SoftWorthy | Projects

CFD UNCERTAINTY

Overview

The importance of uncertainty modeling is clearly recognized in scientific computing, and there has been a growing interest in applications of probabilistic methods. Among the existing methods, polynomial chaos expansion has been shown to work effectively for many problems. It is based on the homogeneous chaos theory of Wiener and has been applied to various practical applications. The classical polynomial chaos expansion employs Hermite orthogonal polynomials in terms of Gaussian random variables to represent stochastic processes and is essentially a spectral expansion of random variables. More recently, a more general framework, called the generalized polynomial chaos, was developed that employs new classes of orthogonal polynomials and is more efficient to represent general non-Gaussian processes. It has a wide spectrum of applications such as stochastic ODEs, PDEs, Navier-Stokes equations, and flow-structure interactions.

In computations, there are many sources of uncertainties such as initial/boundary conditions, modeling parameters, transport coefficient, geometry, etc. Studying the influence of the randomness of input parameters on the output is therefore critical to performance. Spectral stochastic methods based on intrusive polynomials chaos expansions have shown their great potential in numerous applications. The polynomial chaos (PC) is a spectral expansion of the stochastic process in terms of the orthogonal polynomials. PC method is based on the polynomial decomposition of the uncertain variables. Whereas PC method is well established in structural mechanics, its application to CFD is more recent. Normally, CFD methods assume that the geometry of the domain is known in a deterministic sense. However, based on the manufacturing process used, there is always some geometrical uncertainty associated with the domain. Here geometrical uncertainty is represented as a random field via spatial dependence, i.e. any two points in the boundary are partially correlated. Solving the system of stochastic equations, we get the solution for the mean and variance of temperature as shown, when the first order Hermite polynomial chaos is considered for T.

Likewise, input and parametric uncertainty can also be accounted for during CFD analysis. As an example, we compute the time-dependent 2D heat equation using the finite element method in space, and a method of lines in time with the backward Euler approximation for the time derivative. The computational region is a rectangle, with homogenous Dirichlet boundary conditions applied along the boundary. The computational region is first covered with an NX by NY rectangular array of points, creating (NX-1)*(NY-1) subrectangles. Each subrectangle is divided into two triangles, creating a total of 2*(NX-1)*(NY-1) geometric "elements". Because quadratic basis functions are to be used, each triangle will be associated not only with the three corner nodes that defined it, but with three extra midside nodes. If we include these additional nodes, there are now a total of (2*NX-1)*(2*NY-1) nodes in the region. We now assume that, at any fixed time b, the unknown function U(x,y,t) can be represented as a linear combination of the basis functions associated with each node. The time derivative is handled by the backward Euler approximation. Uncertainty is also acconted for, and stochastic equations are solved using either the generalized polynomial chaos or the multi-element generalized polynomial chaos, as chosen by the user.

There are a few variables that are easy to manipulate. In particular, the user can change the variables NX and NY in the main program, to change the number of nodes and elements. The variables (XL,YB) and (XR,YT) define the location of the lower left and upper right corners of the rectangular region, and these can also be changed in a single place in the main program. The uncertainty level and the order of the polynomial chaos can also be supplied by the user.
Articles

Back

LARGE EDDY SIMULATIONS

Overview

Large-Eddy Simulation (LES) techniques with appropriate sub-grid-scale (SGS) turbulent models, combustion models and reaction mechanisms are considered as one of the promising numerical simulation approaches, balancing computational complexity and predictive accuracy. While Direct Numerical Simulations (DNS) resolve all the turbulent length scales and time scales, they are computationally expensive and impractical for high Reynolds number large scale applications. Solving the Reynolds Averaged Navier Stokes Equations (RANS), on the other hand, models the influence of turbulence on the mean flow and hence, cannot capture the unsteady flow.

In LES, rather than averaging the effect of turbulence, the equations are filtered, enabling the larger scales of turbulence to be explicitly resolved, while the effect of the smallest ones on the large scales is modelled. Resolving the large scales enables capturing part of the inherent unsteadiness in the flow, which is particularly important when dynamics at the large scales play an important role, while modelling the sub-grid scale effects ensures that the approach is computationally manageable.

For combustion in a turbulent environment, under most conditions, the chemical reactions are confined to thin layers at scales smaller than those resolved on the LES grid. In RANS models, the averaged chemical source term in the governing equations is modeled in terms of the mean field variables and the modeled fluctuations. In LES models, although finer grid resolution is used and some of the large scales are resolved, the instantaneous flame thickness is still too small to be captured by the LES grid.

In the case of premixed combustion, several approached have been suggested for modeling the filtered reaction rate terms, and models that incorporate finite rate chemistry such as the artificially thickened flame approach are of particular interest.
Articles

Back

MOBILITY SOLUTIONS

Overview

There has been little research done in the past that explicitly deals with the challenge of autonomously assessing the traversability of a vehicle over a given terrain region or obstacle under uncertainty, as most techniques either do not explicitly analyze vehicle mobility on rough terrain or rely on a deterministic study that assumes precise knowledge of vehicle and/or terrain parameters. In field conditions, however, UGVs often only have access to sparse and uncertain parameter estimates drawn from "standard" robotic sensors such as LIDAR. Moreover, significant uncertainties are often associated with estimates of vehicle parameters, due to effects such as loading, wear, fuel consumption, etc. In addition, the issue of mobility through terrain regions with non-geometric hazards (such as highly deformable or slippery regions) has not been addressed in this paradigm. It is therefore imperative to consider these uncertainties when deriving predictions of vehicle mobility. As an example, we show the simulation runs of an unmanned ground vehicle traversing a terrain with fixed motor inputs, while varying the vehicle and soil parameters. As is clearly evident, different outcomes are observed for each simulation run, indicating that parameteric and/or environmental uncertainty can significantly influence the vehicle dynamics and influence vehicle performance.

Various statistical methods for mobility prediction have been developed by the U.S. Army over the past 50 years, including the NATO Reference Mobility Model (NRMM), NRMM II, and others. These are numerical algorithms for predicting cross-country vehicle movement at length scales of several meters to several kilometers. Further, they were developed for vehicles weighing 500 kg or more and are based on empirical results drawn from resource-intensive experimental testing. Thus, these techniques are generally inapplicable to small robotic vehicles, for which extensive empirical test data does not yet exist. Also, the mobility prediction problem also needs to consider movement over particular vehicle-sized terrain regions and obstacles, rather than gross (i.e. km-scale) mobility characteristics.

As an example, we show the results from a simulation study employing the MEgPC approach to study the motion of a vehicle, modeled using linearized differential equations, where uncertainty is considered in the spring stiffness parameter values.

In summary, the ability of autonomous or semi-autonomous unmanned ground vehicles (UGVs) to rapidly and accurately predict terrain negotiability, generate efficient paths online and have effective motion control is a critical requirement for their safety and use in unstructured environments. Most techniques and algorithms for performing these functions, however, either do not explicitly analyze vehicle mobility on rough terrain or rely on deterministic analysis that assumes precise knowledge of vehicle and/or environmental (i.e. terrain) properties. In practical applications, significant uncertainties are associated with the estimation of the vehicle and/or terrain parameters, and these uncertainties must be considered while performing the above tasks.

Computationally inexpensive methods based on the polynomial chaos approach can be employed that consider imprecise knowledge of vehicle and/or terrain parameters while analyzing UGV dynamics and mobility, evaluating safe, traceable paths to be followed and controlling the vehicle motion. Conventional Monte Carlo methods, that are relatively more computationally expensive, can be used as a reference for evaluating the computational efficiency and accuracy of results from the polynomial chaos-based techniques.
Articles

Back

MODAL DECOMPOSITION

Overview

The description of coherent features underlying multi-scale dynamic data-fields is essential to the understanding of the underlying processes. Different methods have been introduced that are able to extract dynamic information from flow data-fields that are either generated by numerical simulation or measured in a physical experiment.

A common approach for identifying coherent structures is the proper orthogonal decomposition (POD) method, which determines the most "energetic" structures in the data-set. However, there are major drawbacks of the technique: (a) the "energy" may not in all circumstances be the correct measure to rank the underlying structures, and (b) due to the choice of second-order statistics as a basis for the decomposition, valuable phase information is lost, which can be critical when classifying the dynamic processes contained in the data-fields.

In this regard, the dynamic mode decomposition (DMD) approach has been introduced. The fundamental principle underlying the approach is to decompose the data into a series of complex Fourier harmonics to obtain the spatial basis functions (dynamic modes) and time-dependent complex coefficients. The frequencies and growth rate of each mode can be obtained by the magnitude and phase of its corresponding eigenvalue. In addition, the modes can be sorted by their "energy content" as well.

It is worth noting that the results are typically sensitive to the time step, the number of snapshots and spatial coverage of the snapshots; it is therefore suggested that the residual be computed in order to obtain an accurate dynamic modal decomposition.
Articles

Back