电磁场数值分析14-15章.ppt

-14,15- Planar triangular surface patch models (TPM) Xie, Yongjun Department of Microwave Telecommunication Engineering 1 Electromagnetic scattering by surface of arbitrary shape (I) Fig. 14.1 2 Electromagnetic scattering by surface of arbitrary shape (II) vertex edge Boundary edge face Fig. 14.2 3 Electric field integral equation (I) Let S denote the surface of an open or closed perfectly conducting scatterer with unit normal . An electric field , denoted to be the field due to an impressed source in the absence of the scatterer, in incident on and induces surface currents on S. The scattered electric field can be computed from the surface current by (14.1) with the magnetic vector potential defined as (14.2) and the scalar potential as (14.3) 4 Electric field integral equation (II) where is the distance between an arbitrarily located observation point and a source point on S. Both and are defined with respect to a global coordinate origin O. The surface charge density is related to the surface divergence of through the equation of continuity, We derive an integrodifferential equation for by enforcing the boundary condition on S, obtaining (14.4) (14.5) on S One notes that the presence of derivatives on the current in (14.4) and on the scalar potential in (14.5) suggests that care should be taken in selecting the expansion functions and testing procedure in the method of moments. 5 Development of basis functions (I) edge O Fig. 14.3 Triangle pair and geometrical parameters associated with interior edge 6 Development of basis functions (II) It is convenient to start the development by noting that each basis function is to be associated with an interior edge (i.e., nonboundary edge) of the patch model and is to vanish everywhere on S except in the two triangles attached to that edge. Points in may be designated either by the position vector defined with respect to O, or by the position vector defined with respect to the free vertex of . Similar remarks apply to the position vector except that it is directed toward the free vertex of . The plus or minus designation of the triangles is determined by the choice of a positive current reference direction for the nth edge, the reference for which is assumed to be from to . 7 Development of basis functions (III) We define the vector basis function associated with the nth edge as in in otherwise (14.6) Where is the length of the edge and is the area of triangle . The basis function is used to approximately represent the surface current, and we list and discuss below some properties which make it uniquely suited to this role. 8 Development of basis functions (IV) - some properties (I) 1) The current has no component normal to the boundary (which excludes the common edge) of the surface formed by the triangle pair and , and hence no line charges exist along this boundary. 2) The component of current normal to the nth edge is constant and continuous across the edge as may be seen with the aid of Fig. 14.4, which shows that the normal component of along edge n is just the height of triangle with edge n as the base and the height expressed as . The latter factor normalizes in (14.6) such that its flux density normal to edge n is unity, ensuring continuity of current normal to the edge. This result, together with 1), implies that all edges of and are free of line charge. 9 Development of basis functions (V) - some properties (II) 3) The surface divergence of , which is proportional to the surface charge density associated with the basis element, is in in otherwise since the surface divergence in is . The charge density is thus constant in each triangle, the total charge associated with the triangle pair and is zero, and the basis functions for the charge evidently have the form of pulse doublets. (14.7) 10 Development of basis functions (VI) - some properties (III) Fig. 14.4 Geometry for construction of component of basis function normal to edge 11 Development of basis functions (VII) - some properties (IV) 4) The moment of is given by where (14.8) and is the vector between the free vertex and the centroid of with directed toward and directed away from the vertex, as shown in Fig. 14.3, and is the vector from O to the centroid of . Equation (14.8) may be easily derived by expressing the integral in terms of area coordinates, to be discussed below. 12 Development of basis functions (VIII) The current on S may be approximated in terms of the as (14.9) Where N is the number of interior (nonboundary) edges. Since the normal component of at the nth edge is unity, each coefficient in (14.9) may be interpreted as the normal component of current density flowing past the nth edge. At surface boundary edges, the sum of the normal components of current on opposite sides of the surface cancel because of current continuity. Therefore we neither define nor include in (14.9) contributions from basis functions associated with such edges. 13 Testing procedure (I) We choose as testing functions the expansion functions developed in the previous section. With a symmetric product defined as (14.10) (14.5) is tested with , yielding (14.11) If one makes use of a surface vector calculus identity and the properties of at the edges of S, the last term in (14.11) can be rewritten as (14.12) 14 Testing procedure (II) With (14.7), the integral in (14.12) may now be written and approximated as follows: (14.13) In (14.13) the average of over each triangle is approximated by the value of at the triangle centroid. 15 Testing procedure (III) With similar approximations, the vector potential and incident field terms in (14.11) may be written as (14.14) Where the