Publisher Summary This chapter discusses the finite element method for solving the neutron transport equation and spatial discretization. It also discusses the numerical approximation of a problem by a finite element method using triangular or quadrilateral elements and other methods for solving the neutron transport equation. The discrete Galerkin method is equivalent to some implicit Runge-Kutta method. The existence and uniqueness of the approximate solution and an algorithm for computing the approximate solution is given. In the chapter, the finite element method is studied which provides an effective way for numerically solving such problems.

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On a Finite Element Method for Solving the Neutron Transport Equation

Light transport algorithms generate realistic images by simulating the emission and scattering of light in an artificial environment. Applications include lighting design, architecture, and computer animation, while related engineering disciplines include neutron transport and radiative heat transfer. The main challenge with these algorithms is the high complexity of the geometric, scattering, and illumination models that are typically used. In this dissertation, we develop new Monte Carlo techniques that greatly extend the range of input models for which light transport simulations are practical. Our contributions include new theoretical models, statistical methods, and rendering algorithms. We start by developing a rigorous theoretical basis for bidirectional light transport algorithms (those that combine direct and adjoint techniques). First, we propose a linear operator formulation that does not depend on any assumptions about the physical validity of the input scene. We show how to obtain mathematically correct results using a variety of bidirectional techniques. Next we derive a different formulation, such that for any physically valid input scene, the transport operators are symmetric. This symmetry is important for both theory and implementations, and is based on a new reciprocity condition that we derive for transmissive materials. Finally, we show how light transport can be formulated as an integral over a space of paths. This framework allows new sampling and integration techniques to be applied, such as the Metropolis sampling algorithm. We also use this model to investigate the limitations of unbiased Monte Carlo methods, and to show that certain kinds of paths cannot be sampled. Our statistical contributions include a new technique called multiple importance sampling, which can greatly increase the robustness of Monte Carlo integration. It uses more than one sampling technique to evaluate an integral, and then combines these samples in a

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Robust Monte Carlo methods for light transport simulation

The concept of species redundancy in ecosystem processes is troublesome because it appears to con- tradict the traditional emphasis in ecology on species singularity. When species richness is high, however, eco- system processes seem clearly insensitive to considerable variation in biodiversity. Some elementary princi- ples from reliability engineering, where engineered redundancy is a valued part of systems design, suggest that we should rethink our stance on species redundancy. For example, a central tenet of reliability engineer- ing is that reliability always increases as redundant components are added to a system, a principle that di- rectly supports redundant species as guarantors of reliable ecosystem functioning. I argue that we should em- brace species redundancy and perceive redundancy as a critical feature of ecosystems which must be preserved if ecosystems are to function reliably and provide us with goods and services. My argument is de- rived from basic principles of reliability engineering which demonstrate that the probability of reliable sys- tem performance is closely tied to the level of engineered redundancy in its design. Empirical demonstrations of the value of species redundancy in ecosystem reliability would provide new insights into the ecology of communities and the value of species conservation.

https://www.cedarcreek.umn.edu/biblio/fulltext/t1866.pdf

Species Redundancy and Ecosystem Reliability

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The EGSnrc Code System: Monte Carlo Simulation of Electron and Photon Transport

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Using Mpi Portable Parallel Programming With The Message Passing Interface

This books presents a balanced overview of the major methods currently available for obtaining numerical solutions in neutron and gamma ray transport. It focuses on methods particularly suited to the complex problems encountered in the analysis of reactors, fusion devices, radiation shielding, and other nuclear systems. Derivations are given for each of the methods showing how the transport equation is reduced to sets of algebraic equations suitable for solution on a digital computer. The limitations of the methods and their suitability for different classes of problems are discussed in terms of computer memory, time requirements, and accuracy.

Computational Methods of Neutron Transport

Probability and Sampling. Continuous Random Variables. Quality and Its Measures. Data and Distributions. Reliability and Rates of Failure. Loads, Capacity, and Reliability. Reliability Testing. Redundancy. Maintained Systems. Failure Interactions. System Safety Analysis. Appendices. Answers to Odd--Numbered Exercises. Index.

Elmer E Lewis

Papers

Computational Methods of Neutron Transport

Introduction to reliability engineering

Monte Carlo simulation of Markov unreliability models

Nuclear Power Reactor Safety

Nuclear power reactor safety