1. Introduction

A majority of studies concerning the dynamic behavior of structures are carried out by carrying out a transient analysis on a modal basis. To calculate these vibration modes, numerous 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 the most robust in practice have been incorporated into the modal calculation operator of*Code_Aster*: CALC_MODES.

The optimal perimeters of use of this operator’s search criterion can be separated. When it comes to determining a few eigenvalues (typically half a dozen) or refining some estimations, the criterion OPTION =” PROCHE “or” SEPARE “or” “or” AJUSTE “is quite appropriate. It combines heuristic algorithms and power algorithms (cf. §4).

On the other hand, to capture a significant part of the spectrum, we use the criterion OPTION =” BANDE “or” CENTRE “or” PLUS_PETITE “or” PLUS_GRANDE “. The latter federates so-called « subspace » methods (Lanczos cf. §5/§6, IRAM §6, §7, Bathe & Wilson §8) which project the work operator in order to obtain an approximated modal problem of smaller size (then treated by a global method such as QR or Jacobi).

This operator also makes it possible to calculate the entire spectrum of the problem in a robust manner using the criterion OPTION =” TOUT “. To do this, a global reference method is used (QZ method cf. §9) which calculates all the modes comprehensively. Given its cost, it should however be reserved for certain uses: a small problem (less than \({10}^{4}\) degrees of freedom) or algorithm benchmark.

The various search criteria can also complement each other because the methods implemented with OPTION =” PROCHE “or” SEPARE “or” AJUSTE “are very efficient in optimizing natural modes that are already almost converged. In one or two iterations, they can thus improve the eigenvectors calculated the first time via OPTION =” BANDE “or” CENTRE “or” PLUS_PETITE “or” “or” PLUS_GRANDE “or” “or” TOUT “. Projection on a modal basis will only be better!

In the first part of the document we summarize the general problem of solving a modal problem, the different classes of methods and their variations in public domain libraries. All things you need to have in mind before discussing, in the second part, the general architecture of a modal calculation in Code_Aster. Then we detail the numerical, computer and functional aspects of each of the approaches available in the code. For each method, its main properties and limitations are given by relating these considerations, which are sometimes complex, to a precise parameterization of Code_Aster operators.

OPTION / Scope of application	Algorithm	Keyword factor SOLVEUR_MODAL	Benefits	Disadvantages
Remarks »
“PROCHE” “SEPARE” “AJUSTE”
1st phase (heuristic)				Only symmetric real (GEP and QEP).
Calculating some modes	Bissection (not applicable in QEP)	“SEPARE”
Calculation of some modes	Bissection + Secant (Müller-Traub method in QEP)	“ AJUSTE “	Better precision	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 (GEP and QEP).
Basic method	Inverse powers	OPTION_INV = “ DIRECT “	Very good eigenvector construction	Not very robust
Acceleration option	Rayleigh quotient (not applicable in QEP)	OPTION_INV = “RAYLEIGH”	Improve convergence	Cost calculation.
“BANDE” “CENTRE” “PLUS_PETITE” “PLUS_GRANDE” “TOUT”
Calculating part of the spectrum	Bathe & Wilson	METHODE = “JACOBI”		Not very robust. Only real symmetric (GEP).
	Lanczos (Newman- Pipano in GEP and Jennings in QEP)	METHODE = “TRI_DIAG”	Specific detection of rigid modes.	Only real symmetric (GEP and QEP).
	IRAM (Sorensen)	METHODE = “ SORENSEN “	Increased robustness. Better computing and memory complexities. Fashion quality control.	Method by default. Worn in: - not real symmetric, - with \(\mathrm{A}\) symmetric complex.
Full spectrum calculation	QZ	METHODE = “QZ”	Robustness. Reference method.	Very expensive in CPU and in memory. To be reserved for the small case (<104degrees of freedom). Worn in: - not real symmetric, - with \(\mathrm{A}\) symmetric complex.
Options” BANDE “ “CENTRE” “PLUS_PETITE” “PLUS_GRANDE” “TOUT” + AMELIORATION =” OUI “	Subspace method selected by SOLVEUR_ MODAL then inverse power method	Implicit addition of OPTION = “PROCHE “on the modes calculated by the 1st step (method of sub**-** space)	Improves the construction of eigenvectors	Additional calculation cost

Table 1-1. Summary of Code_Aster modal solvers (GEP and 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. Examples include F.Chatelin [Cha88], G.H.Golub [GL89], G.H.Golub [], P.Lascaux [LT86], B.N.Parlett [Par80], Y.Saad [Saa80], D.S.Watkins [Watkins [Wat07] and the synthesis committed by J.L.Vaudescal [Vau00].