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Structural optimization under overhang constraints imposed by additive manufacturing processes: an overview of some recent results

Grégoire Allaire

Grégoire Allaire

Centre de Mathématiques Appliquées, Ecole Polytechnique, CNRS, Université Paris-SaclayPalaiseau, France
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Allaire, Grégoire
, 
Charles Dapogny

Charles Dapogny

Laboratoire Jean Kuntzmann, Université Grenoble-Alpes, CNRSGrenoble, France
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Dapogny, Charles
, 
Rafael Estevez

Rafael Estevez

SIMaP, Université Grenoble-Alpes, CNRSGrenoble, France
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Estevez, Rafael
, 
Alexis Faure

Alexis Faure

SIMaP, Université Grenoble-Alpes, CNRSGrenoble, France
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Faure, Alexis
 y 
Georgios Michailidis

Georgios Michailidis

SIMaP, Université Grenoble-Alpes, CNRSGrenoble, France
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Michailidis, Georgios
  
19 sept 2017
Structural optimization under overhang constraints imposed by additive manufacturing processes: an overview of some recent results's Cover Image
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Publicado en línea: 19 sept 2017
Páginas: 385 - 402
Recibido: 06 abr 2017
Aceptado: 19 sept 2017
DOI: https://doi.org/10.21042/AMNS.2017.2.00031
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© 2017 Grégoire Allaire, Charles Dapogny, Rafael Estevez, Alexis Faure and Georgios Michailidis, published by Sciendo
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Fig. 1

Sketch of the slicing procedure, at the beginning of all additive manufacturing processes.
Sketch of the slicing procedure, at the beginning of all additive manufacturing processes.

Fig. 2

Setting of the two-dimensional MBB beam example.
Setting of the two-dimensional MBB beam example.

Fig. 3

(Top) initial and (bottom) optimized shapes for Problem (8) in the two-dimensional MBB Beam test-case of Section 3.2.
(Top) initial and (bottom) optimized shapes for Problem (8) in the two-dimensional MBB Beam test-case of Section 3.2.

Fig. 4

Optimized shapes resulting from Problem (9) in the two-dimensional MBB Beam example, using (top) φ ≡ φa and the threshold angle ν = 45°, and (bottom) φ ≡ φp and the pattern functions ψi defined in (10).
Optimized shapes resulting from Problem (9) in the two-dimensional MBB Beam example, using (top) φ ≡ φa and the threshold angle ν = 45°, and (bottom) φ ≡ φp and the pattern functions ψi defined in (10).

Fig. 5

Intermediate shape Ωh at the height h during the construction of the final structure Ω: the red zone is the lower boundary Γ0and the blue zone is the upper boundaryΓhu. $\Gamma^u_h.$
Intermediate shape Ωh at the height h during the construction of the final structure Ω: the red zone is the lower boundary Γ0and the blue zone is the upper boundaryΓhu. $\Gamma^u_h.$

Fig. 6

Relative errors of the 0th- and 1st-order approximations of (top) Psw(Ω0) and (bottom) its derivative 𝒟Ω0.
Relative errors of the 0th- and 1st-order approximations of (top) Psw(Ω0) and (bottom) its derivative 𝒟Ω0.

Fig. 7

Optimized shapes for the two-dimensional MBB Beam example of Section 5.3: (a) optimized shape Ω*for Problem (8) (i.e. without additive manufacturing constraints), and optimized shapes for Problem (29) using parameters (b)αc = 0:50, (c) αc = 0:30, and (d) αc = 0:10.
Optimized shapes for the two-dimensional MBB Beam example of Section 5.3: (a) optimized shape Ω*for Problem (8) (i.e. without additive manufacturing constraints), and optimized shapes for Problem (29) using parameters (b)αc = 0:50, (c) αc = 0:30, and (d) αc = 0:10.

Fig. 8

Optimized shapes for the two-dimensional MBB Beam example of Section 5.4, solving Problem (29) with the upper-weight manufacturing compliancePsw(Ω) and parameters (a)αc = 0:30, (b) αc = 0:10, (c) αc = 0:05, and (d) αc = 0:03.
Optimized shapes for the two-dimensional MBB Beam example of Section 5.4, solving Problem (29) with the upper-weight manufacturing compliancePsw(Ω) and parameters (a)αc = 0:30, (b) αc = 0:10, (c) αc = 0:05, and (d) αc = 0:03.

Fig. 9

Setting of the three-dimensional bridge test-case.
Setting of the three-dimensional bridge test-case.

Fig. 10

Optimized designs for the three-dimensional bridge example of Section 5.5, (left) without manufacturing constraints, (right) solving Problem (30) withαc = 0:7.
Optimized designs for the three-dimensional bridge example of Section 5.5, (left) without manufacturing constraints, (right) solving Problem (30) withαc = 0:7.

Fig. 11

(Left) Different views of the optimized shape for the three-dimensional bridge example of Section 5.5, solving Problem (30) withαc = 0:1; (right) another view on the three-dimensional bridges for Problem (30) with (top) no manufacturing constraint, (middle)αc = 0:7 and (bottom) ac = 0:1.
(Left) Different views of the optimized shape for the three-dimensional bridge example of Section 5.5, solving Problem (30) withαc = 0:1; (right) another view on the three-dimensional bridges for Problem (30) with (top) no manufacturing constraint, (middle)αc = 0:7 and (bottom) ac = 0:1.
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