Search results

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 introduce a computational time dependent modeling to investigate propagation of elastic-viscoplastic zones in the shock wave loaded circular plates.

Constitutive equations are implemented incrementally by the Von-Kármán finite deflection system which is coupled with a mixed strain hardening rule and physical-base viscoplastic models. Time integrations of the equations are done by the return mapping technique through the cutting-plane algorithm. An integrated solution is established by pseudo-spectral collocation methodology. The Chebyshev basis functions are utilized to evaluate the coefficients of displacement fields. Temporal terms are discretized by the Houbolt marching method. Spatial linearizations are accomplished by the quadratic extrapolation technique.

Results of the center point deflections, effective plastic strain and stress (dynamic flow stress) and temperature rise are compared for three features of the Von-Kármán system. Identifying time history of resultant stresses, propagations of the viscoplastic plastic zones are illustrated for two circumstances; with considering strain rate and hardening effects, and without them. Some of modeling and computation aspects are discussed, carefully. When the results are compared with experimental data of shock wave loadings and finite element simulations, good agreements between them are observed.

This computational approach makes coupling the structural equations with the physical descriptions of the high rate deformation through step-by-step spectral solution of the constitutive equations.

This paper sets out to introduce a numerical method to obtain solutions of Fredholm‐Volterra type linear integral equations.

The flow of the paper uses well‐known formulations, which are referenced at the end, and tries to construct a new approach for the numerical solutions of Fredholm‐Volterra type linear equations.

The approach and obtained method exhibit consummate efficiency in the numerical approximation to the solution. This fact is illustrated by means of examples and results are provided in tabular formats.

Although the method is suitable for linear equations, it may be possible to extend the approach to nonlinear, even to singular, equations which are the future objectives.

In many areas of mathematics, mathematical physics and engineering, integral equations arise and most of these equations are only solvable in terms of numerical methods. It is believed that the method is applicable to many problems in these areas such as loads in elastic plates, contact problems of two surfaces, and similar.

The paper is original in its contents, extends the available work on numerical methods in the solution of certain problems, and will prove useful in real‐life problems.

Purpose — Analysis of nonprismatic members has received a great deal attention from designers and engineers due to their ability in satisfaction of architectural and aesthetic necessities. Using these structural members in complex structures such as aircrafts, turbine blades and space vehicles, exact static and dynamic analyses of these members become more significant. Based on structural/mechanical principles, the purpose of this paper is to present a new method to evaluate exact structural matrices for nonprismatic Euler‐Bernoulli beam elements. Design/methodology/approach — Through introducing the concept of basic displacement functions (BDFs), it is shown that exact shape functions are derived in terms of BDFs. BDFs and their derivatives have structural interpretations; therefore, they are obtained via application of flexibility method. Unlike the conventional methods, which are almost categorized as displacement‐based methods, the flexibility basis of the method ensures the true satisfaction of equilibrium equations at any interior point of the element. Findings — The exact shape functions and consequently structural matrices are derived for general nonprismatic beam elements. Numerical examples are carried out to determine static deflection and natural frequencies, and the results are highly competent with the other methods in literature. Research limitations/implications — The method can be extended to structural analysis of curved beams, plates and shells as well. Moreover, it is possible to derive exact dynamic shape functions via BDFs by solving the governing equation for transverse vibration of beams. Theoretically, the method faces limitation in analysis of nonprismatic beams that converge to a point where cross‐sectional area and moment of inertia are equal to zero. Practical implications — The development of this idea, i.e. BDFs seems to lead to promotive novel approaches for structural analysis and could be a breaking point for developing new elements for plates and shells as it was shown for beam elements. Originality/value — The paper's introduction of special functions, namely BDFs and their application, in both static and dynamic analyses of structures, could be a breaking point in analysis procedures.

Waveform relaxation has potential to overcome problem of excessive computer run times which are necessary for simulation of larger circuits with the use of existing simulators. One of the attractive features of waveform relaxation is its suitability for parallel implementation. Amount of data necessary for interchange between parallel processors after each iteration influences the overall performance of simulation. Method of integration based on Chebyshev series provides for representation of solutions in the most compact form which makes it very attractive for parallel implementations. This paper presents some results of numerical experiments with the spectral integration applied in the relaxation framework to a number of MOS circuits.

The purpose of this paper is to study the steady laminar flow of a pressure driven third‐grade fluid with heat transfer in a horizontal channel. The study serves two purposes: to correct the inaccurate results presented in Siddiqui et al., where the homotopy perturbation method was used, and to demonstrate the computational efficiency and accuracy of the spectral‐homotopy analysis methods (SHAM and MSHAM) in solving problems that arise in fluid mechanics.

Exact and approximate analytical series solutions of the non‐linear equations that govern the flow of a steady laminar flow of a third grade fluid through a horizontal channel are constructed using the homotopy analysis method and two new modifications of this method. These solutions are compared to the full numerical results. A new method for calculating the optimum value of the embedded auxiliary parameter ∼ is proposed.

The “standard” HAM and the two modifications of the HAM (the SHAM and the MSHAM) lead to faster convergence when compared to the homotopy perturbation method. The paper shows that when the same initial approximation is used, the HAM and the SHAM give identical results. Nonetheless, the advantage of the SHAM is that it eliminates the restriction of searching for solutions to the nonlinear equations in terms of prescribed solution forms that conform to the rule of solution expression and the rule of coefficient ergodicity. In addition, an alternative and more efficient implementation of the SHAM (referred to as the MSHAM) converges much faster, and for all parameter values.

The spectral modification of the homotopy analysis method is a new procedure that has been shown to work efficiently for fluid flow problems in bounded domains. It however remains to be generalized and verified for more complicated nonlinear problems.

The spectral‐HAM has already been proposed and implemented by the authors in a recent paper. This paper serves the purpose of verifying and demonstrating the utility of the new spectral modification of the HAM in solving problems that arise in fluid mechanics. The MSHAM is a further modification of the SHAM to speed up converge and to allow for convergence for a much wider range of system parameter values. The utility of these methods has not been tested and verified for systems of nonlinear equations. For this reason as much emphasis has been placed on proving the reliability and validity of the solution techniques as on the physics of the problem.

Multi-term variable-order fractional differential equations (VO-FDEs) are powerful tools in accurate modeling of transient-regime real-life problems such as diffusion phenomena and nonlinear viscoelasticity. In this paper the Chebyshev polynomials of the fourth kind is employed to obtain a numerical solution for those multi-term VO-FDEs.

To this end, operational matrices for the approximation of the VO-FDEs are obtained using the Fourth kind Chebyshev Wavelets (FKCW). Thus, the VO-FDE is condensed into an algebraic equation system. The solution of the system of those equations yields a coefficient vector, the coefficient vector in turn yields the approximate solution.

Several examples that we present at the end of the paper emphasize the efficacy and preciseness of the proposed method.

The value of the paper stems from the exploitation of FKCWs for the numerical solution of multi-term VO-FDEs. The method produces accurate results even for relatively small collocation points. What is more, FKCW method provides a compact mapping between multi-term VO-FDEs and a system of algebraic equations given in vector-matrix form.

The purpose of this paper is to propose a Chebyshev collocation method (CCM) for Hallén’s equation of thin wire antennas.

Since the current induced on the thin wire antennas behaves like the square root of the distance from the end, a smoothed current is used to annihilate this end effect. Then the CCM adopts Chebyshev polynomials to approximate the smoothed current from which the actual current can be quickly recovered. To handle the difficulty of the kernel singularity and to realize fast computation, a decomposition is adopted by separating the singularity from the exact kernel. The integrals including the singularity in the linear system can be given in an explicit formula while the others can be evaluated efficiently by the fast cosine transform or the fast Fourier transform.

The CCM convergence rate is fast and this method is more efficient than the other existing methods. Specially, it can attain less than 1 percent relative errors by using 32 basis functions when a/h is bigger than 2×10−5 where h is the half length of wire antenna and a is the radius of antenna. Besides, a new efficient scheme to evaluate the exact kernel has been proposed by comparing with most of the literature methods.

Since the kernel evaluation is vital to the solution of Hallén’s and Pocklington’s equations, the proposed scheme to evaluate the exact kernel may be helpful in improving the efficiency of existing methods in the study of wire antennas. Due to the good convergence and efficiency, the CCM may be a competitive method in the analysis of radiation properties of thin wire antennas. Several numerical experiments are presented to validate the proposed method.

The purpose of this paper is to develop an efficient and reliable spectral integral equation method for solving two-dimensional unsteady advection-diffusion equations.

In this study, the considered two-dimensional unsteady advection-diffusion equations are transformed into the equivalent partial integro-differential equations via integrating from the considered unsteady advection-diffusion equation. After this stage, by using Chebyshev polynomials of the first kind and the operational matrix of integration, the integral equation would be transformed into the system of linear algebraic equations. Robustness and efficiency of the proposed method were illustrated by six numerical simulations experimentally. The numerical results confirm that the method is efficient, highly accurate, fast and stable for solving two-dimensional unsteady advection-diffusion equations.

The proposed method can solve the equations with discontinuity near the boundaries, the advection-dominated equations and the equations in irregular domains. One of the numerical test problems designed specially to evaluate the performance of the proposed method for discontinuity near boundaries.

This study extends the intention of one dimensional Chebyshev approximate approaches (Yuksel and Sezer, 2013; Yuksel et al., 2015) for two-dimensional unsteady advection-diffusion problems and the basic intention of our suggested method is quite different from the approaches for hyperbolic problems (Bulbul and Sezer, 2011).

The purpose of this paper is to apply rectangular Bézier surface patches directly into the mathematical formula used to solve boundary value problems modeled by Laplace's equation. The mathematical formula, called the parametric integral equation systems (PIES), will be obtained through the analytical modification of the conventional boundary integral equations (BIE), with the boundary mathematically described by rectangular Bézier patches.

The paper presents the methodology of the analytic connection of the rectangular patches with BIE. This methodology is a generalization of the one previously used for 2D problems.

In PIES the paper separates the necessity of performing simultaneous approximation of both boundary shape and the boundary functions, as the boundary geometry has been included in its mathematical formalism. The separation of the boundary geometry from the boundary functions enables to achieve an independent and more effective improvement of the accuracy of both approximations. Boundary functions are approximated by the Chebyshev series, whereas the boundary is approximated by Bézier patches.

The originality of the proposed approach lies in its ability to automatic adapt the PIES formula for modified shape of the boundary modeled by the Bézier patches. This modification does not require any dividing the patch into elements and creates the possibility for effective declaration of boundary geometry in continuous way directly in PIES.