Intrusive Polynomial Chaos for Uncertainty Quantification in CFD using OpenFOAM

Computational Fluid Dynamics (CFD) involves simulating a physical system with a model. In such simulations, uncertainties may arise from various sources, namely, initial and boundary conditions, material properties, model parameters, etc. It is therefore important to compute the confidence intervals in the predictions made using these CFD simulations. To reflect the uncertainty in the numerical solution, we need efficient Uncertainty Quantification (UQ) methods.

Monte Carlo (MC) sampling is one of the simplest approaches to carry out UQ analysis. However, due to its requirement of a large number of samples, the MC method is computationally expensive for application in CFD. As an alternative, we can use Polynomial Chaos (PC) representations to propagate and quantify uncertainty in CFD. This approach is based on the spectral decomposition of the random variables in terms of basis polynomials containing randomness and the unknown deterministic expansion coefficients. There are two methods to determine these coefficients, namely, Intrusive Polynomial Chaos (IPC) and Non-intrusive Polynomial Chaos (NIPC). In NIPC, these coefficients are approximated using quadrature for numerical evaluation of the projection integrals, while in IPC, a reformulation of the original model is performed resulting in governing equations for the PC mode strengths of the model output.

NIPC has an advantage that it uses the original model code without any modifications. However, the NIPC, in general, requires the solution of a much larger set of equations as compared to IPC, particularly for higher dimensional random space. Additionally, the aliasing error in NIPC can grow significantly with the number of random dimensions. This suggests that the IPC approach delivers the most accurate solutions with the least computational expense. Thus, in the present study, we focus on the IPC method for CFD -- fundamentals, implementation and applications.

As the model code, we use OpenFOAM, which is a C++ toolbox to develop numerical solvers, and pre-/ post-processing utilities to solve continuum mechanics problems including CFD. OpenFOAM (a) is a highly templated code, enabling the users to customize the default libraries as needed for their applications, and, (b) gives access to most of the tensor operations (div, grad, laplacian etc.) directly at the top-level code. This avails enough flexibility to implement the IPC frame-work for uncertainty quantification in CFD.

To this end, we have tested the intrusive PC implementation for both the laminar and the turbulent flow problems in CFD. The results are in accordance with the analytical and the non-intrusive approaches. The stochastic solver thus developed, can serve as an alternative to perform uncertainty quantification, especially when the non-intrusive methods are significantly expensive, which is mostly true for a lot of stochastic CFD problems.