Orthogonal Basis Functions Research Articles

A novel matrix technique for multi-order pantograph differential equations of fractional order

The main purpose of this article is to investigate a novel set of (orthogonal) basis functions for treating a class of multi-order fractional pantograph differential equations (MOFPDEs) computationally. These polynomials, denoted bySn(x)and calledspecial polynomials, were first discovered in a study of a certain family of isotropic turbulence fields. They are expressible in terms of the generalized Laguerre polynomials and are related to the Bessel and Srivastava–Singhal polynomials. Unlike the Laguerre polynomials, all coefficients of the special polynomials are positive. We further introduce the fractional order of the special polynomials and use them along with some suitable collocation points in a special matrix technique to treat fractional-order MOFPDEs. Moreover, the convergence analysis of these polynomials is established. Through five example applications, the utility and efficiency of the present matrix approach are demonstrated and comparisons with some existing numerical schemes have been performed in this class.

New Orthogonal Transforms for Signal and Image Processing

In the paper, orthogonal transforms based on proposed symmetric, orthogonal matrices are created. These transforms can be considered as generalized Walsh–Hadamard Transforms. The simplicity of calculating the forward and inverse transforms is one of the important features of the presented approach. The conditions for creating symmetric, orthogonal matrices are defined. It is shown that for the selection of the elements of an orthogonal matrix that meets the given conditions, it is necessary to select only a limited number of elements. The general form of the orthogonal, symmetric matrix having an exponential form is also presented. Orthogonal basis functions based on the created matrices can be used for orthogonal expansion leading to signal approximation. An exponential form of orthogonal, sparse matrices with variable parameters is also created. Various versions of orthogonal transforms related to the created full and sparse matrices are proposed. Fast computation of the presented transforms in comparison to fast algorithms of selected orthogonal transforms is discussed. Possible applications for signal approximation and examples of image spectrum in the considered transform domains are presented.

Efficient space-time Legendre rational spectral method for parabolic problems in unbounded domains

In this paper, we first introduce three series of Legendre rational basis functions on the half/whole line by the diagonalization technique and the matrix decomposition technique. The new basis functions are mutually orthogonal in both L2- and H1-inner products, and lead to diagonal systems for second order problems with constant coefficients. Then we construct efficient space-time spectral methods for parabolic problems in unbounded domains using Legendre rational approximation in space and Legendre-Gauss collocation method in time, which can be implemented in a synchronous parallel fashion. Numerical results demonstrate that the use of simultaneously orthogonal basis functions in space may greatly simplify the implementation of the space-time spectral methods. Using these suggested methods, higher accuracy can also be obtained.

Comparison of spherical cap and rectangular harmonic analysis of airborne vector gravity data for high-resolution (1.5 km) local geopotential field models over Tanzania

SUMMARYThis study examines local geopotential field modelling over a mountainous region in Tanzania using vector airborne gravity data. We use the adjusted spherical cap and rectangular harmonic analyses. Both methods are based on expansion of gravitational potential into a series of orthogonal harmonic basis functions of local support in such a way that the expansion coefficients are determined by gravity observations. All three components of gravity vector are simultaneously inverted to derive the geopotential coefficients. In order to evaluate the accuracy of the local models, independent checkpoints are selected within the study region and around its boundary and the computed gravity vectors are compared with the independent gravity observations. The results show an excellent agreement with root mean square error (RMSE) of < 1.6 mGal over the study area. On the contrary, the RMSEs of global geopotential models against the checkpoints data are 7 mGal for the models up to the maximum degree of 2190 (a resolution of ∼9.1 km) and 5 mGal to 5399 (∼3.7 km). Our local models are significantly more accurate than the state-of-the-art global models and fully exploit the airborne vector data with the measurement error of ∼1 mGal. We also present the regional models constrained only by radial (vertical) or by lateral (horizontal) gravity observations. Those models are considerably less accurate than the one from 3-D gravity data inversion. Lastly, the regional models are validated against topography data. It is found that the gravity–topography correlation is 0.8–0.9 at 100 km, 0.5 at 20 km and higher than the correlations of the global models at all frequencies. The gravity–topography admittances estimated from our regional models indicate ∼130 mGal km−1 and imply the effective density of 2500 kg m−3 for topographical mass.

An Efficient Polynomial Chaos Expansion Method for Uncertainty Quantification in Dynamic Systems

Uncertainty is a common feature in first-principles models that are widely used in various engineering problems. Uncertainty quantification (UQ) has become an essential procedure to improve the accuracy and reliability of model predictions. Polynomial chaos expansion (PCE) has been used as an efficient approach for UQ by approximating uncertainty with orthogonal polynomial basis functions of standard distributions (e.g., normal) chosen from the Askey scheme. However, uncertainty in practice may not be represented well by standard distributions. In this case, the convergence rate and accuracy of the PCE-based UQ cannot be guaranteed. Further, when models involve non-polynomial forms, the PCE-based UQ can be computationally impractical in the presence of many parametric uncertainties. To address these issues, the Gram–Schmidt (GS) orthogonalization and generalized dimension reduction method (gDRM) are integrated with the PCE in this work to deal with many parametric uncertainties that follow arbitrary distributions. The performance of the proposed method is demonstrated with three benchmark cases including two chemical engineering problems in terms of UQ accuracy and computational efficiency by comparison with available algorithms (e.g., non-intrusive PCE).

Numerical study of the boundary layer problem over a flat plate by orthogonal cubic spline basis functions

Numerical study of the boundary layer problem over a flat plate by orthogonal cubic spline basis functions

METHOD OF REFERENCE SIGNALS CREATING IN NON-DESTRUCTIVE TESTING BASED ON LOW-SPEED IMPACT

The article describes the approach to the formation of a simulation model of information signals, which are typical for objects with different types of defects. The dispersive analysis of the signal spectrum components in the bases of the discrete Hartley transform and the discrete cosine transform is carried out. The analysis of the form of the reconstructed information signal is carried out depending on the number of coefficients of the spectral alignment in Hartley bases and cosine functions. The basis of orthogonal functions of a discrete argument is obtained, which can be used for the spectral transformation of information signals of a flaw detector. A method of simulation of information signals has been developed and experimentally investigated, which allows taking into account the deterministic and random components of the characteristics of real information signals. References 24, figures 13, tables 3.

Cutoff Wavenumbers of Multilayered Gyrotropic Circular Waveguides

A coupled-field volume integral equation (CFVIE) method is developed for the calculation of the normalized cutoff wavenumbers of circular metallic walled waveguides having concentric continuously varying highly inhomogeneous gyrotropic (i.e., gyroelectric and gyromagnetic) infill. The normalized cutoff wavenumbers are obtained as the roots of a determinantal equation formed by solving the CFVIEs using the cylindrical Dini series expansion (CDSE) method where the unknown fields inside the waveguide are expanded by entire domain orthogonal Dini-type vectorial basis functions. To account for the electric boundary condition (BC) on waveguide's circular perfect electric conducting (PEC) surface, two modified 2-D tensorial Green's functions (GFs), expanded in cylindrical vector wave functions (CVWFs), are employed in the kernels of the CFVIEs. These modified 2-D tensorial GFs are constructed by enforcing, on their dyadic form, the satisfaction of the electric BC. The CDSE, along with the modified 2-D tensorial GFs, allow for the analytical integration of the volumetric-type integrals and the reduction of the CFVIEs to a set of algebraic equations. We exhaustively demonstrate the validity of the CFVIE-CDSE by a series of comparisons on the normalized cutoff wavenumbers: we first construct the solutions for obtaining the normalized cutoff wavenumbers in homogeneous gyrotropic waveguides by the separation of variables method (SVM), and second, we employ HFSS commercial software for two-layered isotropic and three-layered gyroelectric loaded waveguides. We characterize the type of modes, i.e., TE/TM or hybrid HE/EH, for each configuration presented, and discuss the efficiency of the CFVIE-CDSE method.

Noise Suppression for Vector Magnetic Anomaly Detection by Noise Spatial Characteristics Investigation

This letter reports a noise suppression method for vector magnetic anomaly detection (MAD) based on noise spatial characteristics analysis. The environment noise indicates space anisotropy that is essential for noise depression. To find a projection direction e that allows the projected noise, denoted as Ne = e· N, to have minimum fluctuations, we compute the standard deviation σ of N_e. The projected signal is then processed by a vector orthogonal basis functions (OBFs) detector resulting in a higher signal to noise ratio (SNR) by 3 dB than the traditional OBFs filter. After noise space transformation (NST) processing, the compelling low-frequency noise is suppressed and the weak target signal can be obtained.

Sobolev orthogonal Legendre rational spectral methods for exterior problems

The purpose of this paper is to develop the diagonalized Legendre rational spectral method for exterior problems. We first consider the exterior problems of two-dimensional elliptic and parabolic equations in polar coordinates, construct the Sobolev orthogonal Legendre rational basis functions, and then propose the diagonalized Legendre rational spectral methods. Then we consider the exterior problems of three-dimensional elliptic and parabolic equations in spherical coordinates, construct the Sobolev orthogonal Legendre rational basis functions, and then propose the diagonalized Legendre rational spectral methods. The main advantages of the suggested approaches are that the discrete systems are diagonal and the numerical solutions can be represented as truncated Fourier series. The numerical results show their effectiveness and accuracy.

Variational nodal method for three-dimensional multigroup neutron diffusion equation based on arbitrary triangular prism

With the development of nuclear technology, new concepts of reactor core have been continuously proposed, and new types of geometric assembly and reactor core configuration would be adopted to achieve better characteristics. Traditional nodal method based on rectangle or hexagonal nodal is no longer applicable for complex geometry modeling. Given that triangles have the advantage of constructing arbitrarily complex geometry, in this study, variational nodal method of three-dimensional multigroup neutron diffusion theory based on arbitrary triangular prism was developed. ANSYS code is used to divide calculation region into arbitrary triangle mesh, which is then transformed into regular triangle mesh by coordinate transformation, and a set of orthogonal polynomial basis function is built. Variables such as nodal neutron scalar flux and neutron current on nodal surface are expanded based on those polynomial basis functions. The response relationship between neutron partial neutron current and neutron flux can be obtained by variational principle, and traditional power method is employed to solve those response matrix system. Based on above, TriVNM code was developed, and 4 types of typical benchmarks including rectangle, hexagonal assembly reactor and reactor with curved boundary were employed to verify TriVNM. Both power distributions and keff calculated by TriVNM code agree well with reference results, and the maximum relative error of all the benchmarks’ power doesn’t exceed 1% and error of keff doesn’t exceed 50 pcm. TriVNM code achieves the same or higher level of calculation accuracy compared with other diffusion codes. Mesh sensitivity analysis shows that for large mesh size TriVNM code still has good accuracy.

Magnetic field penetration through a circular aperture in a perfectly conducting plate excited by a coaxial loop

The transmission of an electromagnetic field produced by a current loop of finite radius through a coaxial circular aperture in a perfectly conducting plate is evaluated through a rapidly convergent formulation in an exact form. By applying the equivalence principle, the problem is first formulated in the Hankel transform domain, obtaining a set of dual integral equations in which the equivalent surface magnetic current density defined on the aperture is not known. The set of dual integral equations is regularised in a second-kind Fredholm integral equation by applying the Abel integral-transform technique. The solution is achieved by expanding the unknown in a set of orthogonal basis functions that correctly reproduce the behaviour of the equivalent magnetic current at the edge of the aperture. Finally, under particular assumptions, a low-frequency solution is extracted in a closed form. Numerical results are reported to validate the accuracy and efficiency of the proposed formulations.

A matrix-free isogeometric Galerkin method for Karhunen–Loève approximation of random fields using tensor product splines, tensor contraction and interpolation based quadrature

The Karhunen–Loève series expansion (KLE) decomposes a stochastic process into an infinite series of pairwise uncorrelated random variables and pairwise L2-orthogonal functions. For any given truncation order of the infinite series the basis is optimal in the sense that the total mean squared error is minimized. The orthogonal basis functions are determined as the solution of an eigenvalue problem corresponding to the homogeneous Fredholm integral equation of the second kind, which is computationally challenging for several reasons. Firstly, a Galerkin discretization requires numerical integration over a 2d dimensional domain, where d, in this work, denotes the spatial dimension. Secondly, the main system matrix of the discretized weak-form is dense. Consequently, the computational complexity of classical finite element formation and assembly procedures as well as the memory requirements of direct solution techniques become quickly computationally intractable with increasing polynomial degree, number of elements and degrees of freedom. The objective of this work is to significantly reduce several of the computational bottlenecks associated with numerical solution of the KLE. We present a matrix-free solution strategy, which is embarrassingly parallel and scales favorably with problem size and polynomial degree. Our approach is based on (1) an interpolation based quadrature that minimizes the required number of quadrature points; (2) an inexpensive reformulation of the generalized eigenvalue problem into a standard eigenvalue problem; and (3) a matrix-free and parallel matrix–vector product for iterative eigenvalue solvers. Two higher-order three-dimensional C0-conforming multipatch benchmarks illustrate exceptional computational performance combined with high accuracy and robustness.

Unstable basis function for joint radar-communication systems

• A combined radar and communications system using basis functions derived from an unstable second-order system is described. • Using unstable systems with different frequencies allows for the creation of a set of orthogonal basis functions. • Both communications and radar signals may be decoded with linear analog circuits, leading to simple inexpensive receivers. • The radar properties are not sensitive to the encoded communications method. • Because of simplicity and low cost, the system could be used in applications where it was necessary to piggyback a radar signal on a communications signal. In this paper, we develop a joint radar-communication waveform based on the unstable basis function. Basis functions of a chaotic waveform are uncertain, vary continuously and hence the use of a traditional matched filter is not practical. Corron et al. [1,2] demonstrated a chaotic circuit for which an analytic matched filter could be constructed, solving the signal detection problem. The Corron circuit produced a narrow band signal, which is not optimal for spread spectrum communications and radar applications. In this paper, we adapt Corron’s concept of a chaotic signal composed of a linear combination of basis functions and show how the basis functions are used to design joint radar-communications systems that still have simple analytic matched filters. The form of the matched filter makes it useful for both the radar and communications systems where it is necessary to minimize weight or power consumption.

Discontinuous finite element method for efficient three-dimensional elastic wave simulation

AbstractThe existing discontinuous Galerkin (DG) finite element method (FEM) for the numerical simulation of elastic wave propagation is primarily implemented in two dimensions. Here, a discontinuous FEM (DFEM) for efficient three-dimensional (3D) elastic wave simulation is presented. First, the velocity–stress equations of 3D elastic waves in isotropic media are transformed into first-order coefficient-changed partial differential equations. A DG discretisation method for wave field values on a unit boundary is then defined using the local Lax–Friedrichs flux format. The equations are first transformed into equivalent integral equations, and subsequently into a spatial semi-discrete ordinary differential equation system using a hierarchical orthogonal basis function. The DFEM is extended to an arbitrary high-order accuracy in the time domain using the exponential integrator technique and the explicit optimal strong-stability-preserving Runge–Kutta method. Finally, an efficient method for selecting the calculation area of the geometry of the current shot record is realised. For the computation, a multi-node parallelism with improved resource utilisation and parallelisation efficiency is implemented. The numerical results show that the proposed method can improve both the accuracy of the simulation and the efficiency of the calculation compared with existing methods.

Magnetic anomaly detection based on fast convergence wavelet artificial neural network in the aeromagnetic field

The orthogonal basis functions (OBFs) detector is a detection method widely used in the aerial magnetic measurement. However, OBFs detector works ineffectively under non-Gaussian noise and colored noise. This paper proposes an OBFs detector based on fast convergence wavelet artificial neural network (FC-W-ANN), which can detect abnormal magnetic signals under low SNR. First, the magnetic anomaly signal is modelled. Then, the learning rate is corrected by the iterative error convergence rate under the stability of the network. Finally, the improved network is combined with the OBFs detector to detect magnetic abnormal signals. From the simulation and experimental results, the reconstructed signal of the new method has a higher SNR (SNR = 10.01) compared with OBFs (SNR = -0.21) and OBFs based on the wavelet artificial neural network (W-ANN; SNR = 9.59). Furthermore, the statistical methods are used to analyze FC-W-ANN and W-ANN, showing that FC-W-ANN has higher training accuracy and better stability.

Analytical evaluation of transformation matrix between functional spaces and its application in obtaining exact solutions of thermo-mechanical stresses in disks

Thermo-mechanical stresses developed in a disk, inner boundary of which is subject to the nonstationary thermo-mechanical loadings due to variation in temperature and pressure inside the disk is considered in the presented paper. Problems of theoretical analysis are divided and described as follows: The first problem implies analytical solution of the heat conduction equation with nonhomogeneous boundary conditions of Robin type, and the second problem deals with obtaining analytical solution of dynamic equation of displacement for nonisothermal elastic body (Navier equation) with traction boundary conditions in the plane strain case of uncoupled thermo-elasticity. Finite Hankel transform is used to solve the problems. Two sets of orthogonal basis functions necessary for integral transform are obtained. The first one corresponds to the boundary conditions written for the heat conduction equation, and the second fits the boundary conditions for Navier equation. Integrals defining elements of transformation matrix between the two sets of basis functions are evaluated analytically. In general, convenient method of obtaining analytical solution of linear hyperbolic equation with nonhomogeneous term, which itself is solution of nonhomogeneous parabolic equation, is elaborated within the presented work. Using finite difference method, discrete counterparts of the described problems are constructed and, via the built-in ordinary differential equation (ODE) solvers in MATLAB, numerical solutions are obtained, which exhibit excellent accordance with analytical ones.

Color Discrimination Experiments using Metameric ipRGC Stimuli

In order to investigate the effect of ipRGC on color discrimination, it is necessary to reproduce two metameric light stimuli (we call these stimuli as metameric ipRGC stimuli) that have the same amount of cones and rods but different stimulus amount of ipRGC. However, it is difficult to independently control the amount of only ipRGC stimulation because the spectral sensitivity functions of the cones and rod overlap that of ipRGC in the wavelength band. So far, researchers have not addressed a comprehensive analysis of metameric ipRGC stimuli and color perception experiments for those stimuli. In this study, first, we proposed the calculation method of metameric ipRGC stimulus based on the orthogonal basis functions of human photoreceptors. Then, we clarified the controllable range of metameric ipRGC stimulus in the color gamut. Second, we conducted subjective evaluation experiments for investigating the discriminative colors due to metameric ipRGC stimuli. We showed the effects of ipRGC on color discrimination.

System Identification of Lightly Damped Systems Using Recursive Kautz Functions-Case Study With Coriolis Mass Flow Meter

This manuscript deals with the system identification of lightly (under) damped processes to find a suitable model structure. To identify such processes, the Kautz model, which is a two-parameter representation of orthogonal basis functions (OBF) in state space format, is used. The process parameters are obtained through the subspace method by sequencing the states obtained from the Kautz model. The obtained states are transformed into convex optimization in which the eigen values poles) to be estimated are confined to lie within user-defined regions in the convex plane. To derive the optimal states, an efficient control scheme based on the state observer design is implemented. The viability of the proposed work has been verified on real-time Coriolis Mass Flow Meter (CMFM) under no flow condition and the corresponding responses are plotted using MATLAB.

Adaptive Approximation-Based Feedback Linearization Control for a Nonlinear Smart Thin Plate

This paper proposes feedback linearization control (FBLC) based on function approximation technique (FAT) to regulate the vibrational motion of a smart thin plate considering the effect of axial stretching. The FBLC includes designing a nonlinear control law for the stabilization of the target dynamic system while the closedloop dynamics are linear with ensured stability. The objective of the FAT is to estimate the cubic nonlinear restoring force vector using the linear parameterization of weighting and orthogonal basis function matrices. Orthogonal Chebyshev polynomials are used as strong approximators for adaptive schemes. The proposed control architecture is applied to a thin plate with a large deflection that stimulates the axial loading thus, the plate behaving nonlinearly. The governing partial differential equation for the piezo-plate system is transformed into definite ordinary differential equations (ODEs) using the Galerkin approach; hence, multi-input multi-output ODEs are obtained. Simulation experiments are performed to verify the validity of the proposed control structure.