Shape Optimization for Control and Isolation of Structural Vibrations in Aerospace and Defense Applications

Hardesty, Sean; Kouri, Drew P.; Lindsay, Payton; Ridzal, Denis; Stevens, Brian; Viertel, Ryan

doi:10.2172/1669731

Cited by 4 publications

(4 citation statements)

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“…We have recently applied the strip method to acoustics and elasticity. The report 26 describes our implementation and motivates suitable approaches for extensions to other PDEs. F I G U R E B1 Mapping from unit triangle reference element with coordinates 𝜂 ∈ R 2 to a triangle T with vertices q 1 , q 2 , q 3 ∈ R 2 which can be rearranged into the discrete shape sensitivity equation…”

Section: Discussionmentioning

confidence: 99%

The strip method for shape derivatives

Hardesty

Antil

Kouri

et al. 2022

Numerical Meth Engineering

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A major challenge in shape optimization is the coupling of finite element method (FEM) codes in a way that facilitates efficient computation of shape derivatives. This is particularly difficult with multiphysics problems involving legacy codes, where the costs of implementing and maintaining shape derivative capabilities are prohibitive. The volume and boundary methods are two approaches to computing shape derivatives. Each has a major drawback: the boundary method is less accurate, while the volume method is more invasive to the FEM code. We introduce the strip method, which computes shape derivatives on a strip adjacent to the boundary. The strip method makes code coupling simple. Like the boundary method, it queries the state and adjoint solutions at quadrature nodes, but requires no knowledge of the FEM code implementations. At the same time, it exhibits the higher accuracy of the volume method. As an added benefit, its computational complexity is comparable to that of the boundary method, that is, it is faster than the volume method. We illustrate the benefits of the strip method with numerical examples.

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Section: Discussionmentioning

confidence: 99%

The strip method for shape derivatives

Hardesty

Antil

Kouri

et al. 2022

Numerical Meth Engineering

Self Cite

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“…Recently, it has been shown that the weak formulation has numerical advantages, see, for instance, [8,20,31,56]. In [27,39], practical advantages of volume shape formulations have also been demonstrated.…”

Section: Optimization On Shape Spaces With Steklov-poincaré Metricmentioning

confidence: 99%

“…This deformation vector can be seen as an extension of the stochastic gradient v = v(ω) to the hold-all domain D. This stochastic deformation vector can then be used in the expression (27).…”

Section: End Formentioning

confidence: 99%

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PDE-constrained shape optimization: towards product shape spaces and stochastic models

Geiersbach¹,

Loayza-Romero²,

Welker³

2021

Preprint

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Shape optimization models with one or more shapes are considered in this chapter. Of particular interest for applications are problems in which where a so-called shape functional is constrained by a partial differential equation (PDE) describing the underlying physics. A connection can made between a classical view of shape optimization and the differential-geometric structure of shape spaces. To handle problems where a shape functional depends on multiple shapes, a theoretical framework is presented, whereby the optimization variable can be represented as a vector of shapes belonging to a product shape space. The multi-shape gradient and multi-shape derivative are defined, which allows for a rigorous justification of a steepest descent method with Armijo backtracking. As long as the shapes as subsets of a hold-all domain do not intersect, solving a single deformation equation is enough to provide descent directions with respect to each shape. Additionally, a framework for handling uncertainties arising from inputs or parameters in the PDE is presented. To handle potentially high-dimensional stochastic spaces, a stochastic gradient method is proposed. A model problem is constructed, demonstrating how uncertainty can be introduced into the problem and the objective can be transformed by use of the expectation. Finally, numerical experiments in the deterministic and stochastic case are devised, which demonstrate the effectiveness of the presented algorithms. * This work has been partly supported by the state of Hamburg within the Landesforschungsförderung under project "Simulation-Based Design Optimization of Dynamic Systems Under Uncertainties" (SENSUS) with project number LFF-GK11, and by the German Academic Exchange Service (DAAD) within the program "Research Grants-Doctoral Programmes in Germany, 2017/18."

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