Least-square Error Method Research Articles

A high-order immersed boundary method for acoustic wave scattering and low-Mach number flow-induced sound in complex geometries

A new sharp-interface immersed boundary method based approach for the computation of low-Mach number flow-induced sound around complex geometries is described. The underlying approach is based on a hydrodynamic/acoustic splitting technique where the incompressible flow is first computed using a second-order accurate immersed boundary solver. This is followed by the computation of sound using the linearized perturbed compressible equations (LPCE). The primary contribution of the current work is the development of a versatile, high-order accurate immersed boundary method for solving the LPCE in complex domains. This new method applies the boundary condition on the immersed boundary to a high-order by combining the ghost-cell approach with a weighted least-squares error method based on a high-order approximating polynomial. The method is validated for canonical acoustic wave scattering and flow-induced noise problems. Applications of this technique to relatively complex cases of practical interest are also presented.

Direct and inverse solutions with non-Fourier effect on the irregular shape

This study used the finite element method with linear least-squares error method to construct a simple non-iterative inverse operation procedure, for solving non-Fourier heat transfer effect in irregular geometry. Numerical validation indicated that, irregular geometry would enhance heat wave reflection and thus make temperature distribution more complex, moreover, non-Fourier heat transfer has limited heat wave propagation speed, heat information fails to be reflected in time at measure point of the other side, hence, inverse operation difficulty increases. However, sequential algorithm constructed by this study is able to estimate unknown boundary condition correctly and efficiently at every instant by only one array operation in successive time steps. In addition, by measuring a few future time temperatures, the sensitivity of estimated result to measurement error can be effectively reduced as well.

Estimation for inner surface geometry of furnace wall using inverse process combined with grey prediction model

In this work the inner surface geometry of a cylindrical furnace wall is estimated using inverse process method combined with grey prediction model. In estimating process a virtual area extended from the inner surface of furnace wall is used for analysis. The heat conduction equation and the boundary condition are first discretized by finite difference method to form a linear matrix equation; the inverse model is then optimized by linear least-squares error method and the temperatures of virtual boundary are obtained from a few of measured temperatures in furnace wall using the linear inverse model; and finally the temperature distribution of system is got by direct process and the inner surface geometry of furnace wall can be estimated accordingly. The result shows that using inverse process combined with grey prediction model the geometry can be exactly estimated from relatively small number of measured temperatures. Moreover, the effects of measurement error, location, and number of measured points on the estimation for inner surface geometry of furnace wall are discussed in detail.

PROFILE DESIGN OF PISTON RING USING INVERSE METHOD

This study aims to develop an algorithm for designing the piston ring profile and pressure distribution using an inverse method. The proposed algorithm needs to obtain load and boundary conditions for estimating not only a smooth curve in the piston ring profile, but also in the pressure distribution. The algorithm is developed from Reynolds integral and force balance equations. The least-squares error method, variational method, Gauss-Seidel method and Newton-Raphson method are employed to calculate the piston ring profile. Simulation results reveal that the greater the degree of the polynomial function used, the greater the maximum pressure ( max P ), load and friction coefficient are, and the smaller the minimum film thickness ( min H ) is. However, as the degree of the polynomial function and the number of grid points increase, the estimated piston ring profile and pressure distribution become more accurate. The initial guessed values of the coefficients of the estimated piston ring profile ( j C ) and the position of maximum pressure ( m X ) have more obvious effects upon the present algorithm. The initial guessed value of j C can allow greater error than that of m X when estimating max P and min H using the present algorithm. Consequently, the present algorithm is capable of providing accurate results in terms of piston ring profile and pressure distribution.

Railway Wheelset Parameter Estimation Using Signals From Lateral Velocity Sensor

Abstract A type of parameter estimation technique based on the linear integral filter (LIF) method, the least-absolute error with variable forgetting factor (LAE+VFF) estimation method, is proposed in this paper to estimate the railway wheelset parameters modelled as a time-varying continuous-time (C-T) system. The inputs to the parameter estimator are the control signal and the railway wheelset system output, which is the wheelset’s lateral velocity. The algorithm includes an instrumental variable (IV) element to reduce estimation bias and a variable forgetting factor for good parameter tracking and smooth steady state. Simulation results have shown that the LAE with fixed forgetting factor gives better parameter estimates compared to the recursive least-squares error (RLSE) method, whereas the LAE+VFF offers even better estimation and tracking of system parameters that are subject to abrupt changes, provided that the fs and lf values are chosen accordingly. It has also been proven that the estimation error of the proposed LAE+VFF estimation algorithm is bounded.

A Hybrid Method for the Estimation of Power System Low-Frequency Oscillation Parameters

A hybrid method which combines four different algorithms for low-frequency oscillation parameter estimation using wide-area measurements is presented in this paper. The first algorithm is the discrete Fourier transform which is used to start the low-frequency oscillation estimation process. The second one is the finite-impulse response (FIR) window filter which is employed in filtering unwanted components, and window correction factor is provided to eliminate the side effect of FIR window filter. The third one is the prototype algorithm which is proposed to estimate low-frequency oscillation parameters, and the last one is the least-square error method which is employed to enhance the computational accuracy by varying the window size. This hybrid method has been implemented in wide-area phasor measurements system by Java language and is under test in Taiwan's power system. Matlab/Simulink simulation and field test results are provided in the present paper.

The inverse estimation of the thermal behavior and the viscosity of fluid between two horizontal concentric cylinders with rotating inner cylinder

This study is intended to provide an inverse approach for estimating the viscosity of fluid and thermal behavior of the concentric cylinders. Finite-difference methods are first employed to discredit the problem domain and then a linear inverse model is constructed to identify the condition for the viscosity of fluid and thermal behavior of the concentric cylinders. The present approach is to rearrange the matrix form of differential governing equation and to estimate coefficients of unknown conditions. Then, the linear least-squares error method is adopted to find the solution. Excluding measurement error, the inverse values from the present study are in excellent agreement with the direct solutions regardless of where the measurement points are located. When the measurement errors are considered, more measuring points are needed in order to increase the congruence of the estimated results to exact solutions. The advantage of applying this method in inverse analysis is that no prior information is needed on the functional form of the unknown quantities, and initial guess and iteration are not required.

Geometry estimation of the furnace inner wall by an inverse approach

The geometry of a furnace inner wall is estimated by an inverse method in this work. Based upon the concept of a virtual area, in the analysis process the heat conduction equation with boundary conditions was first discretized by a finite difference method to form a matrix equation. And then the linear least-squares error method was applied to determine the temperature of virtual boundary by inverse process. Finally, the geometry of the furnace inner wall can be obtained by direct process. Furthermore, the effects of the measurement errors, number of measurements and position of measurements on the deviation of geometry prediction are also discussed.

Shape optimum design of slider bearings using inverse method

This study aims to develop an algorithm for designing the optimum shape of slider bearing and pressure distribution using an inverse method. The proposed algorithm only needs to obtain the load and moment conditions in order to simultaneously estimate the slider bearing shape and the pressure distribution. The algorithm is developed from the Reynolds integral, and from force and moment balance equations. The least-squares error method, variational method, Gauss–Seidel method and Newton–Raphson method are employed to calculate the optimum shape of slider bearing. Simulation results reveal that as the degree of the shape polynomial function increases, there are corresponding gains in the maximum pressure, load and torque are, and a corresponding decline in the minimum film thickness. On the other hand, the lower the degree of the objective shape polynomial function is, the more accurate the estimated slider bearing shape and pressure distribution are. With increases in degree of polynomial and number of grid points, the errors in the estimated slider bearing shape and pressure distribution can be reduced. The initial guessed values of the coefficients for the estimated slider bearing shape ( C j ), the position of the maximum pressure ( X m ) and the outlet film thickness ( H 0) have notable effects upon the estimated results for the present algorithm. Moreover, the greatest error of initial guessed value is that of C j , followed by X m , and then H 0. The estimated pressure distributions are more accurate than the estimated values for film thickness. Consequently, the present algorithm is capable of providing accurate results for slider bearing shape and pressure distribution.

Legendre 다항식을 이용한 주파수 응답 함수의 곡선접합과 모드 매개변수 규명

A measured frequency response function can be represented as a ratio of two polynomials. A curve-fitting of frequency responses with Legendre polynomialis suggested in the paper. And the suggested curve-fitting algorithm is based on the least-square error method. Since the Legendre polynomials satisfy the orthogonality condition, the curve-fitting with the polynomials results to more reliable curve-fitting than ordinary polynomial method. Though the proposed curve-fitting with Legendre polynomials cannot cover all frequency range of interest, example shows that the suggested method is quite applicable in a limited frequency band.

The Inverse Prediction of the Geometry of a Two Dimensional Plate with a Rectangular Cavity

The contour of a two dimensional plate with a rectangular cavity was predicted by the linear least-squares error method. In the analysis process the heat conduction equation and the boundary conditions were discretized to form a linear matrix equation. On the basis of the concept of a virtual area, the unknown contour of the two dimensional plate was converted into virtual boundary temperatures that can be definitely expressed. Under the condition that the solution form and the initial values didn’t need to be preset, the contour of the two dimensional plate was predicted from temperatures of some points in the plate by only two inverse processes. The results show that even though the positions of the measured points are different the temperature field of the whole virtual system doesn’t change. When the number of the measured points is decreased or the measurement error is increased, the error of the prediction will increase. With reasonable measurement error, the geometry of the plate can be successfully predicted from a few measured points by the method proposed in this work.

Thermal conductivity investigation for upsetting with a procedure of combining inverse model and the proposed regularization of Tikhonov method

The main purpose of this paper is to establish an inverse method, where by using this inverse method the history of the change in thermal conductivities of the workpiece material and the change in average temperatures of workpiece during upsetting process can be obtained. The inverse method is based on the measurement temperatures of an upsetting experiment and combines the matrix-presentation linear least-squares error method of an inverse heat-transfer finite element model proposed in this paper with the regularization of the Tikhonov method. Since the change of the thermal conductivities between the successive finite element analysis stages is very slight and continuous, the square of the difference of the thermal conductivities between successive finite element analysis stages is adopted as the stabilizing function of the regularization of the Tikhonov method in this paper. From the results, it can be shown and demonstrated that the inverse method proposed in this paper can make the history of the change in thermal conductivities during upsetting process has a better stability.

Broadband VHF sources locating system using arrival-time differences for mapping of lightning discharge process

This paper presents a three-dimensional lightning discharge mapping system using arrival-time difference of broadband VHF electromagnetic pulses (20–200 MHz) methodology. The system consists of four circular plate antennas; three of which are placed to form two perpendicularly crossed-short base lines, with the fourth located at some distance. The directions of the electromagnetic radiation sources are determined using perpendicularly aligned antennas whereas the 3-D positions are obtained using all the four antennas. Broadband pulses arrival time differences are estimated incorporating time-domain signal analysis and the least-squares error method is applied to the detected broadband pulses to estimate the time delay with the quadratic interpolation technique utilized to achieve the resolution less than 1 ns. In this paper, the implementation of the system along with the successful reconstruction of the lightning discharge progression in 2-D and 3-D space is described. Additionally, instances of observation results of the cloud-to-ground and ground-to-cloud lightning discharge have been presented.

A new trust region algorithm for image restoration

The image restoration problems play an important role in remote sensing and astronomical image analysis. One common method for the recovery of a true image from corrupted or blurred image is the least squares error (LSE) method. But the LSE method is unstable in practical applications. A popular way to overcome instability is the Tikhonov regularization. However, difficulties will encounter when adjusting the so-called regularization parameter $\alpha$. Moreover, how to truncate the iteration at appropriate steps is also challenging. In this paper we use the trust region method to deal with the image restoration problem, meanwhile, the trust region subproblem is solved by the truncated Lanczos method and the preconditioned truncated Lanczos method. We also develop a fast algorithm for evaluating the Kronecker matrix-vector product when the matrix is banded. The trust region method is very stable and robust, and it has the nice property of updating the trust region automatically. This releases us from tedious finding the regularization parameters and truncation levels. Some numerical tests on remotely sensed images are given to show that the trust region method is promising.

Optimization of a Semimetallic Friction Material Formulation

A semimetallic friction material formulation consisting of Al2O3,B2O3, brass chips, graphite, NBR (nitrile rubber), steel wool, and benzoxazine (a ring-opening polymerized phenolic) with high friction coefficient ( μ) and low wear was developed using the Golden Section approach to search the optimal volume fraction of each ingredient in the formulations. All raw materials used were divided into three groups: fibers, fillers, and binder. The volume fraction of each ingredient in the formulations was calculated by means of V i = 0.618 n in phase 1, V j = V i ± V i × r m/ V F and V j = V i ± V i × r m/ V f in phase 2 and V j = V i ± V i × r m/( V F + V f) in phase 3. Two exponent equations, w = k0 V1 k1 V2 k2 … V p kp and μ = m0 V1 m1 V2 m2 … V p mp, were proposed to correlate the compositional dependence of wear and μ. The least-squares error method was utilized to calculate the effect of each ingredient on and wear. Experimental and calculated results proved that both brass chips and B2O3 play important roles in enhancing. After doing 18 formulation experiments that preformed using Friction Assessment and Screening Test (FAST), an optimized formulation (BB-17) with high μ (average μ = 0.701) and low wear ( w = 4.74%) was obtained.

An algorithm of image processing for underwater range finding by active triangulation

In this paper, we present an algorithm for range image processing to reduce effects of underwater environments on the quality of range finding. This algorithm of image processing is based on the principle of the least-squares error method, which fits discrete image data to continuous piecewise curves. To simply the fitting of image data, the interval of each row and column is subdivided into several subintervals. Then a straight line is used to fit the image data within the subinterval. To merge two adjacent lines together, a weighting technique with a linear weighting factor is imposed. After the image is processed, it will provide a better imaging quality than the original one if values of the design parameters are properly assigned. Thus, a series of design of experiment process runs are carried out to study effects of the design parameters on quality of range finding. To make the proposed algorithm robust against noises, both environmental illumination and turbidity are forced into the experiments by utilizing an outer array. From the results of the range finding experiments, it was found that the proposed algorithm of image processing in truth has potential of increasing quality of range finding. The results also show that the proposed algorithm can achieve high quality of range finding only having the processing of row elements. Also, the quality of range finding by using the proposed algorithm of image processing is superior to that of using a bandpass optical filter.

An inverse vibration analysis of a tower subjected to wind drags on a shaking ground

A method is presented to estimate the strength of wind drags on an elevated tower and the magnitude of the vibration of the ground on which the tower stands. The governing equations for the motions of the tower are discretized using the finite-difference method. Based on these discretized governing equations, a linear inverse model is constructed to identify the external wind drags and the ground vibration. The optimized solution of the model is determined by the linear least-square error method, which requires no numerical iteration. The uniqueness of the solution can be identified by linear algebra theory. A numerical example is given to demonstrate the feasibility of the method. The results show that the original wind drags and ground vibration may be estimated from the measured deflection at several locations along the tower. The reasonable estimations are achievable even though there exist certain measurement errors. The loading conditions are checked at different locations and the deflection information at different location may be used. The procedure is easy and effective. It may be extended to many inverse applications that the discretized governing equations were derived.

Three-dimensional inverse heat transfer analysis during the grinding process

A three-dimensional inverse analysis is adopted to estimate the unknown conditions on the workpiece surface during a grinding process. The numerical method (linear least-squares error method) requires just one iteration and can solve the inverse problems given only the temperature information at a finite number of locations beneath the working surface within a specified time domain. Results show that the heat source into the grinding zone and the heat transfer coefficient in the cooling region can be obtained by the proposed method even when under the influence of measured errors. Furthermore, it is found that the estimated heat transfer coefficient is more sensitive than the heat source to different measured errors and depths. Analyses of the temperature, heat distribution and heat transfer coefficient of the workpiece will help prevent the occurrence of thermal damage to the workpiece, which are caused by the high temperatures generated during the grinding process.

An Improved Inverse Method for Refractive Index Reconstruction of Ion-exchanged Waveguides

In this work, a new inverse method for the reconstruction of refractive index profiles of the single-mode ion-exchanged planar and channel waveguides from the intensity profiles of the fundamental mode, as measured with near-field (NF) technique, is presented. In the numerical process, a finite-difference method is used to discretize the governing equation and then a linear inverse model is constructed to identify the unknown profiles of refractive index. The present approach is to rearrange the matrix form of the scalar wave equation and then the linear least-squares error method is adopted to find the solutions. The results show that the accuracy of index determination can be accessed even when the measured noise is considered. In comparison with the traditional approach, the advantages of this method are that no prior information is needed on the functional form of the unknown index profiles, no initial guesses are required, no iterations in the calculating process are necessary, and the inverse problem can be solved in a linear domain.

An inverse algorithm to calculate the refractive index profiles of periodically segmented waveguides from the measured near-field intensities

In this paper, the refractive index profiles of the periodically segmented waveguides (PSWs), which are fabricated in soda-lime glasses by the K/sup +/-Na/sup +/ ion-exchanged technique, are reconstructed from the measured transmitted near-field (NF) intensity combined with an inverse method. Through the proposed inverse method, the model /spl Delta/n'=/spl eta//spl Delta/n, which characterizes the behavior of the PSWs, is also verified. In the numerical process, a finite-difference method is used to discretize the governing equation, and then a linear inverse model is constructed to identify the unknown refractive index profiles. The approach used is to rearrange the matrix form of the governing differential equation and estimate the unknown index profiles of the waveguides. Then, the linear least-squares error method is adopted to find the solutions. The results show that the accuracy of index determination can be accessed even when the measured noise is considered. In contrast to the traditional approach, the advantages of this method are that no prior information is needed on the functional form of the unknown index profiles, no initial guesses are required, no iterations in the calculating process are necessary, no intensity smoothing is required in advance, and the inverse problem can be solved in a linear domain. Furthermore, the existence and uniqueness of the solutions can easily be identified.