An approximation-concepts approach to shape optimal design

Braibant, V.; Fleury, Claude

doi:10.1016/0045-7825(85)90002-7

Cited by 89 publications

(12 citation statements)

References 4 publications

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“…Consequently, some cases converged to the low-fidelity solution. 1,2 After realizing that significant differences could arise in the fidelity models, new methods were developed that were proven to converge to the highfidelity solution. Some of these methods used gradient information, 3 and others did not.…”

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confidence: 99%

Hybrid Variable Fidelity Optimization by Using a Kriging-Based Scaling Function

2005

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Solving design problems that rely on very complex and computationally expensive calculations using standard optimization methods might not be feasible given design cycle time constraints. Variable fidelity methods address this issue by using lower-fidelity models and a scaling function to approximate the higher-fidelity models in a provably convergent framework. In the past, scaling functions have mainly been either first-order multiplicative or additive corrections. These are being extended to second order. In this investigation variable metric approaches for calculating second-order scaling information are developed. A kriging-based scaling function is introduced to better approximate the high-fidelity response on a more global level. An adaptive hybrid method is also developed in this investigation. The adaptive hybrid method combines the additive and multiplicative approaches so that the designer does not have to determine which is more suitable prior to optimization. The methodologies developed in this research are compared to existing methods using two demonstration problems. The first problem is analytic, whereas the second involves the design of a supercritical high-lift airfoil. The results demonstrate that the krigingbased scaling methods improve computational expense by lowering the number of high-fidelity function calls required for convergence. The results also indicate the hybrid method is both robust and effective. Nomenclature f = objective function g = inequality constraint h = equality constraint l = design space lower bounds u = design space upper bounds W = hybrid weighting value x = design vector β = multiplicative scaling function γ = additive scaling function = trust region size f = objective function convergence tolerance x = design variable convergence tolerance ρ = trust region ratio ∇ = gradient operator ∇ 2 = Hessian operator Subscripts high = high-fidelity model i, j, k = free indices for elements in a vector or matrix low = low-fidelity model m = number of design variables n = current iteration number scaled = scaled low-fidelity value u = unscaled constraint

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confidence: 99%

Hybrid Variable Fidelity Optimization by Using a Kriging-Based Scaling Function

2005

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“…The first concerns the selection of the design variables and, as a consequence, the possibility to investigate not only the optimal size (Fleury 1973) and shape (Braibant and Fleury 1985;Beckers 1991) but also the optimal topology (Bendsoe and Sigmund 2003) and material properties (Sigmund 1995;Pedersen 1991), as depicted in Fig. 1.…”

Section: Fig 1 Structural Optimization Problemsmentioning

confidence: 99%

Optimization methods for advanced design of aircraft panels: a comparison

et al. 2009

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Advanced nonlinear analyses developed for estimating structural responses for recent applications for the aerospace industry lead to expensive computational times. However optimization procedures are necessary to quickly provide optimal designs. Several possible optimization methods are available in the literature, based on either local or global approximations, which may or may not include sensitivities (gradient computations), and which may or may not be able to resort to parallelism facilities. In this paper Sequential Convex Programming (SCP), Derivative Free Optimization techniques (DFO), Surrogate Based Optimization (SBO) and Genetic Algorithm (GA) approaches are compared in the design of stiffened aircraft panels with respect to local and global instabilities (buckling and collapse). The computations are carried out with software developed for the European aeronautical industry. The specificities of each optimization method, the results obtained, computational time considerations and their adequacy to the studied problems are discussed.

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“…The design variables are, respectively, the coefficients of the polynomial curves and the superposition coefficients. Braibant and Fleury 4, 5 proposed to represent the target structure using spline curves, such as the Bezier and B‐spline curves; the spline curve control points were used as the design variables.…”

Section: Introductionmentioning

confidence: 99%

“…The design variables are, respectively, the coefficients of the polynomial curves and the superposition coefficients. Braibant and Fleury [4,5] proposed to represent the target structure using spline curves, such as the Bezier and B-spline curves; the spline curve control points were used as the design variables.The design-dependent Dirichlet boundary conditions can be imposed without any special numerical techniques in shape optimization methods, since the imposed nodes are obviously the nodes on the finite element edges which represent the surfaces of the target structure. Furthermore, both the nodal coordinate sensitivity [2, 6] and shape sensitivity [7-9] can be computed on the boundaries.…”

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A level set‐based topology optimization method targeting metallic waveguide design problems

Yamasaki

Nomura

Kawamoto

et al. 2011

Numerical Meth Engineering

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SUMMARYIn this paper, we propose a level set-based topology optimization method targeting metallic waveguide design problems, where the skin effect must be taken into account since the metallic waveguides are generally used in the high-frequency range where this effect critically affects performance. One of the most reasonable approaches to represent the skin effect is to impose an electric field constraint condition on the surface of the metal. To implement this approach, we develop a boundary-tracking scheme for the arbitrary Lagrangian Eulerian (ALE) mesh pertaining to the zero iso-contour of the level set function that is given in an Eulerian mesh, and impose Dirichlet boundary conditions at the nodes on the zero iso-contour in the ALE mesh to compute the electric field. Since the ALE mesh accurately tracks the zero iso-contour at every optimization iteration, the electric field is always appropriately computed during optimization. For the sensitivity analysis, we compute the nodal coordinate sensitivities in the ALE mesh and smooth them by solving a Helmholtz-type partial differential equation. The obtained smoothed sensitivities are used to compute the normal velocity in the level set equation that is solved using the Eulerian mesh, and the level set function is updated based on the computed normal velocity. Finally, the utility of the proposed method is discussed through several numerical examples.

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An approximation-concepts approach to shape optimal design

Cited by 89 publications

References 4 publications

Hybrid Variable Fidelity Optimization by Using a Kriging-Based Scaling Function

Hybrid Variable Fidelity Optimization by Using a Kriging-Based Scaling Function

Optimization methods for advanced design of aircraft panels: a comparison

A level set‐based topology optimization method targeting metallic waveguide design problems

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