Fundamental solutions and asymptotic numerical methods for bifurcation analysis of nonlinear bi‐harmonic problems

Abstract

New algorithms, combining asymptotic numerical method (ANM) and method of fundamental solutions, are proposed to compute bifurcation points on branch solutions of a nonlinear bi-harmonic problem. Three methods, mainly based on asymptotic developments framework, are then proposed. The first one consists in exploiting the ANM step accumulation close to the bifurcation points on a solution branch, the second method allows the introduction of an indicator that vanishes at the bifurcation points, and finally the first real root of the Padé approximant denominator represents the third bifurcation indicator. Two numerical examples are considered to analyze the robustness of these algorithms. KEYWORDS asymptotic numerical method, bifurcation branch, bifurcation indicator, bi-harmonic problem, method of fundamental solutions 1 Numer Methods Partial Differential Eq. 2019;35:2091-2102. wileyonlinelibrary.com/journal/num

Bifurcation points and bifurcated branches in fluids mechanics by high‐order mesh‐free geometric progression algorithms

“…To do that, an indicator that takes advantage of the terms of the Taylor series is used to analyze and to detect these bifurcation points if they exist. Several methods to compute numerically this kind of instability in fluid and structure mechanics can be found in recent works as References 24,25,29,42‐45. In this context and using the asymptotic numerical method (ANM), several works have been devoted to the detection of an accumulation zone of step in the vicinity of singular points.…”