A Bernstein Broyden–Fletcher–Goldfarb–Shanno collocation method to solve non-linear beam models

Abstract

A collocation technique based on the use of Bernstein polynomials to approximate the field variable is assessed in Boundary Value Problems (BVPs) of beams with governing non-linear differential equations. The BVPs are transformed into unconstrained optimization problems by means of an extended cost function which leverages the properties of the Bernstein basis to enforce the boundary conditions. The minimization of the squared error cost function is conducted by means of the quasi-Newton Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm. The method is tested in benchmarks of various types of non-linearities, including materials with Ludwick stress–strain curves, follower loads and beams on Winkler foundation. The approach is compared with Isogeometric collocation (IGA-c) and straightforward (pseudospectral) Bernstein collocation in terms of performance and computational effort. Moreover, the accuracy and convergence of the method is discussed to ease its successful application to other non-linear beam problems.