Design methodology of piezoelectric energy-harvesting skin using topology optimization

Abstract

This paper describes a design methodology for piezoelectric energy harvesters that thinly encapsulate the mechanical devices and exploit resonances from higher-order vibrational modes. The direction of polarization determines the sign of the piezoelectric tensor to avoid cancellations of electric fields from opposite polarizations in the same circuit. The resultant modified equations of state are solved by finite element method (FEM). Combining this method with the solid isotropic material with penalization (SIMP) method for piezoelectric material, we have developed an optimization methodology that optimizes the piezoelectric material layout and polarization direction. Updating the density function of the SIMP method is performed based on sensitivity analysis, the sequential linear programming on the early stage of the optimization, and the phase field method on the latter stage of the optimization to obtain clear optimal shapes without intermediate density. Numerical examples are provided that illustrate the validity and utility of the proposed method.

Appendix

Appendix In this appendix, the detailed derivation of the sensitivity in (30) and the adjoint equations in (32)–(35) is outlined. The derivatives of the objective functions with respect to the density function are based on the procedure shown in Chapter 5 of Allaire (2007). The general objective function of piezoelectric problem is defined as $J(\phi ) = \int j(\boldsymbol {u},V) dx$. The derivative of this function in the direction $\theta$ is then

$$\begin{array}{@{}rcl@{}} \langle J'(\phi),\theta \rangle &=& \int j'(\boldsymbol{u})\langle \boldsymbol{u}'(\phi),\theta \rangle dx {}+{} \int j'(V)\langle V'(\phi),\theta \rangle dx \\ &=& \int j'(\boldsymbol{u})\boldsymbol{v} dx + \int j'(V)w dx \end{array} $$

(57)

where $\boldsymbol {v}=\langle \boldsymbol {u}'(\phi ),\theta \rangle ,\ w = \langle V'(\phi ),\theta \rangle$. Setting adjoint states $\boldsymbol {p}$ and q as test functions of the weak-form equations of state in (11)–(18), the Lagrangian is formulated as follows:

$$\begin{array}{@{}rcl@{}} &&{} L(\phi,\boldsymbol{u},V,\boldsymbol{p},q)\\ &&= \int j(\boldsymbol{u},V) dx - m(\boldsymbol{u},\boldsymbol{p}) + a(\boldsymbol{u},\boldsymbol{p}) - b(\boldsymbol{p}, V) \\ &&{\kern8pt} - L_{m}(\boldsymbol{p})+ b(\boldsymbol{u},q) + c(V,q) - L_{e}(q). \end{array} $$

(58)

Using this, the derivative of the objective function can be expressed as

$$\begin{array}{@{}rcl@{}} \left\langle j'(\phi),\theta \right\rangle&=& \left\langle \frac{\partial L}{\partial \phi}(\phi,\boldsymbol{u},V,\boldsymbol{p},q),\theta \right \rangle\\ &&+ \left\langle \frac{\partial L}{\partial \boldsymbol{u}}(\phi,\boldsymbol{u},V,\boldsymbol{p},q), \langle \boldsymbol{u}'(\phi),\theta \rangle \right \rangle\\ &&+ \left\langle \frac{\partial L}{\partial V}(\phi,\boldsymbol{u},V,\boldsymbol{p},q), \langle V'(\phi),\theta \rangle \right \rangle\\ &=& \left\langle \frac{\partial L}{\partial \phi}(\phi,\boldsymbol{u},V,\boldsymbol{p},q),\theta \right \rangle\\ &&+ \left\langle \frac{\partial L}{\partial \boldsymbol{u}}(\phi,\boldsymbol{u},V,\boldsymbol{p},q), \boldsymbol{v} \right \rangle\\ &&+ \left\langle \frac{\partial L}{\partial V}(\phi,\boldsymbol{u},V,\boldsymbol{p},q), w \right\rangle. \end{array} $$

(59)

Consider the case where the second and third terms are zero. These terms are calculated as follows:

$$\begin{array}{@{}rcl@{}} \left\langle \frac{\partial L}{\partial \boldsymbol{u}}, \boldsymbol{v} \right\rangle &=& \int j'(\boldsymbol{u})\boldsymbol{v} dx -m(\boldsymbol{v},\boldsymbol{p})\\ && + a(\boldsymbol{v},\boldsymbol{p}) + b(\boldsymbol{v},q) = 0, \end{array} $$

(60)

$$\begin{array}{@{}rcl@{}}\left\langle \frac{\partial L}{\partial V}, w \right\rangle &=& \int j'(V)w dx - b(p,w) \\ &&+ c(w,q) = 0. \end{array} $$

(61)

When the adjoint states $\boldsymbol {p}$ and q satisfy the above adjoint equations, the second and third terms of (59) can be ignored. On the other hand, the derivatives of (11)–(18) with respect to $\phi$ in the direction $\theta$ are

$$\begin{array}{@{}rcl@{}} &&{}- m'(\boldsymbol{u},\boldsymbol{p}) - m(\boldsymbol{v},\boldsymbol{p}) + a'(\boldsymbol{u},\boldsymbol{p}) + a(\boldsymbol{v},\boldsymbol{p})\\ &&\,\,\,- b'(p,V) - b(p,w) = 0 \end{array} $$

(62)

$$b'(\boldsymbol{u},q) + b(\boldsymbol{p},q) + c'(V,q) + c(w,q) = 0 $$

(63)

where

$$ m'(\boldsymbol{u},\boldsymbol{p}) = {\omega_{\text{input}}}^{2}\int_{\Omega} \rho'(\phi) \boldsymbol{u}\boldsymbol{p} \theta dx $$

(64)

$$ a'(\boldsymbol{u},\boldsymbol{p}) = \int_{\Omega} \boldsymbol{\varepsilon}(\boldsymbol{u})^{T}\boldsymbol{C}'(\phi)\boldsymbol{\varepsilon}(\boldsymbol{p})\theta dx $$

(65)

$$ b'(\boldsymbol{p},V) = \int_{\Omega} \boldsymbol{\varepsilon}(\boldsymbol{p})^{T} \boldsymbol{e}_{s}'(\phi) \boldsymbol{E}(V) \theta dx $$

(66)

$$ c'(V,q) = \int_{\Omega} \boldsymbol{E}(V)^{T}\boldsymbol{\epsilon}'(\phi)\boldsymbol{E}(q)\theta dx. $$

(67)

Substituting (62) and (63) into (60) and (61) and combining them into one equation, the following equation is obtained:

$$\begin{array}{@{}rcl@{}} \int j'(\boldsymbol{u})\boldsymbol{v} dx + \int j'(V)w dx &=&- m'(\boldsymbol{u},\boldsymbol{p}) + a'(\boldsymbol{u},\boldsymbol{p}) \\ && - b'(p,V) + b'(\boldsymbol{u},q) \\ &&+ c'(V,q) \end{array} $$

(68)

Substituting (68) into (57) yields following equation:

$$\begin{array}{@{}rcl@{}} J'(\phi) &=& -{\omega_{\text{input}}}^{2} \rho'(\phi) \boldsymbol{u}\boldsymbol{p} + \boldsymbol{\varepsilon}(\boldsymbol{u})^{T}\boldsymbol{C}'(\phi)\boldsymbol{\varepsilon}(\boldsymbol{p})\\ &&- \boldsymbol{\varepsilon}(\boldsymbol{p})^{T} \boldsymbol{e}_{s}'(\phi) \boldsymbol{E}(V) + \boldsymbol{\varepsilon}(\boldsymbol{u})^{T} \boldsymbol{e}_{s}'(\phi) \boldsymbol{E}(q) \\ &&- \boldsymbol{E}(V)^{T}\boldsymbol{\epsilon}'(\phi)\boldsymbol{E}(q). \end{array} $$

(69)

Next, the adjoint equations are calculated from (60) and (61). When the electric energy $c(V,V)$ is considered as the objective function, it is formulated as follows:

$$\begin{array}{@{}rcl@{}} J(\rho) &=& \int j (\boldsymbol{u},V) dx\\ &=& c(V,V). \end{array} $$

(70)

Thus,

$$ \int j'(\boldsymbol{u})v dx = 0 $$

(71)

$$ \int j'(V)w dx = 2c(V,w) $$

(72)

Substituting (71) and (72) into (60) and (61) respectively gives following equation

$$ m(\boldsymbol{v},\boldsymbol{p}) - a(\boldsymbol{v},\boldsymbol{p}) - b(\boldsymbol{v},q) = 0 $$

(73)

$$ b(p,w) + c(w,q) = 2c(V,w) $$

(74)

When the strain energy $a(\boldsymbol {u},\boldsymbol {u})$ is considered as the objective function, it is formulated as follows:

$$\begin{array}{@{}rcl@{}} J(\rho) &=& \int j (\boldsymbol{u},V) dx\\ &=& a(\boldsymbol{u},\boldsymbol{u}) \end{array} $$

(75)

Thus,

$$ \int j'(\boldsymbol{u})\boldsymbol{v} dx = 2a(\boldsymbol{u},\boldsymbol{v}) $$

(76)

$$ \int j'(V)w dx = 0 $$

(77)

Substituting (76) and (77) into (60) and (61) respectively gives following equation

$$ 2a(\boldsymbol{u},\boldsymbol{v}) - m(\boldsymbol{p},\boldsymbol{v}) + a(\boldsymbol{p},\boldsymbol{v}) + b(\boldsymbol{v},q) = 0 $$

(78)

$$- b(\boldsymbol{p},w) + c(q,w) = 0. $$

(79)