Optimal design of responsive structures

Abstract

With recent advances in both responsive materials and fabrication techniques, it is now possible to construct integrated functional structures, composed of both structural and active materials. We investigate the robust design of such structures through topology optimization. By applying a typical interpolation scheme and filtering technique, we prove existence of an optimal design to a class of objective functions which depend on the compliances of the stimulated and unstimulated states. In particular, we consider the actuation work and the blocking load as objectives, both of which may be written in terms of compliances. We study numerical results for the design of a 2D rectangular lifting actuator for both of these objectives, and discuss some intuition behind the features of the converged designs. We formulate the optimal design of these integrated responsive structures with the introduction of voids or holes in the domain, and show that our existence result holds in this setting. We again consider the design of the 2D lifting actuator now with voids. Finally, we investigate the optimal design of an integrated 3D torsional actuator for maximum blocking torque.

Appendices

Appendix 1: A proof of Lemma 3.2

Here, we provide the proof of Lemma 3.2.

Proof

We will first show by compactness that there exists a $u^{\infty } \in \mathcal {U}$ such that $u_k \rightharpoonup u^{\infty }$ in $W^{1,2}(\Omega )$. Then, we will show that we must have $u^{\infty } = \bar{u}$.

Since $u_k$ is the equillibrium solution corresponding to $\phi _k$ for some fixed S, it satisfies

$$\begin{aligned} u_k = \text {arg}\min _{u \in \mathcal {U}} \ \mathcal {E}_f(\phi _k, u, S), \end{aligned}$$

(65)

and for any $\tilde{u} \in \mathcal {U}$, we have

$$\begin{aligned} \mathcal {E}_f (\phi _k, u_k, S) \le \mathcal {E}_f (\phi _k, \tilde{u}, S). \end{aligned}$$

(66)

Furthermore,

$$\begin{aligned}&\mathcal {E}_f (\phi _k, \tilde{u}, S) \le \int _{\Omega } \frac{1}{2} \left[ \mathbb {C}_s \varepsilon (\tilde{u}) \cdot \varepsilon (\tilde{u}) + \mathbb {C}_r(S) (\varepsilon (\tilde{u}) - \varepsilon ^*(S)) \cdot (\varepsilon (\tilde{u}) - \varepsilon ^*(S)) \right] \ \text {d}x \nonumber \\&\quad - \int _{\partial _f \Omega } f \cdot \tilde{u} \ \text {d}s = M \end{aligned}$$

(67)

where M is some constant, independent of k. So,

$$\begin{aligned} \mathcal {E}_f (\phi _k, u_k, S) \le M. \end{aligned}$$

(68)

Now, expanding the energy functional

$$\begin{aligned} \begin{aligned} \mathcal {E}_f (\phi _k, u_k, S) =&\int _{\Omega } \frac{1}{2} \big [(1 - (F \ast \phi _k)^p) \mathbb {C}_s \varepsilon (u_k) \cdot \varepsilon (u_k) + (F \ast \phi _k)^p \mathbb {C}_r(S) \varepsilon (u_k) \cdot \varepsilon (u_k) \\&+ (F \ast \phi _k)^p \mathbb {C}_r(S)\varepsilon ^*(S) \cdot \varepsilon ^*(S) - 2(F \ast \phi _k)^p \mathbb {C}_r (S) \varepsilon ^*(S) \cdot \varepsilon (u_k) \big ] \ \text{d}x \\&- \int _{\partial _f \Omega } f \cdot u_k \ \text {d}s, \end{aligned} \end{aligned}$$

(69)

and using the ellipticity from Remark 1

$$\begin{aligned} \begin{aligned}&m \left\Vert u_k\right\Vert ^2_{W^{1,2}(\Omega )} - \left| \int _{\Omega } (F \ast \phi _k)^p \mathbb {C}_r (S) \varepsilon ^*(S) \cdot \varepsilon (u_k) \ \text {d}x \right| \\&\quad - \int _{\partial _f \Omega } f \cdot u_k \ \text {d}s \le \mathcal {E}_f (\phi _k, u_k, S), \\&\quad m \left\Vert u_k\right\Vert ^2_{W^{1,2}(\Omega )} - \int _{\Omega } (F \ast \phi _k)^p \left| \mathbb {C}_r (S) \varepsilon ^*(S) \cdot \varepsilon (u_k) \right| \ \text {d}x \\&\quad - \int _{\partial _f \Omega } f \cdot u_k \ \text {d}s \le \mathcal {E}_f (\phi _k, u_k, S), \\&\quad m \left\Vert u_k\right\Vert ^2_{W^{1,2}(\Omega )} - \int _{\Omega } (F \ast 1)^p \left| \mathbb {C}_r (S) \varepsilon ^*(S) \cdot \varepsilon (u_k) \right| \ \text {d}x\\&\quad - \int _{\partial _f \Omega } f \cdot u_k \ \text {d}s \le \mathcal {E}_f (\phi _k, u_k, S), \\&\quad m \left\Vert u_k\right\Vert ^2_{W^{1,2}(\Omega )} - \int _{\Omega } \left| \mathbb {C}_r (S) \varepsilon ^*(S) \cdot \varepsilon (u_k) \right| \ \text {d}x\\&\quad - \int _{\partial _f \Omega } f \cdot u_k \ \text {d}s \le \mathcal {E}(\phi _k, u_k, S), \\&\quad m \left\Vert u_k\right\Vert ^2_{W^{1,2}(\Omega )} - \left\Vert \mathbb {C}_r (S) \varepsilon ^*(S)\right\Vert _{L^2(\Omega )} \left\Vert \varepsilon (u_k)\right\Vert _{L^2(\Omega )}\\&\quad -\int _{\partial _f \Omega } f \cdot u_k \ \text{d}s \le \mathcal {E}_f (\phi _k, u_k, S), \\&\quad m \left\Vert u_k\right\Vert ^2_{W^{1,2}(\Omega )} - c \left\Vert u_k\right\Vert _{W^{1,2}( \Omega )} \\&\quad - \int _{\partial _f \Omega } f \cdot u_k \ \text {d}s \le \mathcal {E}_f (\phi _k, u_k, S) \end{aligned} \end{aligned}$$

(70)

for some constants $m,c > 0$, independent of k. Additionally,

$$\begin{aligned} \begin{aligned}&m \left\Vert u_k\right\Vert ^2_{W^{1,2}(\Omega )} - c \left\Vert u_k\right\Vert _{W^{1,2}( \Omega )} - \left| \int _{\partial _f \Omega } f \cdot u_k \ \text {d}s \right| \\&\quad \le \mathcal {E}_f (\phi _k, u_k, S), \\&\quad m \left\Vert u_k\right\Vert ^2_{W^{1,2}(\Omega )} - c \left\Vert u_k\right\Vert _{W^{1,2}( \Omega )} - \left\Vert f\right\Vert _{L^2(\partial _f \Omega )} \left\Vert u_k\right\Vert _{L^2(\partial _f \Omega )}\\&\quad \le \mathcal {E}(\phi _k, u_k, S), \\&\quad m \left\Vert u_k\right\Vert ^2_{W^{1,2}(\Omega )} - c \left\Vert u_k\right\Vert _{W^{1,2}( \Omega )} - \left\Vert f\right\Vert _{L^2(\partial _f \Omega )} \left\Vert u_k\right\Vert _{L^2(\partial \Omega )}\\&\quad \le \mathcal {E}_f (\phi _k, u_k, S), \\&\quad m \left\Vert u_k\right\Vert ^2_{W^{1,2}(\Omega )} - c \left\Vert u_k\right\Vert _{W^{1,2}( \Omega )} - a \left\Vert u_k\right\Vert _{W^{1,2}(\Omega )}\\&\quad&\le \mathcal {E}_f (\phi _k, u_k, S), \end{aligned} \end{aligned}$$

(71)

for some constant $a > 0$. Then,

$$\begin{aligned} m \left\Vert u_k\right\Vert ^2_{W^{1,2}(\Omega )} - b \left\Vert u_k\right\Vert _{W^{1,2}(\Omega )} \le M \implies \left\Vert u_k\right\Vert _{W^{1,2}(\Omega )} \le d, \end{aligned}$$

(72)

for some constant $b > 0$, where $d > 0$ is a constant independent of k. Thus, $u_k$ is a bounded sequence in $W^{1,2}(\Omega )$, and there exists a $u^\infty \in \mathcal {U}$ such that

$$\begin{aligned} u_{k} \rightharpoonup u^{\infty } \text { in } W^{1,2}(\Omega ) \text { when } k \rightarrow + \infty , \end{aligned}$$

(73)

up to a subsequence. Next, consider $\bar{u} \in \mathcal {U}$ such that

$$\begin{aligned} \bar{u} = \text {arg}\min _{u \in \mathcal {U}} \mathcal {E}_f (\bar{\phi }, u, S). \end{aligned}$$

(74)

Then

$$\begin{aligned} \mathcal {E}_f (\bar{\phi }, \bar{u}, S) \le \mathcal {E}_f (\bar{\phi }, u^{\infty }, S). \end{aligned}$$

(75)

Similarly,

$$\begin{aligned}&\mathcal {E}_f(\phi _k, u_k, S) \le \mathcal {E}_f(\phi _k, \bar{u}, S) = \mathcal {E}_f(\phi _k, \bar{u}, S) - \mathcal {E}_f(\bar{\phi }, \bar{u}, S) + \mathcal {E}_f(\bar{\phi }, \bar{u}, S) \nonumber \\&\quad - \mathcal {E}_f(\bar{\phi }, u_k, S) + \mathcal {E}_f(\bar{\phi }, u_k, S), \end{aligned}$$

(76)

or

$$\begin{aligned} \mathcal {E}_f(\bar{\phi }, u_k, S) \le \mathcal {E}_f(\bar{\phi }, \bar{u}, S) + \mathcal {E}_f(\phi _k, \bar{u}, S) - \mathcal {E}_f(\bar{\phi }, \bar{u}, S) + \mathcal {E}_f(\bar{\phi }, u_k, S) - \mathcal {E}_f(\phi _k, u_k, S). \end{aligned}$$

(77)

Then taking limits, and using the strong convergence of the convolution gives

$$\begin{aligned} \lim _{k \rightarrow \infty } \ \mathcal {E}_f(\bar{\phi }, u_k, S) \le \mathcal {E}_f(\bar{\phi }, \bar{u}, S). \end{aligned}$$

(78)

The convexity of the energy integrand in $\nabla u$ and u for a given $\phi$ and S gives lower semi-continuity of our energy function

$$\begin{aligned} \mathcal {E}_f(\bar{\phi }, u^{\infty }, S) \le \lim _{k \rightarrow \infty } \ \mathcal {E}_f(\bar{\phi }, u_k, S), \end{aligned}$$

(79)

so

$$\begin{aligned} \mathcal {E}_f(\bar{\phi }, u^{\infty }, S) \le \mathcal {E}_f(\bar{\phi }, \bar{u}, S). \end{aligned}$$

(80)

Then from (75),

$$\begin{aligned} \mathcal {E}_f(\bar{\phi }, u^{\infty }, S) = \mathcal {E}_f(\bar{\phi }, \bar{u}, S). \end{aligned}$$

(81)

From the uniqueness of the minimizer of $\mathcal {E}_f(\bar{\phi }, \times , S)$ we have

$$\begin{aligned} u^\infty = \bar{u}. \end{aligned}$$

(82)

Then, as desired,

$$\begin{aligned} u_{k} \rightharpoonup \bar{u} \text { in } W^{1,2}(\Omega ) \text { when } k \rightarrow + \infty . \end{aligned}$$

(83)

$\square$

Appendix 2: Workpiece objective as force in spring

Here we show the workpiece objective is equivalent to maximizing the load of a point spring. This is equivalent to the objective used by Sigmund in an earlier work to study thermal actuators (Sigmund 1998). However, we include the derivation here for completeness. Consider a linear spring in direction $\hat{n}$ of spring constant $\kappa > 0$ connected to the boundary of the domain at some point of interest $x_0 \in \partial _f \Omega$. The aim is to maximize the load carried by this spring upon actuation. Thus, we look to maximize the load in the spring:

$$\begin{aligned} \sup \{ f_0 : f_0 = -\kappa u(x_0) \cdot \hat{n}, \Phi \in \mathcal {D} \} \end{aligned}$$

(84)

where u is the equilibrium solution corresponding to $S = 1$ and $f = f_0 \delta (x - x_0) \hat{n}$. Assuming homogeneous Dirichlet conditions $u_0 = 0$ on $\partial _u \Omega$, it is easy to see using the linearity of the Euler-Lagrange equations

$$\begin{aligned} u = v + u_{S = 0, f_0 \hat{n}} \end{aligned}$$

(85)

where $u_{S = 0, f_0 \hat{n}}$ minimizes the elastic energy (4) with $S = 0$ and $f = f_0 \delta (x - x_0) \hat{n}$. Invoking linearity again gives $u_{S = 0, f_0 \hat{n}} = f_0 u_{S = 0, \hat{n}}$. The displacement can then be written as

$$\begin{aligned} u = v + f_0 u_{S = 0, \hat{n}}. \end{aligned}$$

(86)

Evaluating at $x = x_0$, taking an inner product with $\hat{n}$, and using the constraint that $f_0 = - \kappa u(x_0) \cdot \hat{n}$ gives

$$\begin{aligned} -\frac{f_0}{\kappa } = v(x_0) \cdot \hat{n} + f_0 u_{S = 0, \hat{n}}(x_0) \cdot \hat{n}. \end{aligned}$$

(87)

Rearranging gives

$$\begin{aligned} f_0 = \frac{-\kappa v(x_0) \cdot \hat{n} }{\kappa u_{S = 0, \hat{n}}(x_0) \cdot \hat{n} + 1} = \frac{ - \kappa v(x_0) \cdot \hat{n} - \kappa u_{S = 0, \hat{n}}(x_0) \cdot \hat{n} - 1}{ \kappa u_{S = 0, \hat{n}}(x_0) \cdot \hat{n} + 1} - 1 \end{aligned}$$

(88)

or

$$\begin{aligned} f_0 = \frac{ - \kappa u_{S = 1, \hat{n}}(x_0) \cdot \hat{n} - 1}{\kappa u_{S = 0, \hat{n}}(x_0) \cdot \hat{n} + 1} - 1. \end{aligned}$$

(89)

We recognize $u_{S = 0, \hat{n}}(x_0) \cdot \hat{n}$ and $u_{S = 1, \hat{n}}(x_0) \cdot \hat{n}$ as the unactuated and actuated compliances under loading $f = \delta (x - x_0) \hat{n}$. Thus, the workpiece objective can then be written as a function of compliancies,

$$\begin{aligned} \inf _{\Phi \in \mathcal {D}} \mathcal {O}(\Phi ) = \frac{\kappa \mathcal {C}_{\hat{n}}(\Phi , 1) + 1}{\kappa \mathcal {C}_{\hat{n}}(\Phi , 0) + 1}. \end{aligned}$$

(90)