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.
- 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)
- 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.
- 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.
- 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
- 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).
- 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%