NE 155 - INTRODUCTION TO NUMERICAL SIMULATIONS IN RADIATION TRANSPORT (3 units)

Computational methods used to analyze radiation transport described by various differential, integral, and integro-differential equations. Numerical methods include finite difference, finite elements, discrete ordinates, and Monte Carlo. Examples from neutron and photon transport; numerical solutions of neutron/photon diffusion and transport equations. Monte Carlo simulations of photon and neutron transport. An overview of optimization techniques for solving the resulting discrete equations on vector and parallel computer systems. (Spring) Wirth

Catalog Description

• 155. Introduction to Numerical Simulations in Radiation
  Transport. Computational methods used to analyze radiation transport
  described by various differential, integral, and integro-differential
  equations. Numerical methods include finite difference, finite
  elements, discrete ordinates, and Monte Carlo. Examples from neutron
  and photon transport; numerical solutions of neutron/photon diffusion
  and transport equations. Monte Carlo simulations of photon and
  neutron transport. An overview of optimization techniques for
  solving the resulting discrete equations on vector and parallel
  computer systems.

Course Prerequisite

• Mathematics 53 and 54

Prerequisite knowledge and/or skills

The course uses the following knowledge and skills
from prerequisite and lower-division courses:

• solve linear, first and second order differential
  equations.
• linear algebra, vector calculus
• computer language knowledge (C, C++, FORTRAN)

Textbook(s) and/or other required material

• No required textbook, course notes + handouts

References:

• R.J. Schilling and S.L. Harris, �Applied Numerical
  Methods for Engineers using MATLAB and C�,
  
  Brooks/Cole, CA (2000)
• C. Pozrikidis, �Numerical Computation in Science
  and Engineering�, Oxford University Press, NY (1998)
• T.J. Akai, �Applied Numerical Methods for Engineers�,
  J. Wiley & Sons, Inc, NY (1994
• R.L. Burden and J.D. Faires, �Numerical Analysis�,
  PWS Publishing, MA (1993)
• E.E. Lewis and W.E. Miller, Jr., �Computational
  Methods of Neutron Transport�, American Nuclear
  
  Society, IL (1993)
• J.J. Duderstadt and L.J. Hamilton, �Nuclear Reactor
  Analysis�, J. Wiley & Sons, NY (1976).

Course objectives and outcomes

Course Objectives: It is the instructor's
intention to...

• Review numerical analysis fundamentals (systems
  of linear algebraic equations, linear algebra, eigenvalues and
  eigenvectors of a matrix, spectral radius of a matrix, direct
  and iterative methods for solving linear systems, numerical differentiation
  and integration).
• Introduce the numerical approaches used to solve
  fixed-source and criticality problems in analysis of neutron transport/diffusion
  in nuclear reactor core and other nuclear systems.
• Discuss the basic characteristics of deterministic
  and Monte Carlo approaches to numerical solution of these problems.
• Illustrate, with examples drawn mostly from one-dimensional
  systems, the advantages and disadvantages of various discretization
  schemes and convergence criteria, and their influence on the accuracy
  of particular numerical methodology.
• Introduce the specific features of MCNP � a production
  level Monte Carlo code for simulation of neutron and photon transport
  in complex geometries, and illustrate the use of MCNP in various
  areas of nuclear engineering.
• Develop computational skills that may be required
  for the upper-division design course (NE 170) and/or graduate-level
  reactor physics, reactor design or numerical analysis courses.
• Introduce parallel computing concepts.

Course Outcomes: Students must be able to...

• Write discretized forms of neutron diffusion and
  transport equations in one-dimensional geometries, with full understanding
  of the discretization requirements for spatial, anglular, temporal,
  and energy variables.
• Construct simple numerical models to solve one
  group steady state diffusion and transport equations for simplified
  systems, both non-multiplying and multiplying.
• Construct simple numerical models to solve point
  reactor kinetics equations.
• Evaluate the accuracy of numerical solutions against
  closed-form analytical solutions for simplified examples.
• Prepare MCNP inputs for more complex problems
  (2D/3D) and understand the MCNP outputs.

Topics covered

• Review the basic characteristics of deterministic
  and probabilistic numerical simulations of physical processes.
• Review the fundamentals of numerical analysis:
  systems of linear algebraic equations, direct and iterative methods
  of solving these systems, eigenvalues and eigenvectors, interpolation
  and polynomial approximation, numerical differentiation and integration.
• Numerical solution of initial value problems -
  point-reactor kinetics equation: Taylor, Runge-Kutta, Predictor-Corrector
  numerical methods.
• Review of neutron transport and diffusion theory
• Numerical solutions of the 2 nd order ordinary
  differential equations � neutron diffusion equation in 1D: formulation
  of the finite-difference equations for the "fixed-source"
  problem, direct and iterative solutions, formulation of the finite-difference
  equations for the "eigenvalue-criticality" problem,
  power and "inverse" power iterative methods. Formulation
  of multigroup diffusion equations.
• Numerical solutions of integro-differential equations
  � neutron transport equation in 1D: spatial discretization in
  slab geometry (diamond-difference, step-difference, stepcharacteristic
  methods), angular discretization (discrete-ordinates Sn method),
  solutions of fixed-source problems without scattering, iterative
  methods for solving discretized equations, source iteration for
  k-eigenvalue problems, convergence of source iteration method,
  multidimensional discrete ordinates methods (angular quadrants,
  ray effects, streaming effects). Modern discrete ordinates codes
  for neutron transport. Optimization for vector and parallel processing.
• Probabilistic numerical simulations � Monte Carlo
  method: continuous and discrete probability distributions, probability
  density function, cumulative probability distribution function,
  random numbers, random sampling, complex geometry description
  and ray tracing, analog and non-analog Monte Carlo, importance
  sampling, variance reduction methods, error estimation, Monte
  Carlo simulation of neutron and photon transport, parallel Monte
  Carlo simulations, introduction to MCNP code.

Class/laboratory schedule

• This is primarily a lecture course, meeting two
  times a week for 80-minute lectures. Students are expected to
  spend additional time outside of class developing their own computational
  models.

Contribution of course to meeting the professional
component

• This course contributes primarily to the students'
  knowledge of engineering topics, and does provide design experience.
• Students are required to work on one to two projects
  involving writing their own codes and/or solving more complex
  problems using MCNP (for example � designing critical systems
  � criticality search).

Relationship of course to undergraduate degree
program objectives

• This course primarily serves students in the department.
  The information below describes how the course contributes to
  the undergraduate program objectives.
• This course contributes to the NE program objectives
  by providing education in a fundamental area of numerical simulations
  of radiation transport which is important for a career in nuclear
  engineering. It does not provide students with direct design experience,
  but includes substantial discussion and illustration of design
  issues.

Assessment of student progress toward course objectives

• Homework problem sets: 30%
• Exams: Two midterm and a Final 50%
• Project: 20%