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G.17.3.1 Solution of the Eigenproblem

The eigenproblem is solved for structure frequencies and mode shapes considering a diagonal, lumped mass matrix, with masses possible at all active degrees of freedom (d.o.f.) included. Two solution methods may be used: the subspace iteration method for all problem sizes (default for all problem sizes), and the Arnoldi/Lanczos method for evaluating eigenvectors (Advanced Analysis only). Additionally, load dependant Ritz vectors (LDR) can be used for dynamically loaded structures.

For large scale eigen value problems, the Arnoldi method is very efficient.

Autoshifting of Eigenvectors

For large models having a large number of d.o.f and a large number of modes extracted (i.e., memory-bound), an incremental solver mode called autoshifting can be used by the Advanced Analysis. A mode shift value is specified to indicate the fixed number of modes the solution tries to find in each shift. The main benefit of using the incremental solver is that it is memory efficient (e.g., problems that were not solvable before on 32-bit systems might be solved with that technique). It is recommended to be used only when memory allocation failure takes place during eigen solution using subspace-iteration or Arnoldi/Lanczos methods. When the program fails to extract eigen vectors due to insufficient memory, you can use auto-shifting which significantly reduces memory demand.

Note: Generally, the Arnoldi/Lanczos is more robust than subspace iteration with autoshifting. The Arnoldi/Lanczos method is very efficient at finding large eigen vectors with autoshift. Thus, if memory problems occur with subspace-iteration (regular mode), it is recommended to switch to the Arnoldi/Lanczos method (regular mode). If memory problems occur with this method in regular mode, then autoshift can be applied to reduce the memory demand.

Since autoshifting can provide significant reduction in memory demand in the solution, it can be ideal for the eigenvalue solution of large systems. In this case, the you are required to provide a "targeted" number of eigen modes to be searched in each shift. The solution starts with "0" shift and tries to find targeted number of eigen modes. Once completed, a new shift is applied and it tries to find next set of eigenvalues within the new shift. This continues until all required number of eigenvalues are found.

The subspace iteration method is very sensitive to calculated shift resulting in a partial extraction of mode. It is recommended to use this solution as a supplemental or alternative solution to existing (default mode). Whenever the program extracts partial set of eigen vectors, it issues a warning message.

If the solution return partial results, it means that not all the required number of eigen modes were found. But the solution guarantees that no eigen values are missed among returned results. Partial results can be still useable for dynamic analysis if they satisfy other analysis requirements. For example, "m" modes satisfy 90% mass participation.

Partial results can be also returned by the subspace method (Advanced Analysis) without using auto shifting method.

The program may miss some Eigen values because of applied shift while performing Eigen solution using Subspace-iteration method. In this case, a warning message is given. In this case, the results returned include missing modes. These results should be used with caution and it is strongly advised for further investigation (dynamic contribution from missing modes might be too important to ignore in analysis).

When the Arnoli/Lanczos method is used with autoshifting, an initial frequency shift may also be specified.

Load-Dependent Ritz Vectors

Research has indicated that considering the effect of natural free-vibration mode shapes may not be the most efficient basis for mode-superposition analysis of structures subjected to dynamic loads. This implies that dynamic analyses —like response spectrum and time history analysis— based on a special set of load-dependent Ritz vectors may yield more accurate results than the use of the same number of natural modes.

There are several reasons to consider Ritz vector analysis as a more efficient approach.

  1. For large structural systems, the solution to find free-vibration modes and frequencies may require a significant amount of computational effort.
  2. The Ritz vectors method takes into account the spatial distribution of the dynamic loading, whereas the direct use of natural modes neglects this information. Therefore, many of the natural mode shapes that are calculated may not have significant contribution to the dynamic response.
  3. Ritz vector analysis by default does not include static correction due to higher mode truncation. The command MIS is required to be issued to include missing mass correction.

The Ritz vectors method is recommended where the solution with eigen vectors fails to capture 90% mass participation (a mandatory requirement of most country seismic codes) with a reasonable number of modes.

It is also recommended where eigen vectors capture irrelevant modes. Even though they are real modes, they are not relevant to the structural response due to the applied dynamic loading.

The spatial distribution of the dynamic load vector serves as a starting load vector to initiate the analysis process. The program will automatically generate this starting load or you can specify it using the DEFINE STARTING LOAD command. In the program generated method, the starting load vector is generated using the mass model of the structure. Assuming a mass matrix only has translational components, the resulting load vector will have force components in the directions of all the translational degrees of freedom. and will have zero values for all the rotational degrees of freedom. This will guarantee a static deflection with components in many directions with this initial mode assumption. As long as the initial mode assumption has a component in the direction of interest, it will yield a correct set of Ritz vectors.

If you define a set of starting load vectors, then this load vector will have a force component in one translational degree of freedom in the direction of the dynamic load. This results in a static deflection mainly in the direction of interest. In some models where mass participation is predominant in one translational direction for the initial few modes, this method can achieve 90% mass participation with only a few modes.

Note: A single set of Ritz vectors can be derived from a single starting load. If all starting loads that participate in dynamic loads are defined, a single set of Rtiz vectors can be extracted from all participating starting loads. Therefore, use of the DEFINE STARTING LOAD command is best suited if the structural response is predominant in one translational direction. If response in multiple translational d.o.f. are predominant, then it is recommended to use the program generated starting load vector.

Leger P, Wilson EL, Clough RW., The use of load-dependent Ritz vectors for dynamic and earthquake analyses. Technical Report UC13/EERC86/04, Earthquake Engineering Research Center, University of California Berkeley, Berkeley, CA, 1986.