## Research objective

Using modern mathematical tools and high-performance computing, we can now incorporate elements of uncertainty in simulations to account for lack of knowledge about input parameters, variability in operating conditions, or inappropriate modelling assumptions. The aim of uncertainty quantification (UQ) is to provide more accurate predictions about systems’ behaviour and enable replacement of physical experiments with non-destructive virtual testing.

The objective of our work is to develop such rigorous systematic mathematical methods in both data analytics, modelling and simulation. Errors in computational engineering arise from multiple sources: numerical schemes, boundary conditions, measurements, etc. We focus on non-intrusive approaches which enable use of existing codes and straightforward parallel evaluation for quantities of interest.

## Case study

## Studying in-service degradation

Problem

Advanced gas-cooled reactors are currently the main source of nuclear energy in the UK. Fuel assemblies are composed of fuel pins with helical ribs that are meant make heat transfer more efficient. In extreme environments with high temperatures and pressures, layers of carbon deposits have been observed to form from the coolant on the fuel pin surface. Their presence affects heat transfer due to the insulating properties of carbon and changes to the effective height of the ribs, which detracts from the efficiency of the cooling process. This leads to increased temperature in the steel component, which will result in thermal fatigue and increased pin failures.

Solution

To understand the probability of failures, we are studying heat transfer impairment (HTI), which is obtained from computational fluid dynamics models under different flow conditions. We represent the deposit height and error in thermal conductivity estimates as random variables uniformly and normally distributed, respectively. Applying polynomial chaos expansion, we construct a surrogate model that allows us to calculate the stochastic response of the system’s probability distribution of temperature due to varying input parameters. In addition, to reduce the number of expensive calculations, we can use the model to obtain simulation results for unseen design points. For one-dimensional studies (considering only deposit height), as few as 11 simulations are required to replace the original simulator.

## Case study

## Uncertainty quantification in transient modelling

Problem

Knowledge of a system’s response to unknown operating conditions can be of key importance in engineering design. A standard engineering approach might involve performing a handful of calculations to assess the influence of operating conditions on quantities of interest. However, this simple approach may provide an incomplete picture, potentially leading to poor design decisions. A brute-force approach (performing a large number of simulations over a range of operating conditions) can provide a more complete mapping between inputs and outputs, but at a significantly higher computational cost.

Solution

We apply advanced uncertainty quantification techniques, such as non-intrusive multi-element polynomial chaos, which allow us to maximise the knowledge extracted from the modelled system and, crucially, its response to uncertain input parameters. This is done without changing existing simulation codes. We demonstrate the use of these tools by assessing the propagation of thermal transients within a u-shaped bend. A hot shock is introduced at the inlet and allowed to propagate through the domain. This use case is relevant to nuclear applications, where hot–cold cycles can lead to thermal fatigue and material failure. We perform such a study in a more efficient manner than the brute-force approach — at least 1–2 orders of magnitude lower computational costs with no loss in accuracy.

## Case study

## Uncertainty quantification at multi-scale interface

We looked at parametric uncertainties related to modelling particle interactions through Lenard–Jones potentials and their effect on the substance viscosity which in turns affects the velocity profiles of the fluid dynamics system. With the use of surrogate model techniques computational costs related to propagation of uncertainties through a chain of models can be substantially reduced. In some cases the speedup factor we achieved was as high as 100.

## Case study

## Waves through heterogenous media

Problem

Multi-scale problems are one of the greatest challenges in computational engineering. The objective is to understand how large systems, such as a turbine or pump, are affected by small-scale phonemena such as molecular interaactions. It is very common that the input parameters used in modelling vary or are associated with measurement errors. Therefore, we need to calculate these inaccuracies to obtain realistic predictions. We consider the very challenging problem of uncertainty quantification for acoustic wave propagation in a heterogeneous medium with prescribed covariance.

Solution

A multi-level Monte Carlo (MLMC) method is used along with a novel technique for efficient generation of random fields with a given rank. The strategy of MLMC based on telescoping sums allows for reduced computational cost of sampling with respect to the classical approach or general polynomial chaos, which suffers from the course of dimensionality.

## Ask the experts

### Robert Sawko

### Małgorzata Zimoń

## Publications

[1] S. Antão, E. Bentivegna, W. Lui, C. Moulinec, R. Sawko, A. Skillen, C. Thompson, M. Zimon,

“Uncertainty Quantification of Conjugate Heat Transfer in Stratified Fluid Flow: Application of non-intrusive polynomial chaos”

Technical Report, ePubs, Science & Technology Facilities Council, 2019.

[2] M.J. Zimoń, R. Sawko, D.R. Emerson, C. Thompson,

“Uncertainty Quantification at the Molecular–Continuum Model Interface,”

*Fluids* **2**, 12, 2017.

[3] H.N. Najm,

“Uncertainty Quantification and Polynomial Chaos Techniques in Computational Fluid Dynamics,”

*Annual Review of Fluid Mechanics* **41**, 35-52, 2009.

[4] M. Salloum, K. Sargsyan, R. Jones, H.N. Najm, B. Debusschere,

“Quantifying sampling noise and parametric uncertainty in atomistic-to-continuum simulations using surrogate models,”

*Multiscale Model. Simul.* **13**, 953-976, 2015.