1 / 55

Variational Methods Applied to the Even-Parity Transport Equation

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. Outline.

kamil
Download Presentation

Variational Methods Applied to the Even-Parity Transport Equation

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. 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

  2. Outline • Motivation • Development of the even-parity transport equation • Variationalconcepts • 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

  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 Itcanbeshown.com

  4. 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 Sirajuddin, David Itcanbeshown.com

  5. Outline • Motivation • Development of the even-parity transport equation • Variationalconcepts • 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 • Transport equation: • Boundary conditions • Define even/odd angular-parity components (even) (odd) Sirajuddin, David Itcanbeshown.com

  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 Sirajuddin, David Itcanbeshown.com

  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! Sirajuddin, David Itcanbeshown.com

  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) Sirajuddin, David Itcanbeshown.com

  10. 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  variationalextremum principle Sirajuddin, David Itcanbeshown.com

  11. Outline • Motivation • Development of the even-parity transport equation • Variationalconcepts • 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+ Sirajuddin, David Itcanbeshown.com

  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] Sirajuddin, David Itcanbeshown.com

  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” Sirajuddin, David Itcanbeshown.com

  15. Theory of Calculus of Variations: Stationary solutions • The first variation: • Stationary solutions, , require “first variation” Sirajuddin, David Itcanbeshown.com

  16. Theory of Calculus of Variations: Stationary solutions • Examine term-by-term Sirajuddin, David Itcanbeshown.com

  17. Theory of Calculus of Variations: Stationary solutions • Each term must independently vanish • Examine term-by-term Sirajuddin, David Itcanbeshown.com

  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 Sirajuddin, David Itcanbeshown.com

  19. Theory of Calculus of Variations: Stationary solutions • Examine term-by-term Sirajuddin, David Itcanbeshown.com

  20. 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

  21. Theory of Calculus of Variations: Stationary solutions • Examine term-by-term Sirajuddin, David Itcanbeshown.com

  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) Sirajuddin, David Itcanbeshown.com

  23. Outline • Motivation • Development of the even-parity transport equation • Variationalconcepts • 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 Sirajuddin, David Itcanbeshown.com

  25. Solution of the variation problem: Ritz procedure • Recasting in terms of matrices • Define ,  • Inserting into the Vladimirov functional: • where and Sirajuddin, David Itcanbeshown.com

  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 Sirajuddin, David Itcanbeshown.com

  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 Sirajuddin, David Itcanbeshown.com

  28. Outline • Motivation • Development of the even-parity transport equation • Variationalconcepts • 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 Sirajuddin, David Itcanbeshown.com

  30. 1-D slab methods And  Sirajuddin, David Itcanbeshown.com

  31. 1-D slab methods • Translating the boundary conditions to 1-D Sirajuddin, David Itcanbeshown.com

  32. Outline • Motivation • Development of the even-parity transport equation • Variationalconcepts • 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+1is 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

  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 Sirajuddin, David Itcanbeshown.com

  35. 1-D methods: spatial discretization • Inserting into the functional gives • Where Sirajuddin, David Itcanbeshown.com

  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 Sirajuddin, David Itcanbeshown.com

  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 Itcanbeshown.com

  38. Outline • Motivation • Development of the even-parity transport equation • Variationalconcepts • 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 Itcanbeshown.com

  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 Itcanbeshown.com

  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 Itcanbeshown.com

  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 Itcanbeshown.com

  43. 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

  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 Itcanbeshown.com

  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 Itcanbeshown.com

  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 Sirajuddin, David Itcanbeshown.com

  47. Angular discretization: legendre polynomial expansion • Where • i.e. the angular dependence is contained it these terms Sirajuddin, David Itcanbeshown.com

  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 Sirajuddin, David Itcanbeshown.com

  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) Sirajuddin, David Itcanbeshown.com

  50. Outline • Motivation • Development of the even-parity transport equation • Variationalconcepts • 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

More Related