r5.05.02 Algorithms for directly integrating the DYNA_VIBRA/BASE_CALCUL operator =” PHYS “

Summary:

This document describes time integration schemes that are used to directly solve dynamic problems in linear transient mechanics. The diagrams in NEWMARK and WILSON - \(\theta\) are detailed, as are the « centered differences at a constant step » and « adaptive time step » schemes.

• 1. Introduction
• 2. Methods for the direct, implicit and explicit temporal integration of a dynamic problem
• 3. The diagram WILSON - \(\theta\) [bib1]
  • 3.1. Presentation of the diagram
  • 3.2. Complete algorithm for method WILSON - \(\theta\)
  • 3.3. Stability condition for diagram WILSON - \(\theta\)
• 4. The Newmark schema [bib1], [bib2]
  • 4.1. Presentation of the diagram
  • 4.2. Full Newmark schema algorithm
  • 4.3. Stability conditions of the Newmark schema
• 5. Numerical amortization of implicit schemes
• 6. Diagram of centered differences at a constant step
  • 6.1. Principle
  • 6.2. Stability
  • 6.3. Algorithm
  • 6.4. Diagonal mass matrix
  • 6.5. Checking the time step
  • 6.6. Acceleration calculation
• 7. Adaptive time step diagram
  • 7.1. Principle
  • 7.2. Diagram
  • 7.3. Estimation of the time step according to the required precision
  • 7.4. Choice of the number of steps per apparent period, \(N\)
  • 7.5. Heuristic for the evolution of the time step
  • 7.6. Algorithm
• 8. Conclusion
• 9. Bibliography