Methods Towards a Best Estimate Radiation Transport Capability: Space/Angle Adaptivity and Discretisation Error Control in RADIANT Mark Goffin - EngD Research Engineer Christopher Baker – EngD Research Engineer Dr Andrew Buchan Dr 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.

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

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.

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 k eff

Eigenvalue based adaptivity example Initial mesh Eigenvalue adapted mesh

Usability - Output Output in various formats readable by various free visualisation software. GMSH

Usability - Output MayaVi Paraview

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/Fail Developers 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, k eff ) 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 meshingAnisotropic 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.