Solution Of Two-dimensional Diffusion Equation Research Articles

Modified local singular boundary method for solution of two-dimensional diffusion equation

This paper deals with a new version of the local singular boundary method (LSBM), which will allow the use of irregular node distribution and any shape of the solved area. Unlike previous modifications, it does not impose any requirements on the number of nodes in the support or the number of virtual points. Moreover, it uses radial basis functions for interpolation, not the weighted squares method. The article contains the derivation of the fundamental relations of the steady and unsteady state method and shows its effectiveness in four control examples.

Singular boundary method using dual reciprocity for two-dimensional transient diffusion

In this paper, a meshless numerical algorithm is developed for the solution of two-dimensional diffusion equations with different boundary conditions. The proposed numerical method, which is truly meshless and quadrature-free, is based on the time-marching finite difference scheme, the method of the particular solutions and the singular boundary method. The diffusion equation is transformed into a Poisson-type or a modified Helmholtz equation after the time domain is discretized by the Euler or Houbolt finite difference scheme.

Induction Heating of Aluminum Billets Subjected to a Strong Rotating Magnetic Field Produced by Superconducting Windings

This paper presents an analysis of an original structure of induction heater using superconducting coils. With the proposed structure, two operating modes are possible to heat the aluminum billet. Either the billet is at standstill and placed within a strong rotating magnetic field (around 3.5 T) of low frequency (around 1 Hz), or the billet rotates at low angular speed (around 6 rad/s) within a strong dc magnetic field (around 3.5 T). The analysis is based on the analytical solution of two-dimensional diffusion equations for the magnetic vector potential. A closed-form expression is derived for the eddy-current power loss in the billet. This analytical expression can be used as a tool for design optimization of the induction heater. The analytical results are compared with those obtained by finite-element software. The study of the temperature field in the billet shows that a homogeneous temperature profile is obtained when a strong magnetic field of low frequency (rotating speed or ac current pulsation) is used.

A simple algorithm for the solution of two-dimensional diffusion equations

A simple point iterative method is presented which is based on a splitting scheme natural to the finite difference form of two-dimensional diffusion problems. A proof is given of the unconditional stability and convergence of the algorithm and a comparison is made with other more commonly used iterative methods.

Solution of Two-Dimensional Diffusion Equation by the Finite Fourier Transformation

A two-dimensional diffusion equation is solved by using the finite Fourier transformation. Through applying the Fourier transformation, a one-dimensional Fredholm-type integral equation of the first kind is derived for the flux and its derivative at the boundary. By solving this equation with given boundary conditions, all of the boundary values are determined. The fluxes inside a region are also obtained by solving similar integral equations. The method of this paper differs from the usual Fourier transformation method in that the solutions are obtained without performing the inverse Fourier transforms. Numerical calculations show that the present method gives higher accuracy with less computation time than the usual finite difference method.