An algorithm and a method to search bifurcation points in non‐linear problems

Abstract

SUMMARYThis paper is devoted to a numerical method able to help the determination of the bifurcation threshold in non-linear time-independent continuum mechanic problems. First, some theoretical results about uniqueness are recalled. In the framework of the large-strain assumption, the di erences between the classical ÿnite-step problem and the rate problem are presented. An iterative algorithm able to solve the rate problem is given. Using di erent initializations, it is seen in some numerical experiments that it is possible with this algorithm to get di erent solutions when the underlying mathematical problem solved does not enjoy a uniqueness property. The constitutive equations used have been chosen to be simple enough to deduce some theoretical knowledge about the corresponding uniqueness problems. Finally, a method is given which is able in some case to give an upper bound of the bifurcation threshold.

Switching deformation modes in post-localization solutions with a quasibrittle material

“…If a random initialization is adopted for [ U t n node ], then it is possible to find nonhomogeneous solutions. In fact, for classical continua, our experience (see [Chambon et al 2001b]) is that as soon as uniqueness is lost, the duplication of numerical experiments can yield different solutions, changing only some numerical parameters such as the time step size or the first guess of a given time step. Since all of them are properly converged, this means that they are all different solutions of the same initial boundary value problem defined by the same history of boundary conditions.…”

“…Then the drawback of such a method is that the mode corresponding to the null eigenvalue which allows theoretically to follow the bifurcated solution can correspond for some point of the studied structure to a constitutive branch (loading or unloading) different to the one used to compute the linearized stiffness matrix. In this paper we prefer to follow the ideas initially applied in [Chambon et al 2001b] where the solution for a time step is searched with a Newton-Raphson method with different first estimations which can (if the problem has more than one solution) yield different properly converged solutions.…”

A finite element method for poro mechanical modelling of geotechnical problems using local second gradient models

“…This could be a geometrical default [41], a material imperfection [35] or a small disturbing force [42]. In a different way, Chambon and co-workers [43] proposed an algorithm to search localized solutions in perfect sample using a random initialization of the strain rate at the beginning of the iterative procedure. In the following computations, a material imperfection is introduced in the bottom left finite element.…”

Local Second Gradient Models and Damage Mechanics: 1D Post-Localization Studies in Concrete Specimens

“…In that way, different solutions are possible (an inner hard solution, a hard-soft solution, a soft-hard-soft solution... see figure 3.1). In order to built a patch solution, one has to equate the values of the displacements u, strains u and of the two internal forces N M and M at the ends of the different pieces in order to meet the virtual power equation, and then to check that A 2D second gradient element has been developed [39], [40] and implemented in the finite element code LAGAMINE (Université de Liége). The formulation of the element and the corresponding constitutive equations use a mathematical constraint between the micro kinematics description and the usual macro deformation gradient field.…”

“…The bar is modelled using the 2D second gradient element of the finite element code LAGAMINE (Université de Liége, [39], [40]) under plane deformations. In order to avoid any 2D effects, a zero vertical displacement is applied at the upper and lower boundaries along the bar (u 2 = 0, figure 4.1).…”