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Phase space methods for computing creeping rays
KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
2006 (English)Licentiate thesis, comprehensive summary (Other scientific)
Abstract [en]

This thesis concerns the numerical simulation of creeping rays and their contribution to high frequency scattering problems.

Creeping rays are a type of diffracted rays which are generated at the shadow line of the scatterer and propagate along geodesic paths on the scatterer surface. On a perfectly conducting convex body, they attenuate along their propagation path by tangentially shedding diffracted rays and losing energy. On a concave scatterer, they propagate on the surface and importantly, in the absence of dissipation, experience no attenuation. The study of creeping rays is important in many high frequency problems, such as design of sophisticated and conformal antennas, antenna coupling problems, radar cross section (RCS) computations and control of scattering properties of metallic structures coated with dielectric materials.

First, assuming the scatterer surface can be represented by a single parameterization, we propose a new Eulerian formulation for the ray propagation problem by deriving a set of escape partial differential equations in a three-dimensional phase space. The equations are solved on a fixed computational grid using a version of fast marching algorithm. The solution to the equations contain information about all possible creeping rays. This information includes the phase and amplitude of the ray field, which are extracted by a fast post-processing. The advantage of this formulation over the standard Eulerian formulation is that we can compute multivalued solutions corresponding to crossing rays. Moreover, we are able to control the accuracy everywhere on the scatterer surface and suppress the problems with the traditional Lagrangian formulation. To compute all possible creeping rays corresponding to all shadow lines, the algorithm is of computational order O(N3 log N), with N3 being the total number of grid points in the computational phase space domain. This is expensive for computing the wave field for only one shadow line, but if the solutions are sought for many shadow lines (for many illumination angles), the phase space method is more efficient than the standard methods such as ray tracing and methods based on the eikonal equation.

Next, we present a modification of the single-patch phase space method to a multiple-patch scheme in order to handle realistic problems containing scatterers with complicated geometries. In such problems, the surface is split into multiple patches where each patch has a well-defined parameterization. The escape equations are solved in each patch, individually. The creeping rays on the scatterer are then computed by connecting all individual solutions through a fast post-processing.

We consider an application to mono-static radar cross section problems where creeping rays from all illumination angles must be computed. The numerical results of the fast phase space method are presented.

Place, publisher, year, edition, pages
Stockholm: KTH , 2006. , viii, 20 p.
Series
Trita-CSC-A, ISSN 1653-5723 ; 2006:15
Keyword [en]
Creeping Rays, Phase Space Methods, High Frequency Wave Scattering Problems
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-4146ISBN: 91-7178-467-5 (print)OAI: oai:DiVA.org:kth-4146DiVA: diva2:10915
Presentation
, Lindstedtsvägen 17, Stockholm
Opponent
Supervisors
Note
QC 20101119Available from: 2006-10-12 Created: 2006-10-12 Last updated: 2010-11-19Bibliographically approved
List of papers
1. A fast phase space method for computing creeping rays
Open this publication in new window or tab >>A fast phase space method for computing creeping rays
2006 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 219, no 1, 276-295 p.Article in journal (Refereed) Published
Abstract [en]

Creeping rays can give an important contribution to the solution of medium to high frequency scattering problems. They are generated at the shadow lines of the illuminated scatterer by grazing incident rays and propagate along geodesics on the scatterer surface, continuously shedding diffracted rays in their tangential direction. In this paper, we show how the ray propagation problem can be formulated as a partial differential equation (PDE) in a three-dimensional phase space. To solve the PDE we use a fast marching method. The PDE solution contains information about all possible creeping rays. This information includes the phase and amplitude of the field, which are extracted by a fast post-processing. Computationally, the cost of solving the PDE is less than tracing all rays individually by solving a system of ordinary differential equations. We consider an application to mono-static radar cross section problems where creeping rays from all illumination angles must be computed. The numerical results of the fast phase space method and a comparison with the results of ray tracing are presented.

Keyword
creeping rays, high frequency wave propagation, scattering problems, numerical methods, geometrical theory of diffraction, eikonal equation, finite-difference calculation, high-frequency, wave-propagation, travel-time, level set, computation, equation, optics, rcs
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-16154 (URN)10.1016/j.jcp.2006.03.024 (DOI)000242332500018 ()2-s2.0-33750342435 (Scopus ID)
Note
QC 20100525Available from: 2010-08-05 Created: 2010-08-05 Last updated: 2017-12-12Bibliographically approved
2. A multiple-patch phase space method for computing trajectories on manifolds with applications to wave propagation problems
Open this publication in new window or tab >>A multiple-patch phase space method for computing trajectories on manifolds with applications to wave propagation problems
2007 (English)In: Communications in Mathematical Sciences, ISSN 1539-6746, E-ISSN 1945-0796, Vol. 5, no 3, 617-648 p.Article in journal (Refereed) Published
Abstract [en]

We present a multiple-patch phase space method for computing trajectories on two-dimensional manifolds possibly embedded in a higher-dimensional space. The dynamics of trajectories are given by systems of ordinary differential equations (ODEs). We split the manifold into multiple patches where each patch has a well-defined regular parameterization. The ODEs are formulated as escape equations, which are hyperbolic partial differential equations (PDEs) in a three- dimensional phase space. The escape equations are solved in each patch, individually. The solutions of individual patches are then connected using suitable inter-patch boundary conditions. Properties for particular families of trajectories are obtained through a fast post-processing. We apply the method to two different problems : the creeping ray contribution to mono-static radar cross section computations and the multivalued travel-time of seismic waves in multi-layered media. We present numerical examples to illustrate the accuracy and efficiency of the method.

Keyword
ODEs on a manifold, phase space method, escape equations, high frequency wave, propagation, geodesics, creeping rays, seismic waves, travel-time, partial-differential equations, dielectric coated cylinder, high-frequency, creeping waves, travel-times, computation, rays, rcs
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-16973 (URN)000249723400006 ()2-s2.0-35349024163 (Scopus ID)
Note
QC 20100525Available from: 2010-08-05 Created: 2010-08-05 Last updated: 2017-12-12Bibliographically approved

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