r5.03.14 Implicit and explicit integration of nonlinear behavioral relationships

Summary

This document describes two integration methods (one implicit and one explicit) for solving problems with nonlinear behaviors in operators STAT_NON_LINE [R5.03.01] and DYNA_NON_LINE [R5.05.05].

The implicit numerical method presented is that of Newton, with or without linear search.

The explicit numerical method is that of Runge Kutta of order 2 with adaptive steps.

• 1. Overview of the integration of laws of behavior
  • 1.1. Quasi-static problem solving algorithm
  • 1.2. Purpose of integration
  • 1.3. Resolution methods
• 2. Explicit integration of an incremental behavioral relationship
  • 2.1. Using an explicit method
  • 2.2. Time-step control
  • 2.3. Influence on the overall stage
• 3. Implicit integration of an incremental behavioral relationship.
  • 3.1. Implicit integration by Newton’s method
  • 3.2. Tangent operator
  • 3.3. Implicit integration using Newton’s method with linear search
• 4. Implementation of a new behavioral model
  • 4.1. Possible methods
  • 4.2. Implementation of a new behavioral model using the explicit method
  • 4.3. Complete implementation of a new behavior in PLASTI
  • 4.4. Easy implicit implementation of a new behavior model in the environment PLASTI
• 5. Bibliography
• 6. Description of document versions