Abstract
SummaryThis paper is concerned with a Taylor series–based continuation algorithm, the so‐called Asymptotic Numerical Method (ANM). It describes a generic continuation procedure to apply the ANM principle at best, in other words, that presents a high level of genericity without paying the price of this genericity by low computing performances. The way to quadratically recast a system of equation is now part of the method itself, and the way to handle elementary transcendental function is detailed with great attention. A sparse tensorial formalism is introduced for the internal representation of the system, which, when combined with a block condensation technique, provides a good computational efficiency of the ANM. Three examples are developed to show the performance and the versatility of the implementation of the continuation tool. Its robustness and its accuracy are explored. Finally, the potentiality of this method for complex nonlinear finite element analysis is enlightened by treating 2D elasticity problems with geometrical nonlinearities.
Highlights
The so-called Asymptotic Numerical Method (ANM), first described in [12] and [11], is a continuation technique based on high order Taylor series expansions of the unknowns with respect to a path parameter
Combinig the experiences of twenty years of work on the ANM and this experience on Automatic Differentiation, we present a generic way to solve an algebraic system with the ANM principle applied at best, ie, providing both genericity and efficiency
The decades of experience using the ANM continuation resulted in fourth version of the continuation software Manlab that is used here to treat successfully different systems
Summary
The so-called Asymptotic Numerical Method (ANM), first described in [12] and [11], is a continuation technique based on high order Taylor series expansions of the unknowns with respect to a path parameter. It allows a user to enter its own algebraic system and to interactively draw the bifurcation diagram It works well for systems with a few hundreds of equations but the quadratic recast of elementary transcendental functions was a little abstruse and finite element mechanical models could not be implemented because of the lack of performances. Another very important potentiality of this method is the capacity to implement a FE mechanical model in a quite simple way and, that time, without suffering from poor computing performances. This domain of utility can be computed directly from the knowledge of the series with the formula (6)
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