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**Variational Methods Applied to the Even-Parity Transport Equation**

4/12/2017 8:45 AM Variational Methods Applied to the Even-Parity Transport Equation David Sirajuddin University of Wisconsin - Madison Dept. of Nuclear Engineering and Engineering Physics NEEP 705 – Advanced Reactor Theory, Dr. Douglass L. Henderson Final Presentation December 17, 2010 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

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**Outline Motivation Development of the even-parity transport equation**

Variational concepts Ritz Procedure Statement of the variational problem 1-D slab transport Spatial discretization Angular treatment Discrete Ordinates Collision probability method Legendre polynomial expansion Conclusions References Sirajuddin, David Itcanbeshown.com

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**Outline Motivation Development of the even-parity transport equation**

Variational concepts Ritz Procedure Statement of the variational problem 1-D slab transport Spatial discretization Angular treatment Discrete Ordinates Collision probability method Legendre polynomial expansion Conclusions References Sirajuddin, David Itcanbeshown.com

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**Motivation Sirajuddin, David Itcanbeshown.com**

Computational methods of the raw form of the integrodifferential form of the transport equation are most readily facilitated by: Discrete ordinates Integral equations (e.g. collision probability methods) These methods can become computationally expensive Ray-effects many discrete ordinates must be used Marching scheme, diamond differencing iterative solution Integral equations iterating on a scattering source solving matrix equations with a full coefficient matrices Scattering source approximation method only accurate up to order O(D) detrimental for large system size calculations Computational expense may be reduced by recasting the transport equation into a variational form even-parity transport equation gives rise to a variety of approximation techniques solution requires solving a single matrix equation with sparse coefficient matrices Sirajuddin, David Itcanbeshown.com

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**Outline Motivation Development of the even-parity transport equation**

Variational concepts Ritz Procedure Statement of the variational problem 1-D slab transport Spatial discretization Angular treatment Discrete Ordinates Collision probability method Legendre polynomial expansion Conclusions References Sirajuddin, David Itcanbeshown.com

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**Development of the even-parity transport equation**

Boundary conditions Define even/odd angular-parity components (even) (odd) Sirajuddin, David Itcanbeshown.com

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**Development of the even-parity transport equation**

The angular flux is defined in terms of the even/odd parity fluxes where and Scalar flux Current Sirajuddin, David Itcanbeshown.com

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**Development of the even-parity transport equation**

The even-parity equation is arrived at by considering the transport equation evaluated at W and -W Recalling , subtracting both equations allows a relation between y- and y+ and, by definition Calculation of the even-parity flux allows the computation of the scalar flux and the current! Sirajuddin, David Itcanbeshown.com

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**Development of the even-parity transport equation**

The even-parity equation is arrived at by considering the transport equation evaluated at W and -W Adding and subtracting the above equations produces two new equations that may be combined to eliminate y- Even-parity transport equation (isotropic scattering) Sirajuddin, David Itcanbeshown.com

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**Remarks on the even-parity transport equation**

Even-parity only need to solve half the angular domain Isotropic scattering Cannot be used directly for streaming particles in vacuum (s = 0) Underdense materials (s small) must check computational algorithm is stable The equation is self-adjoint variational extremum principle Sirajuddin, David Itcanbeshown.com

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**Outline Motivation Development of the even-parity transport equation**

Variational concepts Ritz Procedure Statement of the variational problem 1-D slab transport Spatial discretization Angular treatment Discrete Ordinates Collision probability method Legendre polynomial expansion Conclusions References Sirajuddin, David Itcanbeshown.com

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**Theory of Calculus of Variations**

Variational methods aim to optimize functionals: Function: , while a functional: These functionals are often manifest as relevant integrals Examples: minimum energy, Fermat’s principle, geodesics Method: Find an appropriate functional that characterizes y + Introduce a trial function y+ + dy Enforce dy = 0 y+ Sirajuddin, David Itcanbeshown.com

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**Theory of Calculus of Variations: Vladimirov’s functional**

y+ is characterized by the even-parity transport eqn. A relevant functional F[y+] may be computed by the inner product from the self-adjoint extension of the transport operator [2] Sirajuddin, David Itcanbeshown.com

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**Theory of Calculus of Variations: Stationary solutions**

Model as Inputting into the above functional (after much algebra) the terms may be grouped according to Where the zeroeth, first, and second variations depend on , (or ), and , respectively “variation” “zeroeth variation” “first variation” “second variation” Sirajuddin, David Itcanbeshown.com

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**Theory of Calculus of Variations: Stationary solutions**

The first variation: Stationary solutions, , require “first variation” Sirajuddin, David Itcanbeshown.com

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**Theory of Calculus of Variations: Stationary solutions**

Examine term-by-term Sirajuddin, David Itcanbeshown.com

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**Theory of Calculus of Variations: Stationary solutions**

Each term must independently vanish Examine term-by-term Sirajuddin, David Itcanbeshown.com

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**Theory of Calculus of Variations: Stationary solutions**

Examine term-by-term Recall Term 1 vanishes if is a solution to the even-parity transport equation. This is called our Euler-Lagrange Equation Sirajuddin, David Itcanbeshown.com

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**Theory of Calculus of Variations: Stationary solutions**

Examine term-by-term Sirajuddin, David Itcanbeshown.com

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**Theory of Calculus of Variations: Stationary solutions**

Examine term-by-term must satisfy the vacuum boundary condition Or, equivalently, , Modified natural boundary condition Sirajuddin, David Itcanbeshown.com

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**Theory of Calculus of Variations: Stationary solutions**

Examine term-by-term Sirajuddin, David Itcanbeshown.com

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**Theory of Calculus of Variations: Stationary solutions**

Examine term-by-term require no variation = 0 or, on the reflected surface Essential Boundary Condition Modified natural boundary condition (slab geometry) Sirajuddin, David Itcanbeshown.com

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**Outline Motivation Development of the even-parity transport equation**

Variational concepts Ritz Procedure Statement of the variational problem 1-D slab transport Spatial discretization Angular treatment Discrete Ordinates Collision probability method Legendre polynomial expansion Conclusions References Sirajuddin, David Itcanbeshown.com

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**Solution of the variation problem: Ritz procedure**

Suppose we approximate the flux: Where are known even-parity shape functions, , and are unknown coefficients Inputting the approximation into the functional dsafd , and enforcing a matrix equation whose solution gives the coefficients Sirajuddin, David Itcanbeshown.com

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**Solution of the variation problem: Ritz procedure**

Recasting in terms of matrices Define , Inserting into the Vladimirov functional: where and Sirajuddin, David Itcanbeshown.com

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**Solution of the variation problem: Ritz procedure**

Note that is an N x N symmetric matrix, since And : N x 1 column vector : 1 x N row vector : N x N symmetric matrix : N x N symmetrix matrix Sirajuddin, David Itcanbeshown.com

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**Solution of the variation problem: Ritz procedure**

Introducing a variation in the trial function Where , stationary solutions then imply This general procedure is the basis for our solution strategy Sirajuddin, David Itcanbeshown.com

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**Outline Motivation Development of the even-parity transport equation**

Variational concepts Ritz Procedure Statement of the variational problem 1-D slab transport Spatial discretization Angular treatment Discrete Ordinates Collision probability method Legendre polynomial expansion Conclusions References Sirajuddin, David Itcanbeshown.com

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**1-D slab methods Slab geometry: Isotropic source distribution: S(x)**

Reflective boundary at x = 0 Vacuum boundary at x = a The functional Then becomes Sirajuddin, David Itcanbeshown.com

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1-D slab methods And 3-D 1-D Sirajuddin, David Itcanbeshown.com

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**1-D slab methods Translating the boundary conditions to 1-D 3-D 1-D**

Sirajuddin, David Itcanbeshown.com

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**Outline Motivation Development of the even-parity transport equation**

Variational concepts Ritz Procedure Statement of the variational problem 1-D slab transport Spatial discretization Angular treatment Discrete Ordinates Collision probability method Legendre polynomial expansion Conclusions References Sirajuddin, David Itcanbeshown.com

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**1-D methods: spatial discretization**

A spatial mesh is designated Each interval xj < x < xj+1 is a finite element. To discretize, segment the independent variables according to Where the yj approximate y(xj,m), and the hj are shape functions that span the finite the width of one finite element. i.e. [1] Sirajuddin, David Itcanbeshown.com

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**1-D methods: spatial discretization**

Linear piecewise trial functions are used for hj(x) x ≤ xj-1 xj-1 ≤ x ≤ xj xj ≤ x ≤ xj+1 xj+1 < x Sirajuddin, David Itcanbeshown.com

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**1-D methods: spatial discretization**

Inserting into the functional gives Where Sirajuddin, David Itcanbeshown.com

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**1-D methods: spatial discretization**

Inserting into the functional gives Where Each term involves a product of basis Recall, only neighboring finite elements Are nonzero A and B are N x N tridiagonal symmetric matrices Sirajuddin, David Itcanbeshown.com

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**1-D methods: spatial discretization**

Introducing a variation in the trial function into the functional And enforcing stationary solutions gives a single matrix equation A number of angular discretization methods may henceforth be employed to facilitate a solution Sirajuddin, David Itcanbeshown.com

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**Outline Motivation Development of the even-parity transport equation**

Variational concepts Ritz Procedure Statement of the variational problem 1-D slab transport Spatial discretization Angular treatment Discrete Ordinates Collision probability method Legendre polynomial expansion Conclusions References Sirajuddin, David Itcanbeshown.com

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**1-D methods: Angular discretization**

The angular dependence may be handled in a number of ways Discrete ordinates Collision probability methods (integral equations) Legendre polynomials Sirajuddin, David Itcanbeshown.com

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**1-D methods: Angular discretization**

The angular dependence may be handled in a number of ways Discrete ordinates Collision probability methods (integral equations) Legendre polynomials Sirajuddin, David Itcanbeshown.com

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**Angular discretization: discrete ordinates**

Beginning with the result of the spatial discretization N/2 discrete ordinates are imposed where the scalar flux may be approximated by a suitable quadrature rule The solution may be obtained by iterating on the scattering source The solution requires solving N/2 tridiagonal matrix equations due to evenness of the flux function, while the discrete ordinates equations require N tridiagonal matrix equation solutions at each step Sirajuddin, David Itcanbeshown.com

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**1-D methods: Angular discretization**

The angular dependence may be handled in a number of ways Discrete ordinates Collision probability methods (integral equations) Legendre polynomials Sirajuddin, David Itcanbeshown.com

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**Angular discretization: integral equations**

Beginning again with the result of the spatial discretization Isolating the angular flux and integrating over angle: Sirajuddin, David Itcanbeshown.com

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**Angular discretization: integral equations**

Note the similarities Collision probabilty method is of order D, even-parity method is order D2 Both have nonsymmetric, dense coefficient matrices Collision method requires analytic integration over kernels, even-parity could use quadrature rules Collision Probability Method Even-parity integral equations Sirajuddin, David Itcanbeshown.com

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**1-D methods: Angular discretization**

The angular dependence may be handled in a number of ways Discrete ordinates Collision probability methods (integral equations) Legendre polynomials Sirajuddin, David Itcanbeshown.com

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**Angular discretization: legendre polynomial expansion**

In addition to the spatial discretization all ready performed, the angular domain is discretized by a family of known even-parity angular basis functions (e.g. even-order Legendre polynomials) Consider first the angular domain Inserting this into our functional, the following is retrieved Sirajuddin, David Itcanbeshown.com

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**Angular discretization: legendre polynomial expansion**

Where i.e. the angular dependence is contained it these terms Sirajuddin, David Itcanbeshown.com

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**Angular discretization: legendre polynomial expansion**

Enforcing stationary solutions with respect to a variation in the flux gives the spatial Euler-Lagrange operator Which operates on the spatial dependence of the flux giving where Sirajuddin, David Itcanbeshown.com

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**Remarks on the even-parity transport equation**

Writing out the functional shows stationary solutions of the angular flux require solving The zeroeth moment corresponds to the scalar flux {yj}1 = j(xj) Sirajuddin, David Itcanbeshown.com

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**Outline Motivation Development of the even-parity transport equation**

Variational concepts Ritz Procedure Statement of the variational problem 1-D slab transport Spatial discretization Angular treatment Discrete Ordinates Collision probability method Legendre polynomial expansion Conclusions References Sirajuddin, David Itcanbeshown.com

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Conclusions A variational formalism may be used as a basis for developing algorithms on the neutron transport equation Discrete ordinates Integral transport Legendre polynomial expansion The even-parity transport equation was shown to facilitate matrix equations that involved sparse coefficient matrices in certain cases (discrete ordinates, legendre polynomial expansion) Discrete ordinates: Both possess order D2 accuracy The variational structure allowed for the solution of only N/2 tridiagonal matrix equations at each iteration, while the discrete ordinates demanded N tridiagonal matrix equation solution. Discrete ordinates equations involve lower triangular matrices easier to solve. A similar set of integral equations was arrived at with variational methods as was found in collision probability methods. Collision probability methods are of order D, while variational approaches allow accuracy of order D2 Collision probability methods suffer in accuracy for large systems due to the source approximation, while variational methods may be adapted to fit the needed accuracy Sirajuddin, David Itcanbeshown.com

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**Conclusions Legendre polynomial expansions Sirajuddin, David**

Both the PN equations and the variational method allow for a diffusion approximation solution involving tridiagonal coefficient matrices. Higher order PN equations require iterative solutions, higher order variational equations involve more dense matrices The methods presented pertained to isotropic scattering and one-dimension. Anisotropic scattering has also been worked into variational formulations, and algorithms have been to two-dimensions with a variety of finite element types, however the treatment is more involved. Sirajuddin, David Itcanbeshown.com

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**Outline Motivation Development of the even-parity transport equation**

Variational concepts Ritz Procedure Statement of the variational problem 1-D slab transport Spatial discretization Angular treatment Discrete Ordinates Collision probability method Legendre polynomial expansion Conclusions References Sirajuddin, David Itcanbeshown.com

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References Lewis, E.E. and Miller, W.F. Jr. Computational Methods of Neutron Transport. Wiley-Interscience. January 1993. Sirajuddin, David Itcanbeshown.com

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**Additional Slides Vacuum boundary condition**

Subtracting both equations, and using the definitions , Sirajuddin, David Itcanbeshown.com

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