An introduction to Eulerian Geometrical Optics (1992-2002)

This document is an attempt at introducing the different «Eulerian» numerical methods which have recently been developed for the simulation of geometric optics and related model

On the Derivation of Vector Radiative Transfer Equation for Polarized Radiative Transport in Graded Index Media

Light transport in graded index media follows a curved trajectory determined by the Fermat's principle. Besides the effect of variation of the refractive index on the transport of radiative intensity, the curved ray trajectory will induce geometrical effects on the transport of polarization ellipse. This paper presents a complete derivation of vector radiative transfer equation for polarized radiation transport in absorption, emission and scattering graded index media. The derivation is based on the analysis of the conserved quantities for polarized light transport along curved trajectory and a novel approach. The obtained transfer equation can be considered as a generalization of the classic vector radiative transfer equation that is only valid for uniform refractive index media. Several variant forms of the transport equation are also presented, which include the form for Stokes parameters defined with a fixed reference and the Eulerian forms in the ray coordinate and in several common orthogonal coordinate systems.Comment: This paper has been submitted to JQSR

Scaling of Huygens-front speedup in weakly random media

Front propagation described by Huygens' principle is a fundamental mechanism of spatial spreading of a property or an effect, occurring in optics, acoustics, ecology and combustion. If the local front speed varies randomly due to inhomogeneity or motion of the medium (as in turbulent premixed combustion), then the front wrinkles and its overall passage rate (turbulent burning velocity) increases. The calculation of this speedup is subtle because it involves the minimum-time propagation trajectory. Here we show mathematically that for a medium with weak isotropic random fluctuations, under mild conditions on its spatial structure, the speedup scales with the 4/3 power of the fluctuation amplitude. This result, which verifies a previous conjecture while clarifying its scope, is obtained by reducing the propagation problem to the inviscid Burgers equation with white-in-time forcing. Consequently, field-theoretic analyses of the Burgers equation have significant implications for fronts in random media, even beyond the weak-fluctuation limit.Comment: 7 pages, 3 figures, elsart5p. v2: additional discussion of Hamiltonian formalism; v3: clarification of transient behavio

A geometric optics method for high-frequency electromagnetic fields computations near fold caustics—Part II. The energy

AbstractWe present the computation of the amplitudes needed to evaluate the energy deposited by the laser wave in a plasma when a fold caustic forms. We first recall the Eulerian method designed in Benamou et al. (J. Comput. Appl. Math. 156 (2003) 93) to compute the caustic location and the phases associated to the two ray branches on its illuminated side. We then turn to the computation of the amplitudes needed to evaluate the energy. We use the classical geometrical form of the amplitudes to avoid the blow up problem at the caustic. As our proposed method is Eulerian we have to consider transport equations for these geometrical quantities where the advection field depends on the ray flow. The associated vector field structurally vanishes like the square root of the distance to the caustic when approaching the caustic. This introduces an additional difficulty as traditional finite difference scheme do not retain their accuracy for such advection fields. We propose a new scheme which remains of order 1 at the caustic and present a partial theoretical analysis as well as a numerical validation. We also test the capability of our Eulerian geometrical algorithm to produce numerical solution of the Helmholtz equation and attempt to check their frequency asymptotic accuracy

Wave modelling - the state of the art

This paper is the product of the wave modelling community and it tries to make a picture of the present situation in this branch of science, exploring the previous and the most recent results and looking ahead towards the solution of the problems we presently face. Both theory and applications are considered. The many faces of the subject imply separate discussions. This is reflected into the single sections, seven of them, each dealing with a specific topic, the whole providing a broad and solid overview of the present state of the art. After an introduction framing the problem and the approach we followed, we deal in sequence with the following subjects: (Section) 2, generation by wind; 3, nonlinear interactions in deep water; 4, white-capping dissipation; 5, nonlinear interactions in shallow water; 6, dissipation at the sea bottom; 7, wave propagation; 8, numerics. The two final sections, 9 and 10, summarize the present situation from a general point of view and try to look at the future developments

The Zeldovich approximation: key to understanding Cosmic Web complexity

We describe how the dynamics of cosmic structure formation defines the intricate geometric structure of the spine of the cosmic web. The Zeldovich approximation is used to model the backbone of the cosmic web in terms of its singularity structure. The description by Arnold et al. (1982) in terms of catastrophe theory forms the basis of our analysis. This two-dimensional analysis involves a profound assessment of the Lagrangian and Eulerian projections of the gravitationally evolving four-dimensional phase-space manifold. It involves the identification of the complete family of singularity classes, and the corresponding caustics that we see emerging as structure in Eulerian space evolves. In particular, as it is instrumental in outlining the spatial network of the cosmic web, we investigate the nature of spatial connections between these singularities. The major finding of our study is that all singularities are located on a set of lines in Lagrangian space. All dynamical processes related to the caustics are concentrated near these lines. We demonstrate and discuss extensively how all 2D singularities are to be found on these lines. When mapping this spatial pattern of lines to Eulerian space, we find a growing connectedness between initially disjoint lines, resulting in a percolating network. In other words, the lines form the blueprint for the global geometric evolution of the cosmic web.Comment: 37 pages, 21 figures, accepted for publication in MNRA

A fast phase space method for computing creeping rays

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 postprocessing. 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 monostatic 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.

A Survey of Ocean Simulation and Rendering Techniques in Computer Graphics

This paper presents a survey of ocean simulation and rendering methods in computer graphics. To model and animate the ocean's surface, these methods mainly rely on two main approaches: on the one hand, those which approximate ocean dynamics with parametric, spectral or hybrid models and use empirical laws from oceanographic research. We will see that this type of methods essentially allows the simulation of ocean scenes in the deep water domain, without breaking waves. On the other hand, physically-based methods use Navier-Stokes Equations (NSE) to represent breaking waves and more generally ocean surface near the shore. We also describe ocean rendering methods in computer graphics, with a special interest in the simulation of phenomena such as foam and spray, and light's interaction with the ocean surface

Investigation of cutting mechanics in single point diamond turning of silicon

As a kind of brittle material, silicon will undergo brittle fracture at atmospheric pressure in conventional scale machining. Studies in the last two decades on hard and brittle materials including silicon, germanium, silicon nitride and silicon carbide have demonstrated ductile regime machining using single point diamond turning (SPDT) process. The mirror-like surface finish can be achieved in SPDT provided appropriate tool geometry and cutting parameters including feed rate, depth of cut and cutting speed are adopted.The research work in this thesis is based on combined experimental and numerical smoothed particle hydrodynamics (SPH) studies to provide an inclusive understanding of SPDT of silicon. A global perspective of tool and workpiece condition using experimental studies along with localized chip formation and stress distribution analysis using distinctive SPH approach offer a comprehensive insight of cutting mechanics of silicon and diamond tool wear. In SPH modelling of SPDT of silicon, the distribution of von Mises and hydrostatic stress at incipient and steady-state was found to provide the conditions pertinent to material failure, phase transformation, and ductile mode machining. The pressure-sensitive Drucker Prager (DP) material constitutive model was adopted to predict the machining response behaviour of silicon during SPDT. Inverse parametric analysis based on indentation test was carried out to determine the unknown DP parameters of silicon by analysing the loading-unloading curve for different DP parameters. A very first experimental study was conducted to determine Johnson-Cook (J-C) model constants for silicon. High strain rate compression tests using split Hopkinson pressure bar (SHPB) test as well as quasi-static tests using Instron fatigue testing machine were conducted to determine J-C model constants.The capability of diamond tools to maintain expedient conditions for high-pressure phase transformation (HPPT) as a function of rake angle and tool wear were investigated experimentally as well as using SPH approach. The proportional relationship of cutting forces magnitude and tool wear was found to differ owing to wear contour with different rake angles that influence the distribution of stresses and uniform hydrostatic pressure under the tool cutting edge. A new quantitative evaluation parameter for the tool wear resistance performance based on the cutting distance was also proposed. It was also found that the machinability of silicon could be improved by adopting novel surface defect machining (SDM) method.The ductile to brittle transition (DBT) with the progressive tool wear was found to initiate with the formation of lateral cracks at low tool wear volume which transform into brittle pitting damage at higher tool edge degradation. A significant variation in resistance to shear deformation as well as position shift of the maximum stress values was observed with the progressive tool wear. The magnitude and distribution of hydrostatic stress were also found to change significantly along the cutting edge of the new and worn diamond tools.As a kind of brittle material, silicon will undergo brittle fracture at atmospheric pressure in conventional scale machining. Studies in the last two decades on hard and brittle materials including silicon, germanium, silicon nitride and silicon carbide have demonstrated ductile regime machining using single point diamond turning (SPDT) process. The mirror-like surface finish can be achieved in SPDT provided appropriate tool geometry and cutting parameters including feed rate, depth of cut and cutting speed are adopted.The research work in this thesis is based on combined experimental and numerical smoothed particle hydrodynamics (SPH) studies to provide an inclusive understanding of SPDT of silicon. A global perspective of tool and workpiece condition using experimental studies along with localized chip formation and stress distribution analysis using distinctive SPH approach offer a comprehensive insight of cutting mechanics of silicon and diamond tool wear. In SPH modelling of SPDT of silicon, the distribution of von Mises and hydrostatic stress at incipient and steady-state was found to provide the conditions pertinent to material failure, phase transformation, and ductile mode machining. The pressure-sensitive Drucker Prager (DP) material constitutive model was adopted to predict the machining response behaviour of silicon during SPDT. Inverse parametric analysis based on indentation test was carried out to determine the unknown DP parameters of silicon by analysing the loading-unloading curve for different DP parameters. A very first experimental study was conducted to determine Johnson-Cook (J-C) model constants for silicon. High strain rate compression tests using split Hopkinson pressure bar (SHPB) test as well as quasi-static tests using Instron fatigue testing machine were conducted to determine J-C model constants.The capability of diamond tools to maintain expedient conditions for high-pressure phase transformation (HPPT) as a function of rake angle and tool wear were investigated experimentally as well as using SPH approach. The proportional relationship of cutting forces magnitude and tool wear was found to differ owing to wear contour with different rake angles that influence the distribution of stresses and uniform hydrostatic pressure under the tool cutting edge. A new quantitative evaluation parameter for the tool wear resistance performance based on the cutting distance was also proposed. It was also found that the machinability of silicon could be improved by adopting novel surface defect machining (SDM) method.The ductile to brittle transition (DBT) with the progressive tool wear was found to initiate with the formation of lateral cracks at low tool wear volume which transform into brittle pitting damage at higher tool edge degradation. A significant variation in resistance to shear deformation as well as position shift of the maximum stress values was observed with the progressive tool wear. The magnitude and distribution of hydrostatic stress were also found to change significantly along the cutting edge of the new and worn diamond tools