Variation et optimisation de formes

Henrot, Antoine; Pierre, Michel

doi:10.1007/3-540-37689-5

Cited by 516 publications

(757 citation statements)

References 75 publications

(103 reference statements)

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“…, and I D (·) the indicator function of the domain D. With this choice of the cost functional and the flow model, existence of solutions of problem (1)- (2) is in general stated [17,19] for…”

Section: Geometry Description State Equations and System Observationmentioning

confidence: 99%

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Shape optimization for viscous flows by reduced basis methods and free‐form deformation

Manzoni

Quarteroni

Rozza

2011

Numerical Methods in Fluids

107

108

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SUMMARYIn this paper we further develop an approach previosuly introduced in [23] for shape optimization that combines a suitable low-dimensional parametrization of the geometry (yielding a geometrical reduction) with reduced basis methods (yielding a reduction of computational complexity). More precisely, freeform deformation techniques are considered for the geometry description and its parametrization, while reduced basis methods are used upon a finite element discretization to solve systems of parametrized partial differential equations. This allows an efficient flow field computation and cost functional evaluation during the iterative optimization procedure, resulting in effective computational savings with respect to usual shape optimization strategies. This approach is very general and can be applied to a broad variety of problems. In this paper we apply it to find the optimal shape of aorto-coronaric bypass anastomoses based on vorticity minimization in the down-field region. Blood flows in the coronary arteries are modelled using Stokes equations; afterwards, results have been verified in feedback using Navier-Stokes equations.

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Section: Geometry Description State Equations and System Observationmentioning

confidence: 99%

“…where β(µ) and β 0 are the same constants as in (17) and (19). In order to develop both the a posteriori error estimation and the Offline-Online computational procedure, let us explicit the RB …”

Section: Reduced Basis Approximation: Formulation and Main Featuresmentioning

confidence: 99%

Shape optimization for viscous flows by reduced basis methods and free‐form deformation

Manzoni

Quarteroni

Rozza

2011

Numerical Methods in Fluids

107

108

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“…For the shape derivative formulas of the area and of the perimeter, we refer, for instance, to [7]. In order to derive E(Ω) with respect to the domain, we consider a parametrization of ∂Ω:…”

Section: Appendixmentioning

confidence: 99%

Elastic energy of a convex body

Bianchini

Henrot

Takahashi

2015

Mathematische Nachrichten

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In this paper a Blaschke-Santaló diagram involving the area, the perimeter and the elastic energy of planar convex bodies is considered. More precisely we give a description of setwhere A is the area, P is the perimeter and E is the elastic energy, that is a Willmore type energy in the plane. In order to do this, we investigate the following shape optimization problem:where C is the class of convex bodies with fixed perimeter and µ 0 is a parameter. Existence, regularity and geometric properties of solutions to this minimum problem are shown.

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“…Indeed, the minus sign in front of the electrostatic energy introduces additional difficulties. This question is currently under investigation using results and techniques of shape optimization coming from [5,17,21].…”

Section: Remark 12mentioning

confidence: 99%

“…We use in the following the usual notion of shape derivatives. We refer the reader to [17,21] for details on this notion. The idea is to build a differential of a functional defined on domains.…”

Section: Remark 12mentioning

confidence: 99%

Electrowetting of a 3D drop: numerical modelling with electrostatic vector fields

Ciarlet¹,

Scheid²

2010

ESAIM: M2AN

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Abstract. The electrowetting process is commonly used to handle very small amounts of liquid on a solid surface. This process can be modelled mathematically with the help of the shape optimization theory. However, solving numerically the resulting shape optimization problem is a very complex issue, even for reduced models that occur in simplified geometries. Recently, the second author obtained convincing results in the 2D axisymmetric case. In this paper, we propose and analyze a method that is suitable for the full 3D case.Mathematics Subject Classification. 65N12, 65N30, 49Q10.

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Variation et optimisation de formes

Cited by 516 publications

References 75 publications

Shape optimization for viscous flows by reduced basis methods and free‐form deformation

Shape optimization for viscous flows by reduced basis methods and free‐form deformation

Elastic energy of a convex body

Electrowetting of a 3D drop: numerical modelling with electrostatic vector fields

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