DAVID SIRAJUDDIN UNIVERSITY OF WISCONSIN - MADISON DEPT. OF NUCLEAR ENGINEERING AND ENGINEERING...

DAVID SIRAJUDDINU N I V E R S I T Y O F W I S C O N S I N - M A D I S O N

D E P T. O F N U C L E A R E N G I N E E R I N G A N D E N G I N E E R I N G P H Y S I C S N E E P 7 0 5 – A D V A N C E D R E A C T O R T H E O R Y, D R . D O U G L A S S L . H E N D E R S O N

F I N A L P R E S E N T A T I O N

D E C E M B E R 1 7 , 2 0 1 0

Variational Methods Applied to the Even-Parity Transport Equation

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

Itcanbeshown.comSirajuddin, David

Motivation

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

Discrete ordinates

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


Outline






Development of the even-parity transport equation

Transport equation:

Boundary conditions

Define even/odd angular-parity components


(even)

(odd)


The angular flux is defined in terms of the even/odd parity fluxes

where and

Scalar flux

Current



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!


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)

Remarks on the even-parity transport equation

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 principleItcanbeshown.comSirajuddin, David

Outline






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+


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]


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


“first variation”

“variation”

“second variation”

“zeroeth variation”


The first variation:

Stationary solutions, , require


“first variation”

Examine term-by-term



Each term must independently vanish





Recall

Term 1 vanishes if is a solution to the even-parity transport equation.

This is called our Euler-Lagrange Equation




must satisfy the vacuum boundary conditionOr, equivalently,

,



Modified natural boundary condition


require no variation = 0

or,

on the reflected surface



Essential Boundary Condition

Modified natural boundary condition (slab geometry)

Outline






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



Recasting in terms of matrices Define ,

Inserting into the Vladimirov functional:

where


and

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



Introducing a variation in the trial function

Where , stationary solutions then imply

This general procedure is the basis for our solution strategyItcanbeshown.comSirajuddin, David


Outline






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


1-D slab methods

And


3-D 1-D

1-D slab methods

Translating the boundary conditions to 1-D


3-D 1-D

Outline






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]


Linear piecewise trial functions are used for hj(x)


x ≤ xj-1

xj-1 ≤ x ≤ xj

xj ≤ x ≤ xj+1

xj+1 < x


Inserting into the

functional gives

Where



Inserting into the

functional gives

Where


Each term involves a product of basis

Recall, only neighboring finite elementsAre nonzero A and B are N x N tridiagonal symmetric matrices


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


Outline






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


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




number of ways

Discrete ordinates




Angular discretization: integral equations

Beginning again with the result of the spatial discretization

Isolating the angular flux and integrating over angle:


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 rulesItcanbeshown.comSirajuddin, David

Collision Probability Method

Even-parity integral equations



number of ways

Discrete ordinates




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



Where

i.e. the angular dependence is contained it these terms



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

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)


Outline






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.

Integral transport 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 accuracyItcanbeshown.comSirajuddin, David

Conclusions

Legendre polynomial expansions 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.


Outline






References

Lewis, E.E. and Miller, W.F. Jr. Computational Methods of Neutron Transport. Wiley-Interscience. January 1993.


Additional Slides

Vacuum boundary condition

Subtracting both equations, and using the definitions

,


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