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A multiple-patch phase space method for computing trajectories on manifolds with applications to wave propagation problems
KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.ORCID iD: 0000-0002-6321-8619
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.

Place, publisher, year, edition, pages
2007. Vol. 5, no 3, 617-648 p.
Keyword [en]
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: urn:nbn:se:kth:diva-16973ISI: 000249723400006Scopus ID: 2-s2.0-35349024163OAI: oai:DiVA.org:kth-16973DiVA: diva2:335016
Note
QC 20100525Available from: 2010-08-05 Created: 2010-08-05 Last updated: 2017-12-12Bibliographically approved
In thesis
1. Topics in Analysis and Computation of Linear Wave Propagation
Open this publication in new window or tab >>Topics in Analysis and Computation of Linear Wave Propagation
2008 (English)Doctoral thesis, comprehensive summary (Other scientific)
Abstract [en]

This thesis concerns the analysis and numerical simulation of wave propagation problems described by systems of linear hyperbolic partial differential equations.

A major challenge in wave propagation problems is numerical simulation of high frequency waves. When the wavelength is very small compared to the overall size of the computational domain, we encounter a multiscale problem. Examples include the forward and the inverse seismic wave propagation, radiation and scattering problems in computational electromagnetics and underwater acoustics. In direct numerical simulations, the accuracy of the approximate solution is determined by the number of grid points or elements per wavelength. The computational cost to maintain constant accuracy grows algebraically with the frequency, and for sufficiently high frequency, direct numerical simulations are no longer feasible. Other numerical methods are therefore needed. Asymptotic methods, for instance, are good approximations for very high frequency waves. They are based on constructing asymptotic expansions of the solution. The accuracy increases with increasing frequency for a fixed computational cost. Most asymptotic techniques rely on geometrical optics equations with frequency independent unknowns. There are however two deficiencies in the geometrical optics solution. First, it does not include diffraction effects. Secondly, it breaks down at caustics. Geometrical theory of diffraction provides a technique for adding diffraction effects to the geometrical optics approximation by introducing diffracted rays. In papers 1 and 2 we present a numerical algorithm for computing an important type of diffracted rays known as creeping rays. Another asymptotic model which is valid also at caustics is based on Gaussian beams. In papers 3 and 4, we present an error analysis of Gaussian beams approximation and develop a new numerical algorithm for computing Gaussian beams, respectively.

Another challenge in computation of wave propagation problems arises when the system of equations consists of second order hyperbolic equations involving mixed space-time derivatives. Examples include the harmonic formulation of Einstein’s equations and wave equations governing elasticity and acoustics. The classic computational treatment of such second order hyperbolic systems has been based on reducing the systems to first order differential forms. This treatment has however the disadvantage of introducing auxiliary variables with their associated constraints and boundary conditions. In paper 5, we treat the problem in the second order differential form, which has advantages for both computational efficiency and accuracy over the first order formulation.

Finally, paper 6 concerns the concept of well-posedness for a class of linear hyperbolic initial boundary value problems which are not boundary stable. The well-posedness is well established for boundary stable hyperbolic systems for which we can obtain sharp estimates of the solution including estimates at boundaries. There are, however, problems which are not boundary stable but are well-posed in a weaker sense, i.e., the problems for which an energy estimate can be obtained in the interior of the domain but not on the boundaries. We analyze a model problem of this type. Possible applications arise in elastic wave equations and Maxwell’s equations describing glancing and surface waves.

Place, publisher, year, edition, pages
Stockholm: KTH, 2008. xiii, 36 p.
Series
Trita-CSC-A, ISSN 1653-5723 ; 2008:07
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-4715 (URN)978-91-7178-961-7 (ISBN)
Public defence
2008-05-20, Sal D2, KTH, Lindstedtsvägen 5, Stockholm, 10:15
Opponent
Supervisors
Note
QC 20100830Available from: 2008-04-30 Created: 2008-04-30 Last updated: 2010-08-30Bibliographically approved
2. Phase space methods for computing creeping rays
Open this publication in new window or tab >>Phase space methods for computing creeping rays
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
Creeping Rays, Phase Space Methods, High Frequency Wave Scattering Problems
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-4146 (URN)91-7178-467-5 (ISBN)
Presentation
, Lindstedtsvägen 17, Stockholm
Opponent
Supervisors
Note
QC 20101119Available from: 2006-10-12 Created: 2006-10-12 Last updated: 2010-11-19Bibliographically approved

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