1. 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
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2. 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
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3. 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
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4. 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
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5. 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

6. Development of the even-parity transport equation
Boundary conditions
Define even/odd angular-parity components
(even)
(odd)
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7. 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
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8. 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!
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9. 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)
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10. 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
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11. 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

12. 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+
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13. 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]
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14. 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”
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15. Theory of Calculus of Variations: Stationary solutions
The first variation:
Stationary solutions, , require
“first variation”
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16. Theory of Calculus of Variations: Stationary solutions
Examine term-by-term
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17. Theory of Calculus of Variations: Stationary solutions
Each term must independently vanish
Examine term-by-term
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18. 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
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19. Theory of Calculus of Variations: Stationary solutions
Examine term-by-term
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20. Theory of Calculus of Variations: Stationary solutions
Examine term-by-term
must satisfy the vacuum boundary condition
Or, equivalently, ,
Modified natural boundary condition
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21. Theory of Calculus of Variations: Stationary solutions
Examine term-by-term
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22. 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)
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23. 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

24. 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
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25. Solution of the variation problem: Ritz procedure
Recasting in terms of matrices
Define , 
Inserting into the Vladimirov functional:
where
and
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26. 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
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27. 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
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28. 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

29. 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
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30. 1-D slab methods
And 
3-D
1-D
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31. 1-D slab methods Translating the boundary conditions to 1-D 3-D 1-D
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32. 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

33. 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]
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34. 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
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35. 1-D methods: spatial discretization
Inserting into the
functional gives
Where
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36. 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
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37. 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
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38. 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

39. 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
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40. 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
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41. 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
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42. 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
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43. Angular discretization: integral equations
Beginning again with the result of the spatial discretization
Isolating the angular flux and integrating over angle: 
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44. 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
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45. 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
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46. 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
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47. Angular discretization: legendre polynomial expansion
Where
i.e. the angular dependence is contained it these terms
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48. 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
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49. 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)
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50. 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

51. 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
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52. 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
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53. 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

54. References
Lewis, E.E. and Miller, W.F. Jr. Computational Methods of Neutron Transport. Wiley-Interscience. January 1993.
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55. Additional Slides Vacuum boundary condition
Subtracting both equations, and using the definitions 
 ,
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