Orthogonal Basis Functions Research Articles

Reduced model of MIMO systems using Meixner-like functions

This article investigates the use of orthogonal basis functions, particularly Meixner-like functions, to streamline and optimize the parameter complexity in multi-input multi-output (MIMO) system modeling. We introduce a new optimal model for MIMO systems, developed by decomposing the auto-regressive MIMO model with external inputs (ARX) using orthonormal Meixner-like bases. This model, referred to as the MIMO ARX Meixner-like model, enables filtering of the system’s inputs and outputs with Meixner-like functions. Parameter reduction is achieved through the optimization of Meixner-like poles using an iterative technique based on the Newton-Raphson method. The proposed model’s efficacy is demonstrated through simulations on two experimental systems: a two-degrees-of-freedom (DOF) helicopter and a two-tank process. Results reveal that the model effectively reduces approximation errors with fewer parameters. Comparative analysis with the traditional MIMO ARX model confirms the superior performance of the MIMO ARX Meixner-like model.

Robust 3D multi‐material hydrodynamics using discontinuous Galerkin methods

AbstractA high‐order discontinuous Galerkin (DG) method is presented for nonequilibrium multi‐material () flow with sharp interfaces. Material interfaces are reconstructed using the algebraic THINC approach, resulting in a sharp interface resolution. The system assumes stiff velocity relaxation and pressure nonequilibrium. The presented DG method uses Dubiner's orthogonal basis functions on tetrahedral elements. This results in a unique combination of sharp multimaterial interfaces and high‐order accurate solutions in smooth single‐material regions. A novel shock indicator based on the interface conservation condition is introduced to mark regions with discontinuities. Slope limiting techniques are applied only in these regions so that nonphysical oscillations are eliminated while maintaining high‐order accuracy in smooth regions. A local projection is applied on the limited solution to ensure discrete closure law preservation. The effectiveness of this novel limiting strategy is demonstrated for complex three‐dimensional multi‐material problems, where robustness of the method is critical. The presented numerical problems demonstrate that more accurate and efficient multi‐material solutions can be obtained by the DG method, as compared to second‐order finite volume methods.

A cylindrical shell model with distributed springs for a rotating tire

In a companion paper (Vol. 248, Article 108227), we demonstrated the dependence of tire parameters on the modal characteristics of a stationary tire. For a more realistic scenario, when tire rotation is considered, the modal characteristics are veered due to centrifugal, Coriolis, and gyroscopic effects. In this paper, we study the forced vibration response of a rotating tire and also consider the effect of static load on modal characteristics. The tire tread is modeled as a cylindrical shell and the sidewall is modeled using distributed springs in axial, circumferential, and radial directions. The tire inflation pressure is accounted for, which generates pre-stresses in the tire belt in circumferential and axial directions. The material model incorporates an orthotropic composite structure of the tire with multiple stacked layers of steel belt reinforcing. The governing equations of motion of the rotating shell are derived and further reduced to the corresponding eigenvalue problem. Galerkin’s projection with orthogonal basis functions is used to obtain the natural frequencies and associated mode shapes. For the forced vibration, the tire is excited by a harmonic point load at a fixed location, and the response is calculated by superimposing the basis functions of the free vibration problem. The analytical model predictions are compared against the computation finite element (FE) model developed in ABAQUS® and experimental modal testing results. To understand the effects of rotation on wave propagation, dispersion relations were obtained, and the wave number spectrum of the rotating tire was compared with that of the stationary one. Finally, to understand the dependence of natural frequencies on tire rotation and normal load, Campbell diagrams and frequency response functions (FRFs) were obtained from the proposed model. The model shows a fairly good agreement with experimental results and FE simulations.

Airborne magnetic anomaly detection based on Bi-stable stochastic resonance system

In order to address the problem with low signal-to-noise ratio (SNR) detection for airborne magnetic anomaly detection (MAD), an adaptive detection method based on bi-stable stochastic resonance (SR) system is proposed. The method adopts a series structure, S-SR, effectively improving detection speed. Allowing for the difficulty in parameter selection for the SR system, this method can achieve the adaptive selection of system parameters. On this basis, the effect of different sampling frequencies in the Runge–Kutta method on the numerical model of the SR system is investigated. Finally, the simulation and experiment analysis of MAD methods are performed to draw a comparison between the S-SR, orthogonal basis functions (OBF) decomposition, and high-order crossing (HOC) methods. According to the simulation and experiment results, the S-SR method outperforms the OBF and HOC methods in the outcome and range of detection under the same conditions.

A reduced order model formulation for left atrium flow: an atrial fibrillation case

A data-driven reduced order model (ROM) based on a proper orthogonal decomposition-radial basis function (POD-RBF) approach is adopted in this paper for the analysis of blood flow dynamics in a patient-specific case of atrial fibrillation (AF). The full order model (FOM) is represented by incompressible Navier–Stokes equations, discretized with a finite volume (FV) approach. Both the Newtonian and the Casson’s constitutive laws are employed. The aim is to build a computational tool able to efficiently and accurately reconstruct the patterns of relevant hemodynamics indices related to the stasis of the blood in a physical parametrization framework including the cardiac output in the Newtonian case and also the plasma viscosity and the hematocrit in the non-Newtonian one. Many FOM-ROM comparisons are shown to analyze the performance of our approach as regards errors and computational speed-up.

Gyro-kinetic simulations of trapped electron collision effects on low-frequency drift-wave instabilities in tokamak plasmas

Low-frequency drift-wave instabilities play a crucial role in the radial transport of present-day tokamaks, and trapped electron collisions can significantly influence these instabilities. In this paper, the effects of trapped electron collisions on these instabilities are investigated based on linear gyro-kinetic simulations. The basic numerical techniques including dispersion relation integral method and orthogonal basis function expansion are presented in detail with necessary benchmark work. The results demonstrate that in medium gradients, the increase of trapped electron proportions promotes the growth rate and radial heat transport largely for quasi-linear trapped electron modes (TEMs) and ion temperature gradient (ITG) modes. Moreover, trapped electron collisions have strong stabilizing effects, especially for TEMs driven by electron temperature gradients. Two distinctive branches, namely Mode #1 and #2, are investigated in steep gradients. Both behave in a varied instability nature during different ranges of the normalized wave vector k^θ . Mode #1 mainly induces radial heat transport during k^θ<0.5 and is significantly suppressed by the collisions. Mode #2 mainly induces radial heat transport during 0.4<k^θ<0.8 , and is largely enhanced by the collisions. When the collisionality is large enough, Mode #2 has stronger transport capacity than the other. Mode #2 at the medium wave vector, known as dissipative TEM, may provide the mechanism of the edge coherent mode observed in EAST H-mode plasmas, wherein collisionality plays an important role in the mode excitation.

Proper orthogonal descriptors for multi-element chemical systems

We introduce the proper orthogonal descriptors for efficient and accurate interatomic potentials of multi-element chemical systems. The potential energy surface of a multi-element system is represented as a many-body expansion of parametrized potentials which are functions of atom positions, atom types, and parameters. The proper orthogonal decomposition is employed to decompose the parametrized potentials as a linear combination of orthogonal basis functions. The orthogonal basis functions are used to construct proper orthogonal descriptors based on the elements of atoms, thus leading to multi-element descriptors. We compose the multi-element proper orthogonal descriptors to develop linear and quadratic interatomic potentials. We devise an algorithm to efficiently compute the total energy and forces of the interatomic potentials constructed from the proper orthogonal descriptors. The potentials are demonstrated for indium phosphide and titanium dioxide in comparison with the spectral neighbor analysis potential (SNAP) and atomic cluster expansion (ACE) potentials.

A 3-D minimum-enstrophy vortex in stratified quasi-geostrophic flows

Applying a variational analysis, a minimum-enstrophy vortex in three-dimensional (3-D) fluids with continuous stratification is found, under the quasi-geostrophic hypothesis. The buoyancy frequency is held constant. This vortex is an ideal limiting state in a flow with an enstrophy decay while energy and generalized angular momentum remain fixed. The variational method used to obtain two-dimensional (2-D) minimum-enstrophy vortices is applied here to 3-D integral quantities. The solution from the first-order variation is expanded on a basis of orthogonal spherical Bessel functions. By computing second-order variations, the solution is found to be a true minimum in enstrophy. This solution is weakly unstable when inserted in a numerical code of the quasi-geostrophic equations. After a stage of linear instability, nonlinear wave interaction leads to the reorganization of this vortex into a tripolar vortex. Further work will relate our solution with maximal entropy 3-D vortices.

Orthogonal retrieval algorithm of atmospheric temperature profiles from pure rotational Raman lidar signals.

Aimed at the stability of calibration coefficients in a general non-orthogonal retrieval algorithm (NRA) of pure rotational Raman lidars (PRRLs), an orthogonal retrieval algorithm (ORA) of atmospheric temperature profiles based on the orthogonal basis function is proposed. This algorithm eliminates the correlation between the calibration coefficients in the NRA to reduce the influence of the number of calibration points and the selection scheme on the calibration coefficients. In this paper, the stabilities of calibration coefficients in the NRA and ORA are compared and analyzed, and the data analysis for atmospheric temperature profiles with a time resolution of minute-level are given, based on the developed Cloud Precipitation Potential Evaluation (CPPV) lidar data and the parallel radiosonde temperature data. The analysis results show that coefficients of variation (CVs) of ORA calibration coefficients are one order of magnitude smaller than those of NRA coefficients. The mean deviation of the ORA retrieval results is roughly reduced by 16.1% compared with the NRA, and the root-mean-square deviation is roughly reduced by 15.0% compared with the NRA. Therefore, the temperature retrieval performance of the ORA is better than that of the NRA.

Aerodynamic stability analysis of the concrete ceiling reinforced with advanced functionally graded nano-materials: A sustainable approach for construction materials

Proper consideration should be given to the stability of new concrete in order to assure the quality of building engineering, while also demanding its excellent flowability. The inadequate initial state stability negatively impacts the long-term durability of reinforced concrete structures, although this issue has not been well addressed. This work focuses on assessing the aerodynamic response of concrete ceilings reinforced with advanced functionally graded nano-materials. As the reinforcement of the current concrete structure, graphene oxide powders (GOPs) are used with improved material properties than other types of reinforcement. For mathematical modeling of the current structure, Reddy’s Higher-order Shear Deformation Theory is used to model the current work’s displacement fields. Also, the perturbation aerodynamic force is mathematically described using the Bernoulli equation for potential flow. After that using a method that involves separating variables, we may get the answer for the aerodynamic pressure in its ultimate form. Haber-Schaim foundation made of auxetic material in the Cartesian coordinate system is used to model the foundation of the current work with high accuracy. After obtaining the governing equations and associated boundary conditions, a meshless approach with weighted orthogonal basis and Kronecker delta-based shape functions are used to solve the equations. Finally, some suggestions for improving the aerodynamics and flutter velocity of airflow of the presented concrete plate reinforced by GOPs are presented for related aerodynamics industries.

Adaptive Orthogonal Basis Function Detection Method for Unknown Magnetic Target Motion State

Traditional methods of orthogonal basis function decomposition have been extensively used to detect magnetic anomaly signals. However, the determination of the relative velocity between the detection platform and the magnetic target remains elusive in practical detection. And, the non-ideal uniform motion of the magnetic target further complicates the process and adversely impacts the detector’s performance. To address this challenge, this paper introduces an adaptive scale factor method based on orthogonal basis function decomposition. This new method can be used to adjust the relative velocity between the detection platform and the magnetic target and to refine the characteristic time in the basis functions. In this paper, a mathematical relationship between the scale factor and the relative velocity is established, which is subsequently fitted into a Gauss function curve. The optimal scale factor, denoted as β, is adaptively chosen from the fitting curve when the magnetic target moves at a non-ideal uniform velocity with an unknown motion state. The results of simulations indicate that the scale factor improves the signal-to-noise ratio of the magnetic anomaly signals in a non-ideal state. And, this method can improve the energy value of OBF decomposition by 17.7%. Simultaneously, this method ensures that the moment the magnetic target passes the CPA coincides with the energy peak of the orthogonal basis detection, which improves the accuracy by 54.1% compared with the traditional method. The effectiveness and precision of the proposed method are verified using simulations and practical experiments.

Adaptive Basis Function Method for the Detection of an Undersurface Magnetic Anomaly Target

The orthogonal basis functions (OBFs) method is a prevailing choice for the detection of undersurface magnetic anomaly targets. However, it requires the detecting platform or target to move uniformly along a straight path. To circumvent the restrictions, a new adaptive basis functions (ABFs) approach is proposed in this article. It permits the detection platform to search for a possible target at different speeds along any course. The ABFs are constructed using the real-time data of the onboard triaxial fluxgate, GPS module, and attitude gyro. Based on the pseudo-energy of an apparent target signal, the constant false alarm rate (CFAR) method is employed to judge whether a target is present. Moreover, by defining the pixel as a relative possibility for a target at a geographic location, a magnetic anomaly target imaging scheme is introduced by displaying the pixels onto the searching area. On-site experimental data are utilized to demonstrate the proposed approach. Compared with the traditional OBFs method, the present ABFs approach can substantially improve the detection possibility and reduce false alarms.

High-accuracy solution of the Dirac equation for three-dimensional hydrogen atom using a variationally optimized diagonalization method

High-accuracy solution of the Dirac equation for three-dimensional (3D) hydrogen atom is theoretically presented using a variationally optimized diagonalization method. By imposing the asymptotical conditions, a complete set of orthogonal Laguerre basis functions is constructed explicitly for the solution of the Dirac equation, which eliminates the singularity difficulty at the point nucleus and gives a convergent integral. A fully explicit expression of the matrix elements of the Dirac Hamiltonian has been derived from the method of the generating function of Laguerre polynomials. The relative errors of the relativistic energies are ranged from 11 to 14 digits, showing that the calculated relativistic energies are in excellent agreement with the exact energies. These results show that my proposed orthogonal Laguerre basis set is extremely efficient and stable in calculating high-accuracy solution of the Dirac equation for 3D hydrogen atom.

Time-dependent coupled cluster with orthogonal adaptive basis functions: General formalism and application to the vibrational problem.

We derive equations of motion for bivariational wave functions with orthogonal adaptive basis sets and specialize the formalism to the coupled cluster Ansatz. The equations are related to the biorthogonal case in a transparent way, and similarities and differences are analyzed. We show that the amplitude equations are identical in the orthogonal and biorthogonal formalisms, while the linear equations that determine the basis set time evolution differ by symmetrization. Applying the orthogonal framework to the nuclear dynamics problem, we introduce and implement the orthogonal time-dependent modal vibrational coupled cluster (oTDMVCC) method and benchmark it against exact reference results for four triatomic molecules as well as a reduced-dimensional (5D) trans-bithiophene model. We confirm numerically that the biorthogonal TDMVCC hierarchy converges to the exact solution, while oTDMVCC does not. The differences between TDMVCC and oTDMVCC are found to be small for three of the five cases, but we also identify one case where the formal deficiency of the oTDMVCC approach results in clear and visible errors relative to the exact result. For the remaining example, oTDMVCC exhibits rather modest but visible errors.

Trifurcated lined ducts: A comprehensive study on noise reduction strategies.

The present research is centered on analyzing and modeling the scattering characteristics of a trifurcated waveguide that includes impedance discontinuities. A mode-matching method, grounded in projecting the solution onto orthogonal basis functions, is devised for the investigation. The impedance disparities at the interfaces are represented in normal velocity modes, which, when combined with pressure modes, result in a linear algebraic system. This system is subsequently truncated and inverted for numerical experimentation. The convergence of scattering amplitudes is assured by reconstructing matching conditions and adhering to conservation laws. The computational results indicate that optimizing attenuation behavior is achievable through manipulating variation bounding properties and impedance discontinuities.

Adaptive Approximation Tracking Control of a Continuum Robot With Uncertainty Disturbances.

Continuum robot has certain compliance and intrinsic safety, which makes it an excellent substitute for the traditional rigid robot in the various tasks, such as human-robot interaction or medical surgery. However, due to the complex nonlinearities which are induced by the compliance, parameter uncertainties, and unknown disturbances, good performance control scheme for the continuum robot is always a challenging task for the practical applications. Thus, to overcome this challenge of the uncertain dynamics and unknown external disturbances, this article develops a novel adaptive control scheme for a continuum robot using the function approximation technique (FAT). Specifically, for the proposed continuum robot, an adaptive FAT control (AFATC) strategy with no update laws is proposed to handle the uncertain parameters of the robot dynamics and external disturbances. The control law is expressed as a finite linear combination of the orthogonal basis functions by the FAT. The proposed AFATC scheme uses a fixed control structure, and the weight matrices are not updated in time. Then, the stability of the proposed controller is proved based on the Lyapunov function. Afterwards, the simulation results indicate the proposed AFATC scheme has good control performance compared with the regressor-free adaptive control (RFAC) method. Finally, the effectiveness of the proposed AFATC scheme is demonstrated in a group of real-time experiments.

Adaptive passive admittance control of a compliant collaborative robot

Compliance of the collaborative robots plays an important role in human-robot interaction. Due to the complex nonlinearities caused by the compliance, and parameter uncertainties, good performance control method for the compliant collaborative robot is always a challenging task. Therefore, to overcome the challenge of the compliance, and uncertain dynamics, this paper develops an adaptive passive admittance control (APAC) approach for a novel compliant collaborative robot, whose compliant mechanism is constituted of the inerters, springs, and dampers. First, through force-current analogy between mechanical network and electrical network, the compliance of the proposed collaborative robot with the inerters, springs, and dampers, is represented by a novel and simplified admittance matrix to facilitate dynamics model of the proposed collaborative robot. Then, for the proposed compliant collaborative robot, an APAC strategy is presented to dispose of the uncertain parameters of the dynamics. The inertial matrix, Coriolis matrix, admittance matrix, and gravity vector of the compliant collaborative robot are expressed as linear representation of orthogonal basis functions by using function approximation technique (FAT). The boundedness of the robot dynamics is proved by detailed stability analysis which uses Lyapunov function. Finally, simulation results indicate that the proposed APAC of the compliant collaborative robot is effective.

Bernstein spectral method for quasinormal modes and other eigenvalue problems

Spectral methods are now common in the solution of ordinary differential eigenvalue problems in a wide variety of fields, such as in the computation of black hole quasinormal modes. Most of these spectral codes are based on standard Chebyshev, Fourier, or some other orthogonal basis functions. In this work we highlight the usefulness of a relatively unknown set of non-orthogonal basis functions, known as Bernstein polynomials, and their advantages for handling boundary conditions in ordinary differential eigenvalue problems. We also report on a new user-friendly package, called SpectralBP, that implements Berstein-polynomial-based pseudospectral routines for eigenvalue problems. We demonstrate the functionalities of the package by applying it to a number of model problems in quantum mechanics and to the problem of computing scalar and gravitational quasinormal modes in a Schwarzschild background. We validate our code against some known results and achieve excellent agreement. Compared to continued-fraction or series methods, global approximation methods are particularly well-suited for computing purely imaginary modes such as the algebraically special modes for Schwarzschild gravitational perturbations.

Closed-Loop Continuous-Time Subspace Identification with Prior Information

This paper presents a closed-loop continuous-time subspace identification method using prior information. Based on a rational inner function, a generalized orthonormal basis can be constructed, and the transformed noises have ergodicity features. The continuous-time stochastic system is converted into a discrete-time stochastic system by using generalized orthogonal basis functions. As is known to all, incorporating prior information into identification strategies can increase the precision of the identified model. To enhance the precision of the identification method, prior information is integrated through the use of constrained least squares, and principal component analysis is adopted to achieve the reliable estimate of the system. Moreover, the identification of open-loop models is the primary intent of the continuous-time system identification approaches. For closed-loop systems, the open-loop subspace identification methods may produce biased results. Principal component analysis, which reliably estimates closed-loop systems, provides a solution to this problem. The restricted least-squares method with an equality constraint is used to incorporate prior information into the impulse response following the principal component analysis. The input–output algebraic equation yielded an optimal multi-step-ahead predictor, and the equality constraints describe the prior information. The effectiveness of the proposed method is provided by numerical simulations.

Frequency Domain Identification of Continuous-Time Hammerstein Systems With Adaptive Continuous-Time Rational Orthonormal Basis Functions

In this paper, we propose a new identification method for continuous-time Hammerstein systems. This method is based on the adaptive generalized rational orthonormal basis functions. In particular, we use the strategy in which poles of basis functions are selected one by one in terms of a criterion. By doing this, we not only take advantage of the variability of the generalized orthogonal basis functions, but also reduce the computational burden. The proposed algorithm is adaptive. Meanwhile, convergence analysis is performed with considering the sampling frequency, noise level, sampling length, and terms of basis functions. The obtained results guarantee the convergence of the whole Hammerstein system. Finally, a numerical example is presented to illustrate the usefulness of the proposed identification algorithm.