4. Examples

4.1. Example 1

Comparison of the two methods STURMet **** APMsur the GEP standard of**** SDLD02a ** (for STURMles terminals are mentioned in frequency). In the first calculation, the aim is to count the number of modes contained in the frequency band \(\mathrm{[}0;5\mathrm{]}\) with the usual method: STURM. In the second, we do the same thing with method APM, but with the disk centered at the origin (centre=0+0j) and with radius= \({(2\pi .5)}^{2}\) (because there is no unit change with APM). Both results are shown in file MESSAGE.

f1=5.0

nbmod01 = INFO_MODE (MATR_RIGI = MATASSR, MATR_MASS =, = MATASSM, TYPE_MODE =” DYNAMIQUE “,

FREQ =( 0. , f1), COMPTAGE =_F (METHODE =” STURM “),)

w1= (2*pi*f1) **2

nbmod11 = INFO_MODE (MATR_RIGI = MATASSR, MATR_MASS =, = MATASSM, TYPE_MODE =” MODE_COMPLEXE “,

TYPE_CONTOUR =” CERCLE “, CENTRE_CONTOUR =0.0+0.0j,

RAYON_CONTOUR =w1, COMPTAGE =_F (METHODE =” APM “),)

With INFO =1, this causes the following displays in file MESSAGE:

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VERIFICATION OF SPECTRE OF FREQUENCES (METHODE OF STURM)

PAS OF FREQUENCE DANS THE BANDE (0.000E+00, 5.000E+00)

…

(METHODE APM) POUR LES 3 NIVEAUX FROM DISCRETISATION SUIVANTS

— 20 — 40 — 80 —

NOMBRE OF VALEURS PROPRES DETECTEES

— 0 — 0 — 0 — 0 —

(METHODE APM) CONVERGENCE OF THE HEURISTIQUE

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VERIFICATION FROM SPECTRE TO FREQUENCE (METHODE FROM ARGUMENT PRINCIPAL)

PAS OF FREQUENCE DANS THE DISQUE CENTRE EN (0.000E+00, 0.000E+00)

AND RAYON 9.870E+02

Here, Sturm’s method required only two factorizations. Method APM converged immediately on the first iteration. But this one required \(20+40+80\mathrm{=}140\) factorizations. Counting eigenvalues in the complex plan comes at a price (which we can’t currently reduce)!

The number of natural frequencies (0 in this case) as well as the search criteria are saved in a table. Printing, by IMPR_TABLE, the concepts NBMOD01et NBMOD11produits by INFO_MODE in the previous example shows the following composition:

nbmod01

FREQ_MIN FREQ_MAX NB_MODE

0.00000E+00 5.00000E+00 0

…

nbmod11

CENTRE_R CENTRE_I RAYON NB_MODE

0.00000E+00 0.00000E+00 9.86960E+02 0

When INFO_MODE is called with the TYPE_MODE =” MODE_FLAMB “option, the tables produced contain three columns: NB_MODE (the number of eigenvalues) as well as CHAR_CRIT_MIN and CHAR_CRIT_MAX, the search criteria for dynamic problems with linear buckling.

4.2. Example #2

Count for QEP of SDLL123a . This time, only the APM method is legal. Count the number of modes contained in the circle centered at the origin (\(\mathit{centre}\mathrm{=}0+\mathrm{0j}\)) and whose radius is \(\mathit{rayon}\mathrm{=}124\mathrm{\times }2\pi\).

f1=124.*2.*pi

nbmod4= INFO_MODE (MATR_RIGI = RIGIDITE, MATR_MASS =, = MASSE, MATR_C = GYOM, TYPE_MODE =” MODE_COMPLEXE “,

TYPE_CONTOUR =” CERCLE “, CENTRE_CONTOUR =0.0+0.0j,

RAYON_CONTOUR =f1, COMPTAGE =_F (METHODE =” APM “,),)

In INFO =1 this causes the following displays in file MESSAGE:

(METHODE APM) POUR LES 3 NIVEAUX FROM DISCRETISATION SUIVANTS

— 20 — 40 — 80 —

NOMBRE OF VALEURS PROPRES DETECTEES

— 4 — 4 — 4 — 4 —

(METHODE APM) CONVERGENCE OF THE HEURISTIQUE

---

VERIFICATION FROM SPECTRE TO FREQUENCE (METHODE FROM ARGUMENT PRINCIPAL)

THE NOMBRE OF FREQUENCES DANS THE DISQUE CENTRE EN (0.000E+00, 0.000E+00)

AND RAYON 7.791E+02 EST 4

And the printout of the table produced (nbmod4) by IMPR_TABLE gives:

CENTRE_R CENTRE_I RAYON NB_MODE

0.00000E+00 0.00000E+00 7.79115E+02 4