# Presentation: 154700

## Second-Order Neutron

**Second-Order Neutron**
**Transport Methods**
** ***E. E. Lewis*
- Northwestern University
- e-lewis@northwestern.edu

## Scope

**Multigroup Energy Discretization**
**Time-Independent Equation**
**Engineering Software Emphasis**
**Neutronics Emphasis**
**2****nd**** Order & Coupled 1****st**** Order Equations**

- Neutronics & Shielding External Review September 21, 2000

## Outline

**The Equations**
**Some History**
**Coupled & 2****nd**** Order Diffusion Theory**
**Fine Mesh Transport Methods**
**Hybrid and Nodal Transport Methods**
**Integral Transport Methods **
**Present & Future Work**

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## 1. First-Order & Mixed Equations

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## 1. Second-Order Equations

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## 2. Some History

**Analytical Methods & Diffusion Theory**
**Vladimirov Functional (1963)**
**Variational Formulations (~1967)**
**Finite Elements in Diffusion Methods (~1970)**
**Finite Elements in Transport Calculation (~1970’s)**
**Major Engineering Codes ( 1980’s – 1990’s)**
**      ****EVENT, CRONOS, DANTE, VARIANT & more**

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## 3. Diffusion Theory

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## 3. Mixed Equations

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## 3. Second-Order Equation

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## 3. Variational Principle

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## 3. Discretization – Finite Elements

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## 3. Co Spatial Elements

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## 3. Discretized Functional

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## 3. Discretized Equations

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## 4. Even-Parity Transport

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## 4. Anisotropic Scattering

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## 4. Anisotropic Scattering

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## 4. Angular Approximations

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## 4. Angular Approximations

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## 4. Pn vs. Sn – Ray Effects

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## 4.  Funtional & Euler Equations

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## 4.  Spatial Discretization

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## 4. Reduced Functional 

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## 4. Discretized Equations

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## 5. Hybrid & Nodal Methods

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## 5. Hybrid Finite Element

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## 5. Discretized Equations

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## 5. Even-Parity Nodal Transport 

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## 5. Discretized Equations

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## 6. Even-Parity Integral Method

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## 6. Finite Element Discretization

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## 6. Even-Parity Integral Equations

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## 7. Current & Future Work

**Nodal Finite Subelement Method**
**Integral Nodal Finite Subelement Method**
**Treatment of vacuums & thick pure scattering media**
**Curvilinear coordinate systems**
**Improved Response Matrix Solution Algorithms**
**Even-odd coupled parity equation methods**
**Adaptive grid solution algorithms**
**Effective parallelization methods**

- Neutronics & Shielding External Review September 21, 2000