finite-difference time-domain (FDTD)

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A global finite difference time domain analysis of a silicon nonlinear transmission line

A global finite difference time domain analysis of a silicon nonlinear transmission line

This paper presents a global finite difference time domain (FDTD) analysis of a silicon non- linear transmission line (NLTL) using optimized varactors. The simulation is based on the FDTD method [1],[2] also including transmission line losses and the drift-diffusion model for the semi- conductor devices solved by means of finite dif- ferences (FD). The diodes are included in the FDTD scheme as lumped elements. The fall time of 74 ps of a 4 GHz sinewave was com- pressed to approximately 15 ps at the output of a 20 mm long NLTL with 40 diodes. Comparing the measured output signal to the simulation, a good agreement could be achieved.
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Finite-difference time-domain analysis of silicon-based optical microdisk resonators / eingereicht von: Michael Teuchtmann

Finite-difference time-domain analysis of silicon-based optical microdisk resonators / eingereicht von: Michael Teuchtmann

Temporal output is the most commonly used output type in FDTD and consists of a single value for a given quantity recorded as a function of time. This includes one or more field components measured at a single fixed point in space, the energy density in the monitor domain, the power flow through the monitor domain, or the result of an overlap integral. In contrast, spatial output comprises measured values which are recorded as a function of space at fixed values of time. This includes measurements of selected field components, the amplitude of the Pointing vector, the energy density, or the power flow along a specified axis, all in dependence of the spatial coordinates. Spatial output requires much more disk storage space than temporal output, but is often necessary to obtain desired data. [88] To obtain the spectral characteristics of a given index structure, a frequency analysis has to be performed on the recorded time-domain data. If, for example, the frequency response over a broadband spectrum is of interest, a broadband pulse or an impulse excitation can provide this response in a single run. Mathematically, the conversion from the time-domain to the frequency-domain is achieved by means of a Fourier transform (numerical Fourier analysis).
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Energy-stable time-domain finite element methods for the 3D nonlinear Maxwell’s equations

Energy-stable time-domain finite element methods for the 3D nonlinear Maxwell’s equations

The more conventional FDTD methods are regarded as robust simulation schemes for linear and nonlinear models in Optics and Photonics [5]–[13], but also generally in the field of Compu- tational Electromagnetism, although there are considerable limitations in terms of applicability to complicated geometries, less smooth data (e.g. caused by material interfaces), etc. Typically the spatial domain is discretized by regular, structured (quadrilateral or hexahedral), staggered grids. The difference scheme presented in [5] served as the basis for one of the most commonly-used methods to solve the linear Maxwell’s equations. This scheme is of second order in time and ex- hibits a significant numerical dispersion over long time interval of wave propagation simulation [12]. FDTD simulations for the full system of nonlinear Maxwell’s equations have been presented in [7], [8]. Among other things, interacting waves of different frequencies could be treated directly [8]. The auxiliary differential equation (ADE) method along with finite difference time-domain (FDTD) schemes has been originally employed for linear dispersive materials [6] and for the coupling between the polarization vector and the electric field intensity [7], [14]. This scheme was applied to second- and third-order nonlinear phenomena including spatial soliton propagation [14], [15], linear and nonlinear interface scattering [16], and pulse propagation through nonlinear wave guides [17]. A lot of interesting modeling and simulation results for linear and nonlinear Lorentz dispersion with nonlinear Kerr response in case of 1D, 2D and 3D can be found in [15], [18]–[23]. Among non-standard difference methods, pseudospectral spatial domain schemes have been employed for optical carrier shock [24] and linear Lorentz dispersion with nonlinear response [25] simulation. Slowly varying envelope approximations (SVEA) are mostly used to simulate effects in nonlinear Photonics. Using this scheme, the system of Maxwell nonlinear equations transforms into the nonlinear Schrdinger equation. Various nonlinear effects such as self phase modulation and the Kerr effect can be successfully numerically treated [1], [26]. The beam propagation method (BPM) with second-order indices of refraction is employed for modeling of nonlinear optical devices exhibiting on-axis behaviour [27].
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Investigating Finite Volume Time Domain Methods in Computational Electromagnetics

Investigating Finite Volume Time Domain Methods in Computational Electromagnetics

Computational electromagnetics gained its momentum ever since the Finite Element Method (FEM) [1] started being applied to solving the Maxwell’s equations. Exploration of Yee’s work [2] changed this sce- nario. The method proposed by Yee, later named as Finite Difference Time Domain (FDTD), got a huge following mainly due to its simplicity. It is written for the linear non–dispersive media and Cartesian grid. A few years later Weiland proposed [3] another volume discretization method, termed as Finite Integration Technique (FIT). FIT presents a closed theory, which can be applied to the full spectrum of electromagnetics. FIT on a Cartesian grid with appropriate choice of sampling points in the Time Domain (TD) is algebraically equivalent to FDTD. In other words, FIT is a more generalized form of FDTD. All the above mentioned methods got refined and applied to various classes of problems [4–16] over the years and each of these methods enjoys its own set of advantages and disadvantages. FEM, more specifically, continuous FEM, has high flexibility in modeling the curved surfaces when employed on un- structured grids. But the TD formulation of the FEM results in a global implicit time marching scheme which requires global matrix inversion at each time step. This restricts FEM practically to Frequency Domain (FD) simulations. FD methods must solve an algebraic system of equations at all significant frequencies in the desired frequency range individually, making them more costly with increasing band- width. Fast frequency sweep algorithms can be employed in order to avoid calculating the solution at each and every frequency sample. However, these algorithms are memory intensive. When opted for parallel computing, FD methods require significant global operations.
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Application of Finite-Difference Time Domain Method to the Study of Practical Solutions in Room Acoustics

Application of Finite-Difference Time Domain Method to the Study of Practical Solutions in Room Acoustics

In this paper, this numerical method is applied to the study of the sound waves interaction with objects as usually present in concert halls or other types of auditoria. These objects in[r]

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Modelling Boundary Conditions in High-Order, Nodal, Time-Domain Finite Element Methods

Modelling Boundary Conditions in High-Order, Nodal, Time-Domain Finite Element Methods

5.2 Comparison against measurements In order to assess the accuracy of the ER model, a comparison between simulations and measurements is presented. The scenario is a single reflection of a spherical wave from a 5 cm porous absorber backed with a 15 cm air cavity. This material configuration is known to have ER behavior. The measurements were undertaken in an anechoic chamber, for further details on the measurement see Ref. [ 12 ]. The simulation is carried out in a large 3D domain and the resulting response is windowed in time, such that parasitic reflections are removed. A basis order of P = 4 is used and a high spatial resolution is employed in the simulation, roughly 14 PPW at 1 kHz. The initial condition is a Gaussian pulse with spatial variance σ = 0.2 m 2 . The admittance functions are mapped to rational functions who all have the same set of 14 poles, but varying numerical coefficients. The resulting transfer functions can be seen in Fig. 5 . For the small incidence angle case, there is very little difference between the LR and the ER model, as expected. However, as the angle increases, the difference between the two models increases as well. Clearly, the ER model matches the measured transfer functions better than the LR model.
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Time Domain Boundary Integral Equations Analysis

Time Domain Boundary Integral Equations Analysis

Once the transient current on the scatterer are found, the calculation of the resulting ra- diated electric and magnetic fields is a straightforward process [1] as far as the observation point is not located on the surface and most of the functions from the current evaluation stage can be reused. This widely admitted statement has caused direct field calculation routines to be overlooked as a simple post-processing stages in the integral equation-related literature. In many practical problems as those opposed in Section 6.11, accurate near-field calculation is of great importance. Yet there are few well-documented algorithms specifi- cally aimed the direct near-field computations. In [1], in order to calculate the near electric and magnetic fields at an arbitrarily point r(x , y, z ), the curl and gradient operators are ap- proximated by finite difference formulations involving the evaluation of the field quantities, namely the vector and scaler potentials, in six neighboring points r(x ±∆x , y ±∆y, z ±∆z ). This is computationally unacceptable. Besides, a time derivative is taken from (2.2) which in turn imposes an extra numerical integration over time at the final stage of near electric field calculations in [1]. In a relatively similar context, the moment methods themselves, however, invoke the calculation of potential integrals due to the source subdomains at close observation points. This chapter intends to introduce general approaches for direct field calculations exactly at the desired point, without using any approximation for curl and gradient operating on scalar and vector potentials. Unlike the earlier integration methods designed to decouple the spatial and temporal integrations by numerical approximations, analytical approaches are reviewed in this chapter to integrate in space-time concurrently. Eventually, each one of the following sections lead to more efficient and accurate poten- tial calculations which can also be incorporated into the Galerkin’s schemes in the former chapters.
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Meshfree generalized finite difference methods in soil mechanicspart I: theory

Meshfree generalized finite difference methods in soil mechanicspart I: theory

The Finite Pointset Method (FPM) is a meshfree approach to numerically solve (coupled) PDEs based on their strong solution in a sufficiently dense cloud of points carrying the physical information (such as velocity, pressure, etc.). Since a Lagrangian formulation is used, these discrete points move with the occurring velocity field. In order to determine the required approximations of spatial partial derivatives of arbitrary order, the WMLS algorithm (which is described in Section 4) employing the numerical data known at the discrete points of the domain is applied. Due to the compact support of the weighting function, only the values in a defined neighborhood influence the approximations and, thus, accelerate the computation compared to simulations with global weighting functions. Time derivatives are formed by simple finite differences. Thus, FPM is a generalized finite difference method. Over the last ten years, the fields of application have steadily expanded: Computational Fluid Dynamics (CFD), especially gas dynamics and incompressible flows (see, e.g., (Hietel 2005; Iliev 2002; Tiwari 2002a,b, 2004, 2005, 2007b)); fluid structure interaction (see (Tiwari 2007a)); continuum mechanics, in particular plastic and visco-elasto- plastic material behavior (see (Kuhnert 2012)).
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Zeitdiskretisierung höherer Ordnung bei der Finite-Differenzen Simulation seismischer Wellen (Time-Discretization of Higher-Order in the Finite-Difference Simulation of Seismic Waves)

Zeitdiskretisierung höherer Ordnung bei der Finite-Differenzen Simulation seismischer Wellen (Time-Discretization of Higher-Order in the Finite-Difference Simulation of Seismic Waves)

” A new time- space domain high-order finite-difference method for the acoustic wave equation“ [YL09] beschreiben Sie ein Verfahren, das, ¨ahnlich der Methode aus Abschnitt 3.1, Gleichung 3.8 verwendet, jedoch bei der Berechnung der Koeffizienten einen komplett anderen Ansatz w¨ahlt. Die Fehlerordnung kann mit dieser Methode bis auf eine beliebigen Ordnung erh¨oht werden. Laut Yang Liu und Mrinal K. Sen betrifft diese Erh¨ohung sowohl das r¨aumliche als auch das zeitliche Gitter und unterscheidet sich somit deutlich von der Methode 1, die in der Zeit in der Publikation von M. A. Dablain nur bis zu einer Genauigkeit von O(∆t 4 )
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Finite-Difference-Frequency-Domain Simulation of Electrically Large Microwave Structures using PML and Internal Ports

Finite-Difference-Frequency-Domain Simulation of Electrically Large Microwave Structures using PML and Internal Ports

However, the benefits of PML do not come for free. In the frequency-domain case, the material tensors worsen the numerical properties of the system of equations to be solved, which results in increased CPU time. In the time domain, mechanisms are not such clearly to be identified but similar effects are observed. Generally, the deteriorations depend strongly on the number and parameters of the PML layers and occur particularly if the PML layers overlap, e.g., at the edges and corners of the outer boundary of the computational domain. Beyond this, PML gives rise to another problem, which is often overlooked in the literature: during waveguide port simulations, a PML boundary causes appearance of parasitic PML modes with complex propagation constants which must be dealt with attention in order to separate them from the desired physical modes. In [20, 25] a PPP (Power Part in PML) criterion is introduced for this separation. However, this does not give reliable results in all cases and further investigations are necessary to explore these effects.
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CRS-based minimum-aperture Kirchhoff migration in the time domain

CRS-based minimum-aperture Kirchhoff migration in the time domain

Chapter 2 Ray theory In reflection seismics, the propagation of waves through the Earth is usually described in terms of continuum mechanics. The most fundamental equation in this context is the elastodynamic wave equation, a partial di fferential equation of second order which can, in general, not be solved analyti- cally for arbitrary complex media. Common approaches rely either on a direct numerical solution of the wave equation by means of finite-difference schemes or on an approximate asymptotic or itera- tive solution such as the ray theory or the WKBJ method. Although the approximate methods suffer from a limited range of validity and applicability, they are commonly utilised in both forward and inverse seismic problems. Besides their computational efficiency, the asymptotic methods allow to handle different elementary waves like, e. g., primary reflections and converted waves, independently which simplifies the interpretation of the results. Both approximate methods break down under certain conditions, e. g., in focal or shadow regions. Recent extensions like the Gaussian beam method and the Maslov-Chapman method are able to overcome some of the limitations, thus allowing a broader applicability of the approximate solutions.
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Time domain finite element method for Maxwell’s equations

Time domain finite element method for Maxwell’s equations

In the article [19], the Maxwell’s equations are discretized in space by a node-based and edge-based finite element method, and an efficient solver is described both for the frequency and time domain formulations. The paper [24] describes a general way to investigate the stability of temporal discretization schemes such as backward difference, forward difference and central difference methods in electromagnet- ics. The stability is determined by analyzing the root locus map of a characteristic equation and evaluating the spectral radius of finite element system matrix. Stability properties are given in [3] for simulations of transient electromagnetic phe- nomena for the Maxwell’s equations. In the article [25], time domain finite element methods based on Whitney elements are proposed for solving transient response problems on tetra- hedral meshes. One of the proposed schemes is uncondition- ally stable, another scheme is explicit but does not require matrix inversions. An explicit time domain finite element algorithm is presented for the Maxwell’s equations in [30], for complex media in [28] (only numerical result). An energy conserving method for 3D Maxwell’s equations is obtained in [37], based on an exponential operator splitting approach. Many papers have been written about time domain discon- tinuous Galerkin (TDDG) methods in computational electro- magnetics [38]–[49]. Other time domain methods to solve either the Maxwell’s equations or the vector wave equation can be found in [50]–[62]. Several books for electromagnet- ics [63]–[68] are available for analysis and simulation.
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Modeling and forecasting exchange rate volatility in time-frequency domain

Modeling and forecasting exchange rate volatility in time-frequency domain

This paper proposes an enhanced approach to modeling and forecasting volatility using high frequency data. Using a forecasting model based on Realized GARCH with multiple time-frequency decomposed realized volatility measures, we study the influence of di↵erent timescales on volatility forecasts. The decomposition of volatility into several timescales approximates the behaviour of traders at corre- sponding investment horizons. The proposed methodology is moreover able to account for impact of jumps due to a recently proposed jump wavelet two scale realized volatility estimator. We propose a realized Jump-GARCH models esti- mated in two versions using maximum likelihood as well as observation-driven estimation framework of generalized autoregressive score. We compare forecasts using several popular realized volatility measures on foreign exchange rate futures data covering the recent financial crisis. Our results indicate that disentangling jump variation from the integrated variation is important for forecasting per- formance. An interesting insight into the volatility process is also provided by its multiscale decomposition. We find that most of the information for future volatility comes from high frequency part of the spectra representing very short investment horizons. Our newly proposed models outperform statistically the popular as well conventional models in both one-day and multi-period-ahead forecasting.
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A Finite-Time Thermodynamics of Unsteady Fluid Flows

A Finite-Time Thermodynamics of Unsteady Fluid Flows

This assumption is fulfilled by definition for the Proper Orthogonal Decom- position (POD) – also called the Karhunen–LoŒ eve decomposition or Principal Components Analysis (PCA). It is approximately fulfilled also by locally time- periodic shear flows resolved by Fourier modes [34], POD modes [24,26,33], or stability eigenmodes of oscillatory nature [25]. Here, each harmonic fre- quency is resolved by one pair of modes mutually shifted by a quarter period with similar pair-wise energy content. The analysis of this paper remains valid for non-POD modes or non-periodic flows when deviations from Eq. (9) by individual (i,j) pair cancel out approximately in the summation of formula (8) for Q i . The ultimate corroboration in such cases is by numerical experiments,
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2D Navier-Stokes Equations in a Time-Dependent Domain

2D Navier-Stokes Equations in a Time-Dependent Domain

the method which is used in [Fei93]. The proof of existence is based on the implicit time discretisation, proving existence of the solution to the stationary problem, then proving a priori estimates and finally proving the convergence of piecewise-constant or piecewise-linear functions to the weak solution as time steps go to zero. The proof of convergence is based on the compactness of the solution in the appropriate function space (L p (D ×(0, T )) 2 for any p < 4). In this crucial point of our approach, we follow the idea from [AL83] which differs from the method used in [Tem79] in the way of proving the compactness. We also prove existence of the functional ∂( ∂t u ) ∈ X ∗ = L 2 (0, T ; V ∗ ) + L 4/3 (0, T ; L 4/3 (D) 2 ). We extend the result of Feistauer [Fei93] by proving more general uniqueness for the regularised model. Our generalised uniqueness property depends on data (the data describe e.g. the deformation of the domain or the pressure on the boundary). We prove that if two data-sets are sufficiently close to each other, then two corresponding solutions to our problem are also close to each other.
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Working Time Reduction and Employment in a Finite World

Working Time Reduction and Employment in a Finite World

is then independent of the wage level. This property makes, by construction, the equilibrium unemployment rate independent of working time, so that exogenous manipulation of working time cannot affect unemployment (see Layard et al., 1991, p. 503). The key assumption that underlies this result can however be questioned. For instance, Danthine and Kurmann [2004] argue that “the positive incentive effect of a larger own wage is stronger than the negative effect of a higher comparison wage” (p. 112). And yet one can check that in a model ` a la Layard et al. [1991] with an effort function implying a positive effect of the wage level on the in-work effort in equilibrium, work-sharing can reduce unemployment. Before Layard et al. [1991], Hoel and Vale [1986] studied the effects of a WTR in an efficiency wage model with quit behaviour and training costs. The assumptions they make on the quit rate function are mutatis mutandis equivalent to those made by Layard et al. [1991] on their effort function: the quit rate depends negatively on both the unemployment rate and the ratio between the wage paid by the firm and the average wage in the economy. Consequently, the quit behaviour of workers in a symmetric equilibrium depends on the unemployment rate but not on the wage level. Because work-sharing increases training costs, the authors then conclude that a WTR affects (un)employment unfavourably. In the final discussion of their paper, they however consider an alternative specification for the quit behaviour, which leads them to conclude their analysis in a rather nuanced and cautious way.
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The Analysis of Parabolic Equation Model Solutions using Split-Step/ Finite Difference Method

The Analysis of Parabolic Equation Model Solutions using Split-Step/ Finite Difference Method

The general form of PE, the first order differential equation, is obtained by introducing the acoustic field function in Helmholtz equation. It was a simplified differential form that expressed in terms of the operator called the square root operator. The reduced wave equation contains the index of refraction which has the density and its derivative terms. In numerical solutions, the terms related to the density are approached to describe the density discontinuity caused by the vertical density variations at interface between two half space. Various numerical models treat density discontinuity in differential form of wave equation by different approach. The boundary discontinuity leads to phase error with range in SSF numerical solutions and the appropriate approaches can be treated to decrease the numerical errors (9). In PE/SSF methods, the smoothing functions such as the hyperbolic functions are implemented to treat the boundary discontinuity density, but solutions accumulates the phase error with the range, so by Yevick and Thomson (10), the hybrid split-step/finite difference method was developed to treat the density discontinuity issue, and it significantly improved the phase error.
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Survey of Finite Element Method-Based Real-Time Simulations

Survey of Finite Element Method-Based Real-Time Simulations

choice of the reference configuration. Those formulations are characterized by high accuracy, but also by significant numerical effort and convergence issues. Székely et al. [ 54 ] applied total Lagrangian formulation in combination with material nonlinearities in modeling soft tissue deformations for laparoscopic surgery simulation. They addressed the high computational demand of the approach by using a “brute force” method (i.e., by parallelizing the computation on a large, three-dimensional network of high-performance processor units). Despite of the fact that they report on a not highly realistic simulation of uterus tissue deformation. On the other hand, various ideas were implemented to keep the numerical effort in acceptable limits despite the use of rigorous FE formulations. For instance, Heng et al. [ 55 ] proposed an approach denoted as a hybrid condensed FE model. The idea is to partition the model into two regions—a part of the model that is being interacted with and the rest of the model. Hence, a complex nonlinear FE model that can also deal with topological change was used to model a small-scale FE model that represents the operational region, while a linear and topology-fixed FE model was used to model the large-scale nonoperational region. The development was done for a virtual training system for knee arthroscopic surgery. In another field of application, Dulong et al. [ 56 ] focused their work on development of real-time interaction between a designer and a virtual prototype as a promising way to perform optimization of parts design. In the first step, which they refer to as the “training phase,” they performed a set of nonlinear FE computations. Over the course of simulation, deformations for a current load case is determined by interpolating between the results obtained in the “training phase.” One may actually notice here a certain parallel with the approach based on neural networks.
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Precise Depth Image Based Real-Time 3D Difference Detection

Precise Depth Image Based Real-Time 3D Difference Detection

While the next chapters or this thesis will focus on 3D difference detection with a high precision, this section briefly sketches how real-time 3D difference detection can support 3D modeling and aug- mented reality applications [ Kah13 ]. Augmented reality (AR) applications combine real and virtual, are interactive in real time and registered in 3D [ Azu97 ]. Since Azuma first stated these characteristics of AR applications in 1997, augmented reality has matured remarkably [ ZDB08 ]. However, an impor- tant bottleneck remains: The availability of 3D models of real scenes, which correctly model not only sparse point but also the surface of the scene. Such dense 3D models are important for two different augmented reality aspects. First, a 3D model is needed for a smooth and seamless integration of virtual objects into the camera images. Therefore, the virtual objects should be illuminated in a consistent way with the illumination of the real scene, they should cast shadows and they should be occluded by parts of the real scene which are closer than the virtual object. Both occlusion handling and shadow calcula- tion require knowledge about the 3D structure of the real scene [ Hal04 ] [ PSP09 ]. Furthermore, dense 3D models are often used for model-based camera tracking, both for the camera pose initialisation and for model-based frame-to-frame tracking [ LF05 ]. Model-based estimations of the camera pose have the advantage that they overcome the need to prepare the scene with fiducial markers [ GRS06 ].
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Time distributed difference-in-differences estimates of return to training

Time distributed difference-in-differences estimates of return to training

Abstract: This work is devoted to estimating the individual return to worker’s profes- sional training. The research is based on the personnel records of Russian metal- lurgical enterprise (2006–2010). The main factors that distinguish this paper from others are the following: (I) We focused on the internal labour market, concluding that it has common peculiarities of wage setting concerned with training as an open labour market. (II) We show that mobility-friendly training programs give high re- turns, and not only in transition economies. (III) We suggest controlling for mobility by choosing a corresponding control group. (IV) We use a robust new specification that is reactive to different dynamics of the dependent variable in treated and con- trol groups in difference-in-differences estimates. (V) We compared three different kinds of training and our conclusions could have practical application. The best way to raise personal earnings is on-the-job training. The internal mobility caused by retraining courses has the same impact on workers as if they lacked retraining. The wages of workers trained in the same field grow randomly for a few months before and after training. Nevertheless it is difficult to prove the causal effect of this kind of training on wage growth.
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