Polynomial Chaos

Error sensivity analysis in a 1D FEM model of reactor cooling loop

Polynomial Chaos (PC) is a fancy name for the method allowing evaluation of error propagation in a FEM model or any other system described by partial differential equations (PDE). The mathematical model of a physical phenomenon to be solved is usually a set of differential equations where some parameters are uncertain and the output solution is therefore uncertain, too, with its distribution to be determined. Such a simulation may be carried out given the uncertainties of chosen particular model parameters. PC, unlike Monte Carlo (MC), is a fast and reliable algorithm with time of calculations not increasing with dimensionality of the system! Also, its accuracy does not need to be improved with enhanced sampling. These qualities make PC a great candidate for an on-the-fly calculation of errors or uncertainties. A good example of such a place is a nuclear reactor...

In this case, we study a 1D (simplified) nuclear reactor cooling loop. The simple loop model is composed of a heating wall / heat inlet (reactor fuel at fixed temperature), a pressurizer (fixed pressure), a cooling wall / heat outlet (steam generator secondary fluid at fixed temperature), and a pump (adjustable strength).

A linear model of the cooling loop of a nuclear reactor.

The system is described by a set of the PDE for the conservation of mass, momentum and energy, given below:

There are four variables: density, pressure, pressure, and flow speed. The equations are solved with Finite Elements method with periodic boundary conditions. However, because the number of variables is one more than the number of equations, we need so called closure relation, which is a linearized equation of state (EOS) for density:

A specific solution of the PDEs with no uncertainties on the parameters, in a "dry run", give a pressure distribution along the loop as this one:

Pressure distribution along the cooling loop of a nuclear reactor with no uncertainties.

Notice we have there 5 parameters provided by "user" that might have some uncertainties. These are the coolant reference density, viscosity, and heat capacity (in the energy term, e), plus the heat transfer coefficients of the nuclear core / inlet and the steam generator / outlet (present in the heat source term, Q). In this model, we can turn on and off the uncertainty of each parameter. When on, they are Gaussian-distributed with 10% standard deviation.

When we turn on uncertainties, we find out the pressure variation at the inlet is larger than that at the outlet, though the relative standard deviations are quite small.

Pressure distribution at the heat inlet and outlet.

However, what's much more important is the temperature distribution in the core, we don't want our core to melt!

Temperature distribution at the heat inlet and outlet.

One can employ Polynomial Chaos also for a time-dependent profiles, such as the stabilization of pressure in the pressurizer at the launching time. Notice how does the uncertainty vary with time and phase of the oscillation.

Time-dependent uncertainties in the pressurizer.

On the plot below, we can see how turning on different uncertainties is contributing to the distribution of the temperature outlet which can be further used for sensitivity analysis.

Contribution of parameters uncertainties to the outlet temperature distribution.

Now, how does it work? Rather than sampling N-dimensional space of parameters with brute-force Monte Carlo, we represent (expand) each distribution with a Hermite polynomial (hence the name of the method!) of small order (one is enough). The expansion is carried out in one additional dimension or "uncertainty space" (of no physical meaning). So, instead of increasing the dimensionality by N, we do it by 1 only. Now, there are basically two different PC approaches for solving the problem: the Galerkin projection and the Stochastic collocation. The Galerkin method is one of the solution methods that is intrusive, that is to say, the differential equations describing the problem considered need to be modified. The Stochastic Collocation, on the other hand, is non-intrusive. However, the major drawback of this method is that the number of problem solutions needed to be computed can grow expotentially with the number of uncertain parameters (dimension) which often called the curse of dimensionality. Fortunately, this issue can be partially solved by application of sparse integration quadratures. The idea of Smolyak’s sparse quadrature is to remove some points in the tensor grid without loss of accuracy and to modify the remaining weights.

Sparse vs. full tensor grid for Gauss-Hermite quadrature in 2D.

To give you an idea of how efficient in decreasing calculation time this can be: at the dimension N=4, the number of element is reduced from 1,296 to 953, whereas at N=8 the reduction is from 1,679,616 to 15,153! Finally the table below summarizes the effectiveness of Polynomial Chaos for a simple, N=4 case.

Performance of Polynomial Chaos method on a simple case with four uncertainties.

If you would like to know more details about this topic, feel free to contact us or check the provided links and references.

author: Dr. Karol Kulasinski

References & External Links

Karol Kulasinski, Uncertainty Quantification in Flow Models for a Primary Reactor Loop Model, Master Thesis, Institut National des Sciences et Techniques Nucléaires, 2011.

O. L. Maitre, Méthodes spectrales pour la propagation d’incertitudes dans les modèles numériques, Laboratoire de Méchanique et d’Energétique, Université d’Evry, June 2005.

N. Wiener, The Homogenous Chaos, American Journal of Mathematics, 1938.

M. S. Eldred, C. G. Webster, P. G. Constantine, Evaluation of Non-Intrusive Approaches for Wiener-Askey Generalized Polynomial Chaos, American Institute of Aeronautics and Astronautics Paper, 2008.

S. Smolyak, Quadrature and Interpolation Formulas for Tensor Products of Certain Classes of Functions, Doklady Akademii Nauk SSSR, 1964.

F. Heiss, V. Winschel, Quadrature on sparse grids, http://www.sparse-grids.de/.

M. O. Delchini, A preliminary study to assess model uncertainties in fluid flows, Texas A&M University, 2010.

Polynomial Chaos in Wikipedia

Hermite Polynomials