Singular Eigenfunction Expansions in Neutron Transport Theory

McCormíck, N. J.; Kuščer, Ivan

doi:10.1016/b978-0-12-029307-0.50010-x

Cited by 65 publications

(26 citation statements)

References 185 publications

(176 reference statements)

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“…A wide variety of approximate deterministic methods are available for computing BSDFs for general layered materials, including discrete ordinates (SN ) [Thomas and Stamnes 2002], the transfer matrix method [Aronson 1971], the FN method [Siewert 1978], adding-doubling [van de Hulst 1980, analytical discrete ordinates [Siewert 2000], and the singular eigenfunction method [McCormick and Kuscer 1973]. These methods all have different accuracy/cost tradeoffs and stability issues [Chalhoub et al 2003] and in principle any of them could be used to compute transport operators within our general framework.…”

Section: Prior Workmentioning

confidence: 99%

A comprehensive framework for rendering layered materials

Jakob

d’Eon²,

Jakob³

et al. 2014

ACM Trans. Graph.

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Slightly absorbing blue-green dielectric (η = 1.5) α = 0.05 α = 0.05 Textured diffuse layer Clear dielectric (η = 1.5) α = 0.05 Red anisotropic scattering diel. (η = 1.5, g=0.95) α = 0.1 α = 0.1 Purple anisotropic scattering diel. (η = 1.5, g=0.8) Thin blue absorbing dielectric (η = 1.5) α = 0.1 α = 0.1 Conductor (copper) Blue isotropic scattering dielectric (η = 1.5) α = 0.1 α = 0.1 Conductor (Aluminum) White isotropic scattering dielectric (η = 1.5) α = 0.04 α = 0.04 White isotropic scattering diel. half space (η = 1.5) α = 0.1 White isotropic scattering dielectric (η = 1.5) Measured BRDF (Blue metallic paint 2, [Matusik et al.]) α = 0.1 Measured BRDF (Blue metallic paint 2, [Matusik et al.])Conductor (chrome) α = 0.02 Figure 1: All materials in this interior scene were generated and rendered using the techniques described in this paper. The insets on the left and right reveal the corresponding structural descriptions that were used as inputs to our system. AbstractWe present a general and practical method for computing BSDFs of layered materials. Its ingredients are transport-theoretical models of isotropic or anisotropic scattering layers and smooth or rough boundaries of conductors and dielectrics. Following expansion into a directional basis that supports arbitrary composition, we are able to efficiently and accurately synthesize BSDFs for a great variety of layered structures.Reflectance models created by our system correctly account for multiple scattering within and between layers, and in the context of a rendering system they are efficient to evaluate and support texturing and exact importance sampling. Although our approach essentially involves tabulating reflectance functions in a Fourier basis, the generated models are compact to store due to the inherent sparsity of our representation, and are accurate even for narrowly peaked functions. While methods for rendering general layered surfaces have been investigated in the past, ours is the first system that supports arbitrary layer structures while remaining both efficient and accurate. We validate our model by comparing to measurements of real-world examples of layered materials, and we demonstrate an interactive visual design tool that enables easy exploration of the space of layered materials. We provide a fully practical, high-performance implementation in an open-source rendering system.

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Section: Prior Workmentioning

confidence: 99%

A comprehensive framework for rendering layered materials

Jakob

d’Eon²,

Jakob³

et al. 2014

ACM Trans. Graph.

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“…But all in all, it was much more comprehensible to physi cists and engineers than the older methods because it was based on the famil iar concept of separation of variables. A comprehensive review paper cites at least 150 papers directly utilizing the approach of Case by 1973 (McCormick andKuscer, 1973), including the solution of problems not tackled with the Chandrasekhar and Wiener-Hopf approaches; that review paper also links the half-range orthogonality relations (Kuscer et al, 1964) to the approach of Chandrasekhar (1950). As another indication of the prominence of Case's paper, Edward Larsen has estimated, for one of us (PZ), the number of PhD theses it has engendered as being in the hundreds.…”

Section: Ken Case Memoir ] ]mentioning

confidence: 97%

Biographical Memoir of Ken Case

Zweifel

McCormíck

Case

2014

Journal of Computational and Theoretical Transport

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The most important result would seem to be that the techniques which have suc ceeded so well in electrodynamics will be equally successful in obtaining sensible answers from meson theories.This was all pretty speculative since the pion was only identified experi mentally the same year that Case finished his dissertation, and not much was known about its properties yet. In his work the meson was assumed to be a pseudo-scalar boson (known today to be true-the only scalar boson is the pu tative Higgs boson). The coupling between meson and nucleon was taken as a pseudovector, but it turns out that pseudoscalar coupling, the only other pos sibility, would lead to the same results for the magnetic moments (Luttinger, 1948).This scenario raises several interesting questions, the answers to which are the subject of speculation by the authors of this article, as described next. Why did Case publish his dissertation in such abbreviated form? The answer seems to be that J. M. Luttinger had already obtained the same results before Case did. Later Case did in fact present the details of his calculations (Case, 1949a) with an acknowledgment, in a footnote, that he received a preprint of Luttinger's work (1948) prior to publication. He also pointed out that his results agree with those of Luttinger. The rationale for publishing is that it included the electron-neutron interaction, not considered by Luttinger, and that Case's and Luttinger's calculational models were different (Case, 1949a). Case used the Schwinger-Tomonaga formalism while Luttinger used secondquantization in Fock space. (Roughly speaking, this difference in calculational methods is analogous, in a nonrelativistic theory, to the difference between Schrodinger wave mechanics and Heisenberg matrix mechanics. The equiva lence in the nonrelativistic case had been proved much earlier, using the fact that Schrodinger's Hilbert space L2 is isometrically isomorphic to the Fock space, I2 . In fact, all Hilbert spaces of the same dimension are isometrically isomorphic.) So our guess is that Case's showing that Schwinger's methods yielded the same results as second-quantization was an important result at the time, along with his previous observation that the methods already known to be applicable to quantum electrodynamics (i.e., renormalization) could be extended to problems involving mesons.One last point: The fact that pseudoscalar and pseudovector couplings of the meson and nucleon fields are equivalent is literally true only for the calcu lation of the magnetic moment. Other quantities of interest require the intro duction of "equivalent hamiltonians." It was exactly this approximate equiv alence that led to the "Case's theorem" debacle described in Section 3 of this article.

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“…In fact, McConnick and Kuscer [41] have shown that the normal modes satisfy the following half-range orthogonality relations Milne Problem, the number of secondaries per collision and the positive solution of a characteristic equation [15] be restricted to the self-adjoint integral transport equation Marshak [40] finds that the functional,…”

Section: + 11mentioning

confidence: 99%

“…In this formulation [ 41], the equations for one-energy-group are treated one at a time. At each level of the inner iterations, the sources for the chosen group include both the external sources and the contributions from other energy groups, and are considered to be fixed known functions of the independent variables.…”

Section: Introductionmentioning

confidence: 99%

Adjoining appropriate singular elements to transport theory computations

Abu-Shumays

Bareiss

1973

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Singular Eigenfunction Expansions in Neutron Transport Theory

Cited by 65 publications

References 185 publications

A comprehensive framework for rendering layered materials

A comprehensive framework for rendering layered materials

Biographical Memoir of Ken Case

Adjoining appropriate singular elements to transport theory computations

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