Surface effects on shape and topology optimization of nanostructures

Abstract

We present a computational method for the optimization of nanostructures, where our specific interest is in capturing and elucidating surface stress and surface elastic effects on the optimal nanodesign. XFEM is used to solve the nanomechanical boundary value problem, which involves a discontinuity in the strain field and the presence of surface effects along the interface. The boundary of the nano-structure is implicitly represented by a level set function, which is considered as the design variable in the optimization process. Two objective functions, minimizing the total potential energy of a nanostructure subjected to a material volume constraint and minimizing the least square error compared to a target displacement, are chosen for the numerical examples. We present results of optimal topologies of a nanobeam subject to cantilever and fixed boundary conditions. The numerical examples demonstrate the importance of size and aspect ratio in determining how surface effects impact the optimized topology of nanobeams.

Appendix: Derivation of shape derivative

Firstly, the total potential energy objective function is considered. The objective function and its constraints are as follows,

Minimize $J({\varOmega }) = \int \limits _{\varOmega } \mathbf{u}.\mathbf{b}\, d{\varOmega } + \int \limits _{{\varGamma }_N} \mathbf{u}.\mathbf{t}\, d{\varGamma }$

subject to :

$a(\mathbf{u},\delta \mathbf{u},{\varOmega })+a_s(\mathbf{u},\delta \mathbf{u},{\varOmega })= -l_s(\mathbf{u},{\varOmega })+l(\mathbf{u},{\varOmega })$

(i.e.)

$\int \limits _{\varOmega } \epsilon (\delta \mathbf{u}):\mathbb {C}^{bulk}:\varvec{\epsilon }(\mathbf{u})\;d{\varOmega } + \int \limits _{\varGamma } \big (\mathbf{P}\varvec{\epsilon }(\delta \mathbf{u})\mathbf{P}\big ):\varvec{\tau }_\mathbf{s}\;d{\varGamma } + \int \limits _{\varGamma } \big (\mathbf{P}\varvec{\epsilon }(\delta \mathbf{u})\mathbf{P}\big ):\mathbb {C}^s:\big (\mathbf{P}\varvec{\epsilon }(\mathbf{u})\mathbf{P}\big )\;d{\varGamma } = \int \limits _{\varOmega } \mathbf{u}.\mathbf{b}\;d{\varOmega } + \int \limits _{{\varGamma }_N} \mathbf{u}.\mathbf{t}\;d{\varGamma }.$

$$\begin{aligned}&\dot{a}\big (\mathbf{u},\mathbf{w},{\varOmega }\big )+ \dot{a}_s\big (\mathbf{u},\mathbf{w},{\varOmega }\big )\;+\;\dot{l}_s\big (\mathbf{w},{\varOmega }\big )\nonumber \\&\quad = \int \limits _{\varOmega } \varvec{\epsilon }(\mathbf{u}^{\prime }):\mathbb {C}^{bulk}:\varvec{\epsilon }(\mathbf{w}) d{\varOmega } +\int \limits _{\varOmega } \varvec{\epsilon }(\mathbf{u}):\mathbb {C}^{bulk}:\varvec{\epsilon }(\mathbf{w}^{\prime }) d{\varOmega } \nonumber \\&\qquad +\int \limits _{\varGamma } \varvec{\epsilon }(\mathbf{u}):\mathbb {C}^{bulk}:\varvec{\epsilon }(\mathbf{w}) {V_n} d{\varGamma } \nonumber \\&\qquad + \int \limits _{\varGamma } \mathbf{P}\varvec{\epsilon }(\mathbf{w}^{\prime })\mathbf{P} : \varvec{\tau }_\mathbf{s} d{\varGamma } + \int _{\varGamma } \big (\nabla _s(\mathbf{P}\varvec{\epsilon }(\mathbf{w})\mathbf{P} : \varvec{\tau }_\mathbf{s}\big )\cdot \mathbf{n} \nonumber \\&\qquad +\,\kappa \big (\mathbf{P}\varvec{\epsilon }(\mathbf{w})\mathbf{P}:\varvec{\tau }_\mathbf{s})\big )V_n d{\varGamma } \nonumber \\&\qquad +\int \limits _{\varGamma } \big (\mathbf{P}\varvec{\epsilon }(\mathbf{u}^{\prime })\mathbf{P}\big ):\mathbb {C}^s:\big (\mathbf{P}\varvec{\epsilon }(\mathbf{w})\mathbf{P}\big ) d{\varGamma } \nonumber \\&\qquad +\int \limits _{\varGamma } \big (\mathbf{P}\varvec{\epsilon }(\mathbf{u})\mathbf{P}\big ):\mathbb {C}^s:\big (\mathbf{P}\varvec{\epsilon }(\mathbf{w}^{\prime })\mathbf{P}\big ) d{\varGamma } \nonumber \\&\qquad +\int \limits _{\varGamma } \big (\nabla _s (\mathbf{P}\varvec{\epsilon }(\mathbf{u})\mathbf{P}\big ) : \mathbb {C}^s : \big (\mathbf{P}\varvec{\epsilon }(\mathbf{w})\mathbf{P}\big )\cdot \mathbf{n}\,V_n\,d{\varGamma } \nonumber \\&\qquad +\int \limits _{\varGamma } \big (\mathbf{P}\varvec{\epsilon }(\mathbf{u})\mathbf{P}\big ) : \mathbb {C}^s : \nabla _s\big (\mathbf{P}\varvec{\epsilon }(\mathbf{w})\mathbf{P})\big )\cdot \mathbf{n}\,V_n\,d{\varGamma } \nonumber \\&\qquad \int \limits _{\varGamma } \kappa \big (\mathbf{P}\varvec{\epsilon }(\mathbf{u})\mathbf{P}:\mathbb {C}^s:\mathbf{P}\varvec{\epsilon }(\mathbf{w})\mathbf{P}\big )\,V_n\,d{\varGamma }. \end{aligned}$$

(59)

$$\begin{aligned}&\!\!\!\dot{l}(\mathbf{w},{\varOmega }) = \int \limits _{\varOmega } \mathbf{w}^{\prime }.\mathbf{b} d{\varOmega } + \int \limits _{\varGamma } \mathbf{w}.\mathbf{b}\,{V_n} d{\varGamma } + \int \limits _{{\varGamma }_N} \mathbf{w}^{\prime }.\mathbf{t} d{\varGamma } \nonumber \\&\qquad \qquad \quad + \int \limits _{{\varGamma }_N} \big (\nabla (\mathbf{w}.\mathbf{t}).\mathbf{n} + \kappa \mathbf{w}.\mathbf{t}\big )\,{V_n} d{\varGamma }\end{aligned}$$

(60)

$$\begin{aligned}&\!\!\!\dot{J} = \int \limits _{\varOmega } \mathbf{u}^{\prime }.\mathbf{b}\,d{\varOmega } + \int \limits _{\varGamma } \mathbf{u}.\mathbf{b}\,{V_n} d{\varGamma } + \int \limits _{{\varGamma }_N} \big (\nabla (\mathbf{u}.\mathbf{t}).\mathbf{n}+\kappa \mathbf{u}.\mathbf{t}\big )\,V_n\, d{\varGamma }.\nonumber \\ \end{aligned}$$

(61)

The Lagrangian of the objective functional is,

$$\begin{aligned} L= & {} J + l(\mathbf{w},{\varOmega }) - a(\mathbf{u},\mathbf{w},{\varOmega })-a_s(\mathbf{u},\mathbf{w},{\varOmega })\nonumber \\&-\,l_s(\mathbf{w},{\varOmega }). \end{aligned}$$

(62)

The material derivative of the Lagrangian is defined as ,

$$\begin{aligned} \dot{L}= & {} \dot{J} + \dot{l}(\mathbf{w},{\varOmega }) - \dot{a}(\mathbf{u},\mathbf{w},{\varOmega })\nonumber \\&-\,\dot{a}_s(\mathbf{u},\mathbf{w},{\varOmega })-\dot{l}_s(\mathbf{w},{\varOmega }). \end{aligned}$$

(63)

All the terms that contain $\mathbf{w}^{\prime }$ in the material derivative of Lagrangian are collected and the sum of these terms is set to zero, to get the weak form of the state equation,

$$\begin{aligned}&\int \limits _{\varOmega } \mathbf{w}^{\prime }.\mathbf{b}\;d{\varGamma } + \int \limits _{{\varGamma }_N} \mathbf{w}^{\prime }.\mathbf{t}\;d{\varGamma }= \int \limits _{\varOmega } \varvec{\epsilon }(\mathbf{u}):\mathbb {C}^{bulk}:\varvec{\epsilon }(\mathbf{w}^{\prime })\;d{\varOmega }\nonumber \\&+ \int \limits _{\varGamma } \mathbf{P}\varvec{\epsilon }(\mathbf{w}^{\prime })\mathbf{P} : \varvec{\tau }_\mathbf{s}\;d{\varGamma } + \int \limits _{\varGamma } (\mathbf{P}\varvec{\epsilon }(\mathbf{u})\mathbf{P} : \mathbb {C}^s : \mathbf{P}\varvec{\epsilon }(\mathbf{w}^{\prime })\mathbf{P})\;d{\varGamma }.\nonumber \\ \end{aligned}$$

(64)

All the terms that contain $\mathbf{u}^{\prime }$ in the material derivative of Lagrangian are collected and the sum of these terms is set to zero, to get the weak form of the adjoint equation,

$$\begin{aligned}&\int \limits _{\varOmega } \mathbf{u}^{\prime }.\mathbf{b}\;d{\varGamma } + \int \limits _{{\varGamma }_N} \mathbf{u}^{\prime }.\mathbf{t}\;d{\varGamma } = \int \limits _{\varOmega } \varvec{\epsilon }(\mathbf{u}^{\prime }):\mathbb {C}^{bulk}:\varvec{\epsilon }(\mathbf{w})\;d{\varOmega } \nonumber \\&\quad +\int \limits _{\varGamma } (\mathbf{P}\varvec{\epsilon }(\mathbf{u}^{\prime })\mathbf{P} : \mathbb {C}^s : \mathbf{P}\varvec{\epsilon }(\mathbf{w})\mathbf{P})\;d{\varGamma }. \end{aligned}$$

(65)

Considering that ${\varGamma }_N$ and ${\varGamma }_D$ are not modified in the optimization process and assuming that the body forces are zero, the shape derivative of the objective functional can be obtained from Eq. 63,

$$\begin{aligned} J^{\prime } = \int \limits _{{\varGamma }_H} {G}.{V_n} d{\varGamma } \end{aligned}$$

(66)

where,

$$\begin{aligned} {G}= & {} - \int \limits _{\varGamma } \varvec{\epsilon }(\mathbf{u}):\mathbb {C}^{bulk}: \varvec{\epsilon }(\mathbf{w})\;d{\varGamma }\nonumber \\&- \int \limits _{\varOmega } \kappa \Big (\mathbf{P}\varvec{\epsilon }(\mathbf{w})\mathbf{P}:\varvec{\tau }_\mathbf{s})\Big )\; d{\varGamma } \nonumber \\&- \int \limits _{\varGamma } \kappa \Big (\mathbf{P}\varvec{\epsilon }(\mathbf{u})\mathbf{P} : \mathbb {C}^s : \mathbf{P}\varvec{\epsilon }(\mathbf{w})\mathbf{P}\Big )\;d{\varGamma } \end{aligned}$$

(67)

The $G$ obtained can be considered as the negative of velocity, $V_n$ required in order to optimize the level set function. Therefore,

$$\begin{aligned} J^{\prime } = - \int \limits _{{\varGamma }_H} \mathbf{G}^2 d{\varGamma }. \end{aligned}$$

(68)

From the above equation it is evident that the derivative is negative (i.e.) it ensures decrease in the objective function with iterations.

If the objective function is a least square error compared to target displacement as shown below,

$$\begin{aligned}&J({\varOmega }) = \left( \int \limits _{{\varGamma }_N} |\mathbf{u} - \mathbf{u}_\mathbf{0}|^2 d{\varGamma }\right) ^\frac{1}{2}\end{aligned}$$

(69)

$$\begin{aligned}&\dot{J} = {c}_{0} \cdot \left( \int \limits _{{\varGamma }_N} 2|\mathbf{u}-\mathbf{u}_\mathbf{0}| \mathbf{u}^{\prime } d{\varGamma } + \int \limits _{{\varGamma }_N} \left( \nabla (|\mathbf{u}-\mathbf{u}_\mathbf{0}|^2\right) \cdot n \right. \nonumber \\&\qquad \left. +\, \kappa |\mathbf{u}-\mathbf{u}_\mathbf{0}|^2\right) \,{V_n}\,d{\varGamma }. \end{aligned}$$

(70)

Substituting in Eq. 63 and collecting terms with $\mathbf{u}^{\prime }$, the weak form of the adjoint can be obtained as,

$$\begin{aligned} {c}_{0} \int \limits _{{\varGamma }_N} 2|\mathbf{u}-\mathbf{u}_\mathbf{0}|\;\mathbf{u}^{\prime } d{\varGamma }= & {} \int \limits _{\varOmega } \varvec{\epsilon }(\mathbf{u}^{\prime }) \mathbb {C}^{bulk} \varvec{\epsilon }(\mathbf{w}) d{\varOmega } \nonumber \\&+\int \limits _{\varGamma } (\mathbf{P}\varvec{\epsilon }(\mathbf{u}^{\prime })\mathbf{P} : \mathbb {C}^s : \mathbf{P}\varvec{\epsilon }(\mathbf{w})\mathbf{P}) d{\varGamma }\nonumber \\ \end{aligned}$$

(71)

where,

$$\begin{aligned} c_0 = \frac{1}{2} \left( \int \limits _{{\varGamma }_N} |\mathbf{u} - \mathbf{u}_\mathbf{0}|^2 d{\varGamma }\right) ^{-\frac{1}{2}} \end{aligned}$$

(72)