A projection technique for nonlinear adaptive analyses exploring the Generalized Finite Element Method

Resumo

When solving complex engineering problems of structural and solid mechanics, the adoption of lin-
earity hypotheses may not be accurate and the use of nonlinear models can become necessary. Also, numerical

methods are often used to generate approximations to them since analytical solutions are unknown for most of
these problems. In this context, solving the equilibrium equations requires the use of specific strategies, with the
Newton-Raphson Method being one of the most commonly adopted. In standard nonlinear analyses using the
Finite Element Method, an initial solution is obtained for a first discretization. Then, the quality of this solution
is evaluated to decide if further analyses are needed in order to obtain more accurate results, therefore exploring
an improved discretization. If a more refined mesh is necessary, for example, the solution must be recalculated
once again. In this paper, an alternative to this process is presented using a projection technique. Accordingly, all
the information obtained during the analysis using a less refined discretization is transferred to the more refined
one. Thus, the results provided by a first mesh is used as an initial guess for the iterative scheme in order to solve
a second mesh. This also helps improving the convergence within the Newton-Raphson Method. A discussion
related to the performance and effectiveness of this technique, that can be explored in combination with adaptive
procedures for nonlinear analyses, is also presented. Finally, a two-dimensional geometrically nonlinear numerical
problem, using the Generalized Finite Element Method, is shown to validate the presented technique.