Piecewise Basis Research Articles

Analytic Preconditioners for Decoupled Potential Integral Equations and Wideband Analysis of Scattering From PEC Objects

Many integral equations used to analyze scattering, such as the standard combined field integral equation (CFIE), are not well-conditioned for a wide range of frequencies and multi-scale geometries. There has been significant effort to alleviate this problem. A more recent one is using a set of decoupled potential integral equations (DPIE). These equations have been shown to be robust at low frequencies and immune to topology breakdown. But they mimic the ill-conditioning behavior of CFIE at high frequencies. This paper addresses this deficiency through new Calderón-type identities derived from the Vector Potential Integral Equation (VPIE). We construct novel analytic preconditioners for the vector potential integral equation (VPIE) and scalar potential integral equation (SPIE) constrained to perfect electric conductors (PEC). These new formulations are wide-band well-conditioned and converge rapidly for multi-scale geometries. This is demonstrated though a number of examples that use analytic and piecewise basis sets.

A hybrid approach for piecewise fractional reaction–diffusion equations

In this paper, the Caputo and Atangana–Baleanu fractional derivatives are handled to introduce a type of piecewise fractional derivative. More precisely, a linear combination of the Caputo and Atangana–Baleanu fractional derivatives are considered in each sub-interval to define this fractional derivative. It is employed to generate another form of nonlinear reaction–diffusion equations. The orthonormal Legendre polynomials together with the orthonormal piecewise Legendre functions are used to make a hybrid algorithm for this new problem. In this way, an explicit formula for computing the piecewise fractional differentiation of the stated piecewise basis functions is obtained and applied in generating the method. The applicability and validity of the adopted procedure are examined through three examples.

PPP-RTK considering the ionosphere uncertainty with cross-validation

With the high-precision products of satellite orbit and clock, uncalibrated phase delay, and the atmosphere delay corrections, Precise Point Positioning (PPP) based on a Real-Time Kinematic (RTK) network is possible to rapidly achieve centimeter-level positioning accuracy. In the ionosphere-weighted PPP–RTK model, not only the a priori value of ionosphere but also its precision affect the convergence and accuracy of positioning. This study proposes a method to determine the precision of the interpolated slant ionospheric delay by cross-validation. The new method takes the high temporal and spatial variation into consideration. A distance-dependent function is built to represent the stochastic model of the slant ionospheric delay derived from each reference station, and an error model is built for each reference station on a five-minute piecewise basis. The user can interpolate ionospheric delay correction and the corresponding precision with an error function related to the distance and time of each reference station. With the European Reference Frame (EUREF) Permanent GNSS (Global Navigation Satellite Systems) network (EPN), and SONEL (Système d'Observation du Niveau des Eaux Littorales) GNSS stations covering most of Europe, the effectiveness of our wide-area ionosphere constraint method for PPP-RTK is validated, compared with the method with a fixed ionosphere precision threshold. It is shown that although the Root Mean Square (RMS) of the interpolated ionosphere error is within 5 cm in most of the areas, it exceeds 10 cm for some areas with sparse reference stations during some periods of time. The convergence time of the 90th percentile is 4.0 and 20.5 min for horizontal and vertical directions using Global Positioning System (GPS) kinematic solution, respectively, with the proposed method. This convergence is faster than those with the fixed ionosphere precision values of 1, 8, and 30 cm. The improvement with respect to the latter three solutions ranges from 10 to 60%. After integrating the Galileo navigation satellite system (Galileo), the convergence time of the 90th percentile for combined kinematic solutions is 2.0 and 9.0 min, with an improvement of 50.0% and 56.1% for horizontal and vertical directions, respectively, compared with the GPS-only solution. The average convergence time of GPS PPP-RTK for horizontal and vertical directions are 2.0 and 5.0 min, and those of GPS + Galileo PPP-RTK are 1.4 and 3.0 min, respectively.

Operating condition recognition based on temporal cumulative distribution function and AdaBoost-extreme learning machine in zinc flotation process

Bubble size distribution (BSD) is a crucial indicator of operating conditions in froth flotation. However, current BSD-based recognition methods have some limitations in the BSD estimation and don't achieve satisfactory results. Therefore, an alternative operating condition recognition method is proposed. Firstly, the temporal cumulative distribution function (TCDF) is designed to represent the BSD on the froth video. Then, the piecewise basis function is established to fit the TCDF. Secondly, the TCDF-Minimum Redundancy Maximum Relevancy (MRMR) feature is generated by MRMR to increase the sensitivity of TCDF features to the operating conditions. Lastly, the proposed TCDF-MRMR feature together with froth velocity and texture features are fed into an AdaBoost-extreme learning machine (ELM) model to recognize the operating conditions. Experiments in a zinc flotation process demonstrate that the TCDF-MRMR feature is more effective than other BSD-based features, and the AdaBoost-ELM shows better performance than traditional single recognition models.

Structural topology optimization using a level set method with finite difference updating scheme

In conventional level set methods for topology optimization, the evolution of level set function is solved by upwind scheme and needs to be reinitialized frequently throughout the optimization. The upwind scheme is complicated in numerical implementation and the reinitialization is time-consuming and slows down the optimization efficiency. This paper presents a level set method updated with finite difference scheme for structural topology optimization. The velocity field used to update the level set function is interpolated by piecewise basis function in order to incorporate the universal mathematical programming algorithm into the optimization. The diffusion is introduced into the Hamilton-Jacobi equation and the three-step splitting method is adopted to solve the equation. By applying the proposed method, the evolution of the level set can be updated by simple central difference scheme and the procedure of reinitialization is avoided without sacrificing the numerical stability. The classical mean compliance minimization problem with several numerical examples is used to test the validity of the proposed method.

Real-Time Algorithm for Versatile Displacement Sensors Based on Self-Mixing Interferometry

This paper presents a general displacement reconstruction algorithm for sensors based on self-mixing (SM) interferometry suitable to work at different optical feedback conditions. The approach relies on a robust phase observation through the analytic signals for the fringe detection stage, while the motion of the pointed target is retrieved by a self-adapting filter in a piecewise basis. The implementation for the real-time calculations demonstrates the feasibility of robust SM sensors for different usage conditions without the need of modifying the base device configuration.

Bayesian latent structure models with space-time dependent covariates

Spatial-temporal data requires flexible regression models which can model the dependence of responses on space- and time-dependent covariates. In this paper, we describe a semiparametric space-time model from a Bayesian perspective. Nonlinear time dependence of covariates and the interactions among the covariates are constructed by local linear and piecewise linear models, allowing for more flexible orientation and position of the covariate plane by using time-varying basis functions. Space-varying covariate linkage coefficients are also incorporated to allow for the variation of space structures across the geographical location. The formulation accommodates uncertainty in the number and locations of the piecewise basis functions to characterize the global effects, spatially structured and unstructured random effects in relation to covariates. The proposed approach relies on variable selection-type mixture priors for uncertainty in the number and locations of basis functions and in the space-varying linkage coefficients. A simulation example is presented to evaluate the performance of the proposed approach with the competing models. A real data example is used for illustration.

Plünnecke’s Theorem for asymptotic densities

Plünnecke proved that if B ⊆ N B\subseteq \mathbb {N} is a basis of order h > 1 h>1 , i.e., σ ( h B ) = 1 \sigma (hB)=1 , then σ ( A + B ) ⩾ σ ( A ) 1 − 1 h \sigma (A+B)\geqslant \sigma (A)^{1-\frac {1}{h}} , where A A is an arbitrary subset of N \mathbb {N} and σ \sigma represents Shnirel’man density. In this paper we explore whether σ \sigma can be replaced by other asymptotic densities. We show that Plünnecke’s inequality above is true if σ \sigma is replaced by lower asymptotic density d _ \underline {d} or by upper Banach density B D BD but not by upper asymptotic density d ¯ \overline {d} . The result about d _ \underline {d} has some interesting consequences such as the inequality d _ ( A + P ) ⩾ d _ ( A ) 2 / 3 \underline {d}(A+P)\geqslant \underline {d}(A)^{2/3} for any A ⊆ N A\subseteq \mathbb {N} , where P P is the set of all prime numbers, and the inequality d _ ( A + C ) ⩾ d _ ( A ) 3 / 4 \underline {d}(A+C)\geqslant \underline {d}(A)^{3/4} for any A ⊆ N A\subseteq \mathbb {N} , where C C is the set of all cubes of nonnegative integers. The result about B D BD generalizes Theorem 3 of a 2001 work of the author by reducing the requirement of B B being a piecewise basis to the requirement of B B being an upper Banach basis.

MAM—A Multipurpose Admittance Matrix for Antenna Design Via the Finite Element Method

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Further exploitation of matrix-valued transfer functions describing electromagnetic phenomena is proposed. In addition to traditional modal port inclusion in finite element analysis, new computational domain ports are allowed as field expansion in terms of piecewise basis functions is considered. An admittance-type matrix arises and, as a result, a multiport network manages all electromagnetic complexity. Circuit handling of this network efficiently outperforms full-wave analyses for different modifications of the actual structure, thus, making it possible to use this methodology for design purposes. A model order reduction process is also used for fast frequency sweeping and special emphasis is placed on its accuracy. Finally, numerical examples demonstrate the robustness, capability and versatility of the proposed technique. </para>

Method and apparatus for reconstructing and acoustic field

Acoustic field reconstruction is performed by expanding basis function which represent outgoing and incoming waves, without prior knowledge of the source geometry and dimensions. The reconstruction is based on acoustic pressures measured in the field only, and can be accomplished on a piece-wise basis. There are no restrictions on the locations of measurements. Moreover, the present invention can be used to reconstruct acoustic radiation from a plurality of sources or from a single source surrounded by multiple reflecting surfaces.

Adsorption of Volatile Organic Compounds (VOC) Mixtures onto Activated Carbon. Experimental Study and Simulation of Multicomponent Adsorption

In the present study, a simple mathematical model has been developed to simulate multicomponent fixed bed adsorption. The Linear Driving Force model is used to describe the intraparticle mass transfer. The competitive effect is accounted for through the multicomponent adsorption isotherm. This multicomponent equilibrium is described by the Ideal Adsorbed Solution Theory (IAST). To apply this theory, pure component adsorption isotherm data are represented by the Freundlich equation on a piecewise basis. For the prediction, operating conditions (flow rates, bed depths, input concentrations) and some characteristics of activated carbon must be known. It is also necessary to have pure component isotherm data. To demonstrate the applicability of the model, fixed bed experiments were carried out using a ternary mixture, containing methylethylketone, ethylene chloride, methylene chloride. A good agreement between calculations and experiments was obtained.

On the use of Fourier series/FFT's as global basis functions in the solution of boundary integral equations for EM scattering

Current methods for the solution of the surface integral equations of interest to practical electromagnetic scattering appear to have reached an impasse. Plagued by disputes concerning the suitability and performance of various competing choices for piecewise basis functions and facing horribly complicated piecewise quadratures for the construction of influence matrix elements, it has become quite commonplace for these methods to be matrix-build-time limited rather than matrix-solve-time limited. An extremely simple alternative to all this is described below and demonstrated, in an introductory sense, on some 2-D problems. In this procedure, the FFT method, the basis functions are all global Fourier series in a suitably chosen parameter, and one attempts to estimate the coefficients describing the unknown surface current directly from the integral equation. This format circumvents all of the shortcomings discussed above-there are no singular integrals to evaluate. Furthermore, since the FFT algorithm is used throughout, the computation is extremely efficient. The results shown below suggest that, so far, this approach is very effective.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Mach reflection flowfields associated with strong shocks

Abstract : The Mach reflection associated with the passage of a shock wave over a wedge is treated in the limit of an ideal gas and a strong shock. In this limit, flow properties are functions only of wedge angle theta and the ratio of specific heats gamma. Numerical results are presented for gamma = 9/7, 7/5, and 5/3. Wedge angles are noted at which the transition from regular to double-Mach, to complex-Mach, and to simple-Mach reflection occurs. Characteristic velocities in the recirculation region associated with Mach reflection are estimated. Local surface pressure maxima, at the upstream and downstream edge of the recirculation region, are also estimated. The scale of the recirculation region increases with decrease in gamma, in accord with experimental observations. The wedge solution is used on a piecewise basis to estimate height-of-burst (HOB) flow fields. Normalized results are presented for HOB triple-point trajectory and surface pressure variation with range. The present results provide a convenient characterization of Mach reflection flow fields associated with both wedge and HOB flows. Validation of the HOB solution, however, is needed.

Scattering of acoustic waves by rigid cylindrical objects with sharp corners

The T-matrix or Null Field approach is employed for studying scattering of acoustic waves by rigid cylindrical obstacles with sharp corners. A piecewise basis function is used to represent the surface field. The algebraic system of equations obtained from the resulting integral equations is solved for computing the surface and farfield amplitudes. Numerical results are presented for backscattering and bistatic scattering cross sections for square and intersecting circular cross sections and are compared with those obtained for smooth cross section, such as elliptical cross sections. Wavelengths down to one fifth of the maximum dimension of the section are considered. This frequency range is wide enough to definitively conclude that the dominate effect for objects with sharp corners is the interference of waves specularly reflected at such points also known as flash points. For smooth shapes such as the ellipse, the dominant effect is that of creeping waves. Schlieren photographs of a pulse propagating into the shadow of a cylinder with square cross sections are also given.

Dynamic stress concentrations of cylindrical cavities with sharp and smooth boundaries: I. SH waves

This paper demonstrates a method for the calculation of the dynamic stress concentration factor when a time-harmonic elastic wave is incident upon a cylindrical cavity of arbitrary cross-section. We discuss in particular a procedure which enables cross-sectional shapes with corners to be examined, although we also consider how shapes with smooth surfaces may be handled. Our method is a computational one, and is based upon Waterman's T-matrix formulation. It is shown that for scatterers with corners a piecewise basis must be chosen to represent the surface displacement. The algebraic system of equations obtained from the integral equations and appropriate constraints on the basis representation are then sufficient to determine the stress concentration uniquely. Results are given for cylindrical cavities with square, elliptic and intersecting circular cross sections for SH wave incidence.

Surface fields on sound hard and sound soft obstacles with sharp and smooth boundaries

The computation of the surface fields generated on a sound hard or sound soft obstacle when irradiated by a plane harmonic wave is a challenging problem when the obstacle is of arbitrary shape and has corners. Not only is this computation the first step in determining scattered field, but it has other important practical applications such as the dynamic stress concentrations. The integral equation for the field on the surface of the obstacle is solved numerically using null field (T-matrix approach). It will be shown that for scatterers with corners, a piecewise basis must be chosen to represent the surface fields. Results will be presented for cylindrical scatterers with square, elliptical, and intersecting circular cross sections.

Piecewise Fourier Transformation for Picture Bandwidth Compression

Recent advances in digital computer and optical technology have made image spectra determination practical. Pratt and Andrews [1] studied bandwidth compression using the Fourier transform of complete pictures. By treating pictures adaptively on a piecewise basis, picture detail is better represented. Also, subjective preferences of human vision can be used, which result in further improvements in picture quality. The original picture is sampled and then divided into small subsections. Each subsection is expanded in a two-dimensional Fourier series. The L -Fourier coefficients of largest absolute value are determined for each subsection, where L is proportional to the standard deviation of the picture samples in the subsection. The frequencies and complex amplitudes of those L -Fourier coefficients are transmitted. The number of quantization levels used for the Fourier coefficients in each subsection is made dependent on the standard deviation of the picture samples in the subsection, and the size of the quantum steps is made dependent on the magnitude of the largest Fourier coefficient of the subsection, aside from the average value. The frequencies of the coefficients correspond to positions in a twodimensional spatial frequency plane. These positions, or twodimensional frequencies, are transmitted by run-length coding. The process is adaptive in the sense that, its parameters vary from subsection to subsection of the picture in an effort to match the properties of the individual subsections. Subsection size and other important system constants are chosen with knowledge of the properties of human vision. We are able to obtain high-quality reconstructed pictures, using on the average 1.25 bits per picture point.

Automated Optimum Design from Discrete Components

Optimization in structural design using discrete variable can exploit the discrete characteristics of the design problem and avoid artificial and often self-defeating discretation from problem solutions in continuous variables. A method for solving member-sizing problems using wide flange tables, a current design code, and linear elastic behavior due to arbitrary loading conditions is shown to be within reasonable economic bounds. The method consists of a combinatorial algorithm classified as a branch-and-bound type. Starting with a lower-bound infeasible solution the algorithm detects a feasible solution for an upper bound and explores the region for a local optimum, bouncing along the constraints if the value of the objective function can be improved. Reanalysis is performed at each step with an exact method which performs only part of normal analysis steps. Included as part of the STRUDL information system, the programmed algorithm allows an engineer to assist in improving the optimization process and perform optimization on a structure on a piece-wise basis.