New procedure for enumeration of eigenvalues ​​appropriate to the complex plane

20 October 2011

by O. Boiteau, EDF R&D / Sinetics

Provide count the number of eigenvalues ​​in an area of the complex plane is an optional but important ingredient of the modal operators of Code_Aster (MODE_ITER_SIMULT / INV, MACRO_MODE_MECA, CALC_MODAL …). For now a standard method, called "Sturm" meets this need but only for a range of problem (generalized standard issue), when it comes to counting the number of modes belonging to a real segment (see Figure 1). The objective of this study, commissioned by the Project MNAM was to develop a similar technique to any area of the complex plane (see Figure 1). This generalization allows to take into account all the issues proposed by the modal code: widespread problems (GEP) and quadratic (QEP) with hysteretic or viscous damping, gyroscopic effect …

GEP standard test strip
GEP atypical or QEP, test area (disc)
Figure 1 : Two distinct problems: enumeration of eigenvalues ​​in a segment of the real axis and a finite portion of the complex plane

Of some methods proposed in the literature, we have finally chosen as the method called ’APM’ for ’main argument Method’. It seems the only mature enough despite its cost calculation, to enhance the operator IMPR_STURM. At first, we have deliberately limited to the role of pre-treatment diagnostic because this method, just after the academic research is being reliability.

The principle of APM has a geometric description whose simplicity is deceptive. It is to discretize the contour of the area of ​​interest (for example, the disk in Figure 1). Then in each of these points, to assess the main argument of the determinant of the matrix dynamics associated Arg (P (λ)). These argument values ​​trace a sort of "snail shell" around the point of origin (see Figure 2). A mathematical result assures us when you can just count the number of turns for the number of eigenvalues ​​strictly contained in the initial disk.

Figure 2 : Contours in the plane P (λ) on the same modal calculation Aster: the QEP’s case-test sdll123a

But we must know how to compose, as shown in Figures 2, with the contingencies of discretization (which must be fine enough but not too much because it is expensive), calculating distorted by too proximity to an eigenvalue …