The Finite-Difference Analysis and Time Flow

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finite- difference approach

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.。

Advance heat transfer PPT

The numerical solution and exact solutions of the comparison:
Rearrange the equation
The node i-1's finite difference approximation expression can be write as
The finite difference method
The Taylor series expansion:
The finite difference method
Differential equation
Differential equation
Algebraic equation:
Most control equations are differential equations,and there is no analytical solution Determine the number of node of the discrete equations in the calculation region
Determination the position of the variables correspond to the nodes
Differential difference approximation is not the only, du dx can be expressed as follows:
The finite difference method
Using the difference grid instead of continuous domains
ui , jrepresent:

应用力学系_工程有限元分析(共81张PPT)

国均能使用的实用单位制。
2. 1954年第十届国际计量大会决定采用米(m)、千克(kg)、秒(s)、安培(A)、开尔文(K)和坎德拉(cd)作为基本单位。
From: O’ Brien et al.
Accura3te.soluti1on960年第十一届国际计量大会决定将以这六个单位为基本单位的实用计量单位制命名为“国际单位制”，并
迈和码
1. 英“迈”是英制英里mile的音译，1 mile=1.6 km,100迈就是160 km,在速度表上就是160 km/h,比如某人说他在路上开到过180“迈”，换算为公里应该是180*1.6=288KM, 这个速度是在开一级方程式赛车吗?
2. 有的人喜欢说开多少多少"码",这就更不对了,"码"的英文是YARD, 一码=3英尺,1英里=1760码.码与公制的换算关系是: 1 码=0.9144米,就是说每小时开100"码"就是每小时开不到100米,那比蜗牛还跑的慢.更是荒唐之极.
Moaveni, S., Finite Element Analysis – Theory and Application with ANSYS, 2nd Ed., Pearson Education, 2003. Pepper, D.W. and Heinrich, J.C., The Finite Element Method: Basic Concepts and Applications, Hemisphere, 1992. Pao, Y.C., A First Course in Finite Element Analysis, Allyn and Bacon, 1986. Rao, S.S., Finite Element Method in Engineering, 3rd Ed., Butterworth-Heinemann, 1998.

the mathematical theory of finite element method

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.。

Finite-difference time-domain macromodel

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.。

金融衍生品定价理论(同济-英文版)第五章(European Option Prcing —— Black-Scholes Formula)

Basic Assumptions (b) Risk-free interest rate r is a constant
(c) Underlying asset pays no dividend
(d) No transaction cost and no tax
(e) The market is arbitrage-free
Δ-Hedging Technique - Since Vt (St , T ),
where the stochastic process St satisfies SDE*, hence by Ito formula
V 1 2 2 2V V V dVt S S dWt dt S 2 S S S t 2 V 1 2 2 2V V So S S S dt 2 S S t 2
parameter in the underlying asset model , does not appear in the Black-Scholes equation . Instead, the risk-free interest rate r appears it. As we have seen in the discrete model, by the Δ---hedging technique, the Black-Scholes equation puts the investors in a risk-neutral world where pricing is independent of the risk preference of individual investors. Thus the option price arrived at by solving the Black-Scholes equation is a risk-neutral price.

Accuracy of finite-difference and finite-element modeling of the scalar and elastic wave equation

Manuscriptreceivedby the Editor May 20, 1982; revisedmanuscript receivedOctober 14, 1983. *Amoco Production Company, P. 0. Box 591, Tulsa, OK 74102. \(a,1984 Society of Exploration Geophysicists. All rights reserved.
subject to essential boundary conditions, A@,) =
Spatial discretization Numerical techniques such as finite-differences, finiteelements, boundary integral equations, and the method of moments all belong to the more general weighted residual method
Kurt J. Marfurt*
ABSTRACT Numerical solutions of the scalar and elastic wave equations have greatly aided geophysicists in both forward modeling and migration of seismic wave fields in complicated geologic media, and they promise to be invaluable in solving the full inverse problem. This paper quantitatively compares finite-difference and finite-element solutions of the scalar and elastic hyperbolic wave equations for the most popular implicit and explicit time-domain and frequency-domain techniques. In addition to versatility and ease of implementation, it is imperative that one choose the most cost effective solution technique for a fixed degree of accuracy. To be of value, a solution technique must be able to minimize (1) numerical attenuation or amplification, (2) polarization errors, (3) numerical anisotropy, (4) errors in phase and group velocities, (5) extraneous numerical (parasitic) modes, (6) numerical diffraction and scattering, and (7) errors in reflection and transmission coefficients. This paper shows that in homogeneous media the explicit finite-element and finite-difference schemes are comparable when solving the scalar wave equation and when solving the elastic wave equations with Poisson’ s ratio less than 0.3. Finite-elements are superior to finitedifferences when modeling elastic media with Poisson’ s ratio between 0.3 and 0.45. For both the scalar and elastic equations, the more costly implicit time integration schemes such as the Newmark scheme are inferior to the explicit central-differences scheme, since time steps surpassing the Courant condition yield stable but highly inaccurate results. Frequency-domain finiteelement solutions employing a weighted average of consistent and lumped masses yield the most accurate results, and they promise to be the most cost-effective method for CDP, well log, and interactive modeling.

2018年中科院数学SCI期刊分区

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(m) (m) (m) (s) (s) (s)
(s)
(s = 0, 1, . . . , m)
2
¯ k = (xi )k . are known, one can completely restore the initial ﬁnite orbit X i=1 Hence, it follows from (1), we can consider that ﬁnite orbits with length k are given namely in the next special representation
(k ) νs = lim rs k →∞
(= 0.δ1 δ2 δ3 . . .)
(s) (s) (s)
(k ) were rs are the r -coordinates from (2). Let us introduce a thin set B and its subsets Bk , are the base sets, mentioned in Sec. 1. We consider the numbers 0 < x < 1 represented in the form of binary expansion, ∞
(k ) (k ) (k ) ¯k = (r1 ζ , r2 , . . . , rm ; µk,1, µk,2, . . . , µk,m; ρ1 , ρ2 , . . . , ρk−m )
(2) (3)
where
(k ) rs = 0.δ1 δ2 · · · δk−s
(s) (s)
(s)
(1 ≤ s ≤ m)
x = 0.δ1 δ2 δ3 . . .
(=
k =1
2−k δk ,
δk = 0, 1) ;
(4)
further, we also operate with the corresponding binary sequences x ¯, x ¯ = (δ1 , δ2 , δ3 , . . . , δn , . . .) . For a given natural k ≥ 2 we deﬁne the subsets Bk of numerical interval (0, 1): Bk is the set all of those real numbers x ∈ (0, 1) for each of which ni+1 − ni ≤ k (i = 1, 2, 3, . . .)
(s)
i p=1
s) (−1)δp ∆( p )
(s)
(1)

and
s) 0 ∆p+1 ≥ ∆( p 1 ∆(s) < ∆(s) p+1 p
(s)
0
(it is supposed that
1
= 0) ,
µk,s = min{∆i : 1 ≤ i ≤ k − s} . From Eq. (1) it is easy to see, that if for a given m, 0 ≤ m ≤ k all of the ﬁnite binary sequences δ1 , δ2 , . . . , δk−s as well as all of the numbers µk,1, µk,2, . . . , µk,m and ∆1 , ∆2 , . . . , ∆k−m
(m)
are the rational numbers, and µk,1 , µk,2, . . . , µk,m and ρ1 , ρ2 , . . . , ρk−m (where ρi = ∆i )
are some numbers from interval [0, 1]. Here m = mk and mk tend to ∞ as k → ∞. It is easy to see, that after applying the recurrent procedure (1) the ¯k . Now, we can introduce the basic ¯ k is completely recovered by ζ sequence X tool of the method – the notion of conjugate orbit: we say, that numerical sequence ν ¯ = (νs )∞ s=1 is the conjugate orbit, associated with the given orbit ∞ ¯ X = (xi )i=1 , if for each s ≥ 1 the terms νs are deﬁ3;1 − ∆i
(s)
(s−1)
(s−1)
| (i = 1 , 2 , . . . , k − s ).
It is not diﬃcult to obtain ∆i where
(s) δp = (s−1) i−1
= µk,s−1 +
s) (−1)δp ∆( p − min ( p=1 0≤i≤k −s
treatments [2] and in some works on quantum physics [3, 4]. 2. 2.1 Description of the Method Finite diﬀerences and conjugate orbits
We deal with one dimensional, deterministic or stochastic systems, generating ¯ = (xi )∞ (0 ≤ xi ≤ 1). We impose no restrictions on numerical sequences X i=1 the system, and it may also possess diﬀerent inner states changing with time. ¯ , under which it should be Below, we set two conditions on a given orbit X ¯ ) orbit ν regarded as chaotic. We introduce the notion of conjugate (with X ¯, ¯ in terms of which the irregular nature of X can be established and studied. ¯ k = (xi )k , Let us ﬁrst give some special representation of ﬁnite orbit X i=1 ¯ k and which reﬂects its higher order diﬀerential structure. For the sequence X number 1 ≤ s ≤ k − 1 we let ∆i = xi ,
The Finite-Diﬀerence Analysis and Time Flow Ashot Yu. Shahverdian
(Yerevan Physics Institute)
arXiv:chao-dyn/9910005v1 6 Oct 1999
1.
Introduction
The present paper introduces a method of analysis of one dimensional systems. Its application to study of Poincare recurrence time ﬂow is considered. The approach suggested consists of reducing the research of a given system’s ¯ = (xi )∞ to analysis of alternations of the monotone increase and orbits X i=1 ¯ . Through decrease of higher order absolute ﬁnite diﬀerences, taken from X ¯ some nusome special representation of ﬁnite orbits, we associate with X merical sequence, is called the conjugate orbit. We formulate two conditions, ¯ is reduced to analysis of the special orbits’ asympwhen the study of the X totical intersections with some base set from numerical interval (0, 1) of zero Lebesgue measure. Hence, the approach can be characterized as an ”asymptotical” analogy of Poincare’s classical ”section”method. The method distinguishes two cases: the continuous, when some numerical quantities ρ can take arbitrary values from interval (0, 1), and the discrete case, when all of them belong to a ﬁnite set. The continuous was applied in Ref. 1 to study of neural activity. In the present work, we are more interested in the discrete case, which has an additional speciﬁc. Thus, as soon as we have stated the method’s applicability, the mechanism generating time series ¯ , is replaced by the return map R, generating the conjugate series. While X the actual exact description of this mechanism may remain unknown (e.g., for earthquake time series or neuron spike trains), the function R does not depend on it and has a simple analytic shape. This approach is applicable to the study of various time series, arising in modern applied science (e.g., medicine, astronomy). The innovation consists of a new way to measure the fast oscillating time series – this measure is the Hausdorﬀ dimension of some thin spaces A, to which the conjugate series are attracted. As an example of such an application, the computational analysis of the sequences of fractional parts is presented. We show that the results obtained testify on the possibility of Cantorian structure of our usual time ﬂow. In this context, we recall some well known (qualitative) conclusions on irregular and intricate time ﬂow, as indicated in Bergson’s philosophical 1