2.1.6. Non-linear analysis

This section provides a summary of the non-linear theories of solids and structures considered in RamSeries. Several aspects concerning the solution of a non-linear system of equations are addressed. Typical non-linear constitutive models for metals, like the J-2 plasticity theory, are also explained. The main strategy used in the resolution of non-linear problems in RamSeries is based on the Newton-Raphson method (see for instance [Oliver_1990], chapters 1 and 2). More advanced algorithms like the Line-Search and the Arc-Length methods are also implemented (see for instance chapter 9 in [Oliver_1990]). On the other hand, the elasto-plastic models implemented in RamSeries are based on the first three chapters of the classical book by Simo and Hugues [Simo_1998]. In particular, the algorithms presented in chapter 2 of the book by Simo and Hugues are taken as reference.

2.1.7. Non-linear systems of equations

All strategies for the resolution of a non-linear system of equations are related with the decomposition of the non-linear problem in several linear systems, each of them solved using the well-established methods for the solution of linear problems. A well-known of these strategies is the Newton-Raphson method. It consists of an incremental-iterative scheme for which the total applied load is decomposed into several increments. For each increment, an iterative procedure is performed until the desired convergence criteria is achieved. There are several methods to control the fulfilment of the load-displacement curve of the structure. Each one of these methods is actually based on the control of a different parameter, like for instance the load, the displacement or the arc-length increments. Some advanced procedures, like the line-search, automatic incrementation or automatic arc-length switch can be applied to speed-up the calculations.

Deeper explanations can be found in [Bathe_2014] and [Crisfield_2991].