1. Introduction

A majority of studies concerning the dynamic behavior of damped and/or rotating structures are carried out by carrying out a transient analysis on a modal basis. To unearth these vibration modes, a myriad of algorithms have been developed over the past sixty years. In order to deal with the continuous increase in the size of problems and the deterioration of the conditionings of discretized operators, only the most efficient and robust ones, in practice, have been incorporated into the Code_Aster modal operator, CALC_MODES.

The optimal areas of use of the various available methods can be separated. When it comes to determining a few eigenvalues (typically half a dozen) or *refining a few estimations, the option “ PROCHE “or “ AJUSTE “ of CALC_MODES is perfect. This includes heuristic algorithms and power algorithms (cf. §3).

On the other hand, to capture a significant part of the spectrum, we resort to one of the options “ CENTRE “, “ PLUS_PETITE “or “ TOUT “ of CALC_MODES. The latter federates so-called « subspace » methods (Lanczos, IRAM §4) which project the work operator in order to obtain an approximated modal problem of a smaller size (then treated by a global QR-type method).

This operator also makes it possible to unearth the entire spectrum of the modal problem in a robust manner. To do this, a global reference method is used (QZ method cf. §4) which calculates all the modes comprehensively. It is therefore to be reserved for certain uses: small problem (< 1000 degrees of freedom) or algorithm benchmark.

The two families of methods can also complement each other, because the methods implemented with the “PROCHE” or “AJUSTE” option are very efficient in optimizing eigenmodes that are already almost converged. In one or two iterations, they can thus improve the eigenvectors previously estimated via using one of the options” CENTRE “,” PLUS_PETITE “or” TOUT “. Projection on a modal basis will only be better.

In the first part of the document we summarize the problem of solving a quadratic problem and its variation in the general architecture of a Code_Aster modal calculation. Then we detail the numerical, computer and functional aspects of each of the approaches available in the code. We try to give, for each method, its main properties and limitations by relating these considerations, which can sometimes seem a bit « ethereal », to a precise configuration of the*Aster* operators.

The various results, algorithms, or parameters discussed in this document often rely on lower-order modal methods (SEP and GEP) described in document [Boi09]. Reading the latter is therefore a recommended prerequisite!

OPTION / Scope of application	Algorithm	Keyword	Benefits	Disadvantages
Remarks »
“ PROCHE “ “ AJUSTE “
1st phase (heuristic)				Only symmetric real
Calculation of some modes	Müller-Traub	“ AJUSTE “		Cost calculation
Improvement of some estimates	User initialization	“ PROCHE “	Resumption of eigenvalues estimated by another process. Cost: calculation of this phase is almost zero	No multiplicity capture
2nd phase (power method proper)				Only real symmetric
Basic method	Inverse powers (Jennings)	OPTION_INV = “ DIRECT “	Very good eigenvector construction	Not very robust
“ CENTRE “ “ PLUS_PETITE “ “ TOUT “
	Lanczos	METHODE = “TRI_DIAG”		Only symmetric real
	IRAM (Sorensen)	METHODE = “ SORENSEN “	Increased robustness. Better computing and memory complexities. Fashion quality control.	Method by default. Range in non-symmetric and/or with A complex.
Calculating the whole spectrum then filtering a portion	QZ	METHODE = “ QZ “	Robustness. Reference method.	Very expensive in CPU and in memory. To be reserved for the small case (<103 degrees of freedom). Range in non-symmetric and/or with A complex.

Table 1-1. Summary of modal methods for dealing with Code_Aster QEP.

Note:

• The effective implementation and maintenance of modal solvers in Code_Aster is the result of teamwork: D.Séligmann, B.Quinnez, G.Devesa, O.Boiteau, O.Nicolas, O.Nicolas, E.Boyère, I.Nistor…

• We tried to constantly link the various items covered and to limit the use of long mathematical demonstrations to the strict minimum. In any case, the numerous references that dot the text should make it possible to search for precise information.

• The purpose of this document is not to detail all the aspects covered, as complete books have already fulfilled this mission. Numerous references can be found in the note [Boi09]. However, mention should be made of the excellent summary on QEP committed by F.Tiseur and K.Meerbergen [TM01]. Our document uses some of its elements and, in particular, some illustrative examples.