Search results

The Hermite collocation method of discretization can be used to determine highly accurate solutions to the steady‐state one‐dimensional convection‐diffusion equation (which can be used to model the transport of contaminants dissolved in groundwater). This accuracy is dependent upon sufficient refinement of the finite‐element mesh as well as applying upstream or downstream weighting to the convective term through the determination of collocation locations which meet specified constraints. Owing to an increase in computational intensity of the application of the method of collocation associated with increases in the mesh refinement, minimal mesh refinement is sought. Very often this optimization problem is the one where the feasible region is not connected and as such requires a specialized optimization search technique. This paper aims to focus on this method.

An original hybrid method that utilizes a specialized adaptive genetic algorithm followed by a hill‐climbing approach is used to search for the optimal mesh refinement for a number of models differentiated by their velocity fields. The adaptive genetic algorithm is used to determine a mesh refinement that is close to a locally optimal mesh refinement. Following the adaptive genetic algorithm, a hill‐climbing approach is used to determine a local optimal feasible mesh refinement.

In all cases the optimal mesh refinements determined with this hybrid method are equally optimal to, or a significant improvement over, mesh refinements determined through direct search methods.

Further extensions of this work could include the application of the mesh refinement technique presented in this paper to non‐steady‐state problems with time‐dependent coefficients with multi‐dimensional velocity fields.

The present work applies an original hybrid optimization technique to obtain highly accurate solutions using the method of Hermite collocation with minimal mesh refinement.

The Motz problem can be considered as a benchmark problem for testing the performance of numerical methods in the solution of elliptic problems with boundary singularities. The purpose of this paper is to address the solution of the Motz problem using the radial basis function (RBF) method, which is a truly meshfree scheme.

Both the global RBF collocation method (also known as Kansa's method) and the recently proposed local RBF‐based differential quadrature (LRBFDQ) method are considered. In both cases, it is shown that the accuracy of the solution can be significantly increased by using special functions which capture the behavior of the singularity. In the case of global collocation, the functional space spanned by the RBF is enlarged by adding singular functions which capture the behavior of the local singular solution. In the case of local collocation, the problem is modified appropriately in order to eliminate the singularities from the formulation.

The paper shows that the exponential convergence both with increasing resolution and increasing shape parameter, which is typical of the RBF method, is lost in problems containing singularities. The accuracy of the solution can be increased by collocation of the partial differential equation (PDE) at boundary nodes. However, in order to restore the exponential convergence of the RBF method, it is necessary to use special functions which capture the behavior of the solution near the discontinuity.

The paper uses Motz's problem as a prototype for problems described by elliptic partial differential equations with boundary singularities. However, the results obtained in the paper are applicable to a wide range of problems containing boundaries with conditions which change from Dirichlet to Neumann, thus leading to singularities in the first derivatives.

The paper shows that both the global RBF collocation method and the LRBFDQ method, are truly meshless methods which can be very useful for the solution of elliptic problems with boundary singularities. In particular, when complemented with special functions that capture the behavior of the solution near the discontinuity, the method exhibits exponential convergence both with resolution and with shape parameter.

The collocation-based stochastic response surface method (CSRSM) is widely used in geotechnical reliability analyses due to its efficiency and accuracy. Determining the optimal truncated order of the associated polynomial chaos expansion (PCE) is important, as it may strongly affect the practical applicability of CSRSM.

This study investigates the performance of different optimal order selection strategies used in the CSRSM and proposes a new cross-order validation method. First, several methods commonly used for optimal order selection are briefly reviewed, and their merits and limitations for reliability analyses are discussed. Then, an improved optimal order selection method that achieves a better trade-off between efficiency and accuracy is proposed.

In total, ten simple mathematical examples from the literature are employed to perform a preliminary test on the proposed method, and a comparative study is conducted to demonstrate its advantages with respect to some other existing methods.

A total of three typical geotechnical problems are employed to demonstrate the performance of the proposed method in geotechnical practice.

An improved optimal order selection method that achieves a better trade-off between efficiency and accuracy is proposed. The threshold value of the deterministic coefficient used for the proposed method is discussed.

One‐dimensional radiative heat transfer is considered in a plane‐parallel geometry for an absorbing, emitting, and linearly anisotropic scattering medium subjected to azimuthally symmetric incident radiation at the boundaries. The integral form of the transport equation is used throughout the analysis. This formulation leads to a system of weakly‐singular Fredholm integral equations of the second kind. The resulting unknown functions are then formally expanded in Chebyshev series. These series representations are truncated at a specified number of terms, leaving residual functions as a result of the approximation. The collocation and the Ritz‐Galerkin methods are formulated, and are expressed in terms of general orthogonality conditions applied to the residual functions. The major contribution of the present work lies in developing quantitative error estimates. Error bounds are obtained for the approximating functions by developing equations relating the residuals to the errors and applying functional norms to the resulting set of equations. The collocation and Ritz‐Galerkin methods are each applied in turn to determine the expansion coefficients of the approximating functions. The effectiveness of each method is interpreted by analyzing the errors which result from the approximations.

The purpose of this paper is to propose an efficient computational technique, which uses Haar wavelets collocation approach coupled with the Newton-Raphson method and solves the following class of system of Lane–Emden equations:

To deal with singularity, Haar wavelets are used, and to deal with the nonlinear system of equations that arise during computation, the Newton-Raphson method is used. The convergence of these methods is also established and the results are compared with existing techniques.

The authors propose three methods based on uniform Haar wavelets approximation coupled with the Newton-Raphson method. The authors obtain quadratic convergence for the Haar wavelets collocation method. Test problems are solved to validate various computational aspects of the Haar wavelets approach. The authors observe that with only a few spatial divisions the authors can obtain highly accurate solutions for both initial value problems and boundary value problems.

The results presented in this paper do not exist in the literature. The system of nonlinear singular differential equations is not easy to handle as they are singular, as well as nonlinear. To the best of the knowledge, these are the first results for a system of nonlinear singular differential equations, by using the Haar wavelets collocation approach coupled with the Newton-Raphson method. The results developed in this paper can be used to solve problems arising in different branches of science and engineering.

This study attempts to evaluate the effect of the corpus-based inductive teaching approach with multiple academic corpora (PICA, CAEC and Oxford Corpus of Academic English) and conventional deductive teaching approach (i.e., multiple-choice items, filling the gap, matching and underlining) on learning academic collocations by Iranian advanced EFL learners (students learning English as a foreign language).

This is a quasi-experimental, quantitative and qualitative study.

The result showed the experimental group outperformed significantly compared with the control group. The experimental group also shared their perception of the advantages and disadvantages of the corpus-assisted language teaching approach.

Despite growing progress in language pedagogy, methodologies and language curriculum design, there are still many teachers who experience poor performance in their students' vocabulary, whether in comprehension or production. In Iran, for example, even though mandatory English education begins at the age of 13, which is junior and senior high school, students still have serious problems in language production and comprehension when they reach university levels.

To study how the collocational networks method could be used to analyze textual contents in listed companies' annual reports written in Chinese, in an attempt to identify hidden facts that are not released in a listed company's financial statements.

This research extended the collocation network analysis method from English textual contents to Chinese textual contents. The extended collocation network method was used to analyze an Information Technology company, Clever, listed in Shanghai stock exchange of China.

Using the extended collocational networks method, some hidden facts about a Chinese listed company's financial status could be identified, which were not reported in company's officially released financial statements.

The extended collocation network method may supplement the commonly practiced fundamental financial analysis method in helping investors have a better understanding about the financial soundness of listed companies. This is especially important to investors in stock markets of some developing countries, including China. In addition, this method may help regulators of stock market, especially in developing countries, to identify possible loopholes of existing financial regulations as well as some inappropriate practices of some listed companies in disclosing misleading or incorrect financial data in their financial statements.

The first study using the collocational networks method to analyze annual reports of Chinese corporations listed in Shanghai stock exchange, a newly established stock market.

This paper describes the solution of a steady‐state natural convection problem in porous media by the radial basis function collocation method (RBFCM). This mesh‐free (polygon‐free) numerical method is for a coupled set of mass, momentum, and energy equations in two dimensions structured by the Hardy's multiquadrics with different shape parameter and different order of polynomial augmentation. The solution is formulated in primitive variables and involves iterative treatment of coupled pressure, velocity, pressure correction, velocity correction, and energy equations. Numerical examples include convergence studies with different collocation point density and arrangements for a two‐dimensional differentially heated rectangular cavity problem at filtration Rayleigh numbers Ra*=25, 50 and 100, and aspect ratios A=1/2, 1, and 2. The solution is assessed by comparison with reference results of the fine‐mesh finite volume method in terms of mid‐plane velocity components, mid‐plane and insulated surface temperatures, streamfunction minimum, and Nusselt number.

The purpose of this paper is to develop a new local radial basis function collocation method (LRBFCM) for one‐domain solving of the non‐linear convection‐diffusion equation, as it appears in mixture continuum formulation of the energy transport in solid‐liquid phase change systems.

The method is structured on multiquadrics radial basis functions. The collocation is made locally over a set of overlapping domains of influence and the time stepping is performed in an explicit way. Only small systems of linear equations with the dimension of the number of nodes in the domain of influence have to be solved for each node. The method does not require polygonisation (meshing). The solution is found only on a set of nodes.

The computational effort grows roughly linearly with the number of the nodes. Results are compared with the existing steady analytical solutions for one‐dimensional convective‐diffusive problem with and without phase change. Regular and randomly displaced node arrangements have been employed. The solution is compared with the results of the classical finite volume method. Excellent agreement with analytical solution and reference numerical method has been found.

A realistic two‐dimensional non‐linear industrial test associated with direct‐chill, continuously cast aluminium alloy slab is presented.

A new meshless method is presented which is simple, efficient, accurate, and applicable in industrial convective‐diffusive solid‐liquid phase‐change problems with non‐linear material properties.

The purpose of this study is to construct quartic trigonometric tension (QTT) B-spline collocation algorithms for the numerical solutions of the Coupled Burgers’ equation.

The finite elements method (FEM) is a numerical method for obtaining an approximate solution of partial differential equations (PDEs). The development of high-speed computers enables to development FEM to solve PDEs on both complex domain and complicated boundary conditions. It also provides higher-order approximation which consists of a vector of coefficients multiplied by a set of basis functions. FEM with the B-splines is efficient due both to giving a smaller system of algebraic equations that has lower computational complexity and providing higher-order continuous approximation depending on using the B-splines of high degree.

The result of the test problems indicates the reliability of the method to get solutions to the CBE. QTT B-spline collocation approach has convergence order 3 in space and order 1 in time. So that nonpolynomial splines provide smooth solutions during the run of the program.

There are few numerical methods build-up using the trigonometric tension spline for solving differential equations. The tension B-spline collocation method is used for finding the solution of Coupled Burgers’ equation.