Shell Optimization

Rousselet, Bernard; Mehrez, S.; Myśliński, Andrzej; Piekarski, J.

doi:10.1007/978-94-011-0453-1_11

“…1 This method is proved to be e cient and it remains stable at a turning point. The total computing time needed to obtain an optimal shape is seven or eight times the ÿnite element analysis computing time, but the successive shapes do not change a lot after the second shape, and this shape may be su cient in most cases.…”

Section: Resultsmentioning

confidence: 96%

“…For a geometrically non-linear beam in 3-D, equations for G are given by Antman, 13 Simo 14 and Simo and Vu-Quoc; 15 its shape sensitivity has been addressed 16 and justiÿed. 1 In order to describe the shape optimization problem, we introduce a cost functional:…”

Section: Example 22mentioning

confidence: 99%

Sensitivity computation and shape optimization for a non-linear arch model with limit-points instabilities

Aubert¹,

Rousselet

²

1998

Int. J. Numer. Meth. Engng.

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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 1 have to be modiÿed for limit points and simple bifurcation points. These modiÿcations introduce numerical problems which occur at limit points.2 Numerical systems are very sti and the quadratic convergence of Newton-Raphson algorithm vanishes, then higher-order derivatives have to be computed with respect to state variables.3 A geometrically non-linear curved arch is implemented with a ÿnite element method via a formal calculus approach. Thickness and=or shape for di erentiable 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.

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“…In this article, we validate the sensitivity analysis of a geometrically non-linear arch. 1 This method is proved to be e cient and it remains stable at a turning point. The total computing time needed to obtain an optimal shape is seven or eight times the ÿnite element analysis computing time, but the successive shapes do not change a lot after the second shape, and this shape may be su cient in most cases.…”

Section: Resultsmentioning

confidence: 99%

“…ÿnd ' ∈ V such that ∀ ' ∈ V; G( ; ' 0 ; '; ) [ '] = 0 (1) where is a loading parameter, and V is the space of all admissible conÿgurations for the beam midcurve; depending on the model it may be either replacement or position of the midcurve. is the design variable which is ÿxed when G is solved.…”

Section: Classical Equationsmentioning

confidence: 99%

Sensitivity computation and shape optimization for a non‐linear arch model with limit‐points instabilities

Aubert¹,

Rousselet²

1998

Int. J. Numer. Meth. Engng.

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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 1 have to be modiÿed for limit points and simple bifurcation points. These modiÿcations introduce numerical problems which occur at limit points. 2 Numerical systems are very sti and the quadratic convergence of Newton-Raphson algorithm vanishes, then higher-order derivatives have to be computed with respect to state variables. 3 A geometrically non-linear curved arch is implemented with a ÿnite element method via a formal calculus approach. Thickness and=or shape for di erentiable 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.Example 2.1. For a geometrically non-linear shallow beam in 2-D with a small initial curvature 11 and linear change of curvature: ' = (v; w) ∈ V , ' = ( v; w) ∈ V; depending on the boundary conditions, V may be H 1 0 (0; L) × H 2 0 (0; L) for a beam clamped at both ends; V= {(v; w) ∈ H 1 (0; L) × H 2 (0; L) = v(0) = 0; w(0) = 0; w (0) = 0} for a cantilever beam clamped at the left end. V= {(v; w) ∈ H 1 (0; L) × H 2 (0; L) = v(0) = 0; w(0) = 0; v(L) = 0; w(L) = 0} for a simply supported beam.

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