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

- Article in a journal -
 

Area
Mechanical Engineering

Author(s)
Massamaesso Bilasse , Isabelle Charpentier , El Mostafa Daya , Yao Koutsawa

Published in
Computer Methods in Applied Mechanics and Engineering

Year
2009

Abstract
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 Diamant-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.

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BibTeX
@ARTICLE{
         Bilasse2009Aga,
       journal = "Computer Methods in Applied Mechanics and Engineering",
       ad_area = "Mechanical Engineering",
       ad_tools = "Diamant",
       ad_theotech = "Higher Order",
       author = "Bilasse, Massamaesso and Charpentier, Isabelle and Daya, El Mostafa and Koutsawa,
         Yao",
       title = "A generic approach for the solution of nonlinear residual equations. {Part II}:
         {H}omotopy and complex nonlinear eigenvalue method",
       year = "2009",
       volume = "198",
       number = "49--52",
       pages = "3999--4004",
       abstract = "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 Diamant-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.",
       doi = "10.1016/j.cma.2009.09.015",
       issn = "0045-7825",
       id = "Bilasse2009Aga"
}


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