r5.01.01 Modal solvers and generalized problem solving (GEP)

Summary

Whether to study the vibrations of a structure or to search for its buckling modes, the mechanic must often solve a modal problem: either generalized (GEP) or quadratic (QEP) () [R5.01.02]. To do this, Code_Aster offers several methods via the operators CALC_MODES: inverse powers and Rayleigh, Lanczos, IRA,,, Bathe & Wilson and QZ coefficients. They each have their scope of use, their advantages, their disadvantages and their development history.

To deal effectively with large modal problems (in terms of mesh size and/or number of modes sought), it is recommended to use the macro-command CALC_MODES with the option “BANDE” divided into several sub-bands. It breaks down the modal calculation of a standard GEP (symmetric and real), into a succession of independent, less expensive, more robust and more accurate sub-calculations. Just sequentially, the gains can be noteable: factors 2 to 5 in time, 2 or 3 in peak RAM and 10 to 104 on the average error of the modes.

In addition, its multi-level parallelism can provide additional gains of the order of 20 in time and 2 in peak RAM, by reserving around sixty processors.

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.

A specific chapter details the implementation of parallelism and intensive computing in the context of standard GEP modal calculations.

• 1. Introduction
• 2. Overview of modal solvers in structural mechanics
  • 2.1. Modal issues
  • 2.2. Resolution methods
  • 2.3. Linear algebra libraries
  • 2.4. Some results and benchmarks
• 3. background
  • 3.1. Problem
  • 3.2. Taking into account boundary conditions
  • 3.3. Properties of matrices
  • 3.4. Properties of clean modes
  • 3.5. Real spectrum estimation
  • 3.6. Spectral transformation
  • 3.7. Implementation in Code_Aster
• 4. Inverse power method (CALC_MODES with OPTION among [“PROCHE”, “AJUSTE”, “SEPARE”])
  • 4.1. Introduction
  • 4.2. Localization and separation of eigenvalues
  • 4.3. Inverse power method
  • 4.4. Scope of use
  • 4.5. Display in the message file
  • 4.6. Summary of the settings
• 5. Subspace method (CALC_MODES with OPTION among [“BANDE”, “”, “CENTRE”, “PLUS_ *”])
  • 5.1. Introduction
  • 5.2. Rayleigh-Ritz analysis
  • 5.3. Choice of projection space
  • 5.4. Choice of spectral shift
• 6. Lanczos method (SOLVEUR_MODAL = _F (METHODE =” TRI_DIAG “))
  • 6.1. Introduction
  • 6.2. Theoretical Lanczos algorithm
  • 6.3. Practical Lanczos algorithm
  • 6.4. Complementary treatments
  • 6.5. Implementation in Code_Aster
  • 6.6. Scope of use
  • 6.7. Display in the message file
  • 6.8. Setup summary
• 7. Algorithm IRA (SOLVEUR_MODAL =_F (METHODE =” SORENSEN “))
  • 7.1. Introduction
  • 7.2. Arnoldi algorithm
  • 7.3. The challenges
  • 7.4. “Implicit Restarted Arnoldi” Algorithm (IRA)
  • 7.5. Implementation in Code_Aster
  • 7.6. Scope of use
  • 7.7. Display in the message file
  • 7.8. Summary of the settings
• 8. Bathe and Wilson method (SOLVEUR_MODAL = _F (METHODE =” JACOBI “))
  • 8.1. Principle
  • 8.2. Convergence tests
  • 8.3. Implementation in Code_Aster
  • 8.4. Scope of use
  • 8.5. Summary of the settings
• 9. QZ global method (SOLVEUR_MODAL = _F (METHODE =”QZ”))
  • 9.1. Introduction
  • 9.2. The GEP and the QZ method
  • 9.3. The QZ method
  • 9.4. Implementation in Code_Aster
  • 9.5. Scope of use
  • 9.6. Display in the message file
  • 9.7. Summary of the settings
• 10. Parallelism and intensive computing
  • 10.1. Integrated operators
  • 10.2. Operator CALC_MODES, option “BANDE” split into sub-bands
• 11. Bibliography
  • 11.1. Books/articles/proceedings/theses…
  • 11.2. Reports/reports EDF
  • 11.3. Internet resources
• 12. Description of document versions
• 13. Appendix 1. Overview of the QR algorithm
  • 13.1. Principle
  • 13.2. The shift strategy
  • 13.3. Balancing
  • 13.4. The QL method
• 14. Appendix 2. Gram-Schmidt orthogonalization
  • 14.1. Introduction
  • 14.2. Gram-Schmidt algorithm (GS)
  • 14.3. Modified Gram-Schmidt Algorithm (GSM)
  • 14.4. Iterative Gram-Schmidt Algorithm (IGSM)
• 15. Appendix 3. Jacobi method
  • 15.1. Principle
  • 15.2. A few choices