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Gradient-based Approach using Xfem and Cokriging for Optimal Positioning of a Structure in an Acoustic Cavity

Abstract : Reduction of noise is a constraint which is more and more taken into account during design processes. Many devices and technologies have been developed in this context, but the optimal design of such solution remains complicated due to the computation time. Therefore, new efficient technics have to be proposed for studying for instance the noise level in an aircraft cabin with respect with the arrangement within or for finding the optimal design for limiting the noise propagation. Thus, a new gradient-based is proposed. It is based on the use of a finite element analysis combined with surrogate based optimization. The considered mechanical problem couples structural and fluid domain using an acoustic fluid governed by the Helmoltz’s equation, a porous material modeled by the Biot-Allard’s constitutive law [1, 2] and some thin walls arbitrary placed in the fluid domain and governed by elasto-dynamic equation. The air-structure problem is solved using Xfem [3] in order to be able to consider an arbitrary structure placed in the acoustic cavity. The use of the Xfem approach enables us to easily compute the gradient of the pressure field with respect to the design variables which governed the position of the structure in the cavity. Many operators remain indeed independent of the design variables and the main effort consists only of building gradients of the Xfem enrichment operators. Global optimization based on this mechanical problem requires a large number of calls to the mechanical solver. Therefore, a surrogate-based optimization is used. The approach is based on the Efficient Global Optimization [4] composed of two phases: (1) a gradient-based cokriging [5] metamodel is built using only a few sample points and associated responses and gradients and (2) an iterative scheme using the expected improvement [6] allows us to find the global minimum by adding smartly new sample points to the initial surrogate model. The used cokriging metamodel provides better approximation quality than none-gradient-based metamodel by interpolating responses and gradients. The whole strategy has been applied on some 2D and 3D cavity on which the position of a wall is determined in order to minimize the mean quadratic pressure in a control volume. Some examples will be presented for illustrating the performance of the proposed approach. References [1] Maurice A. Biot. Theory of Propagation of Elastic Waves in a Fluid-Saturated Porous Solid. I. Low-Frequency Range. The Journal of the Acoustical Society of America, 28(2):168–178, 1956. [2] Jean-François Allard and Noureddine Atalla. Sound propagation in porous media: modelling sound absorbing materials. Elsevier, London, 1(0):11, 1993. [3] Antoine Legay. An extended finite element method approach for structural-acoustic problems involving immersed structures at arbitrary positions. International Journal for Numerical Methods in Engineering, 93(4):376–399, 2013. [4] Donald R. Jones, Matthias Schonlau, and William J. Welch. Efficient global optimization of expensive black-box functions. Journal of Global optimization, 13(4):455–492, 1998. [5] Luc Laurent, Rodolphe Le Riche, Bruno Soulier, and Pierre-Alain Boucard. An overview of gradient-enhanced metamodels with applications. Archives of Computational Methods in Engineering, 26(1):61– 106, Jan 2019. [6] Matthias Schonlau. Computer Experiments and Global Optimization. PhD thesis, University of Waterloo, 1997.
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https://hal.archives-ouvertes.fr/hal-02136463
Contributor : Luc Laurent Connect in order to contact the contributor
Submitted on : Wednesday, May 22, 2019 - 10:19:41 AM
Last modification on : Wednesday, March 24, 2021 - 2:40:02 PM

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  • HAL Id : hal-02136463, version 1

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Luc Laurent, Antoine Legay. Gradient-based Approach using Xfem and Cokriging for Optimal Positioning of a Structure in an Acoustic Cavity. 13th World Congress of Structural and Multidisciplinary Optimisation, WCSMO 13, May 2019, Beijing, China. ⟨hal-02136463⟩

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