Accurate modeling of dielectric interfaces by the effective permittivities for the fourth-order symplectic finite-difference time-domain method

The fourth-order finite-difference time-domain (FDTD) method using a symplectic integrator propagator can calculate the propagation of the electromagnetic waves with very low dispersion error in the region of a constant or smoothly varying index profile. An additional technique is required for the problem with the discontinuous dielectric interfaces. We derived the third-order effective permittivities at dielectric interfaces for the fourth-order FDTD method in the case of 2D TE polarization. As the required accuracy level is increased, the memory resources used by the fourth-order FDTD method with the effective permittivities are reduced severalfold or more compared with the standard FDTD method. The accurate performance of the proposed method is demonstrated through numerical examples.     

Direct Numerical Simulation of Turbulence at Lower Costs

Direct Numerical Simulation (DNS) is the most accurate, but also the most expensive, way of computing turbulent flow. To cut the costs of DNS we consider a family of second-order, explicit one-leg time-integration methods and look for the method with the best linear stability properties. It turns out that this method requires about two times less computational effort than Adams–Bashforth. Next, we discuss a fourth-order finite-volume method that is constructed as the Richardson extrapolate of a classical second-order method. We compare the results of this fourth-order method and the underlying second-order method for a DNS of the flow in a cubical driven cavity at Re= 10⁴. Experimental results are available for comparison. For this example, the fourth-order results are clearly superior to the second-order results, whereas their computational effort is about twenty times less. With the improved simulation method, a DNS of a turbulent flow in a cubical lid-driven flow at Re = 50,000 and a DNS of a turbulent flow past a square cylinder at Re = 22,000 are performed.     

A new energy and momentum conserving algorithm for the non-linear dynamics of shells

A numerical time-integration scheme for the dynamics of non-linear elastic shells is presented that simultaneously and independent of the time-step size inherits exactly the conservation laws of total linear, total angular momentum as well as total energy. The proposed technique generalizes to non-linear shells recent work of the authors on non-linear elastodynamics and is ideally suited for long-term/large-scale simulations. The algorithm is second-order accurate and can be immediately extended with no modification to a fourth-order accurate scheme. The property of exact energy conservation induces a strong notion of non-linear numerical stability which manifests itself in actual simulations. The superior performance of the proposed scheme method relative to conventional time-integrators is demonstrated in numerical simulations exhibiting large strains coupled with a large overall rigid motion. These numerical experiments show that symplectic schemes often regarded as unconditionally stable, such as the mid-point rule, can exhibit a dramatic blow-up in finite time while the present method remains perfectly stable.     

Finite-difference-time-domain analysis of finite-number-of-periods holographic and surface-relief gratings

The total-field-scattered-field formulation of the finite-difference time-domain method (FDTD) is used to analyze the diffraction of finite incident beams by finite-number-of-periods holographic and surface-relief gratings. Both second-order and fourth-order FDTD formulations are used with various averaging schemes to treat permittivity discontinuities and a comparative study is made with alternative numerical methods. The diffraction efficiencies for gratings of several periods and various beam sizes, for both TE and TM polarization cases, are calculated and the FDTD results are compared with the finite-difference frequency-domain (FDFD) method results in the case of holographic gratings, and with the boundary element method results in the case of surface-relief gratings. Furthermore, the convergence of the FDTD results to the rigorous coupled-wave analysis results is investigated as the number of grating periods and the incident beam size increase.     

Three-dimensional hybrid implicit?explicit finite-difference time-domain method in the cylindrical coordinate system

A novel hybrid implicit--explicit (HIE) finite-difference time-domain (FDTD) method, which is extremely useful for problems with very fine structures along the w-direction in cylindrical coordinate system, is presented. This method has higher computation efficiency than conventional cylindrical FDTD methods, because the time step in this method is only determined by the space discretisations in the radial and vertical directions. The numerical stability of the proposed HIE--FDTD method is presented analytically. Compared with the cylindrical alternating-direction implicit (ADI)--FDTD method, this HIE--FDTD method has higher accuracy, especially for larger time step size. At each time step, the HIE--FDTD method requires the solution of two tridiagonal matrices and four explicit updates. While maintaining the same size of time step, the central processing unit (CPU) time for this weakly conditionally stable FDTD method can be reduced to about 3/5 of that for the ADI--FDTD scheme. The numerical performance of the proposed HIE--FDTD over the conventional cylindrical FDTD method and cylindrical ADI--FDTD method is demonstrated through numerical examples.     

Finite-difference time-domain solution of light scattering by dielectric particles with a perfectly matched layer absorbing boundary condition

A three-dimensional finite-difference time-domain (FDTD) program has been developed to provide a numerical solution for light scattering by nonspherical dielectric particles. The perfectly matched layer (PML) absorbing boundary condition (ABC) is used to truncate the computational domain. As a result of using the PML ABC, the present FDTD program requires much less computer memory and CPU time than those that use traditional truncation techniques. For spheres with particle-size parameters as large as 40, the extinction and absorption efficiencies from the present FDTD program match the Mie results closely, with differences of less than ~1%. The difference in the scattering phase function is typically smaller than ~5%. The FDTD program has also been checked by use of the exact solution for light scattering by a pair of spheres in contact. Finally, applications of the PML FDTD to hexagonal particles and to spheres aggregated into tetrahedral structures are presented.     

High Order Difference Schemes for a Time Fractional Differential Equation with Neumann Boundary Conditions

A compact finite difference scheme is derived for a time fractional differential equation subject to Neumann boundary conditions. The proposed scheme is second-order accurate in time and fourth-order accurate in space. In addition, a high order alternating direction implicit (ADI) scheme is also constructed for the two-dimensional case. The stability and convergence of the schemes are analysed using their matrix forms.     

基于Helmholtz方程的过渡曲面设计

阐述了二阶和四阶Helmholtz方程的一类周期边界问题的差分解法及其在过渡曲面设计中的应用。这类方法不同于传统的PDE方法中的二阶和四阶的偏微分方程,比传统的二阶和四阶偏微分方程有了更多的自由项,因此,在曲面设计的时候,就有更多的形状控制参数可进行调整,文中重点讨论了方程中的系数对曲面形状的影响,并研究了边界切矢条件对曲面形状的影响及其在曲面形状设计中的应用。设计者只需给出边界曲线和边界切矢,并通过对它们的控制就可构造和修改曲面形状。     

Dual-grid finite-difference time-domain scheme for the fast simulation of surrounded antennas

A new finite-difference time-domain (FDTD) multiresolution strategy for surrounded antenna analysis is presented. The dual-grid FDTD (DG-FDTD) is divided into two FDTD simulations using different grids. Indeed, the antenna is firstly simulated without its environment using a finely discretised FDTD in order to determine its main characteristics and save its primary radiation. In a second step, this saved field is used as the excitation of a coarse FDTD to simulate the antenna with its environment. The application of the DG-FDTD to an ultra wide-band problem is discussed, and the DG-FDTD turns out to be an accurate and fast tool to simulate various antenna configurations. Furthermore, this method remains stable along the computational time, and is easy to implement in a classical FDTD scheme     

Finite-difference time-domain solution of light scattering and absorption by particles in an absorbing medium

The three-dimensional (3-D) finite-difference time-domain (FDTD) technique has been extended to simulate light scattering and absorption by nonspherical particles embedded in an absorbing dielectric medium. A uniaxial perfectly matched layer (UPML) absorbing boundary condition is used to truncate the computational domain. When computing the single-scattering properties of a particle in an absorbing dielectric medium, we derive the single-scattering properties including scattering phase functions, extinction, and absorption efficiencies using a volume integration of the internal field. A Mie solution for light scattering and absorption by spherical particles in an absorbing medium is used to examine the accuracy of the 3-D UPML FDTD code. It is found that the errors in the extinction and absorption efficiencies from the 3-D UPML FDTD are less than approximately 2%. The errors in the scattering phase functions are typically less than approximately 5%. The errors in the asymmetry factors are less than approximately 0.1%. For light scattering by particles in free space, the accuracy of the 3-D UPML FDTD scheme is similar to a previous model [Appl. Opt. 38, 3141 (1999)].     

Solution of Maxwell's Equations Using the MQ Method

A meshless time domain numerical method based on the radial basis functions using multiquadrics (MQ) is employed to simulate electromagnetic field problems by directly solving the time-varying Maxwell's equations without transforming to simplified versions of the wave or Helmholtz equations. In contrast to the conventional numerical schemes used in the computational electromagnetism such as FDTD, FETD or BEM, the MQ method is a truly meshless method such that no mesh generation is required. It is also easy to deal with the appropriate partial derivatives, divergences, curls, gradients, or integrals like semi-analytic solutions. For illustration purposes, the MQ method is employed to calculate the propagations of the electric and magnetic waves in the homogeneous, isotropic, and non-lossy 2D rectangular waveguide as well as 3D cavity resonator. Good agreements are obtained as compared to analytical solutions. By directly solving the Maxwell's equations, the MQ scheme provides a very simple and effective numerical scheme for the computational electromagnetism.     

A univariate dimension-reduction method for multi-dimensional integration in stochastic mechanics

This paper presents a new, univariate dimension-reduction method for calculating statistical moments of response of mechanical systems subject to uncertainties in loads, material properties, and geometry. The method involves an additive decomposition of a multi-dimensional response function into multiple one-dimensional functions, an approximation of response moments by moments of single random variables, and a moment-based quadrature rule for numerical integration. The resultant moment equations entail evaluating N number of one-dimensional integrals, which is substantially simpler and more efficient than performing one N-dimensional integration. The proposed method neither requires the calculation of partial derivatives of response, nor the inversion of random matrices, as compared with commonly used Taylor expansion/perturbation methods and Neumann expansion methods, respectively. Nine numerical examples involving elementary mathematical functions and solid-mechanics problems illustrate the proposed method. Results indicate that the univariate dimension-reduction method provides more accurate estimates of statistical moments or multidimensional integration than first- and second-order Taylor expansion methods, the second-order polynomial chaos expansion method, the second-order Neumann expansion method, statistically equivalent solutions, the quasi-Monte Carlo simulation, and the point estimate method. While the accuracy of the univariate dimension-reduction method is comparable to that of the fourth-order Neumann expansion, a comparison of CPU time suggests that the former is computationally far more efficient than the latter.     

Test of the FDTD accuracy in the analysis of the scattering resonances associated with high-Q whispering-gallery modes of a circular cylinder

Our objective is the assessment of the accuracy of a conventional finite-difference time-domain (FDTD) code in the computation of the near- and far-field scattering characteristics of a circular dielectric cylinder. We excite the cylinder with an electric or magnetic line current and demonstrate the failure of the two-dimensional FDTD algorithm to accurately characterize the emission rate and the field patterns near high-Q whispering-gallery-mode resonances. This is proven by comparison with the exact series solutions. The computational errors in the emission rate are then studied at the resonances still detectable with FDTD, i.e., having Q-factors up to 10(3).     

Nonlinear System Identification Using a Subband Adaptive Volterra Filter

This paper presents a flexible and efficient subband adaptive second-order Volterra filter (SBVF) structure for nonlinear system identification. The structure is first described in detail, where the underlying filter-bank scheme and adaptive filtering algorithms are explained, followed by a computational complexity analysis. Simulation results are then presented, showing that the proposed structure can achieve equal system-identification performance compared with that of a fullband second-order Volterra structure at a much-reduced complexity. In addition, the structure provides a more precise system model compared with that of a linear-only structure at a potentially similar computational expense. The results also demonstrate the suggested structure's ability to exploit a priori knowledge of the nature of the system nonlinearity through selectable nonlinear subband filtering, resulting in further complexity savings. The simulation results are experimentally verified under a practical acoustic-echo-cancellation scenario. It is shown that the SBVF structure can achieve up to a 10-dB lower mean-square error than that of a linear-only model at a comparable complexity.     

The green element method

This paper discusses an element-by-element approach of implementing the Boundary Element Method (BEM) which offers substantial savings in computing resource, enables handling of a wider range of problems including non-linear ones, and at the same time preserves the second-order accuracy associated with the method. Essentially, by this approach, herein called the Green Element Method (GEM), the singular integral theory of BEM is retained except that its implementation is carried out in a fashion similar to that of the Finite Element Method (FEM). Whereas the solution procedure of BEM couples the information of all nodes in the computational domain so that the global coefficient matrix is dense and full and as such difficult to invert, that of GEM, on the other hand, involves only nodes that share common elements so that the global coefficient matrix is sparse and banded and as such easy to invert. Thus, GEM has the advantage of being more computationally efficient than BEM. In addition, GEM makes the singular integral theory more flexible and versatile in the sense that GEM readily accommodates spatial variability of medium and flow parameters (e.g., flow in heterogeneous media), while other known numerical features of BEM—its second-order accuracy and ability to readily handle problems with singularities are retained by GEM. A number of schemes is incorporated into the basic Green element formulation and these schemes are examined with the goal of identifying optimum schemes of the formulation. These schemes include the use of linear and quadratic interpolation functions on triangular and rectangular elements. We found that linear elements offer acceptable accuracy and computational effort. Comparison of the modified fully implicit scheme against the generalized two-level scheme shows that the modified fully implicit scheme with weight of about 1·25 offers a marginally better approximation of the temporal derivative. The Newton–Raphson scheme is easily incoporated into GEM and provides excellent results for the time-dependent non-linear Boussinesq problem. Comparison of GEM with conventional BEM is done on various numerical examples, and it is observed that, for comparable accuracy, GEM uses less computing time. In fact, from the numerical simulations carried out, GEM uses between 15 and 45 per cent of the simulation time of BEM.     

CONDIF: A modified central-difference scheme for convective flows

For most practical purposes, the central-difference scheme (CDS) would be ideal only if it were unconditionally stable. It is a simple and second-order scheme which is easy to implement. It does not introduce any second-order ‘diffusion’ like truncation error. However, for grid Peclet numbers larger than 2, the CDS leads to over- and under-shoots and is unstable. This paper presents a method, called CONDIF, which eliminates this undesirable feature of the CDS. It modifies the CDS by introducing a controlled amount of numerical diffusion based on the local gradients. The numerical diffusion can be adjusted to be negligibly low for most problems. CONDIF has been used to solve a number of test problems which have been widely used for comparative study of numerical schemes in the published literature. For all these problems the CONDIF results are significantly more accurate than those obtained from the hybrid scheme when the Peclet number is very high (→∞) and the flow is at large angles (→45 degrees) to the grid. In general the computational effort for CONDIF is comparable (within 20 per cent) to that for the hybrid scheme. However, in one instance the rate of convergence was found to be significantly slower.     

Determination of second-order derivatives of a skew ray with respect to the variables of its source ray in optical prism systems

The second-order derivative of a scalar function with respect to a variable vector is known as the Hessian matrix. We present a computational scheme based on the principles of differential geometry for determining the Hessian matrix of a skew ray as it travels through a prism system. A comparison of the proposed method and the conventional finite difference (FD) method is made at last. It is shown that the proposed method has a greater inherent accuracy than FD methods based on ray-tracing data. The proposed method not only provides a convenient means of investigating the wavefront shape within complex prism systems, but it also provides a potential basis for determining the higher order derivatives of a ray by further taking higher order differentiations.     

A Fourth-Order Compact Finite Difference Scheme for Higher-Order PDE-Based Image Registration

Image registration is an ill-posed problem that has been studied widely in recent years. The so-called curvature-based image registration method is one of the most effective and well-known approaches, as it produces smooth solutions and allows an automatic rigid alignment. An important outstanding issue is the accurate and efficient numerical solution of the Euler-Lagrange system of two coupled nonlinear biharmonic equations, addressed in this article. We propose a fourth-order compact (FOC) finite difference scheme using a splitting operator on a 9-point stencil, and discuss how the resulting nonlinear discrete system can be solved efficiently by a nonlinear multi-grid (NMG) method. Thus after measuring the h-ellipticity of the nonlinear discrete operator involved by a local Fourier analysis (LFA), we show that our FOC finite difference method is amenable to multi-grid (MG) methods and an appropriate point-wise smoothing procedure. A high potential point-wise smoother using an outer-inner iteration method is shown to be effective by the LFA and numerical experiments. Real medical images are used to compare the accuracy and efficiency of our approach and the standard second-order central (SSOC) finite difference scheme in the same NMG framework. As expected for a higher-order finite difference scheme, the images generated by our FOC finite difference scheme prove significantly more accurate than those computed using the SSOC finite difference scheme. Our numerical results are consistent with the LFA analysis, and also demonstrate that the NMG method converges within a few steps.     

On the accuracy of finite-difference solutions for nonlinear water waves

This paper considers the relative accuracy and efficiency of low- and high-order finite-difference discretisations of the exact potential-flow problem for nonlinear water waves. The method developed is an extension of that employed by Li and Fleming (Coastal Engng 30: 235–238, 1997) to allow arbitrary-order finite-difference schemes and a variable grid spacing. Time-integration is performed using a fourth-order Runge–Kutta scheme. The linear accuracy, stability and convergence properties of the method are analysed and high-order schemes with a stretched vertical grid are found to be advantageous relative to second-order schemes on an even grid. Comparison with highly accurate periodic solutions shows that these conclusions carry over to nonlinear problems and that the advantages of high-order schemes improve with both increasing nonlinearity and increasing accuracy tolerance. The combination of non-uniform grid spacing in the vertical and fourth-order schemes is suggested as optimal for engineering purposes.     

First-order and second-order adjoint methods for parameter identification problems with an application to the elasticity imaging inverse problem

We devise a fast and reliable computational framework for the elasticity imaging inverse problem of detecting cancerous tumors in the human body using an output least-squares (OLS) approach. From a mathematical standpoint, this inverse problem requires identifying a parameter in a mixed variational problem. We develop, in a continuous setting, a first-order adjoint method and two second-order adjoint methods. The continuous formulae are then used to devise a scheme for an efficient computation of the gradient and the Hessian of the OLS objective. We give detailed numerical examples.