nonorthogonal Alternating-Direction Implicit Finite-Difference Time-Domain Algorithm for the Full-Wave Analysis of Microwave Monolithic Circuit Devices

An alternating-direction implicit finite-difference time-domain (FDTD) algorithm is applied to the full wave analysis of microwave integrated circuit devices. A 3-D multidomain method is developed in nonorthogonal coordinates. Nonorthogonal grids are only used for the anomalistic regions of a complex structure, whereas the standard FDTD lattice is used for the other regions. By using the Jacobian coordinate transformation, curvilinear coordinates can be converted into conventional FDTD format expediently. The perfectly matched layer is used to truncate the boundary. Accurate griddings using the new scheme are obtained, and the complexity of the algorithm is minimal. To illustrate the theory, a sinusoidal plane wave and a Gaussian pulse that propagate through a localized nonorthogonal grid space is used, and the stability of our code is examined. A newly developed compact microstrip bandpass filter is analyzed using the proposed method. The simulated results agree very well with measurements. As compared to other nonorthogonal FDTD (NFDTD) method, the proposed algorithm is much more efficient than other NFDTD counterpart when complex structures are analyzed.      

A Novel Body-of-Revolution Finite-Difference Time-Domain Method With Weakly Conditional Stability

In this letter, the weakly conditionally stable finite-difference time-domain method (WCS-FDTD) is introduced into the body of revolution finite-difference time-domain (BOR-FDTD) method, resulting in a weakly conditionally stable BOR-FDTD. It inherits the advantages of both WCS-FDTD and BOR-FDTD methods, i.e., not only weakening the restraint of the Courant-Friedrich-Lecy (CFL) condition, with an efficient saving of CPU running time, but also leading to a significant memory reduction in the storage of the field components in comparison with the 3-D FDTD method. The stability condition of proposed BOR-FDTD method is presented analytically and the numerical performance of the proposed method over the alternating-direction implicit BOR-FDTD method is demonstrated through numerical examples.