Project 1

A stabilized stochastic finite element second-order projection method for modeling natural convection in random porous media

Fluid flow through porous media is an ubiquitous process occurring in various applications such as fluidized beds, solidification of alloys, geothermal energy systems and oil recovery. The analysis of flow through a medium with deterministic porosity has been well studied. However, in practice, only limited statistical information is available regarding the structure and material properties of the medium. These statistics are easily extracted and reconstructed from experimental data. The porosity can thus be conveniently described by random fields. This enables us to develop a methodology that treats the porosity as input uncertainty and analyzes the propagation of this uncertainty through the governing equations of thermal and flow transport.

In this project, the porosity is treated as a random field and a Generalized Polynomial Chaos Expansion (GPCE) (stochastic finite element method) is developed. Generalized Polynomial Chaos expansions (GPCE) are an efficient means of representing random processes in stochastic in differential equations to help quantify uncertainty. Stochastic Galerkin (SG) methods based on PC expansions have a number of advantages over traditional uncertainty techniques. SG methods exhibit much faster rates of convergence than traditional Monte-Carlo methods and unlike perturbation methods and second-moment analysis SG is able to deal with highly non-linear systems with large uncertainties in the random inputs. The stochastic projection method is considered for the solution of the high-dimensional stochastic Navier–Stokes equations since it leads to the uncoupling of the velocity and pressure degrees of freedom. Because of the porosity dependence of the pressure gradient term in the governing flow equations, one cannot use the first-order projection method. A stabilized stochastic finite element second-order projection method is presented based on a pressure gradient projection.

The governing equations are the modified Navier-Stokes and the energy equations:

We consider the stochastic natural convection problem in a unit square:

Fig. 1 First-order GPCE modes of temperature.

Figure 1 shows the corresponding GPCE coefficients of the temperature. It is shown that although the flow equations depends on the random porosity, the temperature also exhibits a kind of randomness through the flow temperature coupling.

Fig 2. The standard deviation of the variables. Top left: u velocity; Top right: v velocity; Bottom left: Temperature; Bottom right: Pressure.

However, there are several limitations of this method. As the complexity of the problem or the number of random variables involved in describing the input uncertainties increases, these approaches become highly impractical from implementation and convergence points-of-view. This is especially true in the context of realistic thermal and fluid transport problems, where uncertainties in the topology of the boundary domain, boundary flux conditions and heterogeneous physical properties usually require high-dimensional random descriptors. In addition, when the solution exhibits a discontinuous dependence on the input random parameters, the gPC may converge slowly or even fail to converge. This is due to the global polynomial expansion used in the gPC which cannot resolve the local discontinuity in the random space, the well-known Gibbs phenomenon which occurs in spectral decompositions of discontinuous functions.

An detailed introduction to GPCE written by me can be found here. The paper and PowerPoint presentation can be download from here.