Topology optimization parallel-computing framework based on the inherent strain method for support structure design in laser powder-bed fusion additive manufacturing

Abstract

In this work, a topology optimization parallel-computing framework is developed to design support structures for minimizing deflections in Laser Powder-bed Fusion produced parts. The parallel-computing framework consists of a topology optimization model and an Inherent Strain Method (ISM) model. The proposed framework is used to design stiffer support structures to reduce the before and after-cutting deflections in printed cantilevers. Gravity load and residual stresses calculated from ISM are applied in the topology optimization model. The optimized results were printed and analyzed for validating the effectiveness of the proposed model. Experimental results show that the optimized supports can achieve over 60% reduction in part deflection as well as over 50% material usage reduction compared to the default support structure. In addition, ISM also was used to predict the part deflections and shows good agreement (average error of 6%) between the experimental and simulated results. Lastly, the multi-node parallelization of the proposed framework showed ~ 5 times speedup compared to a single-node implementation.

Appendix: The analytical residual stress model in Sect. 2.5

At each moment, there are two equilibria to be obeyed, the force equilibrium and the moment equilibrium as shown in Eq. (13).

$$ \begin{aligned} \int {\sigma_{x} (z)dz} = 0 \hfill \\ \int {\sigma_{x} (z)zdz} = 0 \hfill \\ \end{aligned} $$

(13)

where σ_x is the stress in the x-direction. The continuity of the deformation is assumed over the combined equilibrated body so that a linear strain profile can be described as,

$$ \varepsilon_{x} (z) = az + b $$

(14)

where a and b are coefficient to be derived and z is the coordinate in the z-direction. Suppose m represents the ratio of the substrate stiffness E_b to the part’s stiffness E_p, and n is the ratio of Young’s modulus between the support E_s and the printed part. In addition, the width ratio between the width of the substrate w_b and the width of the part w_p can also be merged into m, and similarly, support’s width w_s and w_p into n, as illustrated in Eq. (15),

$$ \begin{aligned} m & = \frac{{w_{b} E_{b} }}{{w_{p} E_{p} }} \\ n & = \frac{{w_{s} E_{s} }}{{w_{p} E_{p} }} \\ \end{aligned} $$

(15)

The equilibria in Eq. (13) can then be expressed as follows,

$$ \begin{aligned} & \int_{0}^{{h_{b} }} {m(az + b)E_{p} w_{p} dz} + \int_{{h_{b} }}^{{h_{b} + h_{s} }} {n(az + b)E_{p} w_{p} dz} \\ & \quad + \int_{{h_{b} + h_{s} }}^{{h_{b} + h_{s} + h_{p} - l}} {(az + b)E_{p} w_{p} dz} + \int_{{h_{b} + h_{s} + h_{p} - l}}^{{h_{b} + h_{s} + h_{p} }} {(az + b)\bar{\sigma }E_{p} w_{p} dz} = 0 \\ & \int_{0}^{{h_{b} }} {m(az + b)E_{p} w_{p} zdz} + \int_{{h_{b} }}^{{h_{b} + h_{s} }} {n(az + b)E_{p} w_{p} dz} \\ & \quad + \int_{{h_{b} + h_{s} }}^{{h_{b} + h_{s} + h_{p} - l}} {(az + b)E_{p} w_{p} zdz} + \int_{{h_{b} + h_{s} + h_{p} - l}}^{{h_{b} + h_{s} + h_{p} }} {(az + b)\bar{\sigma }E_{p} w_{p} zdz} = 0 \\ \end{aligned} $$

(16)

where l is the layer thickness, $\overline{\sigma }$ is the ratio between yield stress σ and the part’s stiffness E_p. From Eq. (16), the parameters a and b can be obtained as follows,

$$ \begin{aligned} a & = \frac{{a_{num} }}{{a_{den} }} \\ {\kern 1pt} a_{num} & = - \;6\bar{\sigma }l\left( {mh_{b}^{2} + nh_{s}^{2} + h_{p}^{2} + 2mh_{b} h_{s} + 2mh_{b} h_{p} + 2nh_{s} h_{p} - mh_{b} l - nh_{s} l - h_{p} l} \right) \\ a_{den} & = m^{2} h_{b}^{4} + n^{2} h_{s}^{4} + h_{p}^{4} + l^{4} + 4mnh_{b}^{3} h_{s} + 4mnh_{b} h_{s}^{3} + 4mh_{b}^{3} h_{p} + 4mh_{b} h_{p}^{3} \\ & \quad - 4mh_{b}^{3} l - 4mh_{b} l^{3} + 4nh_{s}^{3} h_{p} + 4nh_{s} h_{p}^{3} - 4nh_{s}^{3} l - 4nh_{s} l^{3} - 4h_{p}^{3} l - 4h_{p} l^{3} \\ & \quad + 6mnh_{b}^{2} h_{s}^{2} + 6mh_{b}^{2} h_{p}^{2} + 6mh_{b}^{2} l^{2} + 6nh_{s}^{2} h_{p}^{2} + 6nh_{s}^{2} l^{2} + 6h_{p}^{2} l^{2} \\ & \quad + 12mh_{b}^{2} h_{s} h_{p} + 12mh_{b} h_{s}^{2} h_{p} + 12mh_{b} h_{s} h_{p}^{2} - 12mh_{b}^{2} h_{s} l - 12mh_{b} h_{s}^{2} l \\ & \quad + 12mh_{b} h_{s} l^{2} - 12mh_{b}^{2} h_{p} l + 12mh_{b} h_{p} l^{2} - 12mh_{b} h_{p}^{2} l - 12nh_{s}^{2} h_{p} l \\ & \quad + 12nh_{s} h_{p} l^{2} - 12nh_{s} h_{p}^{2} l - 24mh_{b} h_{s} h_{p} l \\ \end{aligned} $$

(17)

$$ \begin{aligned} b & = \frac{{b_{num} }}{{b_{den} }} \\ {\kern 1pt} b_{num} & = \bar{\sigma }l\left( {2mh_{b}^{3} + 2nh_{s}^{3} + 2h_{p}^{3} + l^{3} - 3mh_{b}^{2} l - 3nh_{s}^{2} l + 6h_{b} h_{p}^{2} } \right. \\ & \quad + 6h_{s} h_{p}^{2} + 6mh_{b}^{2} h_{s} + 6nh_{b} h_{s}^{2} + 6mh_{b}^{2} h_{p} + 6nh_{s}^{2} h_{p} \\ & \quad + 6nh_{s}^{2} h_{p} \left. { - 6nh_{b} h_{s} l - 6h_{b} h_{p} l - 6h_{s} h_{p} l + 12nh_{b} h_{s} h_{p} } \right) \\ {\kern 1pt} b_{den} & = m^{2} h_{b}^{4} + n^{2} h_{s}^{4} + h_{p}^{4} + l^{4} + 4mnh_{b}^{3} h_{s} + 4mnh_{b} h_{s}^{3} + 4mh_{b}^{3} h_{p} \\ & \quad + 4mh_{b} h_{p}^{3} - 4mh_{b}^{3} l - 4mh_{b} l^{3} + 4nh_{s}^{3} h_{p} + 4nh_{s} h_{p}^{3} - 4nh_{s}^{3} l \\ & \quad - 4nh_{s} l^{3} - 4h_{p}^{3} l - 4h_{p} l^{3} + 6mnh_{b}^{2} h_{s}^{2} + 6mh_{b}^{2} h_{p}^{2} + 6mh_{b}^{2} l^{2} \\ & \quad + 6nh_{s}^{2} h_{p}^{2} + 6nh_{s}^{2} l^{2} + 6h_{p}^{2} l^{2} + 12mh_{b}^{2} h_{s} h_{p} + 12mh_{b} h_{s}^{2} h_{p} \\ & \quad + 12mh_{b} h_{s} h_{p}^{2} - 12mh_{b}^{2} h_{s} l - 12mh_{b} h_{s}^{2} l + 12mh_{b} h_{s} l^{2} \\ & \quad - 12mh_{b}^{2} h_{p} l + 12mh_{b} h_{p} l^{2} - 12mh_{b} h_{p}^{2} l - 12nh_{s}^{2} h_{p} l \\ & \quad + 12nh_{s} h_{p} l^{2} - 12nh_{s} h_{p}^{2} l - 24mh_{b} h_{s} h_{p} l \\ \end{aligned} $$

(18)

By substituting a and b in (17) and (18) into (14), the strain distribution can be derived. Then the approximated residual stress can be derived easily by multiplying the strain with Young’s modulus of the material used.