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J H Hannay - One of the best experts on this subject based on the ideXlab platform.
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Path–linking interpretation of Kirchhoff diffraction: a summary
Philosophical Transactions of the Royal Society of London. Series A: Mathematical Physical and Engineering Sciences, 2001Co-Authors: J H HannayAbstract:The Kirchhoff theory of diffraction of 1882 underpinned the earlier formulations of wave optics of Huygens, Young and Fresnel. Remarkably, it provides an exact explicit solution of the wave equation, with well–defined, though unusual, boundary conditions on the opaque diffracting screen. This prestigious status was not appreciated at the time and, despite the incisive clarification of Kottler in 1923, remains largely unsung to this day. Here I re–present my path–linking interpretation of Kirchhoff diffraction in which the Kirchhoff wave field at any observation position is formally represented as a ‘sum’ of contributions from all possible paths of propagation from the source point, whether or not they pass through the screen. The contributions are shown to be weighted according to the linking number of the path (when closed by an unobstructed return path) with the screen boundary loop(s). Kirchhoff optics is thus shown to be inherently topological in nature.
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Path-linking interpretation of Kirchhoff diffraction
Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences, 1995Co-Authors: J H HannayAbstract:The Kirchhoff-diffraction integral is often used to describe the (scalar) wave field from a monochromatic point source in the presence of ‘opaque’ screens. Despite criticisms that can be made of its ‘derivation’, the Kirchhoff field is an exact solution of the wave equation, and exactly obeys definite, though unusual, boundary conditions (Kottler 1923, 1965). Here, the path-integral picture of wave fields is used to interpret the Kirchhoff-diffraction field in terms of all conceivable propagation paths, whether or not they pass through the opaque screens. Specifically, it is noted that the Kirchhoff field equals Ʃ(1 ─ m )ψ m , where the sum is over all integers m , and ψ m is the wave field due to all paths from the source to the field point for which the number of outward screen crossings minus the number of backwards screen crossings is m . Expressed more topologically, m is the total linking number of a path, when closed by any unobstructed path, with the screen edge lines. Other models of diffraction by screens are compared with Kirchhoff diffraction in the path interpretation.
Michael A Marciniak - One of the best experts on this subject based on the ideXlab platform.
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comparison of microfacet brdf model to modified beckmann Kirchhoff brdf model for rough and smooth surfaces
Optics Express, 2015Co-Authors: Samuel D Butler, Stephen E Nauyoks, Michael A MarciniakAbstract:A popular class of BRDF models is the microfacet models, where geometric optics is assumed. In contrast, more complex physical optics models may more accurately predict the BRDF, but the calculation is more resource intensive. These seemingly disparate approaches are compared in detail for the rough and smooth surface approximations of the modified Beckmann-Kirchhoff BRDF model, assuming Gaussian surface statistics. An approximation relating standard Fresnel reflection with the semi-rough surface polarization term, Q, is presented for unpolarized light. For rough surfaces, the angular dependence of direction cosine space is shown to be identical to the angular dependence in the microfacet distribution function. For polished surfaces, the same comparison shows a breakdown in the microfacet models. Similarities and differences between microfacet BRDF models and the modified Beckmann-Kirchhoff model are identified. The rationale for the original Beckmann-Kirchhoff F(bk)(2) geometric term relative to both microfacet models and generalized Harvey-Shack model is presented. A modification to the geometric F(bk)(2) term in original Beckmann-Kirchhoff BRDF theory is proposed.
Alexander Yarovoy - One of the best experts on this subject based on the ideXlab platform.
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ultra wideband radar imaging using a hybrid of Kirchhoff migration and stolt f k migration with an inverse boundary scattering transform
IEEE Transactions on Antennas and Propagation, 2015Co-Authors: Takuya Sakamoto, Toru Sato, Pascal Aubry, Alexander YarovoyAbstract:In this paper, we propose a fast and accurate radar-imaging algorithm that combines Kirchhoff migration with Stolt’s frequency-wavenumber (F-K) migration. F-K migration is known as a fast-imaging method in the F-K domain, while Kirchhoff migration is reported to be more accurate. However, Kirchhoff migration requires the reflection points to be located as a function of the antenna position and the delay time. This prevents the use of fast Fourier transforms (FFTs) because Kirchhoff migration must be processed in the time domain, and this can be extremely time consuming. The proposed algorithm overcomes this hurdle by introducing the texture angle and the inverse boundary scattering transform (IBST). These two tools enable the locations of the reflection points to be determined rapidly for each pixel of a radar image. The radar signals are then modified according to the Kirchhoff integral, before Stolt F-K migration is applied in the frequency domain to produce an accurate radar image. To demonstrate the performance of the proposed method, the conventional delay-and-sum (DAS) migration, Kirchhoff migration, Stolt F-K migration, and the proposed method are applied to the same measured datasets.
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uwb array based radar imaging using modified Kirchhoff migration
International Conference on Ultra-Wideband, 2008Co-Authors: Xiaodong Zhuge, Alexander Yarovoy, T G Savelyev, L P LigthartAbstract:This paper presents a new modification of Kirchhoff migration algorithm for ultra-wideband (UWB) array-based radar imaging. The developed algorithm is evolved from traditional Kirchhoff migration which is based on the classical integral theorem of Helmholtz and Kirchhoff. The new algorithm is designed for array-based radar imaging with arbitrary multiple input multiple output (MIMO) configuration. The developed algorithm is compared with conventional diffraction stack migration using both synthetic data from numerical simulation and measurement data from landmine detection. The results have shown promising improvements in the aspects of beamwidth, side-lobe rejection ratio and the ability to reconstruct shapes of distributed targets.
Patrizia Pucci - One of the best experts on this subject based on the ideXlab platform.
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LIFESPAN ESTIMATES FOR SOLUTIONS OF POLYHARMONIC Kirchhoff SYSTEMS
Mathematical Models and Methods in Applied Sciences, 2012Co-Authors: Giuseppina Autuori, Francesca Colasuonno, Patrizia PucciAbstract:In mathematical physics we increasingly encounter PDEs models connected with vibration problems for elastic bodies and deformation processes, as it happens in the Kirchhoff–Love theory for thin plates subjected to forces and moments. Recently Monneanu proved in Refs. 26 and 27 the existence of a solution of the nonlinear Kirchhoff–Love plate model. In this paper we treat several questions about non-continuation for maximal solutions of polyharmonic Kirchhoff systems, governed by time-dependent nonlinear dissipative and driving forces. In particular, we are interested in the strongly damped Kirchhoff–Love model, containing also an intrinsic dissipative term of Kelvin–Voigt type. Global non-existence and a priori estimates for the lifespan of maximal solutions are proved. Several applications are also presented in special subcases of the source term f and the nonlinear external damping Q.
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Asymptotic stability for nonlinear Kirchhoff systems
Nonlinear Analysis: Real World Applications, 2009Co-Authors: Giuseppina Autuori, Patrizia Pucci, Maria Cesarina SalvatoriAbstract:Abstract We study the asymptotic stability for solutions of the nonlinear damped Kirchhoff system, with homogeneous Dirichlet boundary conditions, under fairly natural assumptions on the external force f and the distributed damping Q . Then the results are extended to a more delicate problem involving also an internal dissipation of higher order, the so called strongly damped Kirchhoff system. Finally, the study is further extended to strongly damped Kirchhoff–polyharmonic systems, which model several interesting problems of the Woinowsky–Krieger type.
Michael Ortiz - One of the best experts on this subject based on the ideXlab platform.
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Automatically inf − sup compliant diamond-mixed finite elements for Kirchhoff plates
International Journal for Numerical Methods in Engineering, 2013Co-Authors: Luigi E. Perotti, Agustín Bompadre, Michael OrtizAbstract:We develop a mixed finite-element approximation scheme for Kirchhoff plate theory based on the reformulation of Kirchhoff plate theory of Ortiz and Morris [1]. In this reformulation the moment-equilibrium problem for the rotations is in direct analogy to the problem of incompressible two-dimensional elasticity. This analogy in turn opens the way for the application of diamond approximation schemes (Hauret et al. [2]) to Kirchhoff plate theory. We show that a special class of meshes derived from an arbitrary triangulation of the domain, the diamond meshes, results in the automatic satisfaction of the corresponding inf − sup condition for Kirchhoff plate theory. The attendant optimal convergence properties of the diamond approximation scheme are demonstrated by means of the several standard benchmark tests. We also provide a reinterpretation of the diamond approximation scheme for Kirchhoff plate theory within the framework of discrete mechanics. In this interpretation, the discrete moment-equilibrium problem is formally identical to the classical continuous problem, and the two differ only in the choice of differential structures. It also follows that the satisfaction of the inf − sup condition is a property of the cohomology of a certain discrete transverse differential complex. This close connection between the classical inf − sup condition and cohomology evinces the important role that the topology of the discretization plays in determining convergence in mixed problems.
