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Finite Element Implementation

  • Robert M. Hackett
Chapter

Abstract

The state-of-the-art methodology for modeling nonlinear geometrical and material response in the field of solid mechanics is the finite element method. Embedded in any finite element computational model is a material model. The solution of a set of nonlinear equilibrium equations in the finite element analysis of finite deformations is typically achieved through the employment of a Newton-Raphson iteration procedure. This requires the linearization of the equilibrium equations, which necessitates an understanding of the directional derivative. A relatively simple numerical technique for solving nonlinear equations in computational finite elasticity consists of employing the so-called incremental/iterative solution technique of Newton’s type. It is an efficient method with the desirable feature of a quadratic convergence rate near the solution point. It requires a consistent linearization of all of the quantities associated with the nonlinear problem, generating efficient recurrence update expressions. The nonlinear problem is then replaced by a sequence of easily solved linear equations at each iteration. The element stiffness formulation, based upon the “first elasticity tensor,” is carried out.

Keywords

Finite element method Newton-Raphson Linearization Incremental/iterative solution technique Directional derivative Quadratic convergence rate 

References

  1. Bonet J, Wood RD (2008) Nonlinear continuum mechanics for finite element analysis, 2nd edn. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  2. Holzapfel GA (2000) Nonlinear solid mechanics: a continuum approach for engineering. Wiley, ChichesterzbMATHGoogle Scholar
  3. Hughes TJR, Pister KS (1978) Consistent linearization in mechanics of solids and structures. Comput Struct 8:391–397MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Robert M. Hackett
    • 1
  1. 1.Department of Civil EngineeringThe University of MississippiUniversityUSA

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