The Finite-Difference Analysis and Time Flow

Finite-Difference ApproachFinite-difference approach contains two methods such as explicit finite-difference and implicit finite-difference. How can we to distinguish them? We can distinguish them through the solving process. The explicit finite-difference method can compute the solution straightforwardly. Compared with the explicit method, the implicit finite-difference method is more complicated. The implicit method must compute the solution by solving a group of algebraic equations. What are the advantages and disadvantages of these two kinds of method? For the explicit, the advantage is that easy to solving, however, it not stable sometimes. The shortcoming for the implicit is that big amount of calculation but can get a constringency result. In practical these two numerical approaches have a high application value.Definition of calculus of continuous function μ(x), then we get firstderivation dμdx =limδ→0μ(x+δ)−μ(x)(x+δ)−x=limδ→0μ(x+δ)−μ(x)δ. Omit limitationcalculation we get difference approximation dμdx ≈μ(x+δ)−μ(x)δandthis particular finite-difference approximation is called a forwarddifference. We also have dμdx ≈μ(x)−μ(x−δ)δwhich is calledbackward difference. We also define central difference by notingthat dμdx ≈μ(x+δ)−μ(x−δ)2δ.Let say at time t stock price is S t option value is V t, so we get atwo variables function such as V t =V(S t ,t) .So we can express the option profit like this: V (S,t )={max (S T −E,0),(call option );max (E −S T ,0),(put option ).Where E is strike price or exercise price, T is stock expiration time. Let ∆t =T N ⁄ where T is expiration, T is divided to N +1 the same small interval time such as 0, ∆t,2∆t,⋯,T.Let ∆S =S max M ⁄ where S max is assumed as the maximum stock price, so we can get M +1 stock prices such as 0, ∆S,2∆S,⋯,S max .The implicit finite -difference approach: ThisðV ðt+12σ2S 2ð2VðS2+rSðV ðS−rV =0 is Black -Scholes partialdifferential equation, where r is free risk interest and σ is volatility.Now we can use difference approximation instead of derivation.ðV ðt=V i+1,j −V i,j∆t,ðV ðS=V i,j+1−V i,j−12∆S,ð2V ðS 2=V i,j+1−2V i,j +V i,j−1∆S 2. Now putabove difference approximation into Black -Scholes partial differential equation, then we getV i+1,j −V i,j∆t+rj ∆S ×V i,j+1−V i,j−12∆S+12σ2j 2∆S 2×V i,j+1−2V i,j +V i,j−1∆S 2=rV i,j .Then collection of like termsget(1+∆tσ2j 2+∆tr)V i,j +(−12∆trj −12∆tσ2j 2)V i,j+1+(12∆trj −12∆tσ2j 2)V i,j−1=V i+1,j . The explicit finite -difference approach:ðV ðt=V i+1,j −V i,j∆t,ðV ðS=V i+1,j+1−V i,j−12∆S,ð2V ðS 2=V i+1,j+1−2V i+1,j +V i+1,j−1∆S 2.Substitute the derivation,ðVðt ðVðSand ð2VðS2in Black-Scholes partialdifferential equation, we get11+r∆t ×(12∆trj+12∆tσ2j2)V i+1,j−1+1 1+r∆t ×(1−∆tσ2j2)V i+1,j+11+r∆t×(12∆trj+12∆tσ2j2)V i+1,j+1=V i,j.。

the mathematical theory of finiteelement methodThe mathematical theory of the finite element method (FEM) is a branch of numerical analysis that provides a framework for approximating solutions to partial differential equations (PDEs) using discretization techniques. The finite element method is widely used in engineering and scientific disciplines to simulate and analyze physical phenomena.At the core of the FEM is the concept of dividing a domain into a finite number of elements, which are connected at nodes. The unknown solution within each element is approximated using a simple function, referred to as the basis function. These basis functions are usually polynomials of a certain degree, and their coefficients are determined by solving a set of linear equations.The mathematical theory of the FEM involves several key concepts and techniques. One of the fundamental principles is the variational formulation, which transforms the PDE into an equivalent variational problem. This variational problem is then discretized using the finite element approximation, resulting in a system of algebraic equations.Another important aspect is the assembly process, where the contributions from each element are combined to form the global stiffness matrix and right-hand side vector. This assembly is based on the integration of the basis functions and their derivatives over the element domains.Error estimation and convergence analysis are also essential components of the mathematical theory of the FEM. Various techniques, such as the energy method and the posteriori error estimators, are used to assess the accuracy of the finite element solution and to determine the appropriate mesh refinement for achieving convergence.Furthermore, the mathematical theory of the FEM includes the treatment ofboundary conditions, imposition of symmetries, and the development of efficient solvers for the resulting linear systems. It also addresses issues such as numerical stability,并行 computing, and adaptivity.In summary, the mathematical theory of the finite element method provides a comprehensive framework for numerically solving PDEs. It encompasses concepts such as element discretization, variational formulation, assembly, error estimation, and convergence analysis, which collectively enable the accurate and efficient simulation of a wide range of physical problems.。

IEEE TRANSACTIONS ON MAGNETICS, VOL. 41, NO. 1, JANUARY 200565Finite-Difference Time-Domain Macromodel for Simulation of Electromagnetic Interference at High-Speed InterconnectsEn-Xiao Liu, Er-Ping Li, Senior Member, IEEE, Le-Wei Li, Senior Member, IEEE, and Zhongxiang Shen, Senior Member, IEEEAbstract—This paper presents an efficient and systematic approach for transient analysis of hybrid interconnect systems. The approach uses the finite-difference time-domain (FDTD) method to fully characterize the subnetwork of the high-speed interconnects in the form of admittance parameters. It then uses the admittance parameters to construct a macromodel of the subnetwork by the vector fitting method (VFM). The resulting macromodel is ready to be synthesized into a SPICE-compatible circuit simulator to efficiently expedite the transient analysis of the hybrid system with interconnects and linear/nonlinear lumped circuit elements. Numerical examples show the validity of the method. Index Terms—Electromagnetic simulation, FDTD-macromodeling, hybrid circuit system, vector fitting method.I. INTRODUCTIONWITH the rapid advancements in modern very large scale integration (VLSI) technology, the electromagnetic phenomena existing in high-speed and high-density interconnect becomes a dominant factor in determining the system electrical performance [1]. Consequently, the development of an accurate and efficient modeling technique becomes imperative for simulation of electromagnetic interference (EMI) and signal integrity in the high-speed complex hybrid interconnect and lumped circuit systems. However, two major difficulties impede the efficient broadband modeling of the high-speed complex interconnect system [1]. One difficulty is the mixed frequency/time domain problem. At high-frequency regime, the dispersive nature of interconnect requires a representation in frequency domain, whereas the circuit components especially nonlinear ones are ready to be formulated in time domain. A traditional ordinary differential equation solver such as a SPICE-like circuit simulator [2] cannot efficiently handle this mixed domain problem. The other difficulty lies in the central processing unit (CPU) expense. With theManuscript received November 4, 2003; revised September 17, 2004. E.-X. Liu and E.-P. Li are with the Computational Electronics and Electromagnetics Division, Institute of High Performance Computing (IHPC), Singapore 117528, Singapore (e-mail: engp1643@.sg; elelep@.sg). L.-W. Li is with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 119260, Singapore (e-mail: lwli@.sg). Z. Shen is with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore (e-mail: ezxshen@.sg). Digital Object Identifier 10.1109/TMAG.2004.839733continuous increase in the operation speed of the devices and in the complexity of the interconnect structures, modeling and simulation of the interconnect system at both chip and package levels becomes more time consuming. One popular way to circumvent this mixed time/frequency domain problem is to extend the full-wave finite-difference time-domain (FDTD) method to include the lumped circuit elements [3]–[6]. Furthermore, the hybrid FDTD-SPICE method has been proposed and applied to hybrid interconnect problems with more general lumped elements [7], [8]. However, both methods may suffer from the CPU inefficiency and convergence problems. In order to alleviate the disadvantages of the above-mentioned extended FDTD method but at the same time preserve the accuracy of the full wave technique in the analysis of interconnect structures, the conventional FDTD method [9] is employed to extract the network property of the interconnect system, such as its admittance parameters [8]. Thereafter, the macromodeling approach [10]–[11] can be utilized to address the mixed domain problems. The macromodeling approach is intended to construct a lower order model from tabulated data characterizing the network property of the interconnect system. In [10], the complex frequency hopping (CFH) method by moment matching was applied to perform the macromodeling of the interconnect system represented by tabulated scattering parameters. The difficulty of this method is that for every moment, a corresponding derivative of each parameter must be computed using numerical integration across the entire time domain. This process has to be repeated on multiple frequency expansion points, which can be cumbersome for a high order of approximation, or networks with many ports [11]. An efficient approach for macromodeling based on tabulated frequency-domain data is discussed in [11]–[13], which employs the direct rational function approximation instead of a moment-matching approach to solve the mixed domain problem. One of the many rational function approximation methods is the vector fitting method (VFM) developed by Gustavsen and Semlyen [14]. VFM is a robust method and has some advantages over other fitting methodologies [15]. Most fitting methods rely on nonlinear optimization algorithms that tend to be slow and may converge to a local minimum. Instead, VFM relies on the solution of two linear least-square problems, thus obtaining the optimal solution rather directly. At the same time, VFM does not suffer much from the numerical stability problem even when0018-9464/$20.00 © 2005 IEEE66IEEE TRANSACTIONS ON MAGNETICS, VOL. 41, NO. 1, JANUARY 2005the bandwidth of interest is wide. Furthermore, one single run of VFM can achieve the rational function approximation of all the elements in a transfer function matrix with the same set of poles. Therefore, this paper adopts the robust VFM [14] to construct the macromodel of the interconnect system. The organization of this paper is as follows: the formulations of the FDTD method and VFM are briefly presented in Section II. The conventional FDTD method is dedicated to extract the admittance parameters and the VFM is devoted to create the macromodel of the interconnect system. The techniques of macromodel synthesis and passivity check are discussed in Section III. Numerical examples are presented in Section IV to validate the interconnect simulation technique presented in this paper. Section V concludes the paper. II. FORMULATION A. Admittance Parameters Extraction The Maxwell’s equations governing the field in the isotropic and lossless media are given by (1) (2) are the electric field (volts/meter) where the vectors and and magnetic field (amperes/meter), respectively. The constants and are the respective electrical permittivity (farads/meter) and magnetic permeability (henrys/meter). The three-dimensional (3-D) FDTD algorithm is based on the discretization of differential-form Maxwell’s (1) and (2) by using central difference approach and staggering field component arrangement [3], [9]. Here, only the formulations of two and are shown field components, i.e.,are the short-circuited current at port where vectors and and the excitation voltage at port , respectively, which are obtained by the Fourier transform of the FDTD transient waveforms. Although this approach is straightforward, it is quite time consuming. An improved method for the calculation of admittance matrix by using the FDTD method was proposed in [8], where the admittance matrix can alternatively be transformed from its scattering counterparts [12]. This approach is efficient and employed in this paper. B. VFM for Rational Function Approximation The frequency-domain representation of the interconnect subnetwork in (5) cannot be directly inserted into the time-domain simulator for transient simulation. An efficient way to address this problem is approximating each of the elements in with its corresponding lower order model matrix (6) where is the direct coupling constant, is the total number and are the pole-residue pair to be computed. of poles, and As mentioned in Section I, the VFM [14] is adopted to efficiently solve (6). The main procedures of this method are briefly described as follows. First, introducing an unknown function and expanding it with a set of starting poles (7) Then, scaling the original response function and taking the rational function approximation with ,(8) (3) Substituting (7) into (8) and for a given frequency point , the following equation can be derived: (9) (4) where , , and are the lattice space increments in , where , and coordinate directions, respectively. is the time increment of the leapfrog time-stepping. The subscript integers denote a space point in a uniform, rectangular lattice. The superscript indicates the th time step. There are several approaches to obtain the admittance matrix from full-wave FDTD simulations. The fundamental approach to computing the admittance matrix of a network with ports is through the following equation: (5)Once the unknowns in (9) are computed, can be ex, which shows that the poles of pressed as coincide with the zeros of . Substituting the values of the poles into (6) and solving the equation similar to (9), we and constant of . can easily obtain the residues It is to be noted that the selection of starting poles used in (7) is of importance for a successful rational function approx-LIU et al.: FDTD MACROMODEL FOR SIMULATION OF EMI67imation. For transfer functions with many resonant peaks, the starting poles should be chosen as complex conjugates. Furthermore, the imaginary parts of these conjugate pairs shall be linearly distributed over the frequency range of interest and one hundred times larger than the real parts. To assure the stability of the fitting model, a basic requirement is that all the poles of the fitting model must be located in the left-hand side of the . This constraint on the fitting complex plane, i.e., model is often enforced by some simple treatments, e.g., directly deleting the unstable poles or flipping them to the left half-plane [14]. III. EMBEDDING THE MACROMODEL INTO CIRCUIT SIMULATOR A. State-Space Representation From the preceding section, the macromodel of the interconnect subnetwork is created. For a general -port subnetwork characterized real poles and complex conjugate pole pairs, the state-space representation by Jordan-canonical method [1] takes the following form: (10)Fig. 1.Schematics of a microstrip circuit.method in [16] can be applied to enforce the macromodel to be passive. Now it is safe to transform the passive macromodel of the interconnect subnetwork expressed in the form of time-domain differential equations of (10) into an equivalent circuit system, which comprises resistors, capacitors, and controlled sources [1]. This equivalent circuit can be inserted into SPICE-compatible circuit simulator to perform the transient analysis of the original hybrid interconnect and lumped circuit element problems. IV. NUMERICAL RESULTSwhereIn this section, three examples are presented to demonstrate the validity and accuracy of the approach proposed in this paper. A. Two-Port Microstrip Circuit A two-port microstrip loaded by lumped circuit elements is simulated to verify the accuracy of the proposed approach. The schematic circuit diagram is shown in Fig. 1. The 3-D FDTD method is employed to obtain the parameters of the distributed part of the circuit in Fig. 1. For the FDTD simulation, the unit cell size in millimeters is , , and the total grid size is . The dispersive absorbing boundary condition and Gaussian pulse source are used in the FDTD simulation. The parameters of the two-port interconnect are approximated by the vector fitting method to construct its macromodel. Twelve poles including two real poles and ten complex conjugate poles are extracted by the vector fitting method to match the parameters of this two-port interconnect up to 5 GHz. Good agreements can be observed between the FDTD simulated parameters and the macromodel based on the vector fitting method (Fig. 2), which shows that the rational function approximation by vector fitting method is accurate. The interconnect macromodel can pass the passivity check. Therefore, its equivalent circuit is created and inserted into the SPICE circuit simulator to perform the transient analysis of the hybrid circuit system. The transient simulation results of the hybrid circuit are shown in Fig. 3, where a pulse with a 0.5 ns rise and fall time is used. B. Corner Discontinuity With Nonlinear LoadsandMatrices , , and are derived from the pole-residue pairs. The subscripts ( and ) denote the real and complex conjugate pole-residue pairs, respectively. Vectors and contain the port currents and voltages of the interconnect subnetwork. Matrix is directly derived from the constants ’s in (6). B. Passivity Check In Section II-B, the stability of the macromodel is ensured by complying with some simple constraint conditions, i.e., all the poles are located in the left-half complex plane by performing vector fitting approximation. However, stable but not passive macromodels can lead to unstable systems when connected with other passive systems. Therefore, passivity check is essential to identify whether a macromodel is passive or its transient simulation is stable before performing the SPICE simulation. One efficient method to check the passivity of the macromodel was presented in [16] by examining the Hamiltonian matrix. From (10), the macromodel is passive if the following Hamiltonian matrix has no imaginary eigenvalues:(11) If the macromodel is not passive, then the quadratic programming method proposed in [17] or the perturbation of residueIn this example, a corner discontinuity [18] loaded with a nonlinear circuit element as shown in Fig. 4 is simulated to validate the hybrid circuit simulation approach proposed in this paper.68IEEE TRANSACTIONS ON MAGNETICS, VOL. 41, NO. 1, JANUARY 2005Fig. 4. Schematics of a circuit composed of a corner discontinuity and nonlinear loads.Fig. 2. Y parameters of the microstrip simulated by FDTD and integrated FDTD-macromodel methods. (a) Magnitude. (b) Phase.Fig. 5.Yparameters of the corner discontinuity. (a) Magnitude. (b) Phase.Fig. 3.Simulated transient results of the hybrid circuit.The unit cell size of the 3-D FDTD simulation is mm, mm and the total grid size is . Sixteen poles comprising two real polesand 14 complex conjugate poles are extracted by the vector fitting method to match the parameters of this two-port corner discontinuity up to 10 GHz. The approximated values of parameters from the macromodel are compared with that from the FDTD simulation, as shown in Fig. 5. Again, it can be observed that the results obtained by the two methods are in good agreement. Fig. 6 shows the transient simulation results of the overall circuit, where the circuit is excited with a 6-V pulse of 0.1-ns rise/fall time.LIU et al.: FDTD MACROMODEL FOR SIMULATION OF EMI69Fig. 6. Transient response of the hybrid circuit.Fig. 9. Comparison of the Y parameters obtained by FDTD and macromodel based on VFM. (a) y11 and y21. (b) y31 and y41.Fig. 7.Configuration of the four-port microstrip lines with vias.match the parameters up to 15 GHz of this four-port network. The approximated values of parameters from the macromodel agree well with those obtained from the FDTD simulation as illustrated in Fig. 9. Because of the symmetry of this four-port network, only four entries of the matrix are plotted. And for brevity, the plot for phase comparison is omitted. The transient simulation results are shown in Fig. 10. The circuit is excited at port 1 by a pulse having a rise/fall time of 0.05 ns and a pulsewidth of 4 ns.V. CONCLUSION The integrated full-wave FDTD macromodeling method proposed in this paper is an accurate and efficient approach to analyze the hybrid interconnect and circuit systems, in which the electromagnetic field effects are fully considered and the strength of the SPICE circuit simulator is also exploited. The VFM used in this paper provides an accurate way for the rational function approximation of the interconnect subnetwork. The approach of converting the interconnect macromodel into an equivalent circuit can facilitate the transient analysis of the hybrid electromagnetic and circuit problems. Future work will be focused on studying new passivity enforcement methods to construct robust passive macromodels.Fig. 8. Schematic circuit diagram of the four-port network of microstrip lines with vias loaded with lumped circuit elements.C. Four-Port Microstrip Lines With Vias A four-port microstrip lines with vias similar to that in [19] is analyzed. Its schematic geometry diagram and its circuit layout are shown in Figs. 7 and 8, respectively. The unit cell size of the 3-D FDTD simulation is mm and the total grid size is . Twenty-two poles are extracted by the vector fitting method to70IEEE TRANSACTIONS ON MAGNETICS, VOL. 41, NO. 1, JANUARY 2005Fig. 10. Transient voltage waveforms: (a) at port 2 and at the output observation point; and (b) at port 3 and port 4.[9] K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propagat., vol. AP-14, no. 3, pp. 302–307, May 1966. [10] R. Achar and M. S. Nakhla, “Efficient transient simulation of embedded subnetworks characterized by s-parameters in the presence of nonlinear elements,” IEEE Trans. Microwave Theory Tech., vol. 46, no. 12, pp. 2356–2363, Dec. 1998. [11] M. Elzinga, K. L. Virga, and J. L. Prince, “Improved global rational approximation macromodeling algorithm for networks characterized by frequency-sampled data,” IEEE Trans. Microwave Theory Tech., vol. 48, no. 9, pp. 1461–1468, Sep. 2000. [12] T. Mangold and P. Russer, “Full-wave modeling and automatic equivalent-circuit generation of millimeter-wave planar and multilayer structures,” IEEE Trans. Microwave Theory Tech., vol. 47, no. 6, pp. 851–858, Jun. 1999. [13] R. Neumayer, F. Haslinger, A. Stelzer, and R. Wiegel, “Synthesis of SPICE-compatible broadband electrical models from n-port scattering parameter data,” in Proc. IEEE Symp. Electromagn. Compat., Aug. 2002, pp. 469–474. [14] B. Gustavsen and A. Semlyen, “Rational approximation of frequency domain responses by vector fitting,” IEEE Trans. Power Delivery, vol. 14, no. 3, pp. 1052–1061, July 1999. [15] W. Pinello, J. Morsey, and A. Cangelaris, “Synthesis of SPICE-compatible broadband electrical models for pins and vias,” in Proc. 51st IEEE Electronics Components and Technology Conf., Orlando, FL, May 2001, pp. 518–522. [16] D. Saraswat, R. Achar, and M. Nakhla, “Enforcing passivity for rational function based macromodels of tabulated data,” in Proc. 12th IEEE Topical Meeting on Electrical Performance of Electronic Packaging, Princeton, NJ, Oct. 2003, pp. 295–298. [17] B. Gustavsen and A. Semlyen, “Enforcing passivity for admittance matrices approximated by rational functions,” IEEE Trans. Power Syst., vol. 16, no. 1, pp. 97–104, Feb. 2001. [18] Q. Gu, D. M. Sheen, and S. M. Ali, “Analysis of transients in frequencydependent interconnections and planar circuits with nonlinear loads,” Proc. Inst. Elect. Eng.-H, vol. 139, no. 1, pp. 38–44, Feb. 1992. [19] P. C. Cherry and M. F. Iskander, “FDTD analysis of high frequency electronic interconnection effects,” IEEE Trans. Microwave Theory Tech., vol. 43, no. 10, pp. 2445–2451, Oct. 1995.ACKNOWLEDGMENT The authors thank Dr. Y. Weiliang from the Institute of High Performance Computing, Singapore, for his technical discussion.REFERENCES[1] R. Achar and M. S. Nakhla, “Simulation of high-speed interconnects,” Proc. IEEE, vol. 89, no. 5, pp. 693–728, May 2001. [2] Star-HSPice Manual, Avant Corporation, 1998. [3] A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method. Boston, MA: Artech House, 2000. [4] W. Sui, D. A. Christensen, and C. H. Durney, “Extending the two-dimensional FDTD method to hybrid electromagnetic systems with active and passive lumped elements,” IEEE Trans. Microwave Theory Tech., vol. 40, no. 4, pp. 724–730, Apr. 1992. [5] M. Picket-May, A. Taflove, and J. Baron, “FD-TD modeling of digital signal propagation in 3-Dcircuits with passive and active loads,” IEEE Trans. Microwave Theory Tech., vol. 42, no. 8, pp. 1514–1523, Aug. 1994. [6] E. Li, W. Yuan, and S. Wang, “Signal propagation effects on high-speed interconnection lines using time-domain numerical technique,” Microwave Opt. Technol Lett., vol. 35, pp. 416–420, 2002. [7] V. A. Thomas, M. E. Jones, M. Piket-May, A. Taflove, and E. Harrigan, “The use of SPICE lumped circuits as sub-grid models for FDTD analysis,” IEEE Microwave Guided Wave Lett., vol. 4, no. 5, pp. 141–143, May 1994. [8] T. Watanabe and H. Asai, “A framework for macromodeling and mixed-mode simulation of circuits/interconnects and electromagnetic radiations,” IEICE Trans. Fundamentals, vol. E86-A, pp. 252–261, Feb. 2003.En-Xiao Liu received the B.Eng. and M.Eng. degrees in energy and power engineering from Xi’an Jiaotong University, Xi’an, China, in 1996 and 1999, respectively. He is currently pursuing the Ph.D. degree in electrical and computer engineering at the National University of Singapore and the Institute of High Performance Computing, Singapore. From September 1999 to June 2001, he was with the North China Electrical Power Design Institute, Beijing, China, as an Automation Control Design Engineer. His research interests include computational electromagnetics and high-speed interconnect modeling and simulation.Er-Ping Li (M’93–SM’01) received the M.Sc. degree in electrical engineering from Xi’an Jiaotong University, Xi’an, China, in 1986 and the Ph.D. degree in electrical engineering from Sheffield Hallam University, Sheffield, U.K., in 1992. He worked as a Research Fellow from 1989 to 1990 and then as a Lecturer from 1991 to 1992 at Sheffield Hallam University, U.K. Between 1993 and 1999, he was a Senior Research Fellow, Principal Engineer, and Technical Manager/ Director with the Singapore Research Institute and Industry. Since 2000, he has been with A-STAR Institute of High Performance Computing, Singapore, where he is currently a Senior Scientist and Senior R&D Manager of the Computational Electromagnetics and Electronics Division. He has served as Chair of a number of international conferences and is the Technical Advisor to a number of multinational companies in Asia. He has published more than 90 technical papers in international referred journals and conferences and coauthored three book chapters. His research interests include fast and efficient computational electromagnetics, EMC/EMI, high-speed electronic modeling, and computational nanotechnology.LIU et al.: FDTD MACROMODEL FOR SIMULATION OF EMI71Le-Wei Li (S’91–M’92–SM’96) received the of B.Sc. degree in physics from Xuzhou Normal University, Xuzhou, China, in 1984, the M.Eng.Sc. degree in electrical engineering from China Research Institute of Radiowave Propagation (CRIRP), Xinxiang, China, in 1987, and the Ph.D. degree in electrical engineering from Monash University, Melbourne, Australia, in 1992. In 1992, he worked at La Trobe University (jointly with Monash University), Melbourne, as a Research Fellow. Since 1992, he has been with the Department of Electrical and Computer Engineering at the National University of Singapore, where he is currently a Professor. From 1999 to 2004, he was also with High Performance Computations on Engineered Systems (HPCES) Programme of the Singapore-MIT Alliance (SMA) as an SMA Fellow. His current research interests include electromagnetic theory, radio wave propagation and scattering in various media, microwave propagation and scattering in tropical environment, and analysis and design of various antennas. In these areas, he has coauthored a book titled Spheroidal Wave Functions in Electromagnetic Theory (New York: Wiley, 2001), 35 book chapters, more than 190 international refereed journal papers, 25 regional refereed journal papers, and more than 200 international conference papers. Dr. Li is a member of The Electromagnetics Academy based at the MIT, an Editor of Journal of Electromagnetic Waves and Applications, an Associate Editor of Radio Science, and an Editorial Board Member of IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, Electromagnetics, and Chinese Journal of Radio Science. He was the Chairman of IEEE Singapore Section MTT/AP Joint Chapter.Zhongxiang Shen (S’96–M’99–SM’04) received the B.E. degree from the University of Electronic Science and Technology of China, Chengdu, in 1987, the M.S. degree from Southeast University, Nanjing, China, in 1990, and the Ph.D degree from the University of Waterloo, Waterloo, ON, Canada, in 1997, all in electrical engineering. From 1990 to 1994, he was with Nanjing University of Aeronautics and Austronautics, China. From 1994 to 1997, he was a Research Assistant in the Department of Electrical and Computer Engineering, University of Waterloo. He was with Com Dev Ltd., Cambridge, ON, as an Advanced Member of Technical Staff in 1997. He spent six months each in 1998, first with the Gordon McKay Laboratory, Harvard University, Cambridge, MA, and then with the Radiation Laboratory, University of Michigan, Ann Arbor, as a Postdoctoral Fellow. He is presently an Associate Professor in the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. His research interests include microwave/millimeter-wave passive devices and circuits, small and planar antennas for wireless communications, and numerical modeling of various RF/microwave components and antennas. He has authored or coauthored more than 60 journal articles and more than 50 conference papers. Dr. Shen received a Postdoctoral Fellowship from the Natural Sciences and Engineering Research Council of Canada. He received a Best Student Award at the 1997 IEEE AP-S International Symposium.。

有限差分法在一维输运方程定解中的运用2012年2月第12卷第1期廊坊师范学院(自然科学版)JournalofLangfangTeachersCollege(NaturalScienceEdition)Feb.2012V0l_12No.1有限差分法在一维输运方程定解中的运用林喜季(福建江夏学院,福建福州350108)【摘要】有限差分方法就是一种数值解法,在一维输运方程定解中可以巧用它来解题,把表示变量连续变化关系的偏微分方程离散为有限个代数方程,然后利用电子计算机求此线性代数方程组的解.【关键词】一维输运方程;有限差分法;定解问题FiniteDifferenceMethodin0ne—DimensionalTransport EquationintheUseofDefiniteSolutionLINXii【Abstract】Thefinitedifferencemethodisanumericalmethod,one-dimensionaltransportequationinth esolutioncanbeskillfullyusedittosolveproblems,thecontinuousvariationofthatvariablepartialdifferential equationisdiscretizedintoafinitenumberofalgebraicequations,andthenusethecomputertosolveThishnearalgebraice quations.【Keywords】one-dimensionaltransportequation;finitedifferencemethod;definitesolutionproblem[中图分类号]0175.2(文献标识码]A[文章编号]1674—3229(2012)01—0016—03 在数学中,有限差分法的内涵是指用差商代替微商,即用泰勒级数展开式将变量的导数写成变量在不同时间或空间点值的差分形式的方法.它的基本思想是按时间步长和空间步长将时间和空间区域剖分成若干方格网,用泰勒级数展开近似式代替所用偏微分方程中出现的各阶导数,从而把表示变量连续变化关系的偏微分方程离散为有限个代数方程,然后,解此线性代数方程组.l导数用泰勒级数展开近似式导数(微商)y:=m0=±,是无限小的微分m0△),除以无限小的微分是△的商.它可以分别近似为:,,=dxAx=(1)y=Ax=(2)y=dxAx=坐(3)式(1),(2)相当于把泰勒级数y(+~xx)=y()+(Ax)y+1(△)+…(—Ax)=y()一(△)y+1(△)+…截断于(Ax)v项,把(Ax)项以及更高幂次的项全部略去.式(3)相当于把泰勒级数y(+Ax)一Y(一△)=2(Ax)Y+(△)y,-+..截断于2(Ax)项,把(Ax)项以及更高幂次的项全部略去.因此,式(3)的误差小于式(1)和(2).二阶导数类似的可近似为差商的差商,一X[dx…一血dx【一止]:志[(+△)+y(—Ax)一2y()](4)[收稿日期]2011—11—21[作者简介]林喜季(1977一),女,福建江夏学院讲师,研究方向:代数表示论.16?第12卷?第1期林喜季:有限差分法在一维输运方程定解中的运用2012年2月这相当于把泰勒级数Y(+△)一v(一△)=2y()+(△)+(△)Y+..'截断于(△)项,把(△)项以及更高幂次的项全部略去.偏导数也可仿照式(1)一(4)近似为商差.这样一来,偏微分方程就成了差分方程.2一维输运方程的定解问题如,在区间(0,L)上求解一维输运方程"='axx.分析:(1)把整个空间分为.,个"步子",每一步的长度=I/J.于是,自变量以步长跳跃,它的取值是(i=0,1,2,…,.,).把时间步长取为zI,即自变量t取值t=kv(k=0,1,2,…,).(2)仿照式(1)和(4),一维输运方程可近似为(1)=(1—2lM(￡),2+l_!["(+l,t)+u(一1,t)】(5)这样只要知道某个时刻t的u在各个地点的值(,t),代人式(5)就可以得到下个时刻t…的的各个地点的值u(i,t).但这种解法时间t的步长z.不能太大,必须满足条件≤1,否则,由于舍入误差,会在其后各步的计算中产生雪崩影响,以致计算结果完全失去意义. (3)仿照式(2)和(4),一维输运方程可近似为u(,t)一U(i,t一1)r2(+1,t)+u(i一1,t)一2U(,t)即"(¨)=(1+2竿)Ⅱ(),2一旦j三[(+1,t)+"(,t)】(6)这样做可以取消对步长r的限制.但是知道某个时刻t的Ⅱ在各个地点的值(,t),并不能代入式(6)直接得到下个时刻t川的的各个地点的值(,t),且必须把i=1,2,3,…,.,一1的共计J一1个同式(6)的方程联立起来求解u(t,t+1),u(2,t+1),…,u(J一1,t%+1),当然这种联立方程的计算依靠电子计算机还是很方便的.(4)仿照式(3),偏导数近似为u(Xi'tk+1):,从而一维输运方程可近似为M(,t+1)一u(,t):u(Xi+?,tk+1)+u(—t,+.)一2u(,t+吉).上式中(,tk+)可理解为:[配(,t+.)+u(,t)】/2,于是,上例差分方程即为窘u(…)一(+字)"()+au("~i-1+)=一Ⅱ()一(1一窘)"(3Ci~tk)一骞H+1)(7)知道某个时刻t的在各个地点的值M(,t)后,必须把i=1,2,3,…,J一1的共计.,一1个同式(7)的方程联立起来求解"(.,t…),U(2,t+1),…,(J—l,t+1),当然这种联立方程的计算依靠电子计算机还是很方便的.这种解法对时间的步长也有限制,应满足2≤1,但与解法(2)相比限制要宽些.3用近似式求一维波动方程的定解问题如,在区间(0,L)上求解一维波动方程%一a"=0.把整个空间分为.,个"步子",每一步的长度=l/J.把时间步长取为r,仿照式(4),一维波动方程可近似为"(戈,t+1)+(,t一1)一2u(,t)即(.):2(1一譬)u(+旦}[(+1,tk)+(;一1,)一¨(,一1)】.这就是说,只要知道某时刻t及其以前时刻的U在各个地点i的值u(,t),代入上式,就可以得到下个时刻tk+.的的各个地点的值"(,t…).在此f青景下,时间的步长r的限制条件为≤1.17?,J,L2一,J一,L2"r,+,L22012年2月廊坊师范学院(自然科学版)第12卷?第1期从上述例子可以看出,在研究一维输运方程定解问题时,可采用有限差分法,按适当的数学变换把定解问题中的微商换成差商,从而把原问题离散化为差分格式,进而求出数值解.该方法具有简单,灵活,容易在计算机上实现的特点.并且该方法还有很强的通用性,如热传导过程,气体扩散过程这类定解问题,其过程都与时间有关,利用差分法解这类问题,就是从初始值出发,通过差分格式沿时间增加的方向,逐步求出微分方程的近似解.再如弹性力学中的平衡,电磁场及引力场等问题,其特征均为椭圆型方程,利用差分法解这类问题,就是合理选定的差分方格网,建立差分格式,最后求解代数方程组.[参考文献][1]吴顺唐,邓之光.有限差分法方程[M].南京:河海大学出版社.1993.[2]陈祖墀.偏微分方程[M].合肥:中国科学技术大学出版社.2004.[3]王晓东.算法与数据结构[M].北京:电子工业出版, 1998.[4]王震,谢树森.解四阶拟线性波动方程的一类二阶差分格式[J].中国海洋大学,2004,(34).(上接15页)将上述n为奇数与n为偶数两种情况统一起来,可得数列{a)的通项公式为.=1[(口.+.)+(一1)(.一n.)】?r.2)看P≠1,根据文献l3j结论司知,b=+()p,即有an+lan=+【一)p,从而anan_l=+(?一-qp)p.若令一1)=+(一,(n≥2),贝0ana一:f(一1).当p≠±,g≠o时,一)+(g一)p=一p棚=qp(1一p)≠0(n≥2),从而,('一p)≠0(n≥).由%a一.=/(n一1)递推可得:当n为大于1奇数时,(n一1)(一1)厂(//,一3).n':(—.—.::—0n一:je'.——.::—'':e';:—:二—;0n一=?…?f1al;一厂(n一2)厂(一4)厂(3)'()'当ll,为大于2的偶数时,n=:;{—;——{.一:=.一18?=船?…?n一12Ⅱ[,(2Jl})]即当n为大于1奇数时,a=丁_—一a.;Ⅱ厂()Ⅱ()当n为大于2的偶数时,.=T或改写为a=Ⅱ[,(2)]二者可统一为n=—1[(nl+口2)+(一1)(n2一口1)]?{{一吉【3+(一1)】)n一吉[3+(一1)】[Ⅱ[(2)]/Ⅱ厂(+)](-1),nEN+且n≥3,其中f(n)=(1一p),n∈N+.[参考文献][1]劳建祥.递推数列求通项大观[J].数学教学,2005,(3): 41—42.[2]高焕江.也谈二阶线性递推数列的周期?t:ff-[J].廊坊师范学院,2009,9(6):8—10.[3]高焕江.二阶线性递推数列的通项公式[J].保定学院学报,2010,23(3):34—37.。

finite element analysisfinite element analysis, utilising the finite element method (fem), is a product of the digital age, ing to the fore with the advent of digital puters in the1950s. it follows on from matrix methods and finite difference methods of analysis, which had been developed and used long before this time. it is aputer-based analysis tool for simulating and analysing engineering products and systems. fea is an extremely potent engineering design utility, but one that should be used with great care. for example, it is possibleto integrate a system with puter-aided design software, leading to a type of uninformed push-button analysisin the design process. unfortunately, colossal errors can be made at the push of a button, as this warning makes clear.finite element analysis 1fea is an extremely potent engineering design utility, but one which should be used with great care. despite years of research by some of the earth’s most intelligent mathematicians and scientists, it can only answer the questions asked of it. so, as the saying goes, ask a stupid question.the frothy solutioncurrent cad [puter-aided design] vendors are nowselling suites which have cut-down versions of feaengines integrated with puter-aided design software. the notion is to allow ordinary rank-and- to analyseas they design and change and update models to reach workable solutions much earlier in the design process. this kind of approach is monly referred to as thepush-button solution.pensive analysts are petrified of push-button analysis. this is because of the colossal errors that can be made at the push of a button. the errors are usually uncontrollable and often undetectable. some vendorsare even selling fea plug-ins where it is not possible to view the mesh. (this is ludicrous.)the oblivious among us may say that analysts areafraid of push-button solutions due to the job loss factor, or perhaps they are terrified of being castout of the ivory towers in which they reside. such arguments are nonsensical, there will always be real problems and design issues to solve. (would you enter the superbike class isle of man tt on a moped with an objective to win, even if it had the wheels of the latest and greatest superbike?)the temptation to analyse ponents is almostirresistible for the inexperienced, especially in an environment of one-click technology coupled with handsome and forting contour plots. the bottom line is that fea is not a trivial process, no level of automation and pre- and post-processing can make analyses easy, or more importantly, correct.the analysis titanif you have recently been awarded an engineering degree, congratulations, but remember it does not qualify you to carry out fe analyses. if it did, then a sailing course should be adequate to bee captain aboard the blue marli n [the world’s largest transporter vessel at the time of original publication].this is not to say that regular engineers cannot bee top rate analysts without a phd. some analysts have a masters degree, but most have no more than a bachelor’s degree. the k ey to good analyses is knowledge of the limitations of the method and an understanding of the physical phenomena under investigation.superior results are usually difficult to achieve without years of high-level exposure to fields that prise fea technology (differential equations, numerical analysis, vector calculus, etc.). expertise in such disciplines is required to both fully understand the requirements of any particular design circumstance, and to be able to quantify the accuracy of the analysis (or more importantly, inaccuracy) with reasonable success.to concludefinite element puter programs have bee mon tools in the hands of design engineers. unfortunately, many engineers who lack the proper training orunderstanding of the underlying concepts have been using these tools. given the opportunity, fea will confess to anything. the essence of any session should be to interrogate the solver with well-formed and appropriate questions.to summarise, the most qualified person to undertake an fea is someone who could do the analysis without fea.wise words, resisting the temptation to put too much trust in fea puter applications. if, however, puter-based simulations are set up and used correctly,highly plicated mathematical models can be solved to an extent that is sufficient to provide designers with accurate information about how the products will perform in real life, in terms of being able to carry out or sustain the operating conditions imposed upon them. the simulation models can be changed, modified and adapted to suit the various known or anticipated operating conditions, and solutions can be optimised. thus, the designers can be confident that the real products should perform efficiently and safely, and can be manufactured profitably. a few more detailed reasons are given below.the simulations are of continuous field systems subject to external influences whereby a variable, or bination of dependent variables, is described by prehensive mathematical equations. examples include:•stress•strain•fluid pressure•heat transfer•temperature•vibration•sound propagation•electromagnetic fields•any coupled interactions of the above.to be more specific, the fem can handle problems possessing any or all of the following characteristics.•any mathematical or physical problem described by the equations of calculus, e.g. differential,integral, integro-differential or variationalequations.•boundary value problems (also called equilibriumor steady-state problems); eigenproblems(resonance and stability phenomena); and initialvalue problems (diffusion, vibration and wavepropagation).•the domain of the problem (e.g. the region ofspace occupied by the system) may be anygeometric shape, in any number of dimensions.plicated geometries are as straightforward tohandle as simple geometries, with the onlydifference being that the former may require abit more time and expense. for example, a quitesimple geometry would be the shape of a circularcylindrical waveguide for acoustic orelectromagnetic waves (fibre optics). a moreplicated geometry would be the shape of anautomobile chassis, which is perhaps being analysed for the dynamic stresses induced by a rough road surface.•physical properties (e.g. density, stiffness, permeability, conductivity) may also vary throughout the system.•the external influences, generally referred to as loads or loading conditions, may be in any physically meaningful form, e.g. forces, temperatures, etc. the loads are typically applied to the boundary of the system (boundary conditions), to the interior of the system (interior loads) or at the beginning of time (initial conditions).•problems may be linear or non-linear.。