Volume preserving nonlinear density filter based on heaviside functions

Abstract

To prevent numerical instabilities and ensure manufacturability, restrictions should be applied in topology optimization. In this paper, a volume preserving density filter based on Heaviside functions is presented. Different from earlier Heaviside density filters, this filter is volume preserving, which ensures efficiency and stability in optimization. The new filter is compared with four other filters through a compliance minimization problem.

Appendix A: Proof of existence and uniqueness of parameter η

According to (21), we denote f(η) as a monadic function of η:

$$ f(\eta)=\sum\limits_{i=1}^{n} \tilde{\rho}_i {v}_i - \sum\limits_{i=1}^{n} \bar{\rho}_i {v}_i $$

(23)

The goal is to prove that f(η) = 0 has and only has one root in ]0,1[.

First, prove the existence of root. Assume a set of given densities denoted as $\bar{\rho}_1, \bar{\rho}_2, \cdots \bar{\rho}_n$ are arranged from small to large. In these densities, the ones with subscripts less than i are equal to zero. Let $\eta_1 < \bar{\rho}_i$, then for e ≤ i, $\tilde{\rho}_e = \bar{\rho}_e = 0$, and for e > i, $\tilde{\rho}_e > \bar{\rho}_e$, so

$$ \sum\limits_{i=1}^{n} \tilde{\rho}_e {v}_e > \sum\limits_{i=1}^{n}\bar{\rho}_e {v}_e, $$

i.e., when $\eta_1 < \bar{\rho}_i$

$$ f(\eta_1)>0. $$

(24)

Also assume densities with subscripts greater than j are equal to one. Let $\eta_2 > \bar{\rho}_j$, then for e > j, $\tilde{\rho}_e=\bar{\rho}_e=1$, and for e ≤ j, $\tilde{\rho}_e<\bar{\rho}_e$, so

$$ \sum\limits_{i=1}^{n} \tilde{\rho}_e {v}_e < \sum\limits_{i=1}^{n} \bar{\rho}_e {v}_e, $$

i.e., when $\eta_2 > \bar{\rho}_j$

$$ f(\eta_2)<0. $$

(25)

From (24) and (25), it can be concluded that for a given density distribution, when η is small enough, f(η) > 0, and when η is close to one, f(η) < 0. Because f is a continuous function of the parameter η, it must have zero roots in ]0,1[.

Second, prove that the first derivative of f with respect to η is less than zero, i.e.,

$$ \frac{\partial f}{\partial \eta} = \sum\limits_{i=1}^{n} \frac{\partial \tilde{\rho}_i}{\partial \eta} {v}_i <0. $$

(26)

To prove (26), it only has to prove that $\frac{\partial \tilde{\rho}_i}{\partial \eta}$ in (26) is less than zero for arbitrary i.

• a. When $\bar{\rho} \leq \eta$, the first derivative of filtering function with respect to η is

  $$ \frac{\partial \tilde{\rho}}{\partial \eta}= e^{-\beta}\left(e^{\beta \bar{\rho} / \eta}-1-\frac{\beta \bar{\rho}}{\eta}e^{\beta \bar{\rho} / \eta}\right). $$

  (27)

  For fixed $\bar{\rho}, \eta$ and β > 0, it can be proved that (27) is less than or equal to zero. If there exist $\bar{\rho} >0$, then (27) is less than zero.

• b. When $\bar{\rho} > \eta$, the first derivative of filtering function with respect to η is

  $$ \frac{\partial \tilde{\rho}}{\partial \eta} = e^{-\beta \frac{\bar{\rho}-\eta}{1-\eta}} \left(-\frac{\beta(1-\bar{\rho})}{(1-\eta)} + 1-e^{-\beta \frac{1-\bar{\rho}}{1-\eta}}\right). $$

  (28)

  For fixed $\bar{\rho}, \eta$ and β > 0, it can be proved that (28) is less than or equal to zero. If there exist $\bar{\rho}<1$, then (28) is less than zero.

Because $\bar{\rho}$ is the density after linear filtering, there must exist intermediate densities, so the first derivative of filtering function with respect to η is less than zero, and so (26) is true.

Function f(η) is a monotonic function of parameter η, and is greater than zero when η approaches 0, and less than zero when η approaches 1, so it has and only has one root in ]0,1[.