A geometry projection method for shape optimization

Norato, Julián A.; Haber, Robert B.; Tortorelli, Daniel A.; Bendsøe, Martin P.

doi:10.1002/nme.1044

Cited by 117 publications

(81 citation statements)

References 28 publications

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“…(10), the fixed grid finite elements by Kim et al in Ref. (18) or more recently the projection methods as in Norato et al (20). The present work relies on the novel eXtended Finite Element Method (X-FEM) that has been proposed as an alternative to remeshing methods (see Ref.…”

mentioning

confidence: 99%

Stress concentration minimization of 2D filets using X-FEM and level set description

Miegroet

Duysinx

2007

Struct Multidisc Optim

124

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This paper presents and applies a novel shape optimization approach based on the Level Set description of the geometry and the eXtended Finite Element Method (X-FEM). The method benefits from the fixed mesh work using X-FEM and from the curves smoothness of the Level Set description. Design variables are shape parameters of basic geometric features which are described with a Level Set representation. The number of design variables of this formulation remains small whereas global (i.e. compliance) and local constraints (i.e. stress) can be considered. To illustrate the capability of the method to handle stress constraints, numerical applications revisit the minimization of stress concentration in a 2D filet in tension which has been studied previously by Pedersen in (27). Our results illustrate the great interest of using X-FEM and Level Set description together. A special attention is also paid to stress computation and accuracy with the X-FEM.

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confidence: 99%

Stress concentration minimization of 2D filets using X-FEM and level set description

Miegroet

Duysinx

2007

Struct Multidisc Optim

124

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“…This mapping must be differentiable so that we can obtain design sensitivities with respect to the supershape parameters and employ efficient gradient-based optimization methods. To this end, we use the geometry projection method (Norato et al 2004;Bell et al 2012;Norato et al 2015;Zhang et al 2016a). The idea of the geometry projection is simple: the density at a point p in space is the fraction of the volume of the ball B r p of radius r centered at p that intersects the solid geometry, i.e.…”

Section: Geometry Projectionmentioning

confidence: 99%

“…Therefore, this constitutes a shape optimization example. Geometry projection methods have been used for shape optimization in Norato et al (2004;Wein and Stingl 2018).…”

Section: Short Cantilever Beammentioning

confidence: 99%

Topology optimization with supershapes

Norato

2018

Struct Multidisc Optim

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This work presents a method for the continuum-based topology optimization of structures whereby the structure is represented by the union of supershapes. Supershapes are an extension of superellipses that can exhibit variable symmetry as well as asymmetry and that can describe through a single equation, the so-called superformula, a wide variety of shapes, including geometric primitives. As demonstrated by the author and his collaborators and by others in previous work, the availability of a feature-based description of the geometry opens the possibility to impose geometric constraints that are otherwise difficult to impose in density-based or level set-based approaches. Moreover, such description lends itself to direct translation to computer aided design systems. This work is an extension of the author's group previous work, where it was desired for the discrete geometric elements that describe the structure to have a fixed shape (but variable dimensions) in order to design structures made of stock material, such as bars and plates. The use of supershapes provides a more general geometry description that, using a single formulation, can render a structure made exclusively of the union of geometric primitives. It is also desirable to retain hallmark advantages of existing methods, namely the ability to employ a fixed grid for the analysis to circumvent re-meshing and the availability of sensitivities to use robust and efficient gradient-based optimization methods. The conduit between the geometric representation of the supershapes and the fixed analysis discretization is, as in previous work, a differentiable geometry projection that maps the supershapes parameters onto a density field. The proposed approach is demonstrated on classical problems of 2-dimensional compliance-based topology optimization.

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“…In order to circumvent the technical difficulties of the moving mesh problems, a couple of researches have tried to formulate shape optimization with fixed mesh analyses using fictitious domains as in [5], based on fixed grid finite elements in [7] or more recently using projection methods as in [9]. The present work relies on the novel eXtended Finite Element Method (X-FEM) that has been proposed as an alternative to remeshing methods (see [8] or [3] for instance).…”

Section: Introductionmentioning

confidence: 99%

Generalized Shape Optimization Using X-FEM and Level Set Methods

Duysinx

Miegroet

Jacobs

et al. 2006

Solid Mechanics and Its Applications

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Abstract:This paper presents an intermediate approach between parametric shape optimization and topology optimization. It is based on using the recent Level Set description of the geometry and the novel eXtended Finite Element Method (X-FEM). The method takes benefit of the fixed mesh work using X-FEM and of the curves smoothness of the Level Set description. Design variables are shape parameters of basic geometric features. The number of design variables of this formulation is small whereas various global and local constraints can be considered. The Level Set description allows to modify the connectivity of the structure as geometric features can merge or separate from each other. However no new entity can be introduced. A central problem that is investigated here is the sensitivity analysis and the way it can be carried out efficiently. Numerical applications revisit the classical elliptical hole benchmark from shape optimization.

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A geometry projection method for shape optimization

Cited by 117 publications

References 28 publications

Stress concentration minimization of 2D filets using X-FEM and level set description

Stress concentration minimization of 2D filets using X-FEM and level set description

Topology optimization with supershapes

Generalized Shape Optimization Using X-FEM and Level Set Methods

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