A geometry projection method for the topology optimization of plate structures

Abstract

We introduce a topology optimization method for the stiffness-based design of structures made of plates. Our method renders topologies made distinctly of plates, thereby producing designs that better conform to manufacturing processes tailored to plate structures, such as those that employ stock plates that are cut and joined by various means. To force the structural members to be plates, we employ the geometry projection method to project an analytical description of a set of fixed-thickness plates onto a continuous density field defined over a 3-dimensional, uniform finite element grid for analysis. A size variable is assigned to each plate and penalized so that the optimizer can entirely remove a plate from the design. The proposed method accommodates the case where the plates in the topology are rectangular and solid, and the case where the boundaries of the plates can change and holes can be introduced. The latter case is attained by composition with a free density field. We present examples that demonstrate the effectiveness of our method and discuss future work.

A Appendix: Sensitivity analysis

1.1 A.1 Plates geometry projection sensitivities

From (15) and (18), we obtain the design sensitivity of the effective density for all plates as

$$\begin{array}{@{}rcl@{}} \partial_{z} \tilde{\rho} &=& \frac{1-\rho_{\min}^{p}}{\tilde{\rho}^{p-1}} \sum\limits_{q=1}^{N_{q}} \hat{\rho}_{q}^{p-1} \left[ \zeta(\alpha_{q},s) \partial_{d_{q}} \rho_{q} \partial_{z} d_{q}\right.\\ &&\left.+ \rho_{q} \partial_{z} \zeta(\alpha_{q},s) \right] \end{array} $$

(53)

where from (2), the design sensitivity of the density for a single plate is given by

$$ \partial_{d_{q}} \rho_{q}= \left\{\begin{array}{ll} 0 & \text{if} d_{q} > r \\ \frac{3}{4r} \left[ \left( \frac{d_{q}}{r} \right)^{2} -1 \right] & \text{if} -r \leq d_{q} \leq r \\ 0 & \text{if} d_{q} < - r \end{array}\right. $$

(54)

We note that if z corresponds to α _q then $\partial _{\alpha _{q}} d_{q} = 0$ and from (16)

$$ \partial_{x} \zeta(x,s) = \frac{1+s}{[1 + s(1-x)]^{2}} $$

(55)

Otherwise, ∂ _z ζ(α _q , s) = 0 and ∂ _z d _q is given by the results of the following section (and it equals zero if z ∉ z _q ).

An interest insight on the sensitivity of the geometry projection is that if one looks at ∂ _z d _q as the design sensitivity of the closest point on ∂ ω to x, then the first term in square brackets in (53) is a convective term and the second term is an advective term. This is in contrast to level set topology optimization methods which only employ convective sensitivities.

1.2 A.2 Distance function sensitivities

We proceed to derive sensitivities for the distance function with respect to the plate design parameters of (3). For simplicity, in this section we drop the q subscript and note that all quantities refer to a single plate. To simplify the derivation, we first perform the following algebraic manipulation of (4) to eliminate the square root on the right-hand side (RHS) of the equation:

$$\begin{array}{@{}rcl@{}} d^{2} &=& (\| \hat{\mathbf{x}}^{\perp} \|^{2} + H({\Delta}_{1}) {{\Delta}_{1}^{2}} + H({\Delta}_{2}) {{\Delta}_{2}^{2}})\\ &&-2(d+t/2)(t/2) + t^{2}/4 \end{array} $$

(56)

Differentiating with respect to some design variable z, and since we have assumed the plate thickness to be constant, we have

$$ \partial_{z} d \,=\, \frac{1}{2d+t} \left[ \partial_{z} \| \hat{\mathbf{x}}^{\perp} \|^{2} + \partial_{z}(H({\Delta}_{1}) {{\Delta}_{1}^{2}}) + \partial_{z}(H({\Delta}_{2}) {{\Delta}_{2}^{2}}) \right] $$

(57)

The above expression reveals a requirement of our method. If d = −t/2, then the denominator on the RHS of (57) is zero and the derivative is not defined. However, if we require that the sample window for the projection in ( 2 ) is smaller than the plate thickness, i.e. that r < t/2, then ∂ _d ρ from (54) will be 0 when |d| ≥ t/2, hence ∂ _z d need not be computed. This also addresses the issue of the signed distance function potentially being not unique, i.e. if there are two portions of ∂ ω to which the signed distance from x is the same; this would occur, for example, if x lies on the medial surface and d ≥ t/2. This requirement of our method was also shown in Norato et al. (2015).

To obtain the sensitivities with respect to the individual design parameters in (3), we therefore obtain the individual sensitivities on the RHS of (57). Let us start with the sensitivities with respect to the plate center point x _c . We first note that

$$ \| \hat{\mathbf{x}}^{\perp} \|^{2} = (\hat{\mathbf{x}}^{\perp})^{T} \hat{\mathbf{x}}^{\perp} = \hat{\mathbf{x}}^{T} (\mathsf{P}^{\perp})^{T} \mathsf{P}^{\perp} \hat{\mathbf{x}} = \hat{\mathbf{x}}^{T} \mathsf{P}^{\perp} \hat{\mathbf{x}} $$

(58)

where we used the fact that the projection P ^⊥ is symmetric and idempotent. Therefore,

$$ \partial_{z} \| \hat{\mathbf{x}}^{\perp} \|^{2} = 2 (\partial_{z} \hat{\mathbf{x}})^{T} \textsf{P}^{\perp} \hat{\mathbf{x}} + \hat{\mathbf{x}}^{T} \partial_{z} \textsf{P}^{\perp} \hat{\mathbf{x}} $$

(59)

From (5) and (6) we have that $\nabla _{\mathbf {x_{c}}} \hat {\mathbf {x}} = - \textsf {I}$ and $\nabla _{\mathbf {x_{c}}} \textsf {P}^{\perp } = \textsf {0}$ respectively, hence

$$ \nabla_{\mathbf{x_{c}}} \| \hat{\mathbf{x}}^{\perp} \|^{2} = -2 \textsf{P}^{\perp} \hat{\mathbf{x}} = -2 \hat{\mathbf{x}}^{\perp} $$

(60)

To obtain $\nabla _{\mathbf {x_{c}}} (H({\Delta }_{1}){{\Delta }_{1}^{2}})$ we first use the product rule:

$$ \partial_{z} (H({\Delta}_{1}){{\Delta}_{1}^{2}}) = \partial_{z} H({\Delta}_{1}){{\Delta}_{1}^{2}} + H({\Delta}_{1}) \partial_{z} ({{\Delta}_{1}^{2}}) $$

(61)

The derivative of the Heaviside function is given by H ^′(x) = δ(x), where δ(x) = {1 if x = 0, 0 otherwise } is the Dirac delta function. Therefore, ∂ _z H(Δ₁) is non-zero only when Δ₁ = 0. However, since the first term on the RHS is multiplied by ${{\Delta }_{1}^{2}}$, we have that this term is always zero. Hence, using the chain rule we find

$$ \partial_{z} (H({\Delta}_{1}){{\Delta}_{1}^{2}}) = 2 {\Delta}_{1} H({\Delta}_{1}) \partial_{z} {\Delta}_{1} $$

(62)

Similarly, for the third term in (57) we have

$$ \partial_{z} (H({\Delta}_{2}){{\Delta}_{2}^{2}}) = 2 {\Delta}_{2} H({\Delta}_{2}) \partial_{z} {\Delta}_{2} $$

(63)

From (9), we have that

$$\begin{array}{@{}rcl@{}} \partial_{z} {\Delta}_{1} &=& \partial_{z} | \hat{x}^{\parallel}_{1} | - \partial_{z} (l/2) \end{array} $$

(64)

$$\begin{array}{@{}rcl@{}} \partial_{z} {\Delta}_{2} &=& \partial_{z} | \hat{x}^{\parallel}_{2} | - \partial_{z} (w/2). \end{array} $$

(65)

Clearly, $\nabla _{\mathbf {x_{c}}} (l/2) = \vec {0}$, and so we need only look at $\partial _{z} | \hat {x}^{\parallel }_{1} |$. Here we note another requirement of our method: the derivative of the absolute function |x| is not defined at x = 0. However, if l > 0 then H(Δ₁) = 0 if $| \hat {x}^{\parallel }_{1} | = 0 < l/2$, and from (62) and (64), $2 {\Delta }_{1} H({\Delta }_{1}) \partial _{z} | \hat {x}^{\parallel }_{1} | = 0$. Hence, we require in our method that l > 0 and, using a similar argument, that w > 0. This requirement circumvents the problem that when l or w are zero the closest point on ∂ ω to $\hat {\mathbf {x}}$ is no longer unique.

To compute $\partial _{z} | \hat {x}^{\parallel }_{1} |$, we note that

$$ | \hat{x}^{\parallel}_{1} |^{2} = (\hat{x}^{\parallel}_{1})^{2} = (\hat{\mathbf{x}} \cdot \hat{\mathbf{e}}_{1})^{2} = \hat{\mathbf{x}}^{T} \hat{\mathbf{e}}_{1} \hat{\mathbf{e}}_{1}^{T} \hat{\mathbf{x}} = \hat{\mathbf{x}}^{T} (\hat{\mathbf{e}}_{1} \otimes \hat{\mathbf{e}}_{1}) \hat{\mathbf{x}} $$

(66)

Differentiating, we have

$$ 2 | \hat{x}^{\parallel}_{1} | \partial_{z} | \hat{x}^{\parallel}_{1} | = 2 \partial_{z} \hat{\mathbf{x}}^{T} (\hat{\mathbf{e}}_{1} \otimes \hat{\mathbf{e}}_{1})\hat{\mathbf{x}} + \hat{\mathbf{x}}^{T} \partial_{z}(\hat{\mathbf{e}}_{1} \otimes \hat{\mathbf{e}}_{1}) \hat{\mathbf{x}} $$

(67)

Since ∂ _z (a⊗a) = 2Sym(a⊗∂ _z a), where the Sym(A) operator returns the symmetric part of tensor A, we then have

$$ \partial_{z} | \hat{x}^{\parallel}_{1} | = \frac{1}{| \hat{x}^{\parallel}_{1} |} \left[ \partial_{z} \hat{\mathbf{x}} (\hat{\mathbf{e}}_{1} \otimes \hat{\mathbf{e}}_{1}) + \hat{\mathbf{x}}^{T} \text{Sym}(\hat{\mathbf{e}}_{1} \otimes \partial_{z} \hat{\mathbf{e}}_{1}) \right] \hat{\mathbf{x}} $$

(68)

A similar expression follows for $\partial _{z} | \hat {x}^{\parallel }_{2} |$. With $\nabla _{\mathbf {x_{c}}} \hat {\mathbf {e}}_{1} = \textsf {0}$ and using (7) and (8), we have

$$ \nabla_{\mathbf{x_{c}}} | \hat{x}^{\parallel}_{1} | = -\frac{1}{| \hat{x}^{\parallel}_{1} |} (\hat{\mathbf{e}}_{1} \otimes \hat{\mathbf{e}}_{1}) \hat{\mathbf{x}} = -\frac{1}{| \hat{x}_{1} |} \hat{\mathbf{e}}_{1} \hat{x}_{1}= - \text{sgn}(\hat{x}_{1}) \hat{\mathbf{e}}_{1} $$

(69)

Combining (62), (64) and (69) we obtain

$$ \nabla_{\mathbf{x_{c}}} (H({\Delta}_{1}) {{\Delta}_{1}^{2}}) = -2 \text{sgn}(\hat{x}_{1}) {\Delta}_{1}H({\Delta}_{1}) \hat{\mathbf{e}}_{1} $$

(70)

A similar derivation for the third term on the RHS of (57), and using (65), gives

$$ \nabla_{\mathbf{x_{c}}} (H({\Delta}_{2}) {{\Delta}_{2}^{2}}) = -2 \text{sgn}(\hat{x}_{2}) {\Delta}_{2}H({\Delta}_{2}) \hat{\mathbf{e}}_{2} $$

(71)

Collecting (60), (70) and (71), and using (57) we finally obtain the sensitivity of the distance with respect to the center point of the plate as

$$ \nabla_{\mathbf{x_{c}}} d \,=\, -\frac{1}{d+\frac{t}{2}} \left[\hat{\mathbf{x}}^{\perp} + \text{sgn}(\hat{x}_{1}) {\Delta}_{1}H({\Delta}_{1}) \hat{\mathbf{e}}_{1}+ \text{sgn}(\hat{x}_{2}) {\Delta}_{2}H({\Delta}_{2}) \hat{\mathbf{e}}_{2} \right] $$

(72)

To obtain sensitivities of the distance with respect to the Euler angles of the plate, we first note from (10) that

$$\begin{array}{@{}rcl@{}} \partial_{\psi} \hat{\mathbf{e}}_{1} &=& \mathsf{R}_{\psi}^{\prime} \mathsf{R}_{\theta} \mathsf{R}_{\phi} \mathbf{e}_{1} \\ &=& \mathsf{R}_{\psi}^{\prime} \mathsf{R}_{\theta} \mathsf{R}_{\phi} (\mathsf{R}_{\psi} \mathsf{R}_{\theta} \mathsf{R}_{\phi})^{-1} \hat{\mathbf{e}}_{1} \\ &=& \mathsf{R}_{\psi}^{\prime} \mathsf{R}_{\theta} \mathsf{R}_{\phi} \mathsf{R}_{\phi}^{-1} \mathsf{R}_{\theta}^{-1} \mathsf{R}_{\psi}^{-1} \hat{\mathbf{e}}_{1} \\ &=& \underset{A_{\psi}}{\underbrace{\mathsf{R}_{\psi}^{\prime} \mathsf{R}_{\psi}^{T}}} \hat{\mathbf{e}}_{1} \end{array} $$

(73)

$$\begin{array}{@{}rcl@{}} \partial_{\theta} \hat{\mathbf{e}}_{1} &=& \mathsf{R}_{\psi} \mathsf{R}_{\theta}^{\prime} \mathsf{R}_{\phi} \mathbf{e}_{1} \\ &=& \underset{A_{\theta}}{\underbrace{\mathsf{R}_{\psi} \mathsf{R}_{\theta}^{\prime} \mathsf{R}_{\theta}^{T} \mathsf{R}_{\psi}^{{T}}}} \hat{\mathbf{e}}_{1} \end{array} $$

(74)

$$\begin{array}{@{}rcl@{}} \partial_{\phi} \hat{\mathbf{e}}_{1} &=& \mathsf{R}_{\psi} \mathsf{R}_{\theta} \mathsf{R}_{\phi}^{\prime} \mathbf{e}_{1}\\ &=& \underset{A_{\phi}}{\underbrace{\mathsf{R}_{\psi} \mathsf{R}_{\theta} \mathsf{R}_{\phi}^{\prime} \mathsf{R}_{\phi}^{T} \mathsf{R}_{\theta}^{T} \mathsf{R}_{\psi}^{{T}}}} \hat{\mathbf{e}}_{1} \end{array} $$

(75)

where

$$\begin{array}{@{}rcl@{}} \mathsf{R}_{\psi}^{\prime} &=& \left[ \begin{array}{ccc} -\sin \psi & \cos \psi & 0 \\ -\cos \psi & -\sin \psi & 0 \\ 0 & 0 & 0 \end{array} \right] \end{array} $$

(76)

$$\begin{array}{@{}rcl@{}} \mathsf{R}_{\theta}^{\prime} &=& \left[ \begin{array}{ccc} 0 & 0 & 0 \\ 0 & -\sin \theta & \cos \theta \\ 0 & -\cos \theta & -\sin \theta \end{array} \right] \end{array} $$

(77)

$$\begin{array}{@{}rcl@{}} \mathsf{R}_{\phi}^{\prime} &=& \left[ \begin{array}{ccc} -\sin \phi & \cos \phi & 0 \\ -\cos \phi & -\sin \phi & 0 \\ 0 & 0 & 0 \end{array} \right] \end{array} $$

(78)

and we used the fact that R _ψ , R _θ and R _ϕ are orthogonal, hence their inverses equal their transposes. It can be shown the above defined tensors A _β , β ∈ {ψ, θ, ϕ} are skew-symmetric like the R _β , i.e. $\textsf {A}_{\beta }^{T} = -\textsf {A}_{\beta }$. As such, there is an axial vector w = Axial(A) = [A ₃₂, A ₁₃, A ₂₁]^T associated with each of these tensors such that

$$ \textsf{A}_{\beta} \mathbf{a} = \mathbf{w}_{\beta} \wedge \mathbf{a} $$

(79)

for any $\mathbf {a} \in \mathbb {R}^{3}$, where ∧ denotes the wedge or cross product. It can be readily verified that the axial vectors for the A _β of (73)–(75) are given by

$$\begin{array}{@{}rcl@{}} \mathbf{w}_{\psi} &\!:=\!& \text{Axial}(\textsf{A}_{\psi}) \,=\, [0, 0, -1]^{T} \end{array} $$

(80)

$$\begin{array}{@{}rcl@{}} \mathbf{w}_{\theta} &\!:=\!& \text{Axial}(\textsf{A}_{\theta}) |\,=\, [-\cos \psi, \sin \psi, 0]^{T} \end{array} $$

(81)

$$\begin{array}{@{}rcl@{}} \mathbf{w}_{\phi} &\!:=\!& \text{Axial}(\textsf{A}_{\phi}) \,=\, [-\sin \psi \sin \theta, -\cos \psi \sin \theta, -\cos \theta]^{T} \\ \end{array} $$

(82)

For the sensitivity of the first term on the RHS of (57) with respect to β ∈ {ψ, θ, ϕ}, we use (59) and the fact that $\partial _{\beta } \hat {\textbf {x}} = \vec {0}$ to obtain

$$ \partial_{\beta} \| \hat{\textbf{x}}^{\perp} \|^{2} = \hat{\textbf{x}}^{T} \partial_{\beta} \textsf{P}^{\perp} \hat{\textbf{x}} $$

(83)

where from (6) we have

$$ \partial_{\beta} \textsf{P}^{\perp} = -2 \text{Sym}(\partial_{\beta} \hat{\mathbf{e}}_{1} \otimes \hat{\mathbf{e}}_{1} +\partial_{\beta} \hat{\mathbf{e}}_{2} \otimes \hat{\mathbf{e}}_{2} ) $$

(84)

With $\partial _{\beta } \hat {\mathbf {e}}_{1}$ and $\partial _{\beta } \hat {\mathbf {e}}_{2}$ given by one of (73–75), (84) becomes

$$\begin{array}{@{}rcl@{}} \partial_{\beta} \textsf{P}^{\perp} &=& -2 \text{Sym} \left[ \textsf{A}_{\beta} \hat{\mathbf{e}}_{1} \otimes \hat{\mathbf{e}}_{1} + \textsf{A}_{\beta} \hat{\mathbf{e}}_{2} \otimes \hat{\mathbf{e}}_{2} \right]\\ &=& -2 \text{Sym} \left[ \textsf{A}_{\beta} (\hat{\mathbf{e}}_{1} \otimes \hat{\mathbf{e}}_{1} + \hat{\mathbf{e}}_{2} \otimes \hat{\mathbf{e}}_{2}) \right]\\ &=& -2 \text{Sym} (\textsf{A}_{\beta}\textsf{P}^{\parallel}) \end{array} $$

(85)

From the above and from (83) we obtain the sensitivity of the first term on the RHS of (57) with respect to any of the three Euler angles as

$$\begin{array}{@{}rcl@{}} \partial_{\beta} \| \hat{\textbf{x}}^{\perp} \|^{2} &=& -2 \hat{\textbf{x}}^{T} \text{Sym} (\textsf{A}_{\beta} \textsf{P}^{\parallel} ) \hat{\textbf{x}} \\ &=& - \hat{\textbf{x}}^{T} \left( \textsf{A}_{\beta} \textsf{P}^{\parallel} + (\textsf{P}^{\parallel})^{T} \textsf{A}_{\beta}^{T} \right) \hat{\textbf{x}} \\ &=& - \left( \textsf{A}_{\beta}^{T} \hat{\textbf{x}} \right)^{T} \left( \textsf{P}^{\parallel} \hat{\textbf{x}} \right) - \left( \textsf{P}^{\parallel} \hat{\textbf{x}} \right)^{T} \left( \textsf{A}_{\beta}^{T} \hat{\textbf{x}} \right) \\ &=& -2 \hat{\textbf{x}}^{T} \textsf{A}_{\beta} \hat{\textbf{x}}^{\parallel} \end{array} $$

(86)

where we used the fact that P^∥ is symmetric. A similar procedure and Eqns. (62), (63), (64), (65), and (68) render the sensitivities of the second and third terms on the RHS of (57) with respect to any of the three Euler angles as

$$\begin{array}{@{}rcl@{}} \partial_{\beta} (H({\Delta}_{1}){{\Delta}_{1}^{2}}) &=& -\frac{2 H({\Delta}_{1}) {\Delta}_{1}} {| \hat{x}^{\parallel}_{1} |} \hat{\textbf{x}}^{T} \text{sym} \left[ \textsf{A}_{\beta} (\hat{\mathbf{e}}_{1} \otimes \hat{\mathbf{e}}_{1}) \right] \hat{\mathbf{x}} \\ &=& -2 \text{sgn}(\hat{x}_{1}) H({\Delta}_{1}) {\Delta}_{1} \hat{\mathbf{x}}^{T} \textsf{A}_{\beta} \hat{\mathbf{e}}_{1} \end{array} $$

(87)

$$\begin{array}{@{}rcl@{}} \partial_{\beta} (H({\Delta}_{2}){{\Delta}_{2}^{2}}) &=& -\frac{2 H({\Delta}_{2}) {\Delta}_{2}}{| \hat{x}^{\parallel}_{2} |} \hat{\mathbf{x}}^{T} \text{sym} \left[ \textsf{A}_{\beta} (\hat{\mathbf{e}}_{2} \otimes \hat{\mathbf{e}_{2}}) \right] \hat{\mathbf{x}} \\ &=& -2 \text{sgn}(\hat{x}_{2}) H({\Delta}_{2}) {\Delta}_{2} \hat{\mathbf{x}}^{T} \textsf{A}_{\beta} \hat{\mathbf{e}}_{2} \end{array} $$

(88)

Combining (86), (87) and (88), and using (57) and (79) we finally obtain the sensitivity of the distance with respect to any of the Euler angles of a plate as

$$\begin{array}{@{}rcl@{}} \partial_{\beta} d &=& - \frac{1}{d+\frac{t}{2}} \hat{\mathbf{x}} \cdot \big[ \mathbf{w}_{\beta} \wedge \big(\hat{\mathbf{x}}^{\parallel} + \text{sgn}(\hat{x}_{1}) H({\Delta}_{1}) {\Delta}_{1} \hat{\mathbf{e}}_{1}\\ &&+ \text{sgn}(\hat{x}_{2}) H({\Delta}_{2}) {\Delta}_{2} \hat{\mathbf{e}}_{2} \big) \big] \end{array} $$

(89)

with w _β given by one of (80–82).

We now proceed to obtain the sensitivities of the distance with respect to the length and width of the plate. With $\partial _{l} \hat {\textbf {x}} = \partial _{w} \hat {\textbf {x}} = \vec {0}$ from (5) and ∂ _l P^⊥ = ∂ _w P^⊥=0 from (6), we have from (59)

$$ \partial_{l} \| \hat{\textbf{x}}^{\perp} \|^{2} = \partial_{w} \| \hat{\textbf{x}}^{\perp} \|^{2} = 0 $$

(90)

Also, $\partial _{l} \hat {\mathbf {e}}_{1} = \partial _{l} \hat {\mathbf {e}}_{2} = \partial _{w} \hat {\mathbf {e}}_{1} = \partial _{w} \hat {\mathbf {e}}_{2} = \vec {0}$ from (10). Consequently, from (68) we find that $\partial _{l} | \hat {x}^{\parallel }_{1} | = \partial _{w} | \hat {x}^{\parallel }_{1} | = \partial _{l} | \hat {x}^{\parallel }_{2} | = \partial _{w} | \hat {x}^{\parallel }_{2} | = 0$, and so from (64) and (65) we have that ∂ _l Δ₁ = ∂ _w Δ₂ = −1/2 and ∂ _w Δ₁ = ∂ _l Δ₂ = 0. Therefore, from (62) and (63) we find

$$\begin{array}{@{}rcl@{}} \partial_{l} (H({\Delta}_{1}) {{\Delta}_{1}^{2}}) &=& - H({\Delta}_{1}) {\Delta}_{1} \quad\,\,\,\partial_{w} (H({\Delta}_{1}) {{\Delta}_{1}^{2}}) = 0 \end{array} $$

(91)

$$\begin{array}{@{}rcl@{}} \partial_{l} (H({\Delta}_{2}) {{\Delta}_{1}^{2}}) &=& 0 \,\,\,\,\,\quad\qquad \qquad \partial_{w} (H({\Delta}_{2}) {{\Delta}_{2}^{2}}) = - H({\Delta}_{2}) {\Delta}_{2} \end{array} $$

(92)

From (90–91) we finally obtain the sensitivities of the distance with respect to the length and width of the plate as

$$\begin{array}{@{}rcl@{}} \partial_{l} d &=& -\frac{1}{2d+t} H({\Delta}_{1}) {\Delta}_{1} \end{array} $$

(93)

$$\begin{array}{@{}rcl@{}} \partial_{w} d &=& -\frac{1}{2d+t} H({\Delta}_{2}) {\Delta}_{2} \end{array} $$

(94)

Finally, we recall that the distance does not depend on the size variable α, therefore

$$ \partial_{\alpha} d = 0 $$

(95)

To summarize, the sensitivities of the distance with respect to the design parameters of a plate are given by (72), (89), (93), (94) and (95). The derivations in this section not only lead to succinct, elegant expressions, but most importantly they substantially reduce the amount of floating point operations needed to compute the distance sensitivities and therefore they are highly preferred over a naive differentiation.

A final remark is worth noting: the appearance of the Heaviside function in these sensitivities does not render them discontinuous, because H(Δ₁) and H(Δ₂) are multiplied by Δ₁ and Δ₂ respectively wherever they appear in the sensitivities; therefore, even though H(Δ₁) and H(Δ₂) jump from zero to one when Δ₁ and Δ₂ go from being negative to zero, Δ₁ = 0 and Δ₂ = 0, and therefore the corresponding terms in the sensitivities are still zero. This is consistent with Norato et al. (2015), where it was shown that the sensitivities are continuous across branches of the distance function to a bar.

1.3 A.3 Composite Density Sensitivities (Plates with Holes)

From (20), the sensitivities of the composite density are given by

$$ \partial_{z} \breve{\rho} = \frac{1}{2 \breve{\rho}} \left[ \partial_{z} \tilde{\rho} \zeta(\bar{\chi},s) + \tilde{\rho} \partial_{\bar{\chi}} \zeta (\bar{\chi},s) \partial_{z} \bar{\chi} \right] $$

(96)

In the afore expression, $\partial _{z} \tilde {\rho }$ is given by (53) and $\partial _{\bar {\chi }} \zeta (\bar {\chi },s)$ can be obtained from (55). The sensitivities of the filtered free density are more readily expressed in the discretized version of the filter (50). Using the quotient rule and after some simplification, we have

$$ \partial_{z} \bar{\chi} = \frac{1}{D} \sum\limits_{j \in \mathcal{N}_{i}^{\epsilon}} G_{ij} \left[ (\chi_{j} - \bar{\chi}_{i}) \partial_{z} \tilde{\rho}_{j} + \tilde{\rho}_{j} \partial_{z} \chi_{j} \right] $$

(97)

If z corresponds to a plate design parameter (i.e. if z ∈ z _q ), then ∂ _z χ(y) = 0 and $\partial _{z} \tilde {\rho }$ can be computed from (53); on the other hand, if z corresponds to one of the elemental free densities χ _j , then $\partial _{z} \tilde {\rho } = 0$, and since

$$ \partial_{\chi_{j}} \chi_{i} = \left\{\begin{array}{ll} 1 & \text{if}\, i=j \\ 0 & \text{otherwise} \end{array}\right. $$

(98)

then in this case the entire (97) reduces to

$$ \partial_{\chi_{j}} \bar{\chi} =\frac{G_{ij} \tilde{\rho}_{j}}{D} $$

(99)

1.4 A.4 Placement Bound Constraints Sensitivities

The sensitivities of the j-th component of the aggregate placement bound constraints of (33) and (34) are respectively given by

$$\begin{array}{@{}rcl@{}} \partial_{z} KS_{\max}(c_{j}) &=& e^{-\kappa KS_{\max}} \sum\limits_{i} e^{\kappa c_{j}} \partial_{z} c_{j} \end{array} $$

(100)

$$\begin{array}{@{}rcl@{}} \partial_{z} KS_{\min}(c_{j}) &=& e^{\kappa KS_{\min}} \sum\limits_{i} e^{-\kappa c_{j}} \partial_{z} c_{j} \end{array} $$

(101)

where the superscript (i) corresponding to corner i = 1, … , 4 and the subscript q corresponding to plate q have been dropped for brevity, with the understanding that the above expressions lead to 24 equations per plate. The ∂ _z c _j can be obtained from (25) as

$$\begin{array}{@{}rcl@{}} \nabla_{\mathbf{x_{c}}} \mathbf{c}_{q}^{(i)} &=& \textsf{I} \end{array} $$

(102)

$$\begin{array}{@{}rcl@{}} \partial_{\beta} \mathbf{c}_{q}^{(i)}&=& \mathbf{w}_{\beta} \wedge \left[ \frac{l_{q}}{2} \hat{\mathbf{e}}_{1_{q}} + \frac{w_{q}}{2} \hat{\mathbf{e}}_{2_{q}} \right] \end{array} $$

(103)

$$\begin{array}{@{}rcl@{}} \partial_{l} \mathbf{c}_{q}^{(i)} &=& \frac{1}{2} \hat{\mathbf{e}}_{1_{q}} \end{array} $$

(104)

$$\begin{array}{@{}rcl@{}} \partial_{w} \mathbf{c}_{q}^{(i)} &=& \frac{1}{2} \hat{\mathbf{e}}_{2_{q}} \end{array} $$

(105)

with w _β given by one of (80–82). Clearly ∂ _z c _j = 0 if z ∉ z _q since the locations of the corners of the medial surface of a plate only depend on the design parameters of that plate.

1.5 A.5 Compliance and volume fraction sensitivities

Using adjoint sensitivity analysis (cf., for example, Bendsøe and Sigmund (2003)), the compliance sensitivity can be found as

$$ \partial_{z} {\Theta} = -{\int}_{\Omega} \partial_{z} \breve{\rho}(\mathbf{x}, \mathbf{Z}, s, \chi) \nabla \mathbf{u} \cdot \mathbb{C}_{o} \nabla \mathbf{u} \; dv $$

(106)

The corresponding expression on the finite element discretization with element-wise discretized densities is given by

$$ \partial_{z} {\Theta} = - \sum\limits_{i}^{N_{el}} \partial_{z} \breve{\rho}_{i} \mathbf{U}_{i}^{T} \mathbf{K}_{i_{o}} \mathbf{U}_{i} $$

(107)

where N _{e l} is the number of elements in the mesh, U _i and $\mathbf {K}_{i_{o}}$ are the vector of nodal displacements and solid stiffness matrix for element i respectively, and $\partial _{z} \breve {\rho }_{i}$ can be computed from the composite density sensitivity of (96) for design with variable topology plates, or from the effective density of (53) for design with solid, rectangular plates.

The design sensitivity of the volume fraction in (36) is simply given by

$$ \partial_{z} v_{f} = \frac{1}{| {\Omega} |} {\int}_{\Omega} \partial_{z} \breve{\rho}(\mathbf{x}, \mathbf{Z}, 0, \chi) dv $$

(108)

and the corresponding expression on the finite element discretization is

$$ \partial_{z} v_{f} = \frac{1}{| {\Omega} |} \sum\limits_{i}^{N_{el}} \partial_{z} \breve{\rho}_{i} V_{i} $$

(109)

where V _i is the volume of element i. As before, we note the important difference that (109) is computed with s = 0, whereas (107) is computed with s > 0.