Direct Expansion Method of Boundary Condition for solving 3D elliptic equations with small parameters in the irregular domain

Abstract

Direct expansion method 3D rational differential quadrature method Conformal mapping Domain decomposition method Small parameter Irregular domain a b s t r a c t In this article, a new methodology, Direct Expansion Method of Boundary Condition (DEMBC), is developed to solve 3D elliptic equations in the irregular domain. First, the previous Rational Differential Quadrature Method (Rational Spectral Collocation Method in (Berrut et al. 2005) [8]), developed by Berrut et al. (2005) [8], has been generalized to solve 3D elliptic equations. Second, it is showed that Direct Expansion Method of Boundary Condition is capable of handling boundary problems with higher efficiency. Finally, with the help of conformal mapping (Tee and Trefethen, 2006) [9] and domain decomposition method, DEMBC and 3D-RDQM are able solve three kinds of 3D elliptic equations with small parameters in the irregular domain. Numerous test results justify the accuracy and efficiency of our approach.

Mechano-electric Characteristics of the Near-surface Layer of Some Materials

“…In accordance with the fundamentals of electrostatics and nonequilibrium thermodynamics [4,5,7,8] we can show the electric potential Y using Poisson's equation (10) and the stress tensor we can show as part of the equilibrium equation 11:…”

“…The corresponding algorithm for determining , k, b, 0, h we present in three stages. First step, using the equation of equilibrium of  (14) and (15) for j, which follows from the Poisson equations, state equation (16), (17) and also boundary conditions (19), we find five approximations of distributions normal mechanical stresses r,   from coordinate r (in particular, (22)-(24)) using the technique of [4,10] and using method of decomposition  and displacements ur in the ranks by the small parameter b   b0 (20), (21) . At the second step, we direct radius R to infinity and obtain analytical 04005-5 expressions for , x, y (25) depending on the x and the parameter k, not specifying numeric constants for the material.…”

“…; 34  , analytically using the method of small parameter b ?  b0, limiting of the four approximations of the decomposition. Methodics of using the small parameter method for solving problems of mathematical physics is described in [10].…”

Spectral element method for wave propagation on irregular domains

“…In numerical methods, the meshing of the irregular domains remains one of the most cumbersome and important steps in the entire solution process. To deal with problems with irregular domains, some methods have been proposed to overcome this problem, such as meshless local strong form method [9], the Chebyshev tau meshless method based on the integration-differentiation (CMMID) [19], direct expansion method of boundary condition (DEMBC) [24], boundary element method (BEM) [10], the finite element method (FEM) based on the triangular elements [23], the level-set function [21], the fractional step projection scheme [4], the Coons method, the Laplace method and boundary-blending method [3], isoparametric method [13], etc.…”