A generic approach for the solution of nonlinear residual equations. Part II: Homotopy and complex nonlinear eigenvalue method

摘要

Solutions of nonlinear smooth PDE problems may be approximated as higher order truncated Taylor series using the Asymptotic Numerical Method (ANM). The Diamant approach, already presented in Part I, is a generic and efficient Automatic Differentiation implementation of the ANM. In this second Part, a Dia-mant-based ANM driver is designed for the solution of nonlinear problems involving a homotopy, that is an artificial transformation of the original problem into a simpler one whose solutions are known. Complex nonlinear eigenvalue problems are considered as an application, the homotopy being achieved by continuation from the real eigenvalue problem to the complex one. Numerical examples are presented for sandwich beams with frequency dependent viscoelastic cores (PVB and 3M ISD112). Three different constitutive laws - a constant modulus, a power law and a generalized Maxwell model - are presented to enhance the capabilities of this Diamant driver.