# NE 155

Course Title:

Introduction to Numerical Simulations in Radiation Transport

Course Units:

3

Catalog Description:

- 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:

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

Course Objectives:

- 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:

- 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 2nd 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.

Textbook(s) and/or Other Required Materials:

- 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).

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 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%