A new three-dimensional topology optimization method based on moving morphable components (MMCs)

Zhang, Weisheng; Li, Dong; Yuan, Jie; Song, Junfu; Gao, Xu

doi:10.1007/s00466-016-1365-0

Cited by 119 publications

(55 citation statements)

References 33 publications

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“…Under this circumstance, for a typical design variable a , the sensitivities of the objective and constraint functionals with respect to a can be derived as

\frac{\partial C}{\partial a} = - u^{⊤} ((), \sum_{i = 1}^{K} \frac{\partial bold-italicK}{\partial χ^{s_{i}}} \frac{\partial χ^{s_{i}}}{\partial a}) bold-italicu

and

\frac{\partial V_{i}}{\partial a} = \frac{\partial}{\partial a} ((), \int_{D} ((), \prod_{l = 1}^{i - 1} ((), 1 - H_{ϵ} ((), χ^{s_{l}}))) H_{ϵ} ((), χ^{S_{i}}) italicdV),

respectively. It can be seen from Equations and that the central task to calculate the sensitivities is to compute the values

∂ χ^{s_{i}} / ∂ a, i = 1, \dots, K .

We refer the readers to Zhang et al on how to calculate these quantities in an analytical or numerically efficient ways.…”

Section: Numerical Solution Aspectsmentioning

confidence: 99%

Topology optimization with multiple materials via moving morphable component (MMC) method

Zhang

Song

Zhou

et al. 2017

Numerical Meth Engineering

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133

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Summary With the fast development of additive manufacturing technology, topology optimization involving multiple materials has received ever increasing attention. Traditionally, this kind of optimization problem is solved within the implicit solution framework by using the Solid Isotropic Material with Penalization or level set method. This treatment, however, will inevitably lead to a large number of design variables especially when many types of materials are involved and 3‐dimensional (3D) problems are considered. This is because for each type of material, a corresponding density field/level function defined on the entire design domain must be introduced to describe its distribution. In the present paper, a novel approach for topology optimization with multiple materials is established based on the Moving Morphable Component framework. With use of this approach, topology optimization problems with multiple materials can be solved with much less numbers of design variables and degrees of freedom. Numerical examples provided demonstrate the effectiveness of the proposed approach.

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“…Under this circumstance, for a typical design variable a , the sensitivities of the objective and constraint functionals with respect to a can be derived as

\frac{\partial C}{\partial a} = - u^{⊤} ((), \sum_{i = 1}^{K} \frac{\partial bold-italicK}{\partial χ^{s_{i}}} \frac{\partial χ^{s_{i}}}{\partial a}) bold-italicu

and

\frac{\partial V_{i}}{\partial a} = \frac{\partial}{\partial a} ((), \int_{D} ((), \prod_{l = 1}^{i - 1} ((), 1 - H_{ϵ} ((), χ^{s_{l}}))) H_{ϵ} ((), χ^{S_{i}}) italicdV),

respectively. It can be seen from Equations and that the central task to calculate the sensitivities is to compute the values

∂ χ^{s_{i}} / ∂ a, i = 1, \dots, K .

We refer the readers to Zhang et al on how to calculate these quantities in an analytical or numerically efficient ways.…”

Section: Numerical Solution Aspectsmentioning

confidence: 99%

Topology optimization with multiple materials via moving morphable component (MMC) method

Zhang

Song

Zhou

et al. 2017

Numerical Meth Engineering

Self Cite

133

View full text Add to dashboard Cite

show abstract

“…where we note that we have usedd q , defined in Equation 19, as described in the previous section. We also note that w q is a bar parameter, and thus no further differentiation of that term is necessary.…”

Section: Sensitivity Analysismentioning

confidence: 99%

“…This work has recently been extended to three dimensions as well. 19 In these methods, the region adjacent to the sharp phase boundary defined by the level cut requires XFEM basis function enrichment. In the geometry projection method described herein, there is a smeared phase boundary (which becomes sharp in the limit of mesh refinement; see Section 5); it does not require enrichment, although it may benefit from adaptive mesh refinement.…”

Section: Introductionmentioning

confidence: 99%

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

Watts

Tortorelli

2017

Numerical Meth Engineering

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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.

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“…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%

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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A new three-dimensional topology optimization method based on moving morphable components (MMCs)

Cited by 119 publications

References 33 publications

Topology optimization with multiple materials via moving morphable component (MMC) method

Topology optimization with multiple materials via moving morphable component (MMC) method

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

Topology optimization with supershapes

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