Shape optimization of an elastic thin shell under various criteria

Bernadou, Michel; Palma, Francisco; Rousselet, Bernard

doi:10.1007/bf01743485

“…Our shape optimization method for structural mechanics is based on a shape gradient computed through an adjoint state equation. [8][9][10] We illustrate the method with an example from solid structural mechanics.…”

Section: Classical Equationsmentioning

confidence: 99%

“…The condition @G=@ = ∈ Im(@G=@') is equivalent 20 to q T @G=@ = 0 where q is a left eigenvector. With the classical incremental equation (8) we cannot compute ' for a given . In order to remain invertible at turning point, the incremental equation (8) is extended to: 2; 20; 21…”

Section: Sensitivity Analysis At Limit Pointmentioning

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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“…Besides, Lindby and Santos defined the shape parameters based on the generic CAD system. Bernadou considered various criteria for shape optimization of 3D shell structures. In fact, the presence of holes in a thin‐walled shell structure immediately complicates the design process and thus received much attention.…”

Section: Introductionmentioning

confidence: 99%

A bispace parameterization method for shape optimization of thin‐walled curved shell structures with openings

Wang

¹

,

Zhang

²

2012

Numerical Meth Engineering

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SUMMARY Simultaneous shape optimization of thin‐walled curved shell structures and involved hole boundaries is studied in this paper. A novel bispace parameterization method is proposed for the first time to define global and local shape design variables both in the Cartesian coordinate system and the intrinsic coordinate system. This method has the advantage of achieving a simultaneous optimization of the global shape of the shell surface and the local shape of the openings attached automatically on the former. Inherent problems, for example, the effective parameterization of shape design variables, mapping operation between two spaces, and sensitivity analysis with respect to both kinds of design variables are highlighted. A design procedure is given to show how both kinds of design variables are managed together and how the whole design flowchart is carried out with relevant formulations. Numerical examples are presented and the effects of both kinds of design variables upon the optimal solutions are discussed. Copyright © 2012 John Wiley & Sons, Ltd.

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

Aubert¹,

Rousselet

²

1998

Int. J. Numer. Meth. Engng.

Self Cite

View full text Add to dashboard Cite

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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Shape optimization of an elastic thin shell under various criteria

Cited by 12 publications

References 11 publications

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

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

A bispace parameterization method for shape optimization of thin‐walled curved shell structures with openings

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

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