A Users Guide to Eigenvalue Extraction Methods in NASTRAN
NASA Goddard Space Flight Center
November 29, 1995
This document will discuss 3 of the commonly used eigenvalue extraction techniques used in NASTRAN: Modified Givens, Inverse, Lanczos. In addition, 2 methods of automated model dynamic reduction will also be investigated: AUTOOMIT(static condensation or Guyan Reduction) and Generalized Dynamic Reduction.
Since it's beyond the scope of this report (and my scope!) to discuss the theories of the various eigenvalue extraction techniques, this report will stick mainly to a general discription of the technique, the pluses and minuses, and other considerations to be aware of. Following the detailed descriptions of these methods will be a summary table.
This section will discuss three of the NASTRAN methods used to extract eigenvalues for a dynamics analysis. These methods can be used alone, or in conjunction with dynamic reduction techniques.
Modified Givens: (MGIV) This method is, for most problem sizes, the slowest method of eigenvalue extraction. MGIV will solve for ALL eigenvalues in the NASTRAN model (eigenvectors will only be calculated for the range of frequencies specified). As a result, this method is most useful if all the eignevalues are of equal interest. For most problems, MGIV is only practical if the model is small, or if an ASET or OMIT set is provided in the model in order to decrease the number of dynamic degrees of freedom(DOF). When using MGIV, NASTRAN will, by default, use AUTOOMIT to remove all massless degrees of freedom from the ASET DOF in order to compact the matrices.
As stated earlier, for practical purposes, MGIV requires that the model be reduced in size. This can be done by either specifying which degrees of freedom to keep in the model (ASET), or which degrees to remove from the model (OMIT). The problem with reducing the model is that it intorduces error into the results, particularly for higher modes, as not all the structural mass and stiffness is taken into consideration. This can result in the shifting of mode frequencies as well as missing frequencies completely. Deciding what degrees of freedom to keep is an art unto itself. UAI/NASTRAN has automated the process somewhat with the use of AUTOOMIT.
Inverse: (SINV) This method is faster than MGIV for eigenvalue extraction, but still uses the entire mass and stiffness matrices of the model. SINV is very efficient for extracting a specific range of frequencies. Unlike MGIV, SINV only solves for the eigenvalues in the frequency range of interest. Also unlike MGIV, SINV does not need to remove massless DOF's in order to run efficiently. In fact, SINV takes advatage of the sparsness of the matrices caused by massless degrees of freedom. As a result, using AUTOOMIT to remove massless degrees of freedom from an SINV run will actually Increase model run time. Since SINV uses all the DOF's in the model and does not require ASET's for reasonable run times the solution accuracy is very good and, for similar run times, is more accurate than an MGIV model run with an ASET.
Lanczos: (LANCZ) The Lanczos method is, for most applications, the fastest, accurate solution method in NASTRAN. LANCZ is faster than both MGIV and SINV, and is just as accurate for the models tested. LANCZ is also a sparse matrix solver and, similar to SINV, will run slower if the matrices are compacted by removing massless DOF. This method of extraction seems the best choice for most NASTRAN models. Even with compacted matrices, LANCZ can be faster than any of the other methods mentioned. However, if you have large, dense matrices (ie. very few massless DOF), using the MGIV method along with Generalized Dynamic Reduction may be the fastest method.
Dynamic Reduction Techniques
Dynamic reduction is a way of reducing the number of dynamic degrees of freedom in a finite element model and therefore reducing the size of the problem to be solved. This is particulary important when using MGIV as its solution time is dramatically decreased when the ASET can be reduced. One consequence of dynamic reduction, however, is the introduction of error. Since dynamic reduction removes degrees of freedom from the ASET it can cause mode frequencies to shift, and also may miss modes altogether.
AUTOOMIT: AUTOOMIT is a tool for automatically performing dynamic reduction on a model. AUTOOMIT is a case control command that creates an OSET (OMIT cards) automatically, given various parameters. It will automatically remove all massless DOF when using the MGIV method. AUTOOMIT can also be used to remove other DOF's from the ASET. AUTOOMIT elimiates DOF based on their mass to stiffness ratio. A high ratio means a DOF with high mass and relatively low stiffness. In general, this is more likely to be a DOF that will contribute to a lower frequency mode than a DOF that has low mass and high stiffness. AUTOOMIT can be requested to omit DOF's based on a value for the mass to stiffness ratio ("AUTOOMIT(EPS=x)=YES"), based on DOF mass ("AUTOOMIT(MASS=y)=YES"), or a certain percent of DOF's can be requested to be retained ("AUTOOMIT(KEEP=z)=YES").
AUTOOMIT works best in models with good, discreet mass distribution (i.e. - use of concentrated masses). For a model with a more homogeneous mass and stiffness distribution, AUTOOMIT will not produce an ASET with points distributed nicely around the entire model. This can increase errors as some more localized modes may be missed. In general, AUTOOMIT is a good tool to start generating an ASET for a model. In order to improve accuracy of the results, additional DOF's may have to be added later.
Generalized Dynamic Reduction: (DYNRED) Generalized Dynamic reduction (as the name implies) is not a method of eigenvalue extraction, but rather a method to reduce the size of the model other than using ASETs or OMITs. It also produces results much faster than using static condensation. In essence, the way that DYNRED works is by approximating the mode shapes of the model first, and then solving for the actual modes. It generates scalar points, based on the number of modes desired, and uses those scalar points to approximate mode shape. These represent the generalized coordinates for the mode shape approximation. In UAI NASTRAN they are created automatically. DYNRED, used in conjunction with SINV or MGIV, is generally a fast and accurate aid in extracting eigenvalues. For a sparse matrix LANCZ is still faster, but DYNRED is still no slouch. Since it is an approximation of mode shape to begin with, it does not generate answers as accurately as SINV or LANCZ, or MGIV by itself. Generally speaking, the first modes are very accurate, with accuracy tailing off toward the higher frequencies. Even at the higher modes, the accuracies are still rather good, especially when compared to MGIV with ASET or OMIT cards.
In UAI NASTRAN, using DYNRED requires a DYNRED card in the case control to specify the DYNRED ID, and a DYNRED card in the bulk data deck to specify the frquency range of interest. DYNRED still requires a METHOD card in the case control, and an EIGR card.
In order to better understand and compare the various eigenvalue extraction methods, a number of NASTRAN runs were made with a simple plate model. The model was a square plate held along one edge, and also held at one of the opposite corners. The model mesh started as a 5x5 mesh, but meshes of 6x6, 8x8, 10x10, 15x15, 20x20, 30x30, and 50x50 were also created from the same geometry. All of these models were run using the previously mentioned eigenvalue extraction methods to calculate the first 20 modes. The results were then compared. All runs were made in UAI NASTRAN v8, and run on an IBM RS6000-410 computer.
Two specific pieces of data were compared: solution CPU time, and solution accuracy. For the sake of these comparisons, the SINV results were considered to be the baseline, as this method used the entire stiffness and mass matricies. All the methods, with the exception of the MGIV with DYNRED runs, produced exactly the same modal frequencies. The error in the runs with DYNRED occured only in the higher modes, and those errors were negligible. The main differences among the runs were the CPU times. The following table compares the times for the various methods based on the number of DOF in the model ASET.
The results show that for all the cases examined, the Lanczos method was the fastest, followed by the solution using DYNRED. Interestingly, for the smaller size problems, the MGIV method was actually fairly efficient, time-wise. That efficiency, however, drops off dramatically as the size of the problem increases.
As was indicated earlier, LANCZ and SINV depend on sparse matricies in order to run quickly. The following chart shows how the run times change if the matricies are compacted using the AUTOOMIT command to remove all massless DOF (in this case all rotational DOF). This was done automatically for the MGIV runs shown in the previous figure. The solutions of the compacted matrices are indicated by the "_d" suffix.
As the problem size increases, the SINV and LANCZ methods begin to lose their appeal. The DYNRED method times are, essentially, unchanged.
Among dynamic reduction techniques, DYNRED generalized dynamic reduction is much more time efficient, and generally more accurate. The graph below compares AUTOOMIT static condensation to DYNRED for various percentages of DOF retained by AUTOOMIT. This chart is for the 6th mode of the MLS NASTRAN model. The percentage of frequency error for a given percentage of DOF kept by AUTOOMIT is shown in one line, with the CPU time shown on the other. The DYNRED solution time (which had no frequeny error) is shown on the CPU time axis. For comparison, the Lanczos time is also shown:
For most model sizes and applications, the Lanczos eigenvalue extraction method is the most efficient. For smaller models, or models where all the eigenvalues are to be extracted, the MGIV method may be more useful. In addition, for models that do not have sparse matrices, the LANCZ and SINV methods may actually be slower than using other methods. For those cases, the MGIV method with DYNRED may be the best choice.
In addition to the example problems elaborated above, several other, more detailed NASTRAN models were run using the various extraction methods. These were models that had been used for real-world structural analysis. Although there have been some concerns that some solution extraction techniques may miss mechanism or duplicate modes, this was not seen in any of the problems run.
The DYNRED method is also a more acurate and faster aid to model dynamic reduction than using standard static condensation with AUTOOMIT. However, in cases where it's necessary to develop an ASET, or OSET for a model, such as when deciding where to place accelerometers for a vibration test, AUTOOMIT can be a very useful tool. It is, however, most useful for models with discrete, lumped masses and stiffness differences. For more homogeneous mass and stiffness distributions, it is not as efficient at selecting an ASET.
The following table summarizes the eigenvalue extraction methods detailed above:
Summary of Eigenvalue Extraction Methods
||Performance Time vs. Model Size
||small models or when all
eigenvalues are desired
||small to moderate size models when a range of frequencies are desired, and matrices are sparse
||moderate to large models with sparse matrices
||moderate to large models,
particulary with dense matrices
(1) For dense matrices, performance goes to fair
(2) For dense matrices, performance goes to poor
(3) For dense matrices, performance goes to good
November 29, 1995