Numerical Collocation Method Research Articles

Cubic Spline Approximation for Weakly Singular Integral Models

In this paper we propose a numerical collocation method to approximate the solution of linear integral mixed Volterra Fredholm equations of the second kind, with particular weakly singular kernels. The collocation method is based on the class of quasi-interpolatory splines on locally uniform mesh. These approximating functions are particularly suitable to tackle on problems with weakly regular solutions. We analyse the convergence problems and we present some numerical results and comparisons to confirm the efficiency of the numerical model.

Wavelet-based adaptive large-eddy simulation with explicit filtering

Stochastic coherent adaptive large-eddy simulation is a novel approach to the numerical simulation of turbulence, where the coherent energetic eddies are solved for, while modeling the influence of the less energetic coherent/incoherent background flow. The formal separation between resolved and unresolved field is obtained by wavelet threshold filtering that is inherent to the adaptive wavelet collocation numerical method. A new explicit wavelet filtering strategy is introduced and tested, by considering two different filtering levels: the physical level, which controls the turbulence model, and the numerical level that is responsible for the accuracy of numerical simulations. The theoretical basis for wavelet-based adaptive large-eddy simulation with explicit filtering and consistent dynamic modeling is given. Numerical experiments are presented for unsteady homogeneous turbulence, demonstrating the existence of grid-independent solutions.

Computational multiple scattering analysis of elastic waves in unidirectional composites

A numerical procedure is presented in this paper for the two-dimensional, time-harmonic elastodynamic multiple scattering problems for unidirectional fiber-reinforced composites. The proposed procedure is based on the eigenfunction expansion of the displacement potentials and the numerical collocation method to solve the expansion coefficients, and is capable of modeling arbitrary fiber arrangements. To demonstrate the applicability of the procedure, the P and SV wave propagation characteristics in unidirectional fiber-reinforced composites are analyzed for different fiber arrangements and fiber volume fractions. The simulated results are shown to capture the detailed features of the local wave fields in the composites accompanying the mode conversion. From the computed wave fields, the effective phase velocities of the composites are identified as functions of the frequency, and found to be in good agreement with the predictions of a micromechanical model for random composites. The energy transmission spectra of the P and SV waves are also demonstrated, which exhibit the stop-band formation for the composites with regular fiber arrangements.

Subhierarchical method for solving problems of diffraction of electromagnetic waves by a dielectric body in a rectangular waveguide

The problem of diffraction of an external electromagnetic field by a locally inhomogeneous body placed in a rectangular waveguide with perfectly conducting walls is considered. The formulated problem is reduced to a volume singular integral equation. The problem is solved with the use of the numerical collocation method. A subhierarchical method is applied to analyze the problem for structures of a complex geometric shape. Because of a large amount of computation, the problem is solved with the use of parallel algorithms realized on a supercomputer complex.

Interaction of plane elastic harmonic waves with a perfectly bonded elastic inclusion

The interaction of plane harmonic waves with a thin elastic inclusion in the form of a strip in an infinite body (matrix) under plane strain conditions is studied. It is assumed that the bending and shear displacements of the inclusion coincide with the displacements of its midplane. The displacements in the midplane are found from the theory of plates. The priblem-solving method represents the displacements as discontinuous solutions of the Lame equations and finds the unknown discontinuities solving singular integral equations by the numerical collocation method. Approximate formulas for the stress intensity factors at the ends of the inclusion are derived

On direct numerical treatment of hypersingular integral equations arising in mechanics and acoustics

In this paper, we present a treatment of hypersingular integral equations, which have relevant applications in many problems of wave dynamics, elasticity and fluid mechanics with mixed boundary conditions. The main goal of the present work is the development of an efficient direct numerical collocation method. The paper also includes two examples taken from fracture mechanics and acoustics: a single crack in a linear isotropic elastic medium, and diffraction of a plane acoustic wave by a thin rigid screen.

Mixed convection boundary layer flow of non-Newtonian fluids along vertical wavy plates

Mixed convection boundary layer flows of non-Newtonian fluids over the wavy surfaces are studied by the coordinate transformation and the cubic spline collocation numerical method. The effects of the wavy geometry, the buoyancy parameter and the generalized Prandtl number for pseudoplastic fluids, Newtonian fluids and dilatant fluids on the skin-friction coefficient, local and mean Nusselt numbers have been graphically studied. Results show that both higher generalized Prandtl numbers and buoyancy parameters are seen to enhance the influence of wavy surfaces on the local Nusselt number, irrespective of whether the fluids are Newtonian fluids or non-Newtonian fluids. Moreover, the irregular surfaces have higher total heat flux than that of corresponding flats plate for any fluid.

General Collocation Method for Three Dimensional Acoustic Analysis of a Simple Expansion Chamber.

Acoustic analysis of a simple expansion chamber using a three dimensional finite element model is time consuming because of enormous computation. Numerical collocation method utilizing boundary conditions of acoustic pressure and particle velocity, has been developed for an efficient alternative. However, this conventional method cannot be generally applied to a simple expansion chamber with arbitrary inlet/outlet. This paper introduces a general collocation method without any limitation on inlet/outlet, to calculate four-pole parameters of the simple expansion chamber. Acoustic characteristics of simple expansion chamber with rectangular and circular cross section is analyzed using the suggested method and compared with that of conventional collocation method. Results of finite element method reveals that the suggested method has better accuracy compared with the conventional method

A versatile method for analyzing paraxial optical propagation in dielectric structures

This paper presents a fast and accurate quasi-analytic model for studying optical field propagation in weakly guiding dielectric structures. The proposed efficient and versatile computational scheme is obtained by merging the Hermite-Gauss (HG) total field expansion with the numerical collocation method and is particularly suited for longitudinally nonuniform structures. By means of a quasilinearization scheme, the same procedure has also been successfully applied to the analysis of field propagation in Kerr-nonlinear media. The latter achievement gives an indication of the great potentialities offered by this straight forward method. Several examples are discussed in the paper and in all cases the results computed by the proposed method favorably compare with those from alternative methods.

Laminar free convection flow of micropolar fluids from a curved surface

The laminar free convection flow of a micropolar fluid past an arbitrary curved surface has been analysed by using the cubic spline collocation numerical method. It is found that the flow field admits similarity solutions when the power functions for the stream-wise variations of the wall temperature m and the body shape configurations n are constants, and the sum of them is equal to unity. The solutions for the flow and heat transfer characteristics are evaluated numerically for different parameters, such as the body shape parameter n, the vortex viscosity parameter N1, the spin-gradient viscosity parameter N3 and the Prandtl number Pr. Results show that the heat transfer rate increases as the body shape parameter n or the parameter N1 decreases and the Prandtl number increases. The skin friction factor increases as the value of N1 decreases. In addition, an increase in the parameter N3 leads to an increase of the angular velocity boundary layer thickness. The model may solve industrial problems with processing of polymeric liquids, lubricants and molten plastics.

Transient analysis of natural convection along a vertical wavy surface in micropolar fluids

The transient laminar natural convection in the flow of a micropolar fluid along a vertical wavy surface has been investigated by using a simple transposition theorem and cubic spline collocation numerical method. Along with transient Nusselt number and skin friction coefficient at the wavy surface, representative transient velocity and temperature profiles are presented. Numerical results show that the heat transfer rate decreases and the amplitude of the local skin friction increases with increasing amplitude wavelength ratio, and the wavelength of the local skin friction or the couple stress is only half of that of the wavy surface. Comparisons between a Newtonian fluid and a micropolar fluid are also discussed.

Free convection in the boundary layer flow of a micropolar fluid along a vertical wavy surface

A cubic spline collocation numerical method and a simple transposition theorem have been used to study the free convection in the flow of a micropolar fluid along irregular vertical surfaces. A sinusoidal surface is used to elucidate the amplitude wavelength ratio effects on the free convection in a micropolar boundary layer. The effects of micropolar parameterR and geometries on the velocity and temperature fields have been graphically studied. The skin friction stress on the wall has also been studied and discussed. It is observed that the frequency of the local heat transfer rate is twice that of the wavy surface, irrespective of whether the fluid is a Newtonian fluid or micropolar fluid; the same result is also obtained for the skin friction on the wall.

Conjugate forced convection heat transfer from a continuous, moving flat sheet

A cubic spline collocation numerical method has been used to determine the conjugate heat transfer occuring in the laminar boundary layer on a continuous, moving plate. Numerical results for the local heat flux and temperature distribution of both the moving plate and the fluid under the effects of the Peclet number (which determines the relative importance of the motion), the conjugate convection-conduction parameter (which depends on the properties of the fluid and the plate), and the Prandtl number are obtained by solving simultaneously the convective boundary-layer equation of the fluid and the energy equation of the plate. Results show that the Peclet number and the convection-conduction parameter have a significant influence on heat transfer characteristics. In addition, the heat transfer mechanism between the moving plate and the ambient fluid with air (Pr=0.7) as the fluid is substantially different from that with water (Pr=7.0) as the fluid.

First kind integral equations for the numerical solution of the plane Dirichlet problem

We present, in a uniform manner, several integral equations of the first kind for the solution of the two-dimensional interior Dirichlet boundary value problem. We apply a general numerical collocation method to the various equations, and thereby we compare the various integral equations, and recommend two of them. We give a survey of the various numerical methods, and present a simple method for the numerical solution of the recommended integral equations.