Abstract
The paper deals with the numerical computation of difficult problems requiring many steps and many degrees of freedom such as the finite element analysis of wrinkling of film–substrate systems. The asymptotic numerical method (ANM) is well adapted to such computations but with a progressive loss of accuracy during step chaining. Thus, correction phases are necessary, which are rarely carried out within ANM. A convergence acceleration algorithm and a step-length adaptation have been included to limit the growth of computation time and to strengthen the reliability of the procedure. This modified version of the ANM is assessed by simulating the appearance and evolution of sinusoidal wrinkles under uniaxial compression.
Highlights
The asymptotic numerical method (ANM) is a numerical technique for solving nonlinear systems depending on a scalar parameter [1]
The key point is controlling the quality of the solution at each step end and this is achieved by combining a convergence acceleration by minimal polynomial extrapolation (MMPE) with Newton–Riks iterations
The aim of the present paper is to propose simple and efficient ANM algorithms permitting a control of accuracy and including the advantages of a convergence acceleration technique
Summary
The asymptotic numerical method (ANM) is a numerical technique for solving nonlinear systems depending on a scalar parameter [1] It describes the solution branch as a sequence of steps, each one being represented by a truncated Taylor series of vectors a ∈ R → U(a, N ) =. Only the simplest version of the ANM introduced in 1994 is required It associates the calculation of the Taylor series and an estimate of the range of validity, a user parameter permitting us to choose between a strategy of large steps and a strategy of high accuracy [6,10,11]. These types of film–substrate modeling have attracted great interest recently, but they have been rarely studied using the finite element method [26, 27]
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