Design methodology of magnetic fields and structures for magneto-mechanical resonator based on topology optimization

Abstract

Magneto-mechanical resonators—magnetically-driven vibration devices—are used in many mechanical and electrical devices. We develop topology optimization (TO) to configure the magnetic fields of such resonators to enable large vibrations under specified current input to be attained. A dynamic magneto-mechanical analysis in the frequency domain is considered where we introduce the surface magnetic force calculated from the Maxwell stress tensor. The optimization problem is then formulated involving specifically the maximization of the dynamic compliance. This formulation is implemented using the solid-isotropic-material-with-penalization method for TO by taking into account the relative permeability, Young’s modulus, and the mass density of the magnetic material as functions of the density function. Through the 2D numerical studies, we confirm that this TO method works well in designing magnetic field patterns and providing matching between the external current frequency and eigenfrequency of the vibrating structure.

Appendix

We provide details of the derivation of the sensitivity in (24) and the adjoint equation in (25). This sensitivity analysis is based on the procedure shown in Chapter 5 of (Allaire 2007). The general objective function with respect to magnetic potential $({\mathbf {A}})$ and amplitude ${\mathbf {u}}$ is defined as $J(\phi )=\int _{\varOmega }j({\mathbf {A}},\,{\mathbf {u}}) \text {d}x$. The derivative of this function in direction $\theta$ is then:

$$\begin{aligned} \begin{aligned} \langle J'(\phi ),\theta \rangle&= \int j'({\mathbf {A}})\langle {\mathbf {A}}'(\phi ),\theta \rangle \text {d}s +\int j'({\mathbf {u}})\langle {\mathbf {u}}'(\phi ),\theta \rangle \text {d}s\\&= \int j'({\mathbf {A}})\bar{\mathbf{A}}\text {d}s+\int j'({\mathbf {u}})\bar{\mathbf{u}}\text {d}s \end{aligned} \end{aligned}$$

(33)

where $\bar{\mathbf{A}}=\langle {\mathbf {A}}'(\phi ),\theta \rangle ,\ \bar{\mathbf{u}}=\langle {\mathbf {u}}'(\phi ),\theta \rangle$. Given adjoint variables ${\mathbf {p}}$ and ${\mathbf {q}}$ as test functions of the weak-form equations of state in (7) and (11), the Lagrangian is formulated as:

$$\begin{aligned} L(\phi ,{\mathbf {A}},{\mathbf {u}},\bar{\mathbf{A}},\bar{\mathbf{u}}) =\int j({\mathbf {A}},{\mathbf {u}}) \text {d}s+a({\mathbf {A}},{\mathbf {p}})+b({\mathbf {A}},{\mathbf {p}}) -l({\mathbf {p}})-d({\mathbf {u}},{\mathbf {q}})+e({\mathbf {u}},{\mathbf {q}})-m({\mathbf {A}},{\mathbf {q}}) \end{aligned}$$

(34)

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

$$\begin{aligned} \begin{aligned} \langle j'(\phi ),\theta \rangle&=\left\langle \frac{\partial L}{\partial \phi } (\phi ,{\mathbf {A}},{\mathbf {u}},\bar{\mathbf{A}},\bar{\mathbf{u}}),\theta \right\rangle +\left\langle \frac{\partial L}{\partial {\mathbf {A}}} (\phi ,{\mathbf {A}},{\mathbf {u}},\bar{\mathbf{A}},\bar{\mathbf{u}}), \langle {\mathbf {A}}'(\phi ),\theta \rangle \right\rangle \\&+\,\left\langle \frac{\partial L}{\partial {\mathbf {u}}} (\phi ,{\mathbf {A}},{\mathbf {u}},\bar{\mathbf{A}},\bar{\mathbf{u}}), \langle {\mathbf {u}}'(\phi ),\theta \rangle \right\rangle \\&=\left\langle \frac{\partial L}{\partial \phi } (\phi ,{\mathbf {A}},{\mathbf {u}},\bar{\mathbf{A}},\bar{\mathbf{u}}),\theta \right\rangle +\left\langle \frac{\partial L}{\partial {\mathbf {A}}}(\phi ,{\mathbf {A}},{\mathbf {u}}, \bar{\mathbf{A}},\bar{\mathbf{A}}),\bar{\mathbf{A}}\right\rangle \\&+\,\left\langle \frac{\partial L}{\partial {\mathbf {u}}} (\phi ,{\mathbf {A}},{\mathbf {u}},\bar{\mathbf{A}},\bar{\mathbf{u}}), \bar{\mathbf{u}}\right\rangle \\ \end{aligned} \end{aligned}$$

(35)

Consider if the second and third terms are zero. These terms are calculated as

$$\begin{aligned} \left\langle \frac{\partial L}{\partial {\mathbf {A}}},\bar{\mathbf{A}}\right\rangle &= \int j'({\mathbf {A}})\bar{{\mathbf{A}}}\text {d}x+da_A(\bar{\mathbf{A}},{\mathbf {p}}) +a(\bar{\mathbf{A}},{\mathbf {p}})+b(\bar{\mathbf{A}},{\mathbf {p}})-m(\bar{\mathbf{A}},{\mathbf {q}})=0 \end{aligned}$$

(36)

$$\begin{aligned} \left\langle \frac{\partial L}{\partial {\mathbf {u}}},\bar{\mathbf{u}}\right\rangle &= \int j'({\mathbf {u}})\bar{{\mathbf{u}}}\text {d}x-d(\bar{\mathbf{u}},{\mathbf {q}}) +e(\bar{\mathbf{u}},{\mathbf {q}})=0 \end{aligned}$$

(37)

where

$$\begin{aligned} da_A(\bar{\mathbf{A}},{\mathbf {p}}) =-\int _{\varOmega }\left( \nabla \times {\mathbf {p}}\right) ^{\text{T}} \left( \frac{\mu '({\mathbf {A}})}{\mu ^2}\nabla \times {\mathbf {A}}\right) \bar{\mathbf{A}}\text {d}x \end{aligned}$$

(38)

Hence, if the adjoint variables ${\mathbf {p}}$ and ${\mathbf {q}}$ satisfy the above adjoint equations, the second and third terms of (35) can be ignored. Alternatively, the derivatives of (7) and (11) with respect to $\phi$ in the direction $\theta$ are

$$\begin{aligned}&da_\phi ({\mathbf {A}},{\mathbf {p}})+da_A(\bar{\mathbf{A}},{\mathbf {p}})+a(\bar{\mathbf{A}},{\mathbf {p}})+b(\bar{\mathbf{A}},{\mathbf {p}})=0 \end{aligned}$$

(39)

$$\begin{aligned}&\quad - dd({\mathbf {u}},{\mathbf {q}})-d(\bar{\mathbf{u}},{\mathbf {q}})+de({\mathbf {u}},{\mathbf {q}})+e({\mathbf {u}},{\mathbf {q}})-m(\bar{\mathbf{A}},q)=0 \end{aligned}$$

(40)

where

$$\begin{aligned} da_\phi ({\mathbf {A}},{\mathbf {p}}) & = -\int _{\varOmega }\left( \nabla \times {\mathbf {p}}\right) ^{\text{T}}\left( \frac{\mu '(\phi )}{\mu ^2}\nabla \times {\mathbf {A}} \right) \theta \text {d}x \end{aligned}$$

(41)

$$\begin{aligned} dd({\mathbf {u}},{\mathbf {q}}) &= \omega ^2\int _{\varOmega }\rho '(\phi ) {\mathbf {u}}^{\text{T}}{\mathbf {q}} \theta \text {d}x \end{aligned}$$

(42)

$$\begin{aligned} de({\mathbf {u}},{\mathbf {q}}) &= \int _{\varOmega }\varvec{\varepsilon }({\mathbf {u}})^{\text{T}}{\mathbf {C}}'(\phi )\varvec{\varepsilon }({\mathbf {q}})\theta \text {d}x \end{aligned}$$

(43)

Substituting (39) and (40) into (36) and (37) and combining the equations, we then have

$$\begin{aligned} \begin{aligned}&\int j'({\mathbf {A}})\bar{\mathbf{A}} \text {d}s +\int j'({\mathbf {u}})\bar{\mathbf{u}} \text {d}x\\ =\,&da_\rho ({\mathbf {A}},{\mathbf {p}})-dd({\mathbf {u}},{\mathbf {q}})+de({\mathbf {u}},{\mathbf {q}}) \end{aligned} \end{aligned}$$

(44)

Substituting (44) into (33) yields

$$\begin{aligned} J'(\phi )=-\left( \nabla \times {\mathbf {p}}\right) ^{\text{T}}\left( \frac{\mu '(\phi )}{\mu ^2}\nabla \times {\mathbf {A}} \right) -\omega ^2 \rho '(\phi ) {\mathbf {u}}^{\text{T}}{\mathbf {q}}+\varvec{\varepsilon }({\mathbf {u}})^{\text{T}}{\mathbf {C}}'(\phi )\varvec{\varepsilon }({\mathbf {q}}) \end{aligned}$$

(45)

Substituting the dynamic compliance $J=\int _{{\varGamma }_{mf}}{\mathbf {t}}({\mathbf {A}})^{\text{T}}{\mathbf {u}}\text {d} s=m({\mathbf {A}},{\mathbf {u}})$, we obtain the adjoint equations

$$\begin{aligned}&\int _{\varGamma }{\mathbf {t}}'({\mathbf {A}})\bar{\mathbf{A}}{\mathbf {u}}\text {d}s+a(\bar{\mathbf{A}},{\mathbf {p}})+da_A(\bar{\mathbf{A}},{\mathbf {p}})+b(\bar{\mathbf{A}},{\mathbf {p}})-m(\bar{\mathbf{A}},{\mathbf {q}})=0 \end{aligned}$$

(46)

$$\begin{aligned}&\quad m({\mathbf {A}},\bar{\mathbf{u}})-d(\bar{\mathbf{u}},{\mathbf {q}})+e(\bar{\mathbf{u}},{\mathbf {q}})=0 \end{aligned}$$

(47)

As (47) is the same as the state Eq. (11), ${\mathbf {p}}={\mathbf {u}}$. Substituting this into (45), the sensitivity in (24) is obtained whereas substituting the surface magnetic force ${\mathbf {t}}({\mathbf {A}})$ of (16) into (43), the adjoint equation in (25) is obtained.