zPOD Information

This page presents information on ZONA' Proper Orthogonal Decomposition / Response Surface Method (zPOD) software. zPOD is a commercial ready engineering software tool that can be used to perform high-level interpolation on CFD solution sets that are stored in PLOT3D (structured) or Fieldview (unstructured) formats.

Theoretical Background (by example)

zPOD adopts a “snapshot” matrix approach and applies the Proper Orthogonal Decomposition technique to facilitate reduced-order modeling of steady state aerodynamics stored in Computation Fluid Dynamics (CFD) databases.

Let’s say we have two Mach numbers (M) and three Angles-of-Attack (AoA) that together comprise six entries in our CFD table database (shown below). We select the second parameter (namely AoA) to be our POD variable. Further, suppose each CFD solution file contains 2000 nodes (i.e., grids) at which the flow field solutions are given.

First we construct the so-called “snapshot matrix” [S] for the first Mach number for one of the CFD solutions (say non-dimensional u-velocity, u) as in Equation (1), where n = number of AoA’s = number of POD variable entries, and m = number of nodes in the CFD solution file. Then we construct a square POD matrix [A] as shown in Equation (2).

Performing an eigenvalue decomposition on the POD matrix [A] yields an eigenvector [phi] as shown in Equation (3). The eigenvalues from the decomposition can be used to show the energy of the modes. Eigenvalues whose values approach zero (i.e., have little or no contribution) can be ignored. Removing these modes from the total number of POD modes yields a reduced set of modes. It is this reduced set of POD modes that is hereinafter called the “number of reduced order modes”.

Multiplying this eigenvector by the Snapshot matrix [S] yields an eigenmode matrix [PHI] (Equation (4)).

Utilizing the eigenmode matrix [PHI], the solution of the u-velocity for the current Mach number can be written in the form of Equation (5), where [BETA sub Jk] are the so-called “response surface coefficients” and kred is the reduced number of POD modes.

Since [u sub j] and [PHI sub k] are known, the response surface coefficients can be computed Equation (6).

This process is repeated for each Mach number establishing a response surface coefficient matrix for each reduced mode k as shown in Equation (7).

Finally, with the computed response surface coefficients at hand, the desired u-velocity solution to be interpolated is computed by applying Equation (5) for the known neighboring solutions and interpolating the reconstructed solution results to obtain the final desired solution (i.e., those specified on the INTERP bulk data card).

This process is then repeated for each of the remaining seven variables (v, w, Cp, M, RHO, E or S, and P) that are to be interpolated.

Finally note that the above formulation is not restricted to 2 parameters and that zPOD can interpolate on any number of parameters specified.

zPOD Software File Processing

zPOD's computer file processing is shown in the following figure. Upon execution of the software, zPOD reads in a Standard ASCII input deck (in bulk data format) and the referenced CFD Table Database. The output are ASCII text files containing the Standard Output File (containing results), the Log File (containing the execution timing summary), and any user requested Plot Files. zPOD uses ZONA's Engineering Database to store all I/O data and CFD solutions. Being a static database, these stored CFD solutions can then be used in performing re-start runs of zPOD where the CFD Table Database and referenced CFD grids and solution files do not need to be re-processed.

zPOD's Input Data Stream

The input data stream of the Input File to zPOD consists of two sections that contain varying numbers of data packets. These two sections, called the Executive Control Section and the Bulk Data Section along with their data packets, that are input by the user, comprise the Input Data Stream. These sections are shown in the following figure.

Bulk Data Cards

zPOD employs bulk data input that are identical in style to that used within other ZONA commercial products. The following figure shows the bulk data cards used by zPOD and interrelationships of these cards. As can be seen from the figure, zPOD only requires three cards to perform a basic POD/RSM analysis; namely, POD, PODRSP, and INTERP. The remaining bulk data cards are used when processing a CFD Table Database consisting of Fieldview files or to process ATAC data. Because so few bulk data cards are required, zPOD is very easy to set up and execute.

Processing Fieldview CFD Solution Files

Specifying a Reduced Volume Space

Fieldview CFD solution files can contain millions of nodes requiring significant computing time and memory to perform the mapping and interpolation. Occasionally, only a region of the entire flow field solution is of interest to the user (e.g., steady Cp on the wing for a ZAERO transonic analysis), while regions far away from the vehicle are of no interest. Therefore, a VOLUME bulk data card can be used within zPOD to specify a reduced volume space region to perform the interpolation on. By default (i.e., in the absence of a VOLUME bulk data card), zPOD interpolates over the entire Fieldview solution file volume space. The following example shows zPOD extracted Hexahedron elements around the wing tip launcher based on a user input VOLUME card.

Some Hurdles in Processing Fieldview Files

All Fieldview CFD solution files must have the same face elements including the outer boundary face elements (i.e., all the nodes of all face elements must be the same and in the same order).
However, the volume 3-D elements (tetrahedrons, hexahedrons, prisms, and pyramids) can be different as they are generated by the unstructured mesh CFD solvers.
zPOD requires that the number of nodes (i.e., grids) within the solution file be the same among all CFD solution files that are referenced by the CFD table database file. This, then, poses a big problem for unstructured mesh CFD solvers that yield different numbers of flow field nodes for the same CFD input model.
To circumvent this problem, a mapping procedure has been implemented within zPOD to map all Fieldview CFD solutions to a user-defined “static grid.” Note that the static grid mapping is not an interpolation procedure at all, but merely a “which grid is closest approach” to the problem.

Static Grid Mapping Solution

zPOD uses the STATICG bulk data card that allows you to reference a single Fieldview CFD solution file within your CFD Table Database. The default (i.e., specifying no STATICG card) uses the 1st CFD solution file listed in the CFD table database as the grid to be mapped to. Using a single grid; therefore, establishes a fixed number of nodes for the POD/RSM analysis. For each CFD file processed, zPOD maps the solution of the nearest node in the current CFD file to the node being mapped to in the static grid file. Note that mapping on the boundary face elements (which are always the same between different solutions) will be exact. The largest error due to mapping will occur in the flow field volume farthest away from the vehicle (i.e., the boundary face elements).

Cube Discretization Procedure

To perform the Static Grid Mapping, zPOD employs a Cube Discretization procedure which is a completely automated procedure in zPOD. This process greatly expedites the search time required during static grid mapping. The process involves breaking down the current grid volume space into small cubes as shown in the following figure. By default, no. of axis divisions used = cubic root of the number of nodes in the volume space (with a minimum no. of divisions = 10). The CUBEDIV bulk data card can be used to override and specify this value. The algorithm used (in a nutshell):
1. for a given static grid file grid, determine the cube it is located in (per its position, this is instantly known)
2. find all the neighboring cubes within the volume and search through the current and neighboring cubes to find the closest grid
3. if no closest grid is found (i.e., all neighboring cubes are empty), then search through the entire volume to find the closest grid

Grid Mapping Statistics

As shown below, zPOD outputs detailed information about the mapping statistics during the processing of each Fieldview CFD solution file. This provides you a useful gauge in determining the accuracy of the mapped CFD solution within your selected VOLUME space.

Processing PLOT3D CFD Solution Files

Since PLOT3D formatted CFD solution files are stores in structured grid format, the processing of PLOT3D data is a simpler task within zPOD than processing Fieldview files. In this regard, no static grid mapping is required. In processing PLO3D files, the zPOD standard output print includes the file name, location, specified parameters, no. of blocks read, and I,J,K values for each block processed.

INPCFD1 Data File for ZAERO/ZONAIR

zPOD can be used to generate the INPCFD1 input bulk data card's referenced CFD solution file used by ZAERO and ZONAIR to perform a transonic aeroelastic analysis. This is a very nice feature and has the unique benefit in that interpolated solutions can quickly be output so that no additional CFD runs would be required. In addition, the VOLUME space reduction for Fieldview file processing can be used to only output a flowfield region of interest (e.g., over the main wing only). This option also supports mirroring of the output CFD solution files about any of the three principal orthogonal global axes planes. The following example shows the results of a zPOD generated INPCFD1 file that was used to map the F-16 CFD solution (Cp) on the wing and wing-tip launcher.

Interpolated Solutions

zPOD supports two interpolation methods: (1) Local Linear Interpolation, and (2) Radial Basis Function (RBF) Interpolation (A Neural Network Approach). Depending on the CFD table database format used, the following 8 variables are interpolated on.

Reduced Order Model Error (L2_ERROR)

The L2_Error is computed for each CFD solution file listed in the CFD table database for all 8 variables shown above. The L2_Error establishes a measure of the difference between the interpolated data and the original data. A general rule of thumb - if the error is larger than 0.1%, then the interpolation results may be poor (increasing the number of POD modes used for interpolation will usually help (NRDMOD entry of the INTERP card). The zPOD standard output lists the following:

ICASE = case index number referencing the index of the file in the CFD database table

NMODE = number of reduced order modes used to reconstruct the solution

L2_ERROR = summation of the difference between the interpolated solution using the reduced number of pod modes (NMODE) and the original solution for all grids divided by the total number of grids

Response Surfaces

zPOD can generate the Response Surface Plots for any given POD/RSM analysis. This output plot file is always generated in Plot3D format and is a function of two parameters (i.e., the POD variable and the IRSP variable). Only the reduced order modes are retained (i.e., sum of eigenvalues of all ascending modes exceeds 99.999% of unity). An example of such an ouput is shown at right for a sample 30 degree wedge case showing the first three modes.

An F-16 Example Using zPOD

The following example briefly demonstrates the usage of zPOD. An F-16 aircraft with an AIM9 tip missile and 600 gallon under-wing tank is considered. The CFD table database solution files are stored in Fieldview format and computed using an inviscid option. A CFD table database of nine points is set up (3 Mach numbers of 0.85, 0.90, 1.10 and 3 Angles-of-Attack 0.0, 2.0, 4.0 degrees). The test case CFD grid is shown as follows.

A VOLUME bulk data card (whose zPOD extracted boxes are shown below) is used to restrict the volume space to be interpolated to a region encompasing the main wing, tip launcher+missile, and under wing store. A CUBEDIV bulk data card is used to force the number of axis divisions to 25.

A Direct Comparison of the CFD computed Fieldview solution and zPOD Interpolated CFD Solution Results based on Cp contours is shown. As can be seen, the zPOD interpolated solution compares well with the direct CFD computed solution. Mach Number, AoA, and Cp values of the results are not shown due to proprietary issues.