Fast and exact computation of moments using discrete Green's theorem

Green's theorem evaluates a double integral over the region of an object by a simple integration along the boundary of the object. It has been used in moment computation since the shape of a binary object is totally determined by its boundary. By using a discrete analogue of Green's theorem, we present a new algorithm for fast computation of geometric moments. The algorithm is faster than previous methods, and gives exact results. The importance of exact computation is discussed by examining the invariance of Hu's moments. A fast method for computing moments of regions in grey level image, using discrete Green's theorem, is also presented

Scattering and radiation analysis of three-dimensional cavity arrays via a hybrid finite element method

A hybrid numerical technique is presented for a characterization of the scattering and radiation properties of three-dimensional cavity arrays recessed in a ground plane. The technique combines the finite element and boundary integral methods and invokes Floquet's representation to formulate a system of equations for the fields at the apertures and those inside the cavities. The system is solved via the conjugate gradient method in conjunction with the Fast Fourier Transform (FFT) thus achieving an O(N) storage requirement. By virtue of the finite element method, the proposed technique is applicable to periodic arrays comprised of cavities having arbitrary shape and filled with inhomogeneous dielectrics. Several numerical results are presented, along with new measured data, which demonstrate the validity, efficiency, and capability of the technique

Finite element solution of the Helmholtz equation with high wave number Part I: The h-version of the FEM

AbstractThe paper addresses the properties of finite element solutions for the Helmholtz equation. The h-version of the finite element method with piecewise linear approximation is applied to a one-dimensional model problem. New results are shown on stability and error estimation of the discrete model. In all propositions, assumptions are made on the magnitude of hk only, where k is the wavelength and h is the stepwidth of the FE-mesh. Previous analytical results had been shown with the assumption that k2h is small. For medium and high wavenumber, these results do not cover the meshsizes that are applied in practical applications. The main estimate reveals that the error in H1-norm of discrete solutions for the Helmholtz equation is polluted when k2h is not small. The error is then not quasioptimal; i.e., the relation of the FE-error to the error of best approximation generally depends on the wavenumber k. It is noted that the pollution term in the relative error is of the same order as the phase lead of the numerical solution. In the result of this analysis, thorough and rigorous understanding of error behavior throughout the range of convergence is gained. Numerical results are presented that show sharpness of the error estimates and highlight some phenomena of the discrete solution behavior. The h-p-version of the FEM is studied in Part II

An efficient asymptotic extraction approach for the green's functions of conformal antennas in multilayered cylindrical media

Asymptotic expressions are derived for the dyadic Green's functions of antennas radiating in the presence of a multilayered cylinder, where analytic representation of the asymptotic expansion coefficients eliminates the computational cost of numerical evaluation. As a result, the asymptotic extraction technique has been applied only once for a large summation order $n$. In addition, the Hankel function singularity encountered for source and evaluation points at the same radius has been eliminated using analytical integration

Analytical integration of stress field and tangent material moduli over concrete cross-sections

This paper presents a novel stress field and tangent material moduli integration procedure over a cross-section of a biaxially loaded concrete beam. The procedure assumes a sufficiently simple analytical form of the constitutive law of concrete, the polygonal shape of the boundary of the simply- or multi-connected cross-section and the monotonically increasing loading. The area integrals are transformed into the boundary integrals and then integrated analytically. The computational efficiency of the procedure is analyzed by comparing it with respect to the number of floating-point operations needed in various numerical integration-based methods. It is found that the procedure is not only exact, but also computationally effective. (c) 2005 Elsevier Ltd. All rights reserved

Preconditioning the Advection-Diffusion Equation: the Green's Function Approach

We look at the relationship between efficient preconditioners (i.e., good approximations to the discrete inverse operator) and the generalized inverse for the (continuous) advection-diffusion operator -- the Green's function. We find that the continuous Green's function exhibits two important properties -- directionality and rapid downwind decay -- which are preserved by the discrete (grid) Green's functions, if and only if the discretization used produces non-oscillatory solutions. In particular, the downwind decay ensures the locality of the grid Green's functions. Hence, a finite element formulation which produces a good solution will typically use a coefficient matrix with almost lower triangular structure under a "with-the-flow" numbering of the variables. It follows that the block Gauss-Seidel matrix is a first candidate for a preconditioner to use with an iterative solver of Krylov subspace type

Numerical recovery of material parameters in Euler-Bernoulli beam models

A fully Sinc-Galerkin method for recovering the spatially varying stiffness parameter in fourth-order time-dependence problems with fixed and cantilever boundary conditions is presented. The forward problems are discretized with a sinc basis in both the spatial and temporal domains. This yields an approximation solution which converges exponentially and is valid on the infinite time interval. When the forward methods are applied to parameter recovery problems, the resulting inverse problems are ill-posed. Tikhonov regularization is applied and the resulting minimization problems are solved via a quasi-Newton/trust region algorithm. The L-curve method is used to determine an appropriate value of the regularization parameter. Numerical results which highlight the method are given for problems with both fixed and cantilever boundary conditions