Support structure design in additive manufacturing based on topology optimization

Abstract

A support structure design technique for additive manufacturing (AM) is proposed that minimizes the deformation while using the least amount of support material, minimizes the time required to add the supports, and designs supports that are easily removed. This study presents a repulsion index (RI), which satisfies the easy removal requirement and minimizes the number of artifacts left on the specimen surface, and a weighting function, which quantifies the cost incurred by the time taken to build the supports. A multi-objective topological optimization based on the simple isotropic material with penalization method, continuous approximation of material distribution, and method of moving asymptotes is formulated that includes the proposed RI and cost formulation. Numerical simulations demonstrate that rational support layouts can be determined with the proposed cost-based formulation in the topological optimization, allowing designers to find design solutions with a compromise between specimen surface profile error and support structure costs.

Appendix

The sensitivity of the RI used in gradient-based optimizations such as MMA is expressed as follows:

$$ \frac{\partial RI\left({\boldsymbol{\uprho}}_r\right)}{\partial {\boldsymbol{\uprho}}_r}=\frac{\partial {RI}_2\left({\boldsymbol{\uprho}}_2\right)}{\partial {\boldsymbol{\uprho}}_2}+\frac{\partial {RI}_3\left({\boldsymbol{\uprho}}_3\right)}{\partial {\boldsymbol{\uprho}}_3}, $$

(A1)

where

$$ \frac{\partial {RI}_2\left({\rho}_{e1}\right)}{\partial {\rho}_{e1}}=\frac{\partial }{\partial {\rho}_{e1}}\left(\frac{\rho_{e1}^p{\rho}_{e2}^p}{\rho_{e1}^p+{\rho}_{e2}^p}\right)=\frac{\partial }{\partial {\rho}_{e1}}\left(\frac{\prod_{j=1}^{\mathrm{H}}{\rho}_{e j}^p}{\sum_{j=1}^{\mathrm{H}}{\rho}_{e j}^p}\right)=\frac{p{\rho}_{e1}^{p-1}\left({\prod}_{j=2}^{\mathrm{H}}{\rho}_{e j}^p{\sum}_{j=1}^{\mathrm{H}}{\rho}_{e j}^p-{\prod}_{j=1}^{\mathrm{H}}{\rho}_{e j}^p\right)}{{\left({\sum}_{j=1}^{\mathrm{H}}{\rho}_{e j}^p\right)}^2},\mathrm{for}\;{\rho}_{e1}={\rho}_e\in {\boldsymbol{\uprho}}_2, $$

(A2)

$$ \begin{array}{l}\frac{\partial {RI}_3\left({\rho}_{e1}\right)}{\partial {\rho}_{e1}}=\frac{\partial }{\partial {\rho}_{e1}}\left(\frac{RI_{2 L}\left({\rho}_{e1}\right){RI}_{2 R}\left({\rho}_{e1}\right)}{RI_{2 L}\left({\rho}_{e1}\right)+{RI}_{2 R}\left({\rho}_{e1}\right)}\right)\hfill \\ {}=\frac{\partial }{\partial {\rho}_{e1}}\left(\frac{\prod_{j=1}^{\mathrm{H}}{\rho}_{Le j}^p{\prod}_{j=1}^{\mathrm{H}}{\rho}_{Rej}^p}{\prod_{j=1}^{\mathrm{H}}{\rho}_{Le j}^p{\sum}_{j=1}^{\mathrm{H}}{\rho}_{Rej}^p+{\prod}_{j=1}^{\mathrm{H}}{\rho}_{Rej}^p{\sum}_{j=1}^{\mathrm{H}}{\rho}_{Le j}^p}\right)\hfill \\ {}=\frac{p{\rho}_{e1}^{p-1}}{{\left( SP\left({\rho}_{e1}\right)\right)}^2}\left[\left({\prod}_{j=2}^{\mathrm{H}}{\rho}_{Le j}^p{\prod}_{j=1}^{\mathrm{H}}{\rho}_{Rej}^p+{\prod}_{j=1}^{\mathrm{H}}{\rho}_{Le j}^p{\prod}_{j=2}^{\mathrm{H}}{\rho}_{Rej}^p\Big) SP\left({\rho}_{e1}\right)\right.\right.\hfill \\ {}-{\prod}_{j=1}^{\mathrm{H}}{\rho}_{Le j}^p{\prod}_{j=1}^{\mathrm{H}}{\rho}_{Rej}^p\Big({\prod}_{j=2}^{\mathrm{H}}{\rho}_{Le j}^p{\sum}_{j=1}^{\mathrm{H}}{\rho}_{Rej}^p+{\prod}_{j=1}^{\mathrm{H}}{\rho}_{Le j}^p\hfill \\ {}\left.\left.+{\prod}_{j=2}^{\mathrm{H}}{\rho}_{Rej}^p{\sum}_{j=1}^{\mathrm{H}}{\rho}_{Le j}^p+{\prod}_{j=1}^{\mathrm{H}}{\rho}_{Rej}^p\right)\right],\hfill \\ {}\mathrm{for}\;{\rho}_{e1}={\rho}_e\in {\boldsymbol{\uprho}}_3,{\rho}_{Le1}={\rho}_{\mathit{\operatorname{Re}}1}={\rho}_{e1},{\rho}_{Le2}={\rho}_{e2},\mathrm{and}\;{\rho}_{\mathit{\operatorname{Re}}2}={\rho}_{e3},\hfill \end{array} $$

(A3)

and

$$ SP\left({\rho}_{e1}\right)={\prod}_{j=1}^{\mathrm{H}}{\rho}_{Lej}^p{\sum}_{j=1}^{\mathrm{H}}{\rho}_{Rej}^p+{\prod}_{j=1}^{\mathrm{H}}{\rho}_{Rej}^p{\sum}_{j=1}^{\mathrm{H}}{\rho}_{Lej}^p, $$

(A4)

where the subscript _e1 , _e2 and _e3 are the subscript ₁ , ₂ and ₃ of ρ in Fig. 3, H = 2 and Π is the product of a sequence.

The sensitivity of w _te (ρ _e ) with respect to the density of each element is given by

$$ \frac{\partial {w}_{t e}}{\partial {\rho}_e}=\left(1-{p}_t\frac{\partial {RI}_w\left({\rho}_e\right)}{\partial {\rho}_e}\right){w}_e=\left(1-{p}_t\frac{\partial }{\partial {\rho}_e}\left(\frac{RI_{3 H}\left({\rho}_e\right){RI}_{3 V}\left({\rho}_e\right)}{RI_{3 H}\left({\rho}_e\right)+{RI}_{3 V}\left({\rho}_e\right)}\right)\right){w}_e, $$

(A5)

where

$$ \frac{\partial {RI}_w\left({\rho}_e\right)}{\partial {\rho}_e}=\frac{\partial }{\partial {\rho}_e}\left(\frac{RI_{3 H}\left({\rho}_e\right){RI}_{3 V}\left({\rho}_e\right)}{RI_{3 H}\left({\rho}_e\right)+{RI}_{3 V}\left({\rho}_e\right)}\right) $$

$$ =\frac{\partial }{\partial {\rho}_{e1}}\left(1/\left[\frac{\sum_{j=1}^{\mathrm{H}}{\rho}_{Lej}^p}{\prod_{j=1}^{\mathrm{H}}{\rho}_{Lej}^p}+\frac{\sum_{j=1}^{\mathrm{H}}{\rho}_{Rej}^p}{\prod_{j=1}^{\mathrm{H}}{\rho}_{Rej}^p}+\frac{\sum_{j=1}^{\mathrm{H}}{\rho}_{Uej}^p}{\prod_{j=1}^{\mathrm{H}}{\rho}_{Uej}^p}+\frac{\sum_{j=1}^{\mathrm{H}}{\rho}_{Dej}^p}{\prod_{j=1}^{\mathrm{H}}{\rho}_{Dej}^p}\right]\right), $$

(A6)

and

$$ {RI}_{3 V}\left({\rho}_e\right)=\frac{RI_{2 U}\left({\rho}_e\right){RI}_{2 D}\left({\rho}_e\right)}{RI_{2 U}\left({\rho}_e\right)+{RI}_{2 D}\left({\rho}_e\right)}, $$

(A7)

$$ {RI}_{3 H}\left({\rho}_e\right)=\frac{RI_{2 L}\left({\rho}_e\right){RI}_{2 R}\left({\rho}_e\right)}{RI_{2 L}\left({\rho}_e\right)+{RI}_{2 R}\left({\rho}_e\right)}. $$

(A8)