r5.01.02 Solving the modal problem

r5.01.02 quadratic (QEP)

Summary

The study of the dynamic stability of a damped and/or rotating structure leads to the resolution of a modal problem that is higher than the traditional standard (SEP) or generalized (GEP) modal problems.

To understand them, Code_Aster proposes a myriad of methods via the operators CALC_MODES: inverse powers and the Müller, Lanczos, IRA and QZ method. They each have their scope of use, their advantages/disadvantages and their development history.

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. The various results, algorithms, or parameters discussed in this document are often based on lower-order modal methods (SEP and GEP) described in document [R5.01.01]. Reading the latter is therefore a recommended prerequisite.

• 1. Introduction
• 2. background
  • 2.1. Problem
  • 2.2. Properties of clean modes
  • 2.3. Specificities of QEP and elements of theory
  • 2.4. Linearization strategy
  • 2.5. Implementation in Code_Aster
• 3. Inverse power method (OPTION among [“PROCHE”, “AJUSTE”])
  • 3.1. Introduction
  • 3.2. Müller-Traub method
  • 3.3. Inverse power method (Jennings variant)
  • 3.4. Scope of use
  • 3.5. Display in the message file
  • 3.6. Summary of the settings
• 4. Subspace method (METHODE =” TRI_DIAG “/” SORENSEN “)
  • 4.1. From QEP to SEP
  • 4.2. Implementation in Code_Aster
  • 4.3. Scope of use
  • 4.4. Display in the message file
  • 4.5. Summary of the settings
• 5. QZ global method (METHODE =”QZ”)
  • 5.1. From QEP to GEP
  • 5.2. Scope of use
  • 5.3. Impressions in the message file
  • 5.4. Summary of the settings
• 6. Bibliography
  • 6.1. Books/articles/proceedings/theses…
  • 6.2. Reports/reports EDF
  • 6.3. Internet resources
• 7. Description of document versions
• 8. Appendix 1. Interpreting complex eigenvalues