ASSIGNMENT

Assignment

Assignment

1. Introduction

Numerical simulation methods of partial differential equation are roughly classified into the following three, the Finite Difference Method (FDM), the Finite Element Method (FEM) and the Boundary Element Method (BEM). On the other hand, the stokes fields themselves are classified into three, static, quasi-static (eddy current) and high frequency fields, and these fields are governed by different types of partial differential equations, elliptic, parabolic and hyperbolic, respectively. Accordingly, FDM, FEM and BEM are individually formulated for these fields. And then, strictly speaking, there are two kinds of formulations, frequency and time-domain schemes for both quasi-static and high frequency fields. Although every kinds of numerical schemes has been developed up to now, there are still difficulties in the numerical scheme of FEM and BEM for high frequency time-domain stokes fields because these methods are formulated on unstructured meshes. The hyperbolic property often enhances numerical noise which arises in the unstructured meshes, and then simulations result in unphysical solutions owing to the numerical noise. In general, the Finite Difference Time-Domain (FDTD) method is now most popular numerical simulation method for the time-domain simulation of high frequency stokes fields, and the FDTD is widely used in practical industrial applications. The FDTD is classified into a time-domain FDM, and then time-domain BEM is the next issue in numerical simulation technology because difference between the FDTD and the time-domain FEM is just only structured or unstructured meshes, and their target objects are almost same. From this point of view, this paper presented a formulation of the time-domain BEM (TDBEM) for the high frequency stokes fields.

For some special cases, the TDBEM formulation and practical simulations were presented, for example, thin wire materials. This paper especially focus discussion on fundamental part of formulation of TDBEM to make properties of TDBEM scheme clear.

2. Formulation

It is well known that a time-domain boundary integral equation of wave equation is results in so-called Kirchhoff's integral equation. And Kirchhoff's integral equation for the stokes fields was also well known. However, to be clearly state the time-domain property, we start discussion of the time-domain boundary integral equation from the most basic part of the stokes fields theory.

2.1. Maxwell's equation for stokes fields and covariant form [5 and 6]

The stokes fields in vacuum are described by the following Maxwell's equations

(1)

(2)

(3)

(4)

div B=0

where is dielectric constant; µ, the permeability; c is the velocity of light. The charge density ? and current density J satisfy the following continuity equation (the conservation laws of charge and current):

(5)

Maxwell's equations are rewritten by using scalar and vector potentials, f and A as follows

(6)

(7)

where f and A are connected to the stokes fields E and B by the following relations:

(8)

(9)

B=rot A

Then the following Lorentz condition is assumed to remove arbitrariness of the potentials in (8) and (9):

(10)

The Lorentz condition (10) is closely related to the conservation laws of charge and current (5) through (6) and (7).

The covariant form of the stokes field theory gives us compact description of (1), (2), (3), (4), (5), (6), (7), (8), (9) and ...