1. An Adaptive-Stochastic Boussinesq Solver With Safety Critical Applications In Nuclear Reactor Engineering Andrew Hagues PhD Student – KNOO Work Package 4 Supervisors: Prof Chris Pain & Prof Tony Goddard Applied Modelling and Computation Group, Department of Earth Science and Engineering

2. Contents Project aims and motivation Method – Stochastic Finite Elements Example application Future work

3. Project Aims Final deliverable is a fully 3D adaptive-stochastic Boussinesq solver for single phase, incompressible flows. Integrate stochastic solver with current reactor models. Carry out improved safety assessments for various accident scenarios from a thermal hydraulics viewpoint.

4. Why A Stochastic Model? Deterministic models cannot explicitly capture uncertainty in model parameters, limited to many simulations and predicted ‘worst case’ scenarios – computationally inefficient. A stochastic model has the ability to capture uncertainties explicitly. Examples are boundary conditions (fuel element temperatures) and material properties (viscosity)

5. SFEM History The SFEM method was originally developed in the structural mechanics community by Ghanem and Spanos and based upon two approaches – Kahunen-Loeve expansion (similar to POD methods) and Polynomial Chaos expansions. Recently the SFEM has been extended to other fields such as heat transfer, fluid flow and radiation transport. SFEM offers the unique possibility of developing coupled multi-physics uncertainty codes with large or small stochastic variations on multiple variables.

6. SFEM Method Consider a general problem: where Δ is a differential operator. u is the response (dependent) variable whose solution is required. θ is a vector of random variables whose statistics are known. f and ω are functions of known variation. We can write and a similar expression for f. Because the statistics of these functions are known we can expand them using the Karhunen-Loeve (K-L) expansion:

7. Where λ n and g n (x) are the eigenvalues and eigenfunctions of: ξ n (θ) is a set of uncorrelated random variables that are unknown. The covariance function is given by: The K-L expansion requires the covariance function of the random process to be known – acceptable for input random variables. Nothing is known about the output random variables in advance therefore an alternative method is required that is known as the Polynomial Chaos (PC) expansion.

8. Spectral SFEM 3 Where the functions ψ n (θ) are random variables and the b n (x) are unknown expansion functions. The construction of the spectral functions employed in the PC expansion (Legendre, Laguerre, etc..) depends upon the statistics of ξ n (θ) e.g. if the functions ξ n (θ) have Gaussian type statistics then Hermite polynomials would be used to form an orthogonal set of basis functions as these yield optimal convergence rates for Gaussian systems. With all the random variables now parameterised it is possible to assemble the SFEM equations and solve in a similar way to the standard FEM to obtain the PDF of the output response variables.

9. Rayleigh-Bernard Convection Ra = 70000 Ra = 80000 Ra = 90000 t = 0t = 10t = 5t = 30

10. Elements of Future Work After initial familiarisation with the necessary techniques plan is to test on a problem where a standard Gaussian shape is advected through a field where the field velocity is a stochastic variable. Develop more sophisticated model to include such things as buoyancy effects, subgrid turbulence models. Extend to allow a stochastic representation of multiple uncertainties in any desired parameters.

11. Questions?