Higher Order Method Of Moments Research Articles

Lattice Boltzmann model for wave propagation

A lattice Boltzmann model for two-dimensional wave equation is proposed by using the higher-order moment method. The higher-order moment method is based on the solution of a series of partial differential equations obtained by using multiscale technique and Chapman-Enskog expansion. In order to obtain the lattice Boltzmann model for the wave equation with higher-order accuracy of truncation errors, we removed the second-order dissipation term and the third-order dispersion term by employing the moments up to fourth order. The reversibility in time appears owing to the absence of the second-order dissipation term and the third-order dispersion term. As numerical examples, some classical examples, such as interference, diffraction, and wave passing through a convex lens, are simulated. The numerical results show that this model can be used to simulate wave propagation.

A multi‐energy‐level lattice Boltzmann model for two‐dimensional wave equation

AbstractA lattice Boltzmann model for two‐dimensional wave equation is presented. In this model, we used higher‐order moment method, multi‐scale technique and Chapman–Enskog expansion, and multi‐energy‐level to obtain wave equation and energy conservation equation. As numerical examples, the interference and diffraction of wave are simulated. The numerical results show this model can be used to simulate two‐dimensional wave propagation. Copyright © 2009 John Wiley & Sons, Ltd.

A higher-order moment method of the lattice Boltzmann model for the conservation law equation

In this paper, we proposed a higher-order moment method in the lattice Boltzmann model for the conservation law equation. In contrast to the lattice Bhatnagar–Gross–Krook (BGK) model, the higher-order moment method has a wide flexibility to select equilibrium distribution function. This method is based on so-called a series of partial differential equations obtained by using multi-scale technique and Chapman–Enskog expansion. According to Hirt’s heuristic stability theory, the stability of the scheme can be controlled by modulating some special moments to design the third-order dispersion term and the fourth-order dissipation term. As results, the conservation law equation is recovered with higher-order truncation error. The numerical examples show the higher-order moment method can be used to raise the accuracy of the truncation error of the lattice Boltzmann scheme for the conservation law equation.

Treatment of Singular Integrals on Generalized Curvilinear Parametric Quadrilaterals in Higher Order Method of Moments

Singular integrals over the generalized curvilinear parametric quadrilaterals in higher order method of moments (MoM) are treated by a singularity cancellation technique. The singularities are eliminated via coordinate transformation, by which the singular integrals are converted into some regular integrals that can be numerically computed by using the Gauss-Legendre quadrature. Numerical results show that this proposed singular integral treatment approach converges much faster, yet achieves several orders of magnitude higher precision than the widely used singularity subtraction technique.

ANALYSIS OF ELECTROMAGNETIC SCATTERING WITH HIGHER-ORDER MOMENT METHOD AND NURBS MODEL

Abstract—A novel scheme of combining non-uniform rational Bsplines (NURBS) model with higher-order moment method (HOMM) is presented. The mesh precision of conforming to practical object is a major factor for HOMM to yield accurate results. In the present paper, NURBS technique is employed to model complex objects accurately with large curved Bezier patches and no factitious geometric discontinuities are introduced between the adjoining patches. The higher-order modified Legendre basis functions are defined on Bezier patch. As a result of the combination of NURBS model with HOMM, the accuracy of results is greatly improved compared with HOMM on curved parametric quadrilateral (CPQ) model, meanwhile, the number of unknowns is much reduced. Numerical results show that NURBSHOMM is an efficient technique with good potential to solve the electromagnetic (EM) problems of complex electrically large objects.

ANALYSIS OF COMPLEX ANTENNA AROUND ELECTRICALLY LARGE PLATFORM USING ITERATIVE VECTOR FIELDS AND UTD METHOD

A new e-cient technique for the analysis of complex antenna around a scatterer is proposed in this paper, termed the iterative vector flelds with uniform geometrical theory of difiraction (UTD) technique. The complex fleld vector components on the closed surface enclosing the antenna without platform are computed by higher order Method of Moments (MOM), and the scattered flelds from the platform are calculated by UTD method. The process of iteration is implemented according to the equivalence theorem. Based on this approach, an approximation method is outlined, in which the computational time is saved largely, while the accuracy is not reduced. The relative patterns obtained from the present method and the approximation method both show good agreements with that obtained from MOM.

Strong Singularity Reduction for Curved Patches for the Integral Equations of Electromagnetics

A method for weakening strong singularity contributions to the integral equations of electromagnetics is demonstrated. The remaining weakly singular integral derived after the reduction can be computed by any suitable method. Numerical results demonstrate that the singularity reduction method converges and that the new integration method can be used in high-order moment method or Nystrm codes without any adverse effect on their accuracy.

A lattice Boltzmann model for the Korteweg–de Vries equation with two conservation laws

A lattice Boltzmann model for the Korteweg–de Vries (KdV) equation is presented by using the higher-order moment method. In contrast to the previous lattice Boltzmann model to the KdV equation, our method has higher-order accuracy. Two key steps in the development of this model are the addition of a momentum conservation condition, and the construction of a correlation between the first conservation law and the second conservation law. The numerical example shows the higher-order moment method can be used to raise the truncation error of the lattice Boltzmann scheme.

Spectral Elements for the Integral Equations of Time-Harmonic Maxwell Problems

We present a novel high-order method of moments (MoM) with interpolatory vector functions, on quadrilateral patches. The main advantage of this method is that the Hdiv conforming property is enforced, and at the same time it can be interpreted as a point-based scheme. We apply this method to field integral equations (FIEs) to solve time-harmonic electromagnetic scattering problems. Our approach is applied to the first and second Nedelec families of Hdiv conforming elements. It consists in a specific choice of the degrees of freedom (DOF), made in order to allow a fast integral evaluation. In this paper we describe these two sets of DOF and their corresponding quadrature rules. Sample numerical results on FIE confirm the good properties of our schemes: faster convergence rate and cheap matrix calculation. We also present observations on the choice of the discretization method, depending on the FIE selected.

Adaptive Integral Method for Higher Order Method of Moments

The adaptive integral method (AIM) is combined with the higher order method of moments (MoM) to solve integral equations. The technique takes advantage of the low computational complexity and memory requirements of the AIM and the reduced number of unknowns and higher order convergence of higher order basis functions. The classical AIM is appropriately modified to allow larger discretization elements and, consequently, higher basis function expansion orders. Numerical examples based on the higher order hierarchical Legendre basis functions show the advantages of the proposed technique over the classical AIM based on low-order basis functions in terms of memory and computational time.

A higher-order moment method of the lattice Boltzmann model for the Korteweg–de Vries equation

In this paper, a lattice Boltzmann model for the Korteweg–de Vries (KdV) equation with higher-order accuracy of truncation error is presented by using the higher-order moment method. In contrast to the previous lattice Boltzmann model, our method has a wide flexibility to select equilibrium distribution function. The higher-order moment method bases on so-called a series of lattice Boltzmann equation obtained by using multi-scale technique and Chapman–Enskog expansion. We can also control the stability of the scheme by modulating some special moments to design the dispersion term and the dissipation term. The numerical example shows the higher-order moment method can be used to raise the accuracy of truncation error of the lattice Boltzmann scheme.

Fast Algorithm to Extract the Singularity of Higher Order Moment Method

The higher order moment method has far fewer unknowns compared to the low order methods. However, the computation of the self-term matrix elements is extremely time-consuming. This paper presents an algorithm to extract the singularity by dividing the integrand into two parts on account of Taylor's formula. The integrand of the first part with a removable discontinuity is easy to be integrated. The rest part consists of three principal singular functions. Their singularities are canceled by the Jacobian of a simple transformation. This singularity extraction leads to a rapid non-redundant integration. Numerical results prove that this algorithm is more than 20 times faster than the old method.

Detection of the recent 2000a climatic extreme anomalies based on a new kind of approach-the high-order moment method

Based on the normalized high-resolution climatic proxy data, high-order moment method is used to detect the climate extreme anomalies in recent 2000 years in this paper. Combining with filtering method to extract the dominant components at physical backgrounds from the normalized data, we analyzed the information and examined the contributions of each component of the climate extreme events. The results show that: 1) On the timescale of more than 100 years, climatic oscillation with the period of about 1000 years may exits; besides, the 20th century witnessed the most active climate extreme anomalous phenomenon in recent 2000 years, so it may correspond to an active period of climate extreme anomaly phenomenon. 2) On 20—60year timescale, climate extreme anomaly phenomenon in the period of A.D. 300—1100a is relatively distinct and that in time span of A.D.1100—1980a is comparatively mitigative. Possibly, this level makes no considerable contribution to the climatic anomaly of 20th century. This level and the century scale both reflect that for climatic change in recent 2000 years, the year around A.D. 1100a maybe the key period of climate transition. 3) On the interannual timescale (less than 20years), the years of occurrence of climatic extreme anomalies reflected in Beijing Shihua cave stalagmite proxy records has good correspondence with that of the E1 Ni?o or La Ni?a events (in this paper, only the period of 1960—1980a was considered). 4) high-order moment method has a good prospect in detecting the climatic extreme events.

Double Higher Order Method of Moments for Surface Integral Equation Modeling of Metallic and Dielectric Antennas and Scatterers

A novel double higher order Galerkin-type method of moments based on higher order geometrical modeling and higher order current modeling is proposed for surface integral equation analysis of combined metallic and dielectric antennas and scatterers of arbitrary shapes. The technique employs generalized curvilinear quadrilaterals of arbitrary geometrical orders for the approximation of geometry (metallic and dielectric surfaces) and hierarchical divergence-conforming polynomial vector basis functions of arbitrary orders for the approximation of electric and magnetic surface currents within the elements. The geometrical orders and current-approximation orders of the elements are entirely independent from each other, and can be combined independently for the best overall performance of the method in different applications. The results obtained by the higher order technique are validated against the analytical solutions and the numerical results obtained by low-order moment-method techniques from literature. The examples show excellent accuracy, flexibility, and efficiency of the new technique at modeling of both current variation and curvature, and demonstrate advantages of large-domain models using curved quadrilaterals of high geometrical orders with basis functions of high current-approximation orders over commonly used small-domain models and low-order techniques. The reduction in the number of unknowns is by an order of magnitude when compared to low-order solutions.

Higher order hierarchical discretization scheme for surface integral equations for layered media

This paper presents an efficient technique for the analysis of electromagnetic scattering by arbitrarily shaped perfectly conducting objects in layered media. The technique is based on a higher order method of moments (MoM) solution of the electric field, magnetic field, or combined-field integral equation. This higher order MoM solution comprises higher order curved patches for the geometry modeling and higher order hierarchical basis functions for expansion of the electric surface current density. Due to the hierarchical property of the basis functions, the order of the expansion can be selected separately on each patch depending on the wavelength in the layer in which the patch is located and the size of the patch. In this way, a significant reduction of the number of unknowns is achieved and the same surface mesh can be reused in a wide frequency band. It is shown that even for fairly large problems, the higher order hierarchical MoM requires less memory than existing fast multipole method (FMM) or multilevel FMM implementations.