Shape optimization for the generalized Graetz problem

International audienceThis article is devoted to the study of two extremal problems arising naturally in heat conduction processes. We look for optimal configurations of thermal axisymmetric fins and model this problem as the issue of (i) minimizing (for the worst shape) or (ii) maximizing (for the best shape) the first eigenvalue of a selfadjoint operator having a compact inverse. We impose a pointwise lower bound on the radius of the fin, as well as a lateral surface constraint. Using particular perturbations and under a smallness assumption on the pointwise lower bound, one shows that the only solution is the cylinder in the first case whereas there is no solution in the second case. We moreover construct a maximizing sequence and provide the optimal value of the eigenvalue in this case. As a byproduct of this result, and to propose a remedy to the non-existence in the second case, we also investigate the well-posedness character of another optimal design problem set in a class enjoying good compactness properties

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“…The model used combines a Fluid Mechanics partial differential equation with a parabolic equation involving a transport term (see e.g. [8,14]).…”

Section: Motivations In Convection-conduction Theorymentioning

confidence: 99%

An extremal eigenvalue problem arising in heat conduction

Nadin

Privat

2016

Journal de Mathématiques Pures et Appliquées

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“…Regarding inverse shape optimization problems, remedies to address the ill-possing include mapping the physical domain onto a fixed computational domain [16,17,18,19,20,21] and redistribution of ill-ordered nodal points [12,13,14,15], Tikhonov regularization [22], homogenization [23], or transforming the problem into a parameter estimation by expanding the shape in terms of a small number of parameters, e.g. adaptive mesh [24], eigenfunction expansion [10], and mesh-morphing [25]. Recently, a method that proved effective is to use the boundary element method and define the variables of the optimization as the angles between adjacent elements [9].…”

Section: Introductionmentioning

confidence: 99%

Shape optimization of conductive-media interfaces using an IGA-BEM solver

Κώστας

Fyrillas

Politis

et al. 2018

Computer Methods in Applied Mechanics and Engineering

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In this paper, we present a method that combines the Boundary Element Method (BEM) with IsoGeometric Analysis (IGA) for numerically solving the system of Boundary Integral Equations (BIE) arising in the context of a 2-D steady-state heat conduction problem across a periodic interface separating two conducting and conforming media. Our approach leads to a fast converging solver that achieves the same level of accuracy for fewer degrees of freedom, when compared with low-order BEM. Additionally, an optimization procedure is developed and test cases of interface's shape optimization, with the aim of heat transfer maximization under given constraints, are demonstrated.

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Pseudo spectral solution of extended Graetz problem for combined pressure-driven and electroosmotic flow in a triangular micro-duct

Alshammari

Akyıldız

2020

Computers & Mathematics with Applications

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Shape optimization for the generalized Graetz problem

Cited by 9 publications

References 32 publications

An extremal eigenvalue problem arising in heat conduction

An extremal eigenvalue problem arising in heat conduction

Shape optimization of conductive-media interfaces using an IGA-BEM solver

Pseudo spectral solution of extended Graetz problem for combined pressure-driven and electroosmotic flow in a triangular micro-duct

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