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Nonlinear Iterative Solution Procedures

Linear analyses enforce the assumption that the stiffness of the structure does not change throughout the solution. This means that the stiffness matrix need only be evaluated and inverted once, the displacements for any number of loadcases being computed directly from the static finite element equilibrium equation:

d = F K-1

For nonlinear analyses, it is no longer possible to obtain a solution that equilibrates a set of external loads with the internal stress and strain field in such a direct "one-step" manner. This is because the stiffness of the structure may change at any stage. In this case a solution procedure is adopted in which the load is applied gradually over a number of increments, enabling a continual "tracking" of the structural response.

Within each load increment a linear prediction of the nonlinear response is initially made and subsequent iterative corrections are performed in order to restore equilibrium by the elimination of the residual or "out of balance" forces. The iterative corrections are referred to various convergence criteria that control to what extent equilibrium has been achieved. Such a solution procedure is commonly referred to as an incremental-iterative (or predictor-corrector) method.

In LUSAS, the nonlinear solution is based on the Newton-Raphson iterative procedure.

Specification of nonlinear procedures (via the loadcase properties) does not generate, of itself, a nonlinear response in a structure. It simply invokes the solution procedures. Thus if no material, geometric or boundary condition nonlinearity is specified, a linear solution will be produced.

Newton-Raphson Iterative Procedure

In this procedure the initial prediction of the incremental solution is based on the tangent stiffness (KT1) from which incremental displacement (Da1) and the corresponding iterative force corrections [y(a1)] may be computed. Subsequent iterative calculations use the current tangent stiffness. In the figure below, the initial tangent stiffness (KT1) is evaluated and the incremental load (R) is applied. This load-stiffness combination produces an iterative displacement (Da1) that does not generally lead to force equilibrium in the structure and an out of balance residual force is created [y(a1)]. The new tangent stiffness is then evaluated (KT2) and, in conjunction with both the applied load (R) and the previous residual force, used to predict the next iterative displacement (Da2). The next residual force may then be evaluated and the procedure continued until the convergence criteria are satisfied.

Only the tangent stiffness (KT1) necessarily corresponds to the elastic modulus, subsequent evaluations will differ according to the degree of nonlinearity present in the analysis.

Newton-Raphson procedures assume that a displacement solution may be found for a given load increment and that, within each load increment, the load level remains constant. This method is, therefore, often referred to as a constant load level incrementation procedure.

Although the Newton-Raphson method generally converges rapidly, the continual formation and inversion of the tangent stiffness matrix at the start of each iteration is often expensive. The need for a robust yet inexpensive procedure therefore leads to the development of the family of modified Newton-Raphson methods.

Nonlinear_Iterative_Solution_Procedures.gif (7816 bytes)

Iterative Acceleration

A slow convergence rate may be significantly improved by employing an iterative acceleration technique. In cases of severe and often localised nonlinearity, typically encountered in materially nonlinear or contact problems, some form of acceleration may be a prerequisite to convergence.

In LUSAS, iterative acceleration may be performed by applying line searches. Essentially, the line search procedure involves additional optimisation iterations in which the potential energy associated with the residual forces at each iterative step are minimised. The added expense of their use is usually more than offset by their effectiveness in reducing the number of iterations required for convergence. They are particularly useful for material nonlinear problems having localised nonlinearity - especially when using the modified Newton-Raphson methods.

Line search application is controlled via parameters in the Solution Strategy> Advanced... section of the Nonlinear Control properties.

Line searches are carried out if the absolute value of the line search tolerance factor (otherwise know as epsln) in the nonlinear iterative log file output exceeds the line search tolerance factor (toline). The aim is to make the line search tolerance parameter reasonably small on each iteration to speed convergence and prevent divergence.

See Section in Theory Manual for more details.

Separate Iterative Loops

In problems where both material and slideline contact nonlinearities are present, convergence difficulties can arise when evaluating material nonlinearities with contact conditions that are invalid because the solution is not in equilibrium. To avoid this situation, contact equilibrium can firstly be established using elastic properties from the previous load increment before the material nonlinearity is resolved.

This option only has effect in the presence of nonlinear materials and contact. Since the procedure is designed to deal with contact and nonlinear material interaction it only applies to those elements that can be used with slidelines (i.e. not bar, beam or plate elements, etc.). If these other element types have nonlinear material models assigned, the materially nonlinear behaviour will be computed for both the contact only iterative loop as well as the subsequent contact and material iterative loop. This parameter is otherwise known as isilcp and the default is for no separate iterative loops to be invoked.

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