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Sensitivity computation and shape optimization for a non‐linear arch model with limit‐points instabilities
Author(s) -
Aubert P.,
Rousselet B.
Publication year - 1998
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/(sici)1097-0207(19980515)42:1<15::aid-nme347>3.0.co;2-n
Subject(s) - mathematics , limit point , finite element method , differentiable function , sensitivity (control systems) , newton's method , numerical analysis , limit (mathematics) , computation , quadratic equation , rate of convergence , mathematical analysis , convergence (economics) , limit load , bifurcation , nonlinear system , geometry , algorithm , computer science , physics , quantum mechanics , economics , thermodynamics , economic growth , electronic engineering , engineering , computer network , channel (broadcasting)
A shape optimization method for geometrically non‐linear structural mechanics based on a sensitivity gradient is proposed. This gradient is computed by means of an adjoint state equation and the structure is analysed with a total Lagrangian formulation. This classical method is well understood for regular cases, but standard equations have to be modified for limit points and simple bifurcation points. These modifications introduce numerical problems which occur at limit points. Numerical systems are very stiff and the quadratic convergence of Newton–Raphson algorithm vanishes, then higher‐order derivatives have to be computed with respect to state variables. A geometrically non‐linear curved arch is implemented with a finite element method via a formal calculus approach . Thickness and/or shape for differentiable costs under linear and non‐linear constraints are optimized. Numerical results are given for linear and non‐linear examples and are compared with analytic solutions. © 1998 John Wiley & Sons, Ltd.