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Most studies of power‐law fluids are carried out using stress‐based system of Navier‐Stokes equations; and least‐squares finite element models for vorticity‐based equations of power‐law fluids have not been explored yet. Also, there has been no study of the weak‐form Galerkin formulation using the reduced integration penalty method (RIP) for power‐law fluids. Based on these observations, the purpose of this paper is to fulfill the two‐fold objective of formulating the least‐squares finite element model for power‐law fluids, and the weak‐form RIP Galerkin model of power‐law fluids, and compare it with the least‐squares finite element model.

For least‐squares finite element model, the original governing partial differential equations are transformed into an equivalent first‐order system by introducing additional independent variables, and then formulating the least‐squares model based on the lower‐order system. For RIP Galerkin model, the penalty function method is used to reformulate the original problem as a variational problem subjected to a constraint that is satisfied in a least‐squares (i.e. approximate) sense. The advantage of the constrained problem is that the pressure variable does not appear in the formulation.

The non‐Newtonian fluids require higher‐order polynomial approximation functions and higher‐order Gaussian quadrature compared to Newtonian fluids. There is some tangible effect of linearization before and after minimization on the accuracy of the solution, which is more pronounced for lower power‐law indices compared to higher power‐law indices. The case of linearization before minimization converges at a faster rate compared to the case of linearization after minimization. There is slight locking that causes the matrices to be ill‐conditioned especially for lower values of power‐law indices. Also, the results obtained with RIP penalty model are equally good at higher values of penalty parameters.

Vorticity‐based least‐squares finite element models are developed for power‐law fluids and effects of linearizations are explored. Also, the weak‐form RIP Galerkin model is developed.

The purpose of the paper is to examine the impact of infrastructure investment and development on economic growth in Brazil, Russia, India, China and South Africa (BRICS) countries. The effect is examined for each country separately and also collectively by combining each country.

Ordinary least square regression method is applied to examine the effects of infrastructure investment and development on economic growth for each country. Panel data techniques such as panel least square method, panel least square fixed-effect model and panel least square random effect model are used to examine the collective impact by combining all countries in BRICS. The dynamic panel model is also incorporated for analysis in the study.

The results of the study are mixed. The association between infrastructure investment and development and economic growth for countries within BRICS is not robust. There is an insignificant relationship between infrastructure investment and development and economic growth in Brazil and South Africa. Energy and transportation infrastructure investment and development lead to economic growth in Russia. Telecommunication infrastructure investment and development and economic growth have a negative relationship in India, whereas there is a negative association between transport infrastructure investment and development and economic growth in China. Panel data results conclude that energy infrastructure investment and development lead to economic growth, whereas telecommunication infrastructure investment and development are significant and negatively linked with economic growth.

The study is novel as time series analysis and panel data analysis are used, taking the time span for 38 years (1980–2017) to investigate the influence of infrastructure investment and development on economic growth in BRICS Countries. Time-series regression analysis is used to test the impact for individual countries separately, whereas panel data regression analysis is used to examine the impact collectively for all countries in BRICS.

The purpose of this paper is to present an original mixed least squares method for the numerical solution of vector wave equations.

The proposed approach involves two different types of unknowns: velocities and stresses. The approximate solution to the dynamic elasticity equations is obtained via a minimization of a least squares functional, consisting of two terms: a term, which includes the squared residual of a weak form of the time rate of the constitutive relationships, expressed in terms of velocities and stresses, and a term, which depends on the squared residual of the equations of motion. At each time step the functional is minimized with respect to the velocities and stresses, which for the purpose of this study, are approximated by equal order piece-wise linear polynomial functions. The time discretization is based upon the backward Euler scheme. The displacements are computed from the obtained velocities in terms of a finite difference interpolation.

To test the performance of the method, it has been implemented in original computer codes, using object-oriented logic. One model problem has been solved: propagation of Rayleigh waves. The performed convergence study suggests that the method is convergent for both: velocities and stresses. The obtained results show excellent agreement between the exact and analytical solutions for displacement modes, velocities and stresses. It is observed that this method appears to be stable for the different mesh sizes and time step values.

The mixed least squares formulation, described in this paper, serves as a basis for interesting future developments and applications.

The great value of offline identification of machine parameters is when the machine manufacturer does not provide its parameters. Most machine control strategies require parameter values, and some circumstances in the industrial sector only require offline identification. This paper aims to present a new offline method for estimating induction motor parameters based on least squares and a salp swarm algorithm (SSA).

The central concept is to use the classic least squares (LS) method to acquire the majority of induction machine (IM) constant parameters, followed by the SSA method to obtain all parameters and minimize errors.

The obtained results showed that the LS method gives good results in simulation based on the assumption that the measurements are noise-free. However, unlike in simulations, the LS method is unable to accurately identify the machine’s parameters during the experimental test. On the contrary, the SSA method proves higher efficiency and more precision for IM parameter estimation in both simulations and experimental tests.

After performing a primary identification using the technique of least squares, the initial intention of this study was to apply the SSA for the purpose of identifying all of the machine’s parameters and minimizing errors. These two approaches use the same measurement from a simple running test of an IM, and they offer a quick processing time. Therefore, this combined offline strategy provides a reliable model based on the identified parameters.

Large sparse least-squares problems arise in different scientific disciplines such as optimization, data analysis, machine learning and simulation. This paper aims to propose a two-level hybrid direct-iterative scheme, based on novel block independent column reordering, for efficiently solving large sparse least-squares linear systems.

Herewith, a novel block column independent set reordering scheme is used to separate the columns in two groups: columns that are block independent and columns that are coupled. The permutation scheme leads to a two-level hierarchy. Using this two-level hierarchy, the solution of the least-squares linear system results in the solution of a reduced size Schur complement-type square linear system, using the preconditioned conjugate gradient (PCG) method as well as backward substitution using the upper triangular factor, computed through sparse Q-less QR factorization of the columns that are block independent. To improve the convergence behavior of the PCG method, the upper triangular factor, computed through sparse Q-less QR factorization of the coupled columns, is used as a preconditioner. Moreover, to further reduce the fill-in, then the column approximate minimum degree (COLAMD) algorithm is used to permute the block consisting of the coupled columns.

The memory requirements for solving large sparse least-squares linear systems are significantly reduced compared to Q-less QR decomposition of the original as well as the permuted problem with COLAMD. The memory requirements are reduced further by choosing to form larger blocks of independent columns. The convergence behavior of the iterative scheme is improved due to the chosen preconditioning scheme. The proposed scheme is inherently parallel due to the introduction of block independent column reordering.

The proposed scheme is a hybrid direct-iterative approach for solving sparse least squares linear systems based on the implicit computation of a two-level approximate pseudo-inverse matrix. Numerical results indicating the applicability and effectiveness of the proposed scheme are given.

The purpose of this paper is to present extended least squares support vector machines (LS‐SVM) where data selection methods are used to get sparse LS‐SVM solution, and to overview and compare the most important data selection approaches.

The selection methods are compared based on their theoretical background and using extensive simulations.

The paper shows that partial reduction is an efficient way of getting a reduced complexity sparse LS‐SVM solution, while partial reduction exploits full knowledge contained in the whole training data set. It also shows that the reduction technique based on reduced row echelon form (RREF) of the kernel matrix is superior when compared to other data selection approaches.

Data selection for getting a sparse LS‐SVM solution can be done in the different representations of the training data: in the input space, in the intermediate feature space, and in the kernel space. Selection in the kernel space can be obtained by finding an approximate basis of the kernel matrix.

The RREF‐based method is a data selection approach with a favorable property: there is a trade‐off tolerance parameter that can be used for balancing complexity and accuracy.

The paper gives contributions to the construction of high‐performance and moderate complexity LS‐SVMs.

The purpose of this paper is to focus on the performance of three typical nonlinear least-squares estimation algorithms in atmospheric density model calibration.

The error of Jacchia-Roberts atmospheric density model is expressed as an objective function about temperature parameters. The estimation of parameter corrections is a typical nonlinear least-squares problem. Three algorithms for nonlinear least-squares problems, Gauss–Newton (G-N), damped Gauss–Newton (damped G-N) and Levenberg–Marquardt (L-M) algorithms, are adopted to estimate temperature parameter corrections of Jacchia-Roberts for model calibration.

The results show that G-N algorithm is not convergent at some sampling points. The main reason is the nonlinear relationship between Jacchia-Roberts and its temperature parameters. Damped G-N and L-M algorithms are both convergent at all sampling points. G-N, damped G-N and L-M algorithms reduce the root mean square error of Jacchia-Roberts from 20.4% to 9.3%, 9.4% and 9.4%, respectively. The average iterations of G-N, damped G-N and L-M algorithms are 3.0, 2.8 and 2.9, respectively.

This study is expected to provide a guidance for the selection of nonlinear least-squares estimation methods in atmospheric density model calibration.

The study analyses the performance of three typical nonlinear least-squares estimation methods in the calibration of atmospheric density model. The non-convergent phenomenon of G-N algorithm is discovered and explained. Damped G-N and L-M algorithms are more suitable for the nonlinear least-squares problems in model calibration than G-N algorithm and the first two algorithms have slightly fewer iterations.

The purpose of this paper is to present a method to accurately forecast the tendency of the gross amount of energy sources consumption of the country and construct a new kind of algorithm for forecasting that synthesizes the advantages of the grey model, Markov chains, and least square method.

With the application of this new algorithm, this paper have forecasted the trend of the gross amount of energy sources consumption of the country and come to the conclusions that the new algorithm is more precise than the grey model. It is proved that the improved grey‐Markov chain algorithm is effective and can be used by authorities to make decision.

It was found that combining the grey model, Markov chains, and least square method, can be a new algorithm for forecasting the trendency of the gross amount of energy sources consumption.

The new algorithm is only suitable for the short‐term forecast.

The grey forecasting method reflects the overall tendency of primitive data sequence of the gross amount of energy source, and the Markov chain forecasting method reflects the effect of the random fluctuation. The least square method reflects the tendency of increase. The new algorithm is more precise than the grey model.

Considering typical self‐adjoint and non‐self‐adjoint problems that are governed by differential equations with predominant lower order derivatives, a comparison is presented of their finite element solutions by Ritz, Galerkin, least square (LSQ) and discrete least square (DLSQ) methods.

The purpose of this paper is to present a novel gradient‐based iterative algorithm for the joint diagonalization of a set of real symmetric matrices. The approximate joint diagonalization of a set of matrices is an important tool for solving stochastic linear equations. As an application, reliability analysis of structures by using the stochastic finite element analysis based on the joint diagonalization approach is also introduced in this paper, and it provides useful references to practical engineers.

By starting with a least squares (LS) criterion, the authors obtain a classical nonlinear cost‐function and transfer the joint diagonalization problem into a least squares like minimization problem. A gradient method for minimizing such a cost function is derived and tested against other techniques in engineering applications.

A novel approach is presented for joint diagonalization for a set of real symmetric matrices. The new algorithm works on the numerical gradient base, and solves the problem with iterations. Demonstrated by examples, the new algorithm shows the merits of simplicity, effectiveness, and computational efficiency.

A novel algorithm for joint diagonalization of real symmetric matrices is presented in this paper. The new algorithm is based on the least squares criterion, and it iteratively searches for the optimal transformation matrix based on the gradient of the cost function, which can be computed in a closed form. Numerical examples show that the new algorithm is efficient and robust. The new algorithm is applied in conjunction with stochastic finite element methods, and very promising results are observed which match very well with the Monte Carlo method, but with higher computational efficiency. The new method is also tested in the context of structural reliability analysis. The reliability index obtained with the joint diagonalization approach is compared with the conventional Hasofer Lind algorithm, and again good agreement is achieved.