A generic and efficient Taylor series–based continuation method using a quadratic recast of smooth nonlinear systems

• Published in: International journal for numerical methods in engineering Vol. 119; no. 4; pp. 261 - 280
• Main Authors: Guillot, Louis, Cochelin, Bruno, Vergez, Christophe
• Format: Journal Article
• Language: English
• Published: Bognor Regis Wiley Subscription Services, Inc 27.07.2019; Wiley
• Subjects: Algorithms; Asymptotic methods; asymptotic numerical method; Computer Science; continuation; Continuation methods; Elasticity; Engineering Sciences; Finite element method; Mathematical Physics; Mathematical Software; Mathematics; Mechanics; Nonlinear analysis; Nonlinear systems; Numerical methods; quadratic recast; Robustness (mathematics); Structural mechanics; Taylor series; Transcendental functions; Two dimensional analysis; asymptotic numerical method; nonlinear systems; Taylor series; quadratic recast; finite element method; continuation
• ISSN: 0029-5981; 1097-0207
• DOI: 10.1002/nme.6049

Summary This 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.