1 Introduction
Method, Year | Description of Structure | Design variables | Combination of shapes | Connection geometry-analysis | Topological changes | Analysis |
---|---|---|---|---|---|---|
Bubble method, 1994 (Eschenauer et al. 1994) | All boundaries are splines | Control points | None (splines do not intersect) | Remeshing | Hole insertion via indicator function | Standard FEA on conforming mesh |
Solid envelope with implicitly defined circular, rectangular and triangular holes | Scale, position and orientation of holes | Composition via R-functions | Modulus is function of smooth Heaviside of implicit function | Hole merging via R-composition, hole insertion via TD | Standard FEA on fixed grid | |
Isogeometric TO with trimmed splines, 2010 (Seo et al. 2010) | All boundaries are splines | Control points | Uses ’trimmed surface analysis’ to merge curves | Through isogeometric analysis | Hole merging via spline trimming, hole insertion via TD | Isogeometric analysis |
Material mask overlay, 2011 (Saxena 2011) | Solid envelope with circular, elliptical and rectangular holes | Location, orientation and size of holes | Through cell membership test | Cells removed from analysis if inside of holes | Holes randomly added and removed via zero-order opt. | Hexagonal elements + mesh smoothing |
Material mask overlay with smooth projection, 2012 (Wang et al. 2012) | Solid envelope with circular holes | Locations and radii of holes | Product of smooth Heavisides | Elemental density as a function of Heavisides (ersatz material) | Holes merge though Heavisides product; no hole insertion | Standard FEA on fixed grid |
Position, orientation, and a penalized size variable for each geometric element | Union through p-norm | Projection to density field | Elements merge through aggregation function, and can be removed through size variable; no insertion | Standard FEA on fixed grid | ||
Level set TO with geom. constraints, 2014 (Liu et al. 2014) | Solid envelope with primitive-shaped holes approximated via RBFs | Coefficients of RBFs | Composition via R-functions | XFEM to follow level set | Hole merging via composition; no hole insertion | XFEM |
Maximum of Heaviside function of individual geometric components | Through union of Heavisides (bars merge, separate and are engulfed in other bars); no component insertion | |||||
ISOCOMP, 2015 (Lin et al. 2015) | Solid structure with holes described by NURBS | Control points | Holes do not merge | Ersatz material from NURBS interpolation | Hole insertion via TD; update via simulated annealing | Isogeometric analysis |
MMC with thickness control, 2017 (Hoang and Jang 2016) | Union of rectangular bars with semicircular ends | Positions of endpoints of bars medial axes | Product of regularized Heavisides | Geometry projection (ersatz material) | Bars merge through product of Heavisides; no bar insertion | Standard FEA on fixed grid |
2 Supershapes
3 Geometry projection
4 Signed distance calculation
5 Optimization problem
6 Computer implementation
6.1 Signed distance and projected density calculation
6.2 Finite element primal and sensitivity analyses
6.3 Optimization
6.4 Design variables bounds
7 Results
7.1 Short cantilever beam
7.2 MBB beam
7.3 Long cantilever beam
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2 | 0.556 | 137 | 2 (ni ≥ 7.999) | 8 (ni ≤ 2.039) |
3 | 0.569 | 92 | 1 (ni = 7.998) | 9 (ni ≤ 2.117) |
4 | 0.509 | 168 | 2 (ni ≥ 7.999) | 8 (ni ≤ 2.124) |
5 | 0.575 | 121 | 10 (ni ≥ 7.848) | 0 |
6 | 0.503 | 239 | 10 (ni ≥ 7.863) | 0 |
7 | 0.590 | 119 | 10 (ni ≥ 7.898) | 0 |
8 | 0.582 | 214 | 10 (ni ≥ 7.854) | 0 |