Structural optimization under overhang constraints imposed by additive manufacturing technologies

TLDR: This article proposes a new mechanical constraint functional, which mimics the layer by layer construction process featured by additive manufacturing technologies, and thereby appeals to the physical origin of the difficulties caused by overhangs.

About: This article is published in Journal of Computational Physics.The article was published on 2017-12-15 and is currently open access. It has received 157 citations till now. The article focuses on the topics: Shape optimization.

Figures

Figure 21. Optimized shapes for the two-dimensional MBB Beam example of Section 5.3.3: (a) optimized shape Ω∗ for Problem (5.3) (i.e. without additive manufacturing constraints), and optimized shapes for Problem (5.9) using parameters (b) αc = 0.50, (c) αc = 0.30, and (d) αc = 0.10.

Figure 34. (Top) negative subdomain of the level set function associated to the optimized shape in Figure 33, (d), and (bottom) the distribution of the Young’s modulus in the computational domain D.

Table 4. Values of the compliance, the volume and the mechanical constraint Psw(Ω) for the optimized designs obtained in the 2d MBB Beam example of Section 5.3.3.

Table 1. Values of the compliance, the volume and the geometric constraints Pa(Ω) and Pp(Ω) for the optimized designs obtained in the 2d cantilever example of Section 5.2.1.

Figure 7. Setting of the two-dimensional cantilever beam test case.

Figure 15. Optimized two-dimensional cantilever beams under the self-weight manufacturing compliance constraint Psw(Ω), in the setting of Section 5.3.1: (a) initial shape Ω 0, (b) optimized shape Ω∗ for Problem (5.3) (i.e. without additive manufacturing constraints), and optimized shapes for Problem (5.9) using the parameter (c) αc = 0.80; (d) αc = 0.60; (e) αc = 0.50.

Frequently asked questions

Q1. What are the contributions mentioned in the paper "Structural optimization under overhang constraints imposed by additive manufacturing technologies" ?

This article addresses one of the major constraints imposed by additive manufacturing processes on shape optimization problems that of overhangs, i. e. large regions hanging over void without sufficient support from the lower structure. After revisiting the ‘ classical ’ geometric criteria used in the literature, based on the angle between the structural boundary and the build direction, the authors propose a new mechanical constraint functional, which mimics the layer by layer construction process featured by additive manufacturing technologies, and thereby appeals to the physical origin of the difficulties caused by overhangs. 

Q2. What is the shape functional Pg() defined by (3.1)?

The shape functional Pg(Ω) defined by (3.1) is shape differentiable at any admissible shape Ω ∈ Uad when deformations θ are in Θk, k ≥ 1. 

Q3. Why do the authors initialize Problem (5.14) with the full computational domain D?

Due to the existence of some nearly flat overhanging parts in the results of section 5.4.2 close to the anchor points, the authors initialize Problem (5.14) with the full computational domain D for this example. 

Q4. What is the common quantity used in the literature to detect and constrain the presence of overhangs?

As the authors have mentioned in the introduction, the prevalent quantity used in the literature to detect and constrain the presence of overhangs is the angle between the normal vector nΩ to the structural boundary ∂Ω and the negative build direction −ed, where ed is th drh vector in the canonical basis of Rd. 

Q5. How is the communication between the optimization algorithm and the Finite Element solver achieved?

the communication between the optimization algorithm and the Finite Element solver is achieved via file exchange, which is a notorious source of inefficiency. 

Q6. What is the a priori definition of the normal vector n?

Although the normal vector n is a priori defined only on the boundary ∂Ω, it can always be extended in a neighborhood of the boundary (at least when Ω is sufficiently smooth). 

Q7. How do the authors perform the aforementioned evaluations?

The intuitive way to perform the aforementioned evaluations consists in discretizing the height interval (0, H) into N small subintervals(4.6) 

Q8. What is the a priori non trivial evaluation of Psw()?

The constraint Psw(Ω) and its shape derivative P ′ sw(Ω)(θ) (or equivalently the integrand DΩ) bring into play a continuum of shapes, and so their numerical evaluation is a priori non trivial. 

Q9. What is the thickness of the regions where g is applied in the definition of the body force?

The thickness δ of the regions where g is applied in the definition (4.13) of the body force gh is of the order of the mesh size: δ = ∆x. 

Q10. What is the th-order procedure used for the calculations of Puw()?

the 0 th-order procedure is used for the calculations of Puw(Ω) and its derivative, with N = 40 evenly distributed layers.