A multi-step linearization technique for a class of boundary value problems in non-linear mechanics

Non-linear finite element analyses of structures (such as beams) involve construction of weak solutions for the governing equations. While a weak approach weakens the differentiability requirements of the so-called shape functions, the governing equations are only satisfied in an integral sense and not point-wise, or, even path-wise. Moreover, use of a finite mesh leads to a stiffening of the numerical model. While strong solutions obtained through some of the existing mesh-free collocation methods overcomes some of these lacunae to an extent, the quality of the numerical solutions would be considerably improved if the computational algorithm were able to faithfully reproduce (or approximate or preserve) certain geometrical features of the response surfaces or manifolds. This paper takes the first step towards realizing this objective and proposes a multi-step transversal linearization (MTL) technique for a class of non-linear boundary value problems, which are treated as conditionally dynamical systems. Numerical explorations are performed, to a limited extent, through applications to large deflection analyses of planar beams with or without plastic deformations.