Non-orthogonal Transformation Research Articles

Non-orthogonal multiple-relaxation-time lattice Boltzmann method for vorticity-streamfunction formulation

In this work, we propose a non-orthogonal multiple-relaxation-time (MRT) lattice Boltzmann method (LBM) for the two-dimensional (2D) incompressible Navier-Stokes equations in vorticity-stream function formulation. In this method, the vorticity and the streamfunction equations are calculated by two separate LB equations, which are constructed by using the non-orthogonal transformation matrix. Most importantly, to eliminate the residual term that may appear in the recovered vorticity equation, the convection term is handled as a source term in the LB equation rather than being directly recovered by the evolution equation. In addition, due to the convection term in our method is computed with a local scheme, it retains the parallel feature of the standard LBM. We then test the accuracy and the stability of the present LBM by considering various benchmark examples. The results show that the present method has a second-order convergence rate in space. Our numerical results indicate that the present LBM is a good candidate for solving streamfunction-vorticity formulation.

Nonorthogonal wavelet transformation for reconstructing gravitational wave signals

Detections of gravitational-wave signals from compact binary coalescences have enabled us to study extreme astrophysical phenomena and explore fundamental physics. A crucial requisite for these studies is to have accurate signal models with characteristic morphologies, which have been challenging for many decades, and researchers are still endeavoring to incorporate important physics. Therefore, morphology-independent methods have been developed for identifying a signal and its reconstruction. The reconstructed signal allows us to test the agreement between the observed signal and the waveform posterior samples from parameter estimation. These methods model observed signals using a nearly orthogonal wavelet basis in the frame of continuous wavelet transformation. Here, we propose log-uniform scales to construct the wavelets, which are are highly redundant (nonorthogonal) compared to the conventional octave scales but more efficient for reconstructing the signals at high frequencies. And we introduce a semi-model-dependent reconstruction method using the posterior samples of the events, where we model the signal using Gabor-Morlet wavelets with log-uniform scales. We demonstrate the ability to detect deviation using a numerical simulation of an eccentric binary black hole merger, where the signal in the data does not belong to the search template waveform manifold. Finally, we apply this method to each binary black hole merger event in GWTC-1. We have found that the signal produced by the GW150914 event has 96% agreement with the waveform posterior samples. As the detector sensitivity improves and the detected population of black hole mergers grows, we expect the proposed method will provide even stronger tests.

Reduction of Torque Ripples in PMSM using Fuzzy Controlled based Driving Converter

In this paper we have introduce a four switch PWM (Pulse Width Modulation) inverter for controlling the operation of PMSM (Permanent Magnet Synchronous Motor). The inverter is controlled with 49 rule base fuzzy controller controlling the current component in the feedback loop. The torque of the machine is reduced with the help of second order filter stabilizing the output from the fuzzy controller. The second order torque harmonics are produced by DC capacitor voltage fluctuations are demonstrated first, and a very simple method for compensation is presented by introducing a novel non-orthogonal coordinate transformation. The design of the three phase four switch PWM inverter is modeled in MATLAB software with all graphical representations and reports generated with dynamic characteristics.

On the Electromagnetic Propagation Paths and Wavefronts in the Vicinity of a Refractive Object

AbstractMaxwell's equations are transformed from a Cartesian geometry to a Riemannian geometry. A geodetic path in the Riemannian geometry is defined as the raypath on which the electromagnetic energy efficiently travels through the medium. Consistent with the spatial behavior of the Poynting vector, the metric tensor is required to be functionally dependent on the refractive index of the medium. A symmetric nonorthogonal transformation is introduced, in which the metric is a function of an electromagnetic tension. This so‐called refractional tension determines the curvature of the geodetic line. To verify the geodetic propagation paths and wavefronts, a spherical object with a refractive index not equal to one is considered. A full 3‐D numerical simulation based on a contrast‐source integral equation for the electric field vector is used. These experiments corroborate that the geodesics support the actual wavefronts. This result has consequences for the explanation of the light bending around the Sun. Next to Einstein's gravitational tension there is room for an additional refractional tension. In fact, the total potential interaction energy controls the bending of the light. It is shown that this extended model is in excellent agreement with historical electromagnetic deflection measurements.

A high‐order compact scheme for solving the 2D steady incompressible Navier‐Stokes equations in general curvilinear coordinates

SummaryIn this paper, a high‐order compact finite difference algorithm is established for the stream function‐velocity formulation of the two‐dimensional steady incompressible Navier‐Stokes equations in general curvilinear coordinates. Different from the previous work, not only the stream function and its first‐order partial derivatives but also the second‐order mixed partial derivative is treated as unknown variable in this work. Numerical examples, including a test problem with an analytical solution, three types of lid‐driven cavity flow problems with unusual shapes and steady flow past a circular cylinder as well as an elliptic cylinder with angle of attack, are solved numerically by the newly proposed scheme. For two types of the lid‐driven trapezoidal cavity flow, we provide the detailed data using the fine grid sizes, which can be considered the benchmark solutions. The results obtained prove that the present numerical method has the ability to solve the incompressible flow for complex geometry in engineering applications, especially by using a nonorthogonal coordinate transformation, with high accuracy.

Simulating disturbance frequency effect on heat transfer enhancement in wavy condensate films

It is of great significance to investigate microscopic mechanisms of heat transfer enhancement in wavy condensate films which are frequently encountered in nuclear power plants. A numerical model, in which the wavy interface is tracked by a non-orthogonal coordinate transformation, is developed to study the disturbance frequency effect on heat transfer enhancement along a vertical wall. A sinusoidal disturbance is introduced by a fictitious body force to sustain waves. A disturbance frequency of 16 Hz associated with the critical film Reynolds number is numerically determined by analyzing the neutral stability curve. Compared with other disturbance frequencies, the wave effect on local heat transfer coefficients is the most significant at 16 Hz. The minimum time-averaged thickness and the maximum proportion of substrate film length in the wave length at 16 Hz proves that the film thinning effect is determined by the length of the film substrate rather than the substrate film thickness. The more notable variation of the normal velocity close to the wall at the wave front at 16 Hz accounts for the convection effect of heat transfer enhancement.

Multiple-relaxation-time lattice Boltzmann model for simulating axisymmetric thermal flows in porous media

In this paper, a multiple-relaxation-time (MRT) lattice Boltzmann (LB) model is developed for simulating axisymmetric thermal flows in porous media at the representative elementary volume scale. In the present model, the flow field is solved by a D2Q9 MRT-LB equation based on the generalized non-Darcy model, while the temperature field is solved by a thermal MRT-LB equation using a non-orthogonal transformation matrix with the D2Q5 lattice. The present model has the following features. First, the forcing and source terms are simple and contain no velocity or temperature gradient terms. Second, the thermal MRT-LB equation is within the framework of the standard LB method and is consistent with the philosophy of the LB method. Owing to these features, the present model retains the inherent advantages of the standard LB method and is easy to implement. The present model is tested by simulating several typical axisymmetric thermal problems in porous media, and the numerical results indicate that the present model can serve as an accurate and efficient numerical tool for studying axisymmetric thermal flows in porous media.

Microscopic mechanisms of wave effect on heat transfer enhancement in condensate films

In order to deeply understand the inherent links between hydrodynamic behaviors and heat transfer enhancement in wavy condensate films, a numerical model using a non-orthogonal coordinate transformation was developed to simulate the evolution of wavy films. A sinusoidal disturbance was introduced by a fictitious body force to sustain waves. Two major pressure gradients are established at the wave front. One parallel to the stream-wise direction accelerates fluids and the other normal to the wall increases the normal velocity. The temperature profiles prove that both convection and conduction contribute microscopic mechanisms to heat transfer enhancement in the condensate films. The normal velocity close to the wall significantly influences the temperature gradient at the wall. The predicted heat transfer coefficient for the wavy film was found to be 30% greater than the Nusselt solution for the given parameters. Under the given conditions, the effects of film thinning and convection account for 6% and 24% of the heat transfer enhancement, respectively.

The connection between quadratic bent–negabent functions and the Kerdock code

In this paper we prove that all bent functions in the Kerdock code, except for the coset of the symmetric quadratic bent function, are bent–negabent. In this direction, we characterize the set of quadratic bent–negabent functions and show some results connecting quadratic bent–negabent functions and the Kerdock code. Further, we note that there are bent–negabent preserving nonsingular transformations outside the well known class of orthogonal ones that might provide additional functions in the bent–negabent set. This is the first time we could identify non-orthogonal (nonsingular) linear transformations that preserve bent–negabent property for a special subset.

Ellipsoidal Inclusion with a Shell in an Anisotropic Medium Subjected to a Uniform Electric Field

An electrostatic problem has been solved for a dielectric inclusion consisting of an anisotropic core and a shell immersed in a homogeneous anisotropic dielectric medium (matrix) subjected to a uniform electric field. The outer boundaries of the core and shell are assumed to be ellipsoids, which are confocal after a linear nonorthogonal transformation that eliminates the anisotropy of the dielectric properties of the shell. Analytical expressions have been obtained for the potential and the electric field strength in the matrix, in the shell and core, and an expression for the inclusion polarizability tensor. A special case of inclusion with an isotropic shell is considered. The expressions obtained are applied to the case of an anisotropic sphere with an isotropic shell immersed in an anisotropic medium. It is also shown that in the limiting case of a homogeneous ellipsoidal inclusion in an anisotropic medium, the obtained result agrees with known solutions.

An analytic solution for the noise generated by gust–aerofoil interaction for plates with serrated leading edges

This paper presents an analytic solution for the sound generated by an unsteady gust interacting with a semi-infinite flat plate with a serrated leading edge in a background steady uniform flow. Viscous and nonlinear effects are neglected. The Wiener–Hopf method is used in conjunction with a non-orthogonal coordinate transformation and separation of variables to permit analytical progress. The solution is obtained in terms of a modal expansion in the spanwise coordinate; however, for low- and mid-range incident frequencies only the zeroth-order mode is seen to contribute to the far-field acoustics, therefore the far-field noise can be quickly evaluated. The solution gives insight into the potential mechanisms behind the reduction of noise for plates with serrated leading edges compared to those with straight edges, and predicts a logarithmic dependence between the tip-to-root serration height and the decrease of far-field noise. The two mechanisms behind the noise reduction are proposed to be an increased destructive interference in the far field, and a redistribution of acoustic energy from low cut-on modes to higher cut-off modes as the tip-to-root serration height is increased. The analytic results show good agreement in comparison with experimental measurements. The results are also compared against nonlinear numerical predictions where good agreement is also seen between the two results as frequency and tip-to-root ratio are varied.

Analytic solution for aerodynamic noise generated by plates with spanwise-varying trailing edges

This paper presents an analytic solution for aerodynamic noise generated by an unsteady wall pressure gust interacting with a spanwise-variable trailing edge in a background steady uniform flow. Viscous and nonlinear effects are neglected. The Wiener–Hopf method is used in conjunction with a non-orthogonal coordinate transformation and separation of variables to permit analytical progress. The solution is obtained in terms of a tailored modal expansion in the spanwise coordinate; however, only finitely many modes are cut-on, therefore the far-field noise can be quickly evaluated. The solution gives insight into the potential mechanisms behind the reduction of noise for plates with serrated trailing edges compared to those with straight edges. The two mechanisms behind the noise reduction are an increased destructive interference in the far field, and a redistribution of acoustic energy from low cut-on modes to higher cut-off modes. Five different test-case trailing-edge geometries are considered. The analytic solution identifies which geometries are most effective in different frequency ranges: geometries which promote destructive interference are best at low frequencies, whilst geometries which promote a redistribution of energy are better at high frequencies.

Demonstrating that the nonorthogonal orbital optimized coupled cluster model converges to full configuration interaction

Coupled cluster (CC) methods are among the most accurate methods in quantum chemistry. However, the standard CC linear response formulation is not gauge invariant, resulting in errors when modelling properties like optical rotation and electron circular dichroism. Including an explicit unitary orbital rotation in the CC Lagrangian makes the linear response function gauge invariant, but the resulting models are not equivalent to full configuration interaction (FCI) in the untruncated limit. In this contribution, such methods are briefly discussed and it is demonstrated that methods using a nonorthogonal orbital transformation, such as nonorthogonal orbital optimized CC, can converge to FCI in the untruncated limit. This has been disputed in the literature.

Quantum solution of coupled harmonic oscillator systems beyond normal coordinates

Normal coordinates can be defined as orthogonal linear combinations of coordinates that remove the second order couplings in coupled harmonic oscillator systems. In this paper we go further and explore the possibility of using linear although non-orthogonal coordinate transformations to get the quantum solution of coupled systems. The idea is to use as non-orthogonal linear coordinates those which allow us to express the second-order Hamiltonian matrix in a block diagonal form. To illustrate the viability of this treatment, we first apply it to a system of two bilinearly coupled harmonic oscillators which admits analytical exact solutions. The method provides in this case, as an extra mathematical result, the analytical expressions for the eigenvalues of a certain type of symmetrical tridiagonal matrices. Second, we carry out a numerical application to the Barbanis coupled oscillators system, which contains a third order coupling term and cannot be solved in closed form. We demonstrate that the non-orthogonal coordinates used, named oblique coordinates, are much more efficient than normal coordinates to determine the energy levels and eigenfunctions of this system variationally.

Comprehensive Analysis and Reduction of Torque Ripples in Three-Phase Four-Switch Inverter-Fed PMSM Drives Using Space Vector Pulse-Width Modulation

As a result of their reduced number of switches, three-phase four-switch (TPFS) inverters are generally applied as cost-reduction topologies for permanent magnet synchronous motor (PMSM) drives. However, the torque ripples of PMSM severely deteriorate the performance and reliability of the entire system. Hence, comprehensive considerations for torque ripple reduction, including high- and low-frequency torque ripples, are elaborated considering TPFS inverter-fed PMSM drives. The second-order torque harmonics produced by dc-capacitor voltage fluctuations are first demonstrated, and a very simple compensation method is presented by introducing a novel nonorthogonal coordinate transformation. Then, to evaluate the effects on the high-frequency torque ripples of space vector modulation (SVM) schemes, three SVM schemes for TPFS inverter-fed PMSM drives are assessed based on the torque ripple root-mean-square value. Consequently, the preferred SVM scheme is obtained for high-frequency torque ripple minimization. Moreover, the linear modulation range of the TPFS inverter-fed PMSM drive is derived considering capacitor voltage fluctuations, therein avoiding the low-frequency torque ripples caused by overmodulation. Meanwhile, an adaptive capacitor voltage offset suppression method is proposed to fully exploit the dc-link voltage. The experimental results demonstrate the validation and effectiveness of the proposed analysis and methods for torque ripple reduction.

Anisotropic ellipsoidal inclusion with an anisotropic shell in an isotropic medium subjected to a uniform electric field

An electrostatic problem has been solved for a dielectric inclusion that consists of an anisotropic core and an anisotropic shell. The inclusion is immersed in a uniform isotropic medium (matrix) subjected to a uniform electric field. It is assumed that the outer boundaries of the core and shell are ellipsoidal and become confocal after a linear nonorthogonal transformation that removes the anisotropy of the dielectric properties of the shell. Analytical expressions have been derived for the potential and strength of the electric field in the matrix and also in the shell and core of the inclusion, and an expression for the polarizability tensor of the inclusion has been deduced. It has been shown that the results agree with the well-known solutions in partial (limiting) cases.

Double MRT lattice Boltzmann model for axisymmetric convective flow in porous media

A double multiple-relaxation-time lattice Boltzmann (MRT LB) model for the axisymmetric thermal flow in porous media at the representative element volume, which uses the Darcy-Brinkmann-Forchheimer equation to describe the flow in porous region, is presented. The present model utilizes non-orthogonal transformation matrixes instead of the classical orthogonal ones. Validations are carried out in term of two numerical tests, including natural convection in a vertical annulus without and with a porous medium, respectively, and the results show good agreement with the previous studies.

Non-orthogonal multiple-relaxation-time lattice Boltzmann method for axisymmetric thermal flows

Axisymmetric thermal flows in cylindrical systems are widely encountered in engineering practices. Typically, axisymmetric thermal flows belong in three-dimensional (3D) problems. However, taking advantage of the axisymmetric condition, the 3D axisymmetric flows can be reduced to quasi two-dimensional (2D) problems in the meridian plane, which significantly reduces the computational requirements and avoids treating the curved boundary. In recent years, various 2D lattice Boltzmann (LB) models, including single relaxation time LB (SRT-LB, or LBGK) and multiple relaxation time LB (MRT-LB) models, for axisymmetric thermal flows have been proposed. In the LB community, it is well accepted that the MRT-LB is superior to the LBGK in terms of numerical stability. The existing MRT-LB model for axisymmetric thermal flows are developed based on orthogonal basis vectors obtained from the combination of the lattice velocity components, i.e., the transform matrix in the existing MRT-LB is an orthogonal one. Unlike the existing MRT-LB model, in this paper, a non-orthogonal multiple-relaxation-time lattice Boltzmann (MRT-LB) method of simulating axisymmetric thermal flows is proposed. In the proposed MRT-LB method, the velocity field is solved by a D2Q9 discrete velocity set while the temperature by a D2Q5 discrete velocity set. The main advantage of the present MRT-LB model is that the transform matrix of the model is a non-orthogonal one, which is comprised of some proper non-orthogonal basis vectors obtained from the combination of the lattice velocity components. The non-orthogonal transform matrix of the present MRT-LB model contains more zero elements than the classical orthogonal transform matrix, and thus the present MRT-LB model is expected to be more efficient than the existing orthogonal-based MRT-LB model. The equilibrium velocity and temperature moments of the present MRT-LB model are expressed by mapping the equilibrium distribution functions onto their moment spaces through using the non-orthogonal transformation matrix. Also the vectors in the forcing term are modified according to the matrix mapping. Through the Chapman-Enskog analysis, it is demonstrated that the macroscopic governing equations in the cylindrical coordinate can be recovered from the present MRT-LB model. Then several numerical tests, including thermal Womersley flow, Rayleigh-Bnard convection in a vertical cylinder and natural convection in a vertical annulus, are conducted to validate the present model. It is found that the present numerical results are in good agreement with the analytical solutions and/or other numerical results reported in the literature. Numerical stability is also tested, and the results suggest that the present MRT model shows better numerical stability than its LBGK counterpart. Moreover, the numerical results also indicate that the present MRT-LB model is more computationally efficient than the existing MRT-LB model for axisymmetric thermal flow. These findings indicate that the present MRT-LB model can serve as a powerful method of computing the axisymmetric thermal flows.

Quantum treatment of Hénon–Heiles systems using oblique coordinates

In this work we investigate the possibility of using oblique coordinates to determine the energy levels and wave functions of Hénon–Heiles coupled oscillator systems. Oblique coordinates are constructed by making a non-orthogonal linear transformation which permits the expression of the matrix representation of the second-order Hamiltonian operator of the system in a block-diagonal form. They are especially indicated for the treatment of Hénon–Heiles systems with two dissociative channels, which are representative of the stretching vibrational motions of symmetrical triatomic molecules. The ability of oblique coordinates to facilitate the solution of the vibrational Schrödinger equation versus normal coordinates and orthogonal rotated coordinates is analyzed by carrying out linear variational calculations with anharmonic basis function sets. It is shown that oblique coordinates require a considerably smaller number of basis functions than normal and rotated coordinates to variationally converge all the bound energy levels of the Hénon–Heiles potential with two dissociative channels to a given accuracy.

Analysis of a two-way tension-shear coupling in woven fabrics under combined loading tests: Global to local transformation of non-orthogonal normalized forces and displacements

Dependence of shear rigidity of woven fabrics on yarn pre-tension has been reported in the recent literature, however some conflict regarding the trend of this effect is observed. Sources of this conflict are discussed and resolved in the present article using a new characterization framework and a custom-design combined loading fixture. It is shown that in order to correctly characterize the tension-shear coupling behavior in woven fabrics, instead of using global measured data, local normalized forces and displacements should be driven via a non-orthogonal transformation procedure, while considering kinematic force coupling in the setup. In addition, the effect of fabric shear on the tensile behavior of yarns has been investigated, suggesting that the coupling under question is in fact two-way. In particular, results revealed that applying yarn pre-tension increases the shear resistance of the fabric reinforcement, while the tensile behavior of the material becomes more compliant when undergoing shear deformation.