Combined length scale and overhang angle control in minimum compliance topology optimization for additive manufacturing

Abstract

This paper focuses on topology optimization for additive manufacturing. In order to ensure that the optimized design is immediately manufacturable, it is essential to take into account the appropriate geometric constraints during the optimization. Two important constraints are minimum length scale and maximum overhang angle. A minimum length scale is needed to ensure that the condition on minimal printable feature sizes is satisfied, while an imposed overhang angle eliminates the need for a temporary support structure. This paper first shows that both constraints cannot simultaneously be met by a straightforward coupling of existing methods for length scale and overhang angle control. Next, a new filtering scheme is introduced, based on a specific combination of spatial filters, which allows direct control over these constraints in a minimum compliance topology optimization problem. A 2D benchmark problem and a complex 3D case study are presented to demonstrate that the proposed filtering scheme successfully imposes a target length scale in both the solid and the void phase of the design domain, while simultaneously allowing control over the overhang angle.

Appendix: Sensitivity analysis

The sensitivity analysis of the different filtering schemes considered in this paper are elaborated in the following subsections, where the derivative of a vector y with respect to a vector x is denoted as $\frac {\partial \mathbf {y}}{\partial \mathbf {x}}$, which represents a matrix where the (i,j)^th elements equals:

$$ \begin{array}{llllllllllll} \left[\frac{\partial\mathbf{y}}{\partial\mathbf{x}}\right]_{\text{ij}} &= \frac{\partial y_{i}}{\partial x_{j}}\\ \end{array} $$

(23)

1.1 A.1 Minimum compliance topology optimization

The relation between the design variables ρ and the compliance c is given by the filtering scheme in (7), the sensitivity $\frac {\partial c}{\partial {\mathbf {\rho }}}$ of the objective function c(ρ) with respect design variables ρ for minimum compliance topology optimization is computed by applying the chain rule 2 times:

$$ \begin{array}{lllllllllll} \frac{\partial c}{\partial \mathbf{\rho}} = {\frac{\partial c}{\partial{\bar{\mathbf{\rho}}}}} {\frac{\partial {\bar{\mathbf{\rho}}}}{\partial \tilde{\mathbf{\rho}}}} {\frac{\partial {\tilde{\mathbf{\rho}}}}{\partial \mathbf{\rho}}}, \end{array} $$

(24)

The sensitivity $\frac {\partial c}{\partial \bar {\mathbf {\rho }}}$ is computed using the adjoint variable method:

$$ \frac{\partial c}{\partial \bar{\rho}_{e}} = - \mathbf{\lambda}^{T} \frac{\partial{\mathbf{K}}}{\partial{\bar{\rho}_{e}}} \mathbf{u}, $$

(25)

where the adjoint variable λ is obtained by solving Kλ = f, this results in λ = u for the minimum compliance problem.

The sensitivity $\frac {\partial {\bar {\mathbf {\rho }}}}{\partial \tilde {\mathbf {\rho }}}$ of the Heaviside projection, (6), is given by the following:

$$ \frac{\partial {\bar{\mathbf{\rho}}}}{\partial \tilde{\mathbf{\rho}}} = \text{diag} \left( \frac{\beta \left( \text{sech} \left( \beta \left( \tilde{\mathbf{\rho}} - \eta \right) \right) \right)}{\tanh\left( \beta\eta\right) + \tanh \left( \beta \left( 1- \eta \right) \right)} \right) $$

(26)

The density filter in (4) is a linear operator that can be expressed as following:

$$ \tilde{\mathbf{\rho}} = {\mathbf{H}^{R} \mathbf{\rho}}, $$

(27)

where the coefficient matrix H^R consists of elements $H_{\text {ij}}^{R} = \frac {h_{\text {ij}}^{R}}{{\sum }_{j\in \mathbb {N}_{e}} {h_{\text {ik}}^{R}}}$, resulting in a sensitivity $\frac {\partial {\tilde {\mathbf {\rho }}}}{\partial \mathbf {\rho }}$ of:

$$ \frac{\partial {\tilde{\mathbf{\rho}}}}{\partial \mathbf{\rho}} = \mathbf{H}^{R} $$

(28)

The sensitivity $\frac {\partial {V}}{\partial {\mathbf {\rho }}}$ of the constraint function with respect to the design variables ρ is obtained in a similar way as the sensitivities of the compliance.

1.2 A.2 Length scale control

Following the filtering scheme presented in (11), the sensitivity $\frac {\partial c}{\partial {\mathbf {\rho }}}$ of the objective function c(ρ) with respect to the design variables ρ for minimum compliance topology optimization with length scale control is computed by applying the chain rule two times:

$$ \begin{array}{lllllllllll} &\frac{\partial c}{\partial \mathbf\rho} = {\frac{\partial c}{\partial {\hat{\check{{\mathbf{\rho}}}}}}} {\frac{\partial {\hat{\check{{\mathbf\rho}}}}}{\partial \check{\mathbf\rho}}} {\frac{\partial {\check{\mathbf\rho}}}{\partial \mathbf\rho}}, \end{array} $$

(29)

where the sensitivity $\frac {\partial c}{\partial \hat {\check {{\mathbf \rho }}}}$ of the objective function with respect to the design variables $\hat {\check {{\mathbf \rho }}}$ is computed using the adjoint variable method described in (25), and

$$ \frac{\partial {\hat{\check{\mathbf\rho}}}}{\partial \check{\mathbf\rho}} = \text{diag} \left( \frac{\beta \left( \text{sech} \left( \beta \left( \check{\mathbf\rho} - \eta_{\mathrm{d}} \right) \right) \right)}{\tanh \left( \beta \eta_{\mathrm{d}} \right) + \tanh \left( \beta \left( 1- \eta_{\mathrm{d}} \right) \right)} \right) \mathbf{H}^{R}, $$

(30)

$$ \frac{\partial {\check{\mathbf\rho}}}{\partial {\mathbf\rho}} = \text{diag} \left( \frac{\beta \left( \text{sech} \left( \beta \left( {\mathbf\rho} - \eta_{\mathrm{e}} \right) \right) \right)}{\tanh \left( \beta \eta_{\mathrm{e}} \right) + \tanh \left( \beta \left( 1- \eta_{\mathrm{e}} \right) \right)} \right) \mathbf{H}^{R} $$

(31)

1.3 A.3 Overhang angle control

Following the filtering scheme presented in (16), the sensitivity $\frac {\partial c}{\partial {\mathbf \rho }}$ of the objective function c(ρ) with respect to the design variables ρ for minimum compliance topology optimization with overhang angle control can be obtained via direct differentiation by applying the chain rule two times:

(32)

where $\frac {\partial \bar {\mathbf \rho }}{\partial \mathbf \rho }$ is calculated along the same lines as (30) and (31) with a threshold η = 0.5. However, is a densely populated matrix, as the design variables in a specific layer depend on the blueprint densities $\bar {\mathbf \rho }$ of all underlying elements. Alternatively, the sensitivity $\frac {\partial c}{\partial \mathbf \rho }$ is obtained as follows:

$$ \begin{array}{lllllllllll} \frac{\partial c}{\partial{\mathbf \rho}} = \frac{\partial c}{\partial \bar{\mathbf\rho}} \frac{\partial \bar{\mathbf\rho}}{\partial \mathbf\rho}, \end{array} $$

(33)

where $\frac {\partial c}{\partial \bar {\mathbf \rho }}$ is efficiently obtained using an adjoint formulation proposed by Langelaar (2016a). Collecting the blueprint densities of all elements of layer k in a vector $\bar {\mathbf \rho }_{k}$ and the printed densities in a vector , the latter can be expressed as follows:

(34)

where, for k > 1, the definition of the operator $\breve {\mathbf s}_{k}$ immediately follows from (14) and (15):

(35)

For k = 1, $\breve {\mathbf s}_{k}$ simply returns $\bar {\mathbf \rho }_{k}$, so that $\frac {\partial \breve {\mathbf s}_{1}}{\partial \bar {\mathbf \rho }_{1}} = \mathbf {I}$ and .

The compliance can now be expressed as follows:

(36)

where λ_k is a vector with Lagrange multipliers. Differentiation of (36) with respect to the design variables $\bar {\mathbf \rho }_{l}$ results in:

(37)

where n_k is the number of layers. As printed densities only depend on blueprint densities in underlying layers, for k < l, thus, terms in the summations with k < l will disappear. Taking terms with k = l outside of the summations, and using , gives the following:

(38)

Next, the last term in the summation is written as a separate sum and the first term (k = l + 1) is taken out as follows:

(39)

By reindexing, the last sum can be changed into a summation from k = l + 1 → n_k − 1. Both sums get the same limits by taking the last term k = n_k out of the summation. Using and recombining summations gives the following:

(40)

The computation of the densely populated matrix can be avoided if the Langrange multipliers are choses as follows:

(41)

Each multiplier depends on the one associated with the layer above. This means that the evaluation starts at the top layer and proceeds downwards. With Lagrange multipliers defined by (41), the sensitivities of the response c follow from (40) as follows:

(42)

The sensitivity of the objective function c with respect to the printed densities is calculated using the adjoint approach described by (25), and the sensitivity $\frac {\partial \breve {\mathbf s}}{\partial {\bar {\mathbf {\rho }}}}$ follows immediately from the differentiation of (14) and (15).

Finally, the sensitivity $\frac {\partial c}{\partial \bar {\mathbf \rho }}$ needed in (33) is obtained by concatenating the derivatives $\frac {\partial c}{\partial \bar {\mathbf \rho }_{l}}$ for all individual layers.

1.4 A.4 Combined length scale and overhang angle control

Following the filtering scheme presented in (22), the sensitivity $\frac {\partial c}{\partial \mathbf \rho }$ of the objective function c with respect to the design variables ρ for minimum compliance topology optimization with length scale and overhang angle control (strategy 3) can be obtained via direct differentiation by applying the chain rule 6 times:

(43)

where $\frac {\partial \check {\mathbf \rho }}{\partial \mathbf \rho }$ is calculated following the same principle as (31). In this case, the term is a densely populated matrix, as the design variables in a specific layer depend on the blueprint densities $\check {\mathbf \rho }$ of all underlying elements. In order to avoid the calculation of this densely populated matrix, the sensitivity $\frac {\partial c}{\partial \mathbf \rho }$ is determined as follows:

$$ \begin{array}{lllllllllll} \frac{\partial c}{\partial{\mathbf \rho}} = \frac{\partial c}{\partial \check{\mathbf\rho}} \frac{\partial \check{\mathbf\rho}}{\partial \mathbf\rho}, \end{array} $$

(44)

To calculate $\frac {\partial c}{\partial \check {\mathbf \rho }}$, the adjoint approach described in the previous section is used as follows:

(45)

where

(46)

and is calculated via direct differentiation using (30) and (31):

(47)

where is calculated using the adjoint approach described in (25).

The sensitivity $\frac {\partial c}{\partial \check {\mathbf \rho }}$ needed in (44) is obtained by concatenating the derivatives $\frac {\partial c}{\partial \check {\mathbf \rho }_{l}}$ for all individual layers.