International Journal of Computational Methods

Two cases of a rectangular plate having moment singularities at the ends of a partial internal line support are analytically investigated. The bending of the plate by uniform loading is formulated in terms of dual-series equations. Application of the finite Hankel integral transform reduces the dual-series equations to a Fredholm integral equation of the second kind that can be solved by standard techniques. Numerical results are given for the deflections and bending moments along the line outside of an internal line support and the change in strain energy due to the presence of a partial support.

In this paper, we develop a B-spline collocation method using artificial viscosity for solving a class of singularly perturbed reaction–diffusion equations. We use the artificial viscosity to capture the exponential features of the exact solution on a uniform mesh, and use the B-spline collocation method, which leads to a tridiagonal linear system. The convergence analysis is given and the method is shown to have uniform convergence of second order. The design of an artificial viscosity parameter is confirmed to be a crucial ingredient for simulating the solution of the problem. Known test problems have been studied to demonstrate the accuracy of the method. Numerical results show the behavior of the method, with emphasis on treatment of boundary conditions. Results shown by the method are found to be in good agreement with the exact solution.

This paper presents a matrix–vector computational library written in the C++ language. Effective matrix–vector manipulation is an essential aspect of scientific numerical modeling, especially while using the finite element method. Even though there are a lot of matrix libraries written in C++ which are freely or commercially available on the Internet, most of them are either too complicated or too simple, i.e. with very limited capabilities. Therefore, the main idea is to develop a reliable matrix library, as part of the computer code in C++, which will be, at the same time, as simple as possible and as complex as possible. Consequently, the basic motto of the library, which is called SMV++, is "Simple and effective" and it is meant to be of use to researchers dedicated to reliable obtained results and not to the "beauty of programming." The library is developed without use of the template concept and inheritance, with the purpose of understanding it easily and of extending it easily with new possibilities and algorithms. Last but not least, it is aimed at having a simple and intuitive user interface. The bottom line is that the SMV++ is not meant to be of use to someone who is in the software development business. Rather, it is oriented to the user who has a modest programming knowledge (e.g. a Ph.D. student) and who is basically concerned about how to obtain the solution to the considered problem, for which he has to do some "homemade programming," because the available commercial software cannot be applied for all of his/her needs. Consequently, as mentioned, the motto of the library is "Simple and effective"; the effectiveness is in the sense of complexity of possibilities, rather than computational efficiency. However, if one is forced to choose between the two in the development of the library, having in mind the target users, priority is given to simplicity of use.

Faster convergence, better accuracy and improved stability of the solutions to fluid flow and heat transfer problems in CFD reduce the computational cost and time. The numerical solutions to partial differential equations governing the physical flow and heat phenomena, using computer software and hardware, have been obtained by various techniques which have been refined over the years. The numerical techniques have obtained the base in finite difference method (FDM) approximations derived from Taylor series expansion. Because of linearization, FDM approximations have truncation error creeping into the values of the partial derivatives, which projects an unrealistic picture of the final outcome of results in terms of accuracy, convergence and stability. As the prime objective of this paper, the minimization of truncation error is attempted with the aid of the interface theory (briefly described in the appendix) used as a computational treatment tool. In simple terms, the interface theory provides an optimal solution to all variables in a linear indeterminate system with redundancy in unknowns. The effort has converged in the form of Hegde's interface numerical technique (HINT), which is demonstrated on a quasi-one-dimensional nozzle flow, the physical behavior of which is described by the Navier–Stokes equation considered specific to the said case. HINT could successfully match the results of MacCormack's predictor–corrector method as far as the accuracy is concerned, but with less computational effort and higher productivity. To the knowledge of the authors, HINT may be considered both original and different for its kind in the vast developments in CFD.

An enhancement of the FEM using Kriging interpolation (K-FEM) was recently proposed. This method offers advantages over the conventional FEM and mesh-free methods. With Kriging interpolation, the approximated field over an element is influenced not only by its own element nodes but also by a set of satellite nodes outside the element. This results in incompatibility along interelement boundaries. Consequently, the convergence of the solutions is questionable. In this paper, the convergence is investigated through several numerical tests. It is found that the solutions of the K-FEM with an appropriate range of parameters converge to the corresponding exact solutions.

The one-dimensional nonlinear Schrödinger equation (NLSE) has a wide range of applications in nonlinear optical and laser physics. The design of new optical systems requires analysis of the propagation characteristics of pulses in the fiber media. In this paper, the Sobolev gradient technique has been used for numerical solution of the NLSE. We show that gradients in a suitably chosen Sobolev space (Sobolev gradients) can be used in finite difference settings in a computationally efficient way to approximate the profile of a pulse given by the NLSE.

This paper presents an effective test pattern generation approach for FPGA circuits by applying quantum computing algorithms. A prototypical new algorithm named QFPGA is developed utilizing the properties of quantum theory, such as quantum superposition and quantum parallelism. The effectiveness of this technique in terms of result quality, CPU requirements, fault detection and number of iterations is experimentally compared with some of the existing classical approaches, like exhaustive search, simulated annealing and genetic algorithms. The algorithm developed is so efficient that it requires only √N (N is the total number of vectors) iterations to find the desired test vector, whereas in classical computing it takes N/2 iterations. Simulation results on various benchmark circuits are also covered in this paper. The extendability of the new approach enables users to easily find the test vector from FPGA circuits and can be adapted for testing FPGA chips.

This paper presents a normed G¹ space and a weakened weak (W²) formulation of a cell-based smoothed point interpolation method (CS-PIM) for 2D solid mechanics problems using three-node triangular cells. Displacement fields in the CS-PIM are constructed using the point interpolation method (polynomial PIM or radial PIM) and hence the shape functions possess the Kronecker delta property facilitating the easy enforcement of Dirichlet boundary conditions. The edge-based T-schemes are introduced for selecting supporting nodes for creating the PIM shape functions and an adaptive coordinate transformation (CT) technique is proposed to solve the singularity problem for the moment matrix. Smoothed strains are obtained by performing the generalized smoothing operation over each triangular background cell. Because the nodal PIM shape functions can be discontinuous, a W² formulation of generalized smoothed Galerkin (GS-Galerkin) weak form is then used to create the discretized system equations. Numerical examples including static, free and forced vibration problems have been studied to examine the present method in terms of accuracy, convergence, efficiency and temporal stability.