A Method for Improving The Eigenvalue In Transport A Method for Improving The Eigenvalue In Transport ProblemsProblems

Simon R. Merton, Prof. Richard Smedley-Stevenson Computational Physics Group, AWE plc

Prof. Chris PainDepartment of Earth Science and Engineering, Imperial College London

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ContentsContents

● Motivation

● Transport Equation

● Discretisation

● Eigenvalue and Normalisation of Eigenvector

● Correction Procedures

● Error Improvement

● Conclusions

MotivationMotivation

●The need to obtain high fidelity eigenvalues on coarse meshes whose native accuracy is poor - fine meshes have desired accuracy but are too expensive

●The practicality of obtaining eigenvalues on a coarse mesh and improving them with an in-line defect iteration

●Convergence rate of eigenvalues is improved considerably using adjoint methods to recover the error they contain

●Indication of where in phase space adaption is needed and the potential thishas for automating mesh optimisation, AMR and grid generation

●Accurate functional estimates are important in other radiation/neutron transport problems – applications include dose rate, pin power, K

eff

●Extends to other types of functional such as detector response and shielding where error recovery in fixed source calculations would be useful

MotivationMotivationAdjoint based error measures have been successfully applied to the diffusionequation in eigenvalue problems and give significant improvement to meshconvergence rate in the K-eigenvalue

MotivationMotivationAdjoint based methods have been used successfully for the transport equationfor mesh adaption/anisotropic mesh optimisation (ref. Baker, Goffin, Lathouwers, Giles, Pierce,Venditti, Darmofal.)

Anisotropic Mesh Optimisation

(courtesy C. Baker)

Transport EquationTransport EquationThe transport equation describes the advective transport of neutral particles such as photons and neutrons through a material

The source term generally comprises a number of terms:

Interested in eigenvector solution so eigenvalues can be derived, typically viainverse power iteration on the fission term.Usefulness of these quantities depends on accurate discretisation of the equation

1g

∂r , , E

∂ t⋅∇r , , Et r ,E r , , E−qr , ,E =0

qr , , E=s r E ∫ f r , E r , , Ed d E

∫ sr , , EE r , E , d d E

DiscretisationDiscretisation

●Galerkin-weighted discontinuous finite elements (integrated across element)

●Using x-y steady state geometry

●Integrating advection term by parts to couple the elements

●Spherical Harmonics (PN), Discrete Ordinate (SN) and Linear Octahedral waveletbases in direction of particle travel. Code is referred to as arbitrary angle as the methodsare independent of the choice of this basis function

●Linear, quadratic and cubic polyhedral element type available in code (higher ordershave been useful for benchmarking error recovery schemes)

DG ImplementationDG ImplementationDiscontinuous Galerkin (DG) method first proposed for computational neutron transport in1973by Read & Hill (LANL). Use basis functions N

i as the weight functions and integrate by parts

Continuity of solution is not enforced on element boundaries. This allows capture of radiation fieldsthat are very poorly behaved. However, DG uses more spatial unknowns on the grid. Incominginformation is usually taken from the upwind element:

∫−1

1

N iA⋅∇H−BdV=∫−1

1

−A⋅∇ N iN iH−BdV∫

A⋅nN id

DG methods have a rapid convergence rate but sharp gradients can cause oscillations. Someform of stabilisation/error control is needed. DG has been used successfully for both spatial andtemporal discretisation. Compared to continuous FE transport, DG works extremely well.

i

2i

1i

2i−1

n⋅0 n⋅0

2i

1i

1i1i

EigenvalueEigenvalueDiscrete equation for ith eigenvector may be written in discrete residual form. Eigenvector isobtained via inverse power iteration on the fission source

R i=A−iB i r =0

ith eigenvalue may be written as

i=bT A i

bTB i

In which is the unit vector =bT 1,1,1,...1T

Improvement to the eigenvalue is obtained by enriching the eigenvector, in a post-processingstep. Approximations to the error in the eigenvalue from the power iteration can then be derivedand used to improve it.

Eigenvector NormalisationEigenvector Normalisation

Simplifications arise if certain normalisations are applied to the eigenvector, e.g.

bTB i=1

Eigenvector normalisation iT B i

T=1

Using the fission source normalisation, equation (3) simplifies

Fission source normalisation

i=bT A i

bTB i

i=bT A i

Eigenvector NormalisationEigenvector Normalisation

Eigenvector normalisation gives rise to a saddle point:

Therefore higher derivatives exist and so the Hessian matrix needs to be computed.

i=1 ,2 ,3

3Choose that satisfies a normalisation condition and obtain associated eigenvalue.

Each normalisation can be plotted: Let

Choose and arbitrarily12

Fission Source NormalisationFission Source Normalisation

Fission source normalisation produces zero derivatives greater than first order.

Therefore first order Taylor expansion is exact as all higher terms are zero.

Correction ProcedureCorrection ProcedureExpand residual and eigenvalue in a first order Taylor series

i i−i i=

∂i

∂ i

i− i

R i−R i≈−R i=∂R∂ i

i− i

Derive from these two equations an approximation to the defect in i i

i i−i i=

−∂i∂ i

∂R i

∂ i

−1

R i

=R T ∂R i

∂ i

−T

−∂i

∂ i

T

=R T i

*

(exact if fission source norm is used)

(repeat for residual)

(Refs: Venditti, Darmofal, Giles & Pierce, Ainsworth & Oden, Becker, Rannacher)

Bicubic SolutionBicubic Solution

Fit bicubic surface across continuous component, using the gradients and cross-derivativeof the continuous component to define the coefficients local to each element:

Finally, derive eigenvalue based on the “enriched” bicubic eigenvector

i=bT A i

bc

bTB ibc

Solve low-order system for eigenvector then compute gradients. This can be difficult to doproperly for DG, and obtaining these gradients accurately remains a topic of research.However, one can decompose the finite element solution into a continuous component anda discontinuous component:

This has been shown to be better, in many cases, than the solution obtained using a linearelement type but without the computational cost of a higher order element type. However, there is no information in the above equation on how the error is distributed.

i= iCi

ibc=∑

i=0

3

ixi×∑

j=0

3

j yj

21

3 443

1 2

Local Quadratic/Cubic SolutionLocal Quadratic/Cubic SolutionA second method has been implemented that enriches the linear element eigenvector byinserting quadratic/cubic elements. This involves prolongating the linear eigenvector onto aquadratic or cubic mesh, then using the prolongated values that are on the edge of anelement as the boundary condition for a full quadratic/cubic solve on that element, treatingeach element on the mesh in turn. This replaces the 4 local values with either 9 or 16 values.

linear value

prolongated value

local q/c value

21

3 4

21

3 4

This method has the advantage that gradients are not required; it may have similarities withcertain sub-cell/split cell approaches already in use for transport/advection problems.

Eigenvalue ImprovementEigenvalue Improvement

Eigenvalue ImprovementEigenvalue Improvement

Eigenvalue ImprovementEigenvalue Improvement

Eigenvector ImprovementEigenvector Improvement

ConclusionsConclusionsA sensitivity (goal-based) error metric has been defined for eigenvalue correction in problems containing a fission source. It has been shown how this applies when the eigenvector is normalised to the fission source.

Attempts have been made to improve the eigenvalue by post-process enrichment of the eigenvector, using higher order smoothing locally across each element. This has been tested in a couple of simple problems.

This method has been applied to the transport equation in x-y geometryusing discontinuous finite elements and spherical harmonics in direction ofparticle travel.

Initial results, are encouraging, although may not be as reliable as for diffusion. The potential for mesh adaption should be investigated, as should methods for accurately calculating the eigenvector gradient with DG.

This work has scope for further investigation and collaboration between AWE plc and Imperial College, and collaboration with other academic groups pursuing similar strategies.

AcknowledgementsAcknowledgements

Prof. Richard Smedley-Stevenson(AWE plc)

Prof. Chris PainDr. Andrew BuchanAhmed El Sheikh

(Department of Earth Science and Engineering, Imperial College London)