Computer Methods in Applied Mechanics and Engineering
Extrapolation locking and its sanitization in Koiter's asymptotic analysis
Introduction
It is our opinion that in the analysis of a non-linear elastic structure the imperfection sensitivity analysis plays a central role. It is composed of a large number of re-analyses, and consequently, the time consumed by each single analysis has to be very low. This is one of the major reasons for the interest in Liapunov–Smidth–Koiter's asymptotic algorithms [1], whose effectiveness, when searching for analytical solutions, is well known in the literature.
Recently, a lot of work has been done to prove that such an approach produces a coherent and general conceptual frame to recognize and reconstruct simple and multiple bifurcations, snapping and complex post-critical phenomena such as mode-jumping and post-critical attractiveness [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13]. Other works have been published to show that these perturbation algorithms are easy to be implemented in a very general finite element context and that, in cases of weakly non-linear pre-critical behaviours, they provide very accurate results, if care is taken with both the continuum model assumption and its finite element discretization [14], [15], [16].
It seems worth further discussion, because among FEM stability people it is widely held that Koiter's perturbation algorithms work well only in very simple cases, characterized by a banal pre-critical behaviour, and are unreliable when applied to structures or loading paths that imply a sensible non-linear pre-critical behaviour, that is, in all cases which are denoted in the literature as non-linear buckling [17], [18], [19], [20].
On the other hand, if we require that an asymptotic method based on linear, or at most, quadratic extrapolations of two single paths (the fundamental path, starting from the rest configuration, and the bifurcated path, starting from the first bifurcation point of the fundamental path) be able not only to reconstruct accurately very non-linear equilibrium paths of imperfect structures between the rest and the first limit load configurations but also to furnish the initial post-critical behaviour, we need to deeply understand how some description choices can affect the representation non-linearity of the equilibrium path. This is, in fact, the principal cause of loss of accuracy and reliability of the method.
In the sequel, we will show that to represent the equilibrium path by using the displacements as primary variables to extrapolate (compatible formulation) has to be considered the least suitable choice, because, due to a perverse interaction between rigid rotations and high stiffness ratios of the elements, it implies hard non-linearities in the path description and causes a pathological dependence of the results of the bifurcation and the post-critical analysis on the axial/flexural stiffness ratios. We name this phenomenon extrapolation locking, whose occurrence has already been shown within a different context, that of step-by-step solution algorithms, where it comes to light like convergence difficulties of the iteration process [21]. Later on in the paper, we will show that, in the more sensitive case of the asymptotic analysis, such a phenomenon can completely destroy the accuracy of the solution.
In our opinion, these representation problems are the primary cause of the frequent use of a bifurcation analysis characterized by zeroed pre-critical displacements (frozen configuration hypothesis), while usually being motivated by the need to reduce the bifurcation search to a linear eigenvalue problem [6], [7], [22], [23], [24], [25], [26], [27], [28], [29], [30]. However, a more natural and general solution to this problem exists, that is, to use a mixed representation of the equilibrium path which implies independent extrapolations of displacement and stress fields (mixed formulation). As emphasized in [21] and as will be shown in detail, that simple device allows us to both avoid inessential non-linearities in the path representation and obtain reliable results.
It is worth mentioning that the use of both displacement and stress variables increases the dimension of the problem, but the computational extra-cost is less than one might expect. In fact and as will be shown, a non-linear analysis based on mixed formulation can be performed in terms of displacement variables alone, simply by performing static condensation of the stresses in terms of the displacements and using the “compatible” stiffness matrix, banded and positive definite (in the configurations of interest), as iteration matrix. In addition to this, referring to a less non-linear description of the fundamental path appreciably simplifies the bifurcation search and noticeably reduces the overall cost of the analysis. In fact, while a quite fast and reliable algorithm for non-linear eigenvalue analysis is available (see [16], [31]), a coherent use of the mixed formulation noticeably speeds up the solution process, especially in the case of multimodal analysis, where a local linearization is needed in the Jacobi orthogonalization phase of the algorithm (see [7], [16]).
All these assertions will be justified by referring to plane beam and frame test problems, using a slight extension and a mixed generalization of the compatible non-linear finite element model described in [15] (see also [14], [32]), which uses axial stresses as additional primary variables. Such a structural type is a good compromise between simplicity and generality, and in any case, is adequate to make manifest all the relevant aspects of the extrapolation locking phenomenon. The possibility of obtaining reference “exact” results for comparisons is a further advantage.
The paper is planned as follows: (i) we start by recalling the asymptotic method and its main formulae; (ii) extrapolation locking is shown using some test cases; (iii) a very simple test problem is used to fully understand the phenomenon; (iv) the proposed mixed finite element of a two-dimensional beam for the analysis of plane frames is then described in some details; (v) some insights on the numerical strategies are given; (vi) a series of numerical tests are reported, all showing a reliable behaviour and excellent agreement with the exact results; (vii) further comments and suggestions are given in the conclusions.
Section snippets
A quadratic asymptotic algorithm
This section recalls an essential description, specifically for the single buckling mode case, of the asymptotic algorithm proposed in [4], [5], [6], [7], [8], [9], [10], [11], [12], [13]. The interested reader can refer to these papers for further details (see, in particular, [16] for the implementation aspects of the algorithm).
The algorithm provides an asymptotic approximation for the equilibrium path of a non-linear elastic structure subjected to a proportional loading, defined by the
An extrapolation locking phenomenon
Even if the described algorithm has been extensively tested, proving to be capable of providing excellent results and theoretically characterized by a very small error (at least by an asymptotic point of view), its implementation, in the solution of problems characterized by a non-negligible non-linear pre-critical behaviour, sometimes furnishes unexpectedly bad results. This unpredictable behaviour is actually the main reason for the widely held view that the very nature of this asymptotic
The role of primary extrapolation variables
The unreliability of the results provided by the asymptotic algorithm, when analysing structures characterized by high stiffness ratios combined with moderate pre-critical rotations, necessitates further considerations.
Some preliminary remarks are necessary. When we substitute a curve with its Taylor expansion of order n, the obtained extrapolation is affected by an infinitesimal error O(ξn), if ξ is the distance from the extrapolation point. However, we do not get any a priori quantitative
A mixed formulation for a planar beam
As has been shown in Fig. 13, the mixed version of the algorithm presented in Section 2 can be considered the most reliable. Therefore, within the context of the post-critical analysis of plane frames, we need a mixed formulation of a non-linear beam model, by “mixed formulation” meaning nothing but a Hellingher–Reissner formulation of the beam strain energy.
The non-linear beam model just summarized here is the mixed generalization of the model published in [15], to which the interested reader
The non-linear finite element
The finite element used to obtain the numerical results presented in Section 8 is the “mixed version” of the FE described in [15], being based on the same interpolation functions while reproducing the “mixed” strain energy variations , , . To say by other words, the two finite elements are different in formulation, but equal with respect to approximation. Finally, with respect to the FE tested in [15], the contribution of the shear deformation has been explicitly added.
The pseudo-compatible solution strategy
It is obvious that the use of the mixed formulation incurs some computational extra-costs due to the increase in the number of variables. An other drawback is the loss of the advantage of using the compatible matrix, which is banded and positive definite in the rest configuration (λ=0), as iteration matrix in the solution of the linear systems and the eigenvalue problem generated by the perturbation algorithm.
However, it is well known that a mixed problem can be conveniently solved by using a
Numerical results
The numerical results shown in the present section use the format introduced in Section 4, by which the effectiveness of a representation of the equilibrium path is measured by the minimum and maximum eigenvalues of the normalized tangent stiffness operator. In examining the simple test of Section 4, we have compared the results furnished by compatible, mixed formulations and frozen configuration strategy (FC). Now, in the analysis of examples with more complex behaviour, we will also introduce
Concluding remarks
A locking phenomenon, that we called extrapolation locking, has been shown. This form of locking arises in buckling and postbuckling analysis of slender structures characterized by high stiffness ratios (membranal/flexural) and not negligible pre-critical rotations (shells, reticulated domes etc.), when a compatible formulation is used for describing the path evolution. It usually has a very dangerous effect which can destroy the accuracy of the asymptotic method and render its results
Acknowledgements
This research was supported by MURST and Brite–Euram APRICOS (EC BE95-1017).
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