A geometry projection method for the topology optimization of plate structures

Zhang, Shanglong; Norato, Julián A.; Gain, Arun L.; Lyu, Naesung

doi:10.1007/s00158-016-1466-6

Cited by 105 publications

(64 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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%

“…A unique feature of the method presented in Bell et al (2012), Norato et al (2015), and Zhang et al (2016a) is that in the ersatz material this variable is penalized as in density-based topology optimization. Therefore, the elasticity tensor at a point p is given by…”

Section: Geometry Projectionmentioning

confidence: 99%

“…Even though supershapes are given in polar coordinates, the computation of the signed distance is not as straightforward as in the case of the geometric representations used in our previous works, whereby the discrete geometric components are determined by a medial axis (in the case of approximately cylindrical bars, Norato et al 2015) or a medial surface (in the case of plates, Zhang et al 2016a). In these geometric representations, which in effect are implicit, distance-constrained offset surfaces (Bloomenthal 1990), the signed distance can be readily computed in closed form as the distance to the medial axis (or surface) minus the offset (e.g., the half-width of bars or half-thickness of plates).…”

Section: Signed Distance Calculationmentioning

confidence: 99%

“…Among the methods listed in Table 1, two methods represent the structure as the union of primitive-shaped geometric components with fixed shape but variable dimensions and position: the moving morphable components method (Guo et al 2014a(Guo et al , 2016Zhang et al 2016b;2017b), and the geometry projection method (Bell et al 2012;Norato et al 2015;Zhang et al 2016a). The former method performs the topology optimization of a) 2d-structures, by employing as geometric components rectangles, approximated via superellipses, and quadrilateral shapes with two opposite sides described by polynomial and trigonometric curves; and b) 3d-structures, by using cuboids.…”

Section: Introductionmentioning

confidence: 99%

“…Section 3 describes the projection of the supershapes onto a density field for the analysis. The computation of the signed distance to (Bell et al 2012(Norato et al 2015, 2016 (Zhang et al 2016a) Union of rectangular bars with straight (Bell et al 2012) or semicircular (Norato et al 2015) (Guo et al 2014b), 2016a (Zhang et al 2016b), 2016b (Guo et al 2016), 2017(Zhang et al 2017b Union of rectangles approximated as superellipses (Guo et al 2014b), or curved quadrilaterals with polynomial or trigonometric sides (Zhang et al 2016b;Guo et al 2016) Positions and orientation of discrete elements, dimensions (Guo et al 2014b) and curve parameters (Zhang et al 2016b;Guo et al 2016) Maximum of Heaviside function of individual geometric components XFEM to follow level set (Guo et al 2014b;Zhang et al 2016b), ersatz material from average of nodal smooth Heaviside (Guo et al 2016) Through union of Heavisides (bars merge, separate and are engulfed in other bars); no component insertion XFEM (Guo et al 2014b;…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Topology optimization with supershapes

Norato

2018

Struct Multidisc Optim

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Geometry Projectionmentioning

confidence: 99%

Section: Geometry Projectionmentioning

confidence: 99%

Section: Signed Distance Calculationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Topology optimization with supershapes

Norato

2018

Struct Multidisc Optim

Self Cite

View full text Add to dashboard Cite

show abstract

A geometric projection method for designing three‐dimensional open lattices with inverse homogenization

Watts

Tortorelli

2017

Numerical Meth Engineering

View full text Add to dashboard Cite

Summary Topology optimization is a methodology for assigning material or void to each point in a design domain in a way that extremizes some objective function, such as the compliance of a structure under given loads, subject to various imposed constraints, such as an upper bound on the mass of the structure. Geometry projection is a means to parameterize the topology optimization problem, by describing the design in a way that is independent of the mesh used for analysis of the design's performance; it results in many fewer design parameters, necessarily resolves the ill‐posed nature of the topology optimization problem, and provides sharp descriptions of the material interfaces. We extend previous geometric projection work to 3 dimensions and design unit cells for lattice materials using inverse homogenization. We perform a sensitivity analysis of the geometric projection and show it has smooth derivatives, making it suitable for use with gradient‐based optimization algorithms. The technique is demonstrated by designing unit cells comprised of a single constituent material plus void space to obtain light, stiff materials with cubic and isotropic material symmetry. We also design a single‐constituent isotropic material with negative Poisson's ratio and a light, stiff material comprised of 2 constituent solids plus void space.

show abstract

A geometry projection method for the topology optimization of curved plate structures with placement bounds

Zhang

Gain

Norato

2018

Numerical Meth Engineering

Self Cite

View full text Add to dashboard Cite

Summary Single‐curvature plates are commonly encountered in mechanical and civil structures. In this paper, we introduce a topology optimization method for the stiffness‐based design of structures made of curved plates with fixed thickness. The geometry of each curved plate is analytically and explicitly represented by its location, orientation, dimension, and curvature radius, and therefore, our method renders designs that are distinctly made of curved plates. To perform the primal and sensitivity analyses, we use the geometry projection method, which smoothly maps the analytical geometry of the curved plates onto a continuous density field defined over a fixed uniform finite element grid. A size variable is ascribed to each plate and penalized in the spirit of solid isotropic material with penalization, which allows the optimizer to remove a plate from the design. We also introduce in our method a constraint that ensures that no portion of a plate lies outside the design envelope. This prevents designs that would otherwise require cuts to the plates that may be very difficult to manufacture. We present numerical examples to demonstrate the validity and applicability of the proposed method.

show abstract

A geometry projection method for the topology optimization of plate structures

Cited by 105 publications

References 27 publications

Topology optimization with supershapes

Topology optimization with supershapes

A geometric projection method for designing three‐dimensional open lattices with inverse homogenization

A geometry projection method for the topology optimization of curved plate structures with placement bounds

Contact Info

Product

Resources

About