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Methods Towards a Best Estimate Radiation Transport Capability: Space/Angle Adaptivity and Discretisation Error Control in RADIANT
Mark Goffin - EngD Research EngineerChristopher Baker – EngD Research Engineer
Dr Andrew BuchanDr Matthew Eaton
Prof. Chris Pain
Contents
• Introduction• RADIANT– Spatial discretisation – Spatial adaptivity– Angular discretisation– Angular adaptivity– Goal based adaptivity
• Automated verification and validation• Future goals and objectives
Introduction
• The Boltzmann transport equation is used extensively in both reactor physics, nuclear criticality and reactor shielding calculations.
• RADIANT (RADIAtion Non-oscillatory Transport) is a deterministic transport code developed at Imperial College.
SH
Spatial Discretisation – Multi(sub-grid)scale Method
• Combines continuous and discontinuous finite elements to produce stable solutions to the transport equation.
• The method does not result in the large number of unknowns associated with a pure discontinuous solution.
• Enables rigorous coupling of ‘assembly level’ and ‘whole core calculations’ with reduced computational complexity.
• Enables a mathematical framework to be developed for multiscale uncertainties.
Comparison of spatial discretisation schemes
Continuous Galerkin
Even parity
Streamline Upwind Petrov-Galerkin (SUPG)
Non-linear SUPG
Discontinuous Galerkin
Multi (sub-grid) scale
C5G7 Benchmark Example
RADIANT
refeff
codeeff
k
k
Anisotropic Spatial Adaptivity
• The mesh is adapted anisotropically.• The error metric used is based on the
interpolation error of the mesh:
where H is the Hessian of the flux and ε is the desired interpolation error.
HM
2
222
2
2
22
22
2
2
zyzxz
zyyxy
zxyxx
H
Supermeshing
• Typically the transport equation is solved on a single spatial mesh.
• This is inefficient in areas where the flux needs refining for only a single energy group.
• RADIANT has the capability to use different spatial meshes for each energy group.
• Supermeshing is the process of interpolation from one mesh to the other.
+ =
Angular Discretisation
• RADIANT has the capability to implement one of three angular discretisations for the calculation:– Spherical harmonics expansion– Discrete ordinates– Angular wavelets
Angular Adaptivity using Wavelets
Dog legged duct
example
Wavelet resolution
Angular flux
Goal Based Adaptivity
• The Hessian based error metric adapts the whole mesh regardless of a regions importance (only based upon curvature of solution/flux).
• Goal based adaptivity refines regions that are of greater importance to a given variable (“goal”).
• This reduces the error to the goal under consideration.
Example “goal” functionals
• Such examples of goals are:– Reaction rates in a given region
– Multiplication factor keff
)(
)()( eff
A
FkJ
drdEdErErJ ),,(),()( det
Eigenvalue based adaptivity example
Initial mesh Eigenvalue adapted mesh
Automated Verification and validation: the future (currently implemented in our CFD codes and used by Serco)
Commit to source
Automated build
Validation
Unit tests
Parallel simulations
Serial simulations
Profiling data
collected
Pass/FailDevelopers
notified
Analytical benchmarks
Takeda benchmarks
ICSBEP
IRPhEP
Anisotropic adaptivity
Project Objectives
• Develop error measures appropriate for adaptivity in both space and angle simultaneously. Implemented within RADIANT.
• Develop the capability for the code to produce a solution for a given user input discretisation error for a specific field/value (e.g. flux, reaction rates, keff)
• Combination with work of D. Ayres and J. Dyrda to produce an uncertainty from deterministic codes that encompass discretisation error, nuclear data uncertainty and problem model uncertainty through data assimilation/model calibration methods.
Total uncertainty = Discretisation error + data uncertainty + model uncertainty
Eventual Goal of AMCG Reactor Physics Methods
Fully adaptive RT methods tailoring themselves to the physics of the problem (to a given resolution scale) capable of assessing effects of multiple uncertainties and performing inversion
Fully adaptive, fast, robust uncertainty propagating RT framework (with inversion and
appropriate adjoint error metrics)
Adaptive spatial meshing Anisotropic adaptivity in angle
Adaptivity in energy Adaptivity in time
Hierarchical solvers
Sub-grid scale
stabilisation
Multiscale model
reduction
SFEM uncertainty methods + covariance
data
Thank you for listening. Any questions…?
Acknowledgements & Questions
I would like to express my thanks to Serco, EPSRC and the Royal Academy for support.