Файл: Doicu A., Wriedt T., Eremin Y.A. Light scattering by systems of particles (OS 124, Springer, 2006.pdf

216 3 Simulation Results

z

2a

x

Fig. 3.24. Integration surfaces for a cube

Fig. 3.25. Normalized di erential scattering cross-sections of a dielectric cube using the symmetry properties of the transition matrix

3.3.3 Triangular Surface Patch Model

Some nonaxisymmetric particle shapes such as ellipsoids, quadratic prisms and regular polyhedral prisms are directly included in the Fortran program

Fig. 3.26. Normalized di erential scattering cross-sections of a chiral cube using the symmetry properties of the transition matrix

Scattering Angle (deg)

Fig. 3.27. Normalized di erential scattering cross-sections of a perfectly conducting cube using the symmetry properties of the transition matrix

as it is provided on the CD-ROM with the book. To handle arbitrary particle geometries, the program can also read particle shape data from an input file.

The particle shape data is based on a surface description using a triangular surface patch model. There are various 3D object file formats suitable for a meshed particle shape. We decided for the Wavefront .obj file format but the

218 3 Simulation Results

program will only support the polygonal format subset and not the free-form geometry (also included in the .obj file format). In this case, a triangular surface patch model of the particle surface has to be generated by an adequate software. For shapes given by an implicit equation, the HyperFun polygonizer [106, 183] supporting high-level language functional representations is a possible candidate for surface mesh generation. Function representation is a generalization of traditional implicit surfaces and constructive solid geometry, which allows the construction of complex shapes such as isosurfaces of realvalued functions [184]. The HyperFun polygonizer generates a VRML output of a su ciently regular triangular patch model, which is then converted to the

.obj file format by using the 3D Exploration program.

To compute the T -matrix elements by surface integrals we employ a modified midpoint or centroid quadrature. The integral over each surface patch is approximated by multiplying the value of the integrand at the centroid by the patch area [79]

Si

where, vi,1, vi,2, vi,3 are the vertices spanning a triangle and vi,c denotes the mass center of the triangle [vi,1, vi,2, vi,3]

j=1

For a su ciently large number of surface patches, the use of this centroid integration is satisfactorily accurate. In convergence checks versus the number of triangular faces, we found that this centroid quadrature is quite stable and the computational results are not much influenced by the number of integration elements.

As an example, we consider a sphere which has been cut at a quarter of its diameter on the z-axis as shown in Fig. 3.28. The cut sphere has been meshed

Fig. 3.28. Geometry of a cut sphere with 10,132 faces

Scattering Angle (deg)

Fig. 3.29. Normalized di erential scattering cross-sections of a sphere and a cut sphere

Fig. 3.30. Geometry of a cube shaped particle with 31,284 faces

with 10132 faces, the Mie parameter computed from the radius is 10, the refractive index is 1.5 and the direction of the incident plane wave is along the z-axis. In Fig. 3.29 the scattering pattern is plotted together with the scattering pattern of a sphere with the same parameters. For better visibility the curves are shifted and it is apparent that there are pronounced di erences in the scattering diagrams of a sphere and a cut sphere.

There are di erent ways to compute scattering by arbitrary particle shapes reconstructed from a number of measured data points. Next we present an example using compactly supported radial basis functions (CS-RBF) for scattered data points processing. For this application we used the software toolkit by Kojekine et al. [121] available from www.karlson.ru. The data points represent a cube shaped particle as depicted in Fig. 3.30 and a number of 378