Level set-based topology optimization for the design of a peltier effect thermoelectric actuator

Abstract

Thermoelectric actuators are a type of thermal actuator that generates motion through the input of thermal energy by thermoelectric devices. Thermoelectric actuators utilize thermal expansion and contraction effects, achieved by heating and cooling appropriate parts of the mechanism, which enables specified motions to be carried out and can provide quicker response times than those of typical thermal compliant mechanisms that rely on thermal expansion effects alone. However, the need to consider both thermal expansion and contraction effects makes the design process more complex. This paper proposes a topology optimization method, especially appropriate for the conceptual design of thermoelectric actuators, that uses a level set function to represent structural shape profiles so that optimized configurations have clear structural boundaries. Several numerical examples of thermoelectric actuator design problems are presented to confirm the effectiveness and utility of the proposed method.

Appendix: Sensitivity analysis

In this Appendix, we provide the details of the sensitivity analysis. First, we note that the design sensitivity, $\bar {F}^{\prime }$, is obtained as

$$ \langle \bar{F}^{\prime},\delta \tilde{\chi}_{\phi} \rangle= \frac{\text{d}}{\text{d} \eta} \bar{F}(\tilde{\chi}_{\phi} + \eta \delta \tilde{\chi}_{\phi}) \mid_{\eta=0}. $$

(44)

We expand above equation as follows:

$$\begin{array}{@{}rcl@{}} \langle \bar{F}^{\prime},\delta \tilde{\chi}_{\phi} \rangle &=& {\int}_{D} (\epsilon (\delta {u}) : \mathbf{E} : \epsilon (\tilde{{u}}) \delta \tilde{\chi}_{\phi} - \alpha_{T} T \textbf{I} :\mathbf{E} : \epsilon(\tilde{{u}}) \delta \tilde{\chi}_{\phi} \\ &+& \kappa \nabla T \cdot \nabla \tilde{T} \delta \tilde{\chi}_{\phi} + \lambda \delta\tilde{\chi}_{\phi} )\text{d}{\Omega} \! +\! {\int}_{{\Gamma}_{t}} (k \delta {u} \cdot \tilde{{u}}- {t} \cdot \delta {u})\text{d} {\Omega}\\ & +& {\int}_{D} \epsilon (\delta {u}) : \mathbf{E} : \epsilon (\tilde{{u}})\tilde{\chi}_{\phi} \text{d} {\Omega} + {\int}_{D} (-\alpha_{T} \delta T \textbf{I} : \mathbf{E} : \epsilon (\tilde{{u}})\chi \\ &+& \kappa \nabla \delta T \cdot \nabla \tilde{T} \tilde{\chi}_{\phi} ) \text{d} {\Omega} + {\int}_{{\Omega}_{p}} \epsilon (\delta {u}) : \mathbf{E} : \epsilon (\tilde{{u}}) \text{d} {\Omega} \\ &+& {\int}_{{\Omega}_{p}} (-\alpha_{T} \delta T \textbf{I} : \mathbf{E} : \epsilon (\tilde{{u}}) + \kappa (T) \delta T \nabla T \cdot \nabla \tilde{T} \\ &-& \frac{\text{d} }{\text{d} T}(\alpha(T) \gamma(T) T) \nabla \tilde{T} \cdot \nabla V\\ &-& \frac{\text{d} }{\text{d} T}(\kappa(T) +\alpha(T)^{2}\gamma(T) T) \nabla \tilde{T} \cdot \nabla T \\ &-& (\kappa(T)+ \alpha(T)^{2}\gamma(T) T) \nabla \tilde{T} \cdot \nabla \delta T\\ &+& \frac{\text{d} \gamma (T)}{\text{d} T} \tilde{T} \nabla V \cdot \nabla V + \frac{\text{d}}{\text{d} T} (\alpha(T) \gamma(T) )\tilde{T} \nabla T \cdot \nabla V \\ &+& \alpha(T) \gamma(T) \tilde{T} \nabla \delta T \cdot \nabla V +\frac{\text{d} \gamma (T)}{\text{d} T} \nabla \tilde{V} \cdot \nabla V\\ &-&\frac{\text{d}}{\text{d} T}(\alpha(T) \gamma(T) )\nabla \tilde{V} \cdot \nabla T \! -\! \alpha(T) \gamma(T) \nabla \tilde{V} \cdot \nabla \delta T) \text{d} {\Omega}\\ & +& {\int}_{{\Omega}_{p}} (-\alpha(T)\gamma(T)T \nabla \tilde{T} \cdot \nabla \delta V + 2 \gamma (T) \tilde{T} \nabla \delta V \cdot \nabla V \\ &+&\alpha(T) \gamma(T) \tilde{T} \nabla T \cdot \nabla \delta V - \gamma(T) \nabla \tilde{V} \cdot \nabla \delta V) \text{d}{\Omega}. \end{array} $$

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Based on the adjoint variable method (Errico 1997), the adjoint equations are now defined as follows:

$$\begin{array}{@{}rcl@{}} {\int}_{D} \epsilon (\delta {u}) : \mathbf{E} : \epsilon (\tilde{{u}}) \tilde{\chi}_{\phi} \text{d}{\Omega} &+& {\int}_{{\Omega}_{p}} \epsilon (\delta {u}) : \mathbf{E} : \epsilon (\tilde{{u}}) \text{d}{\Omega} \\ & +& {\int}_{{\Gamma}_{t}} (k \delta {u} \cdot \tilde{{u}} - {t} \cdot \delta {u})\text{d}{\Gamma} = 0,\\ \end{array} $$

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$$\begin{array}{@{}rcl@{}} {\int}_{D} (&-& \alpha_{T} \delta T \textbf{I} : \mathbf{E} : \epsilon({u}) \tilde{\chi}_{\phi} + \kappa \nabla \delta T \cdot \nabla \tilde{T} \tilde{\chi}_{\phi}\text{d}{\Omega}) \text{d} {\Omega}\\ & +& {\int}_{{\Omega}_{p}} (-\alpha_{T} \delta T \textbf{I} : \mathbf{E} : \epsilon({u}) +\frac{\text{d} \kappa (T)}{\text{d} T} \nabla T \cdot \nabla \tilde{T} \\ &+&\kappa(T) \nabla \delta T \cdot \nabla \tilde{T} - \frac{\text{d} }{\text{d} T}(\alpha(T)\gamma(T)T ) \nabla\tilde{T} \cdot \nabla V \\ & -&\frac{\text{d} }{\text{d} T}(\kappa(T) + \alpha(T)^{2}\gamma(T)T ) \nabla \tilde{T} \cdot \nabla T\\ & -&(\kappa(T) + \alpha(T)^{2}\gamma(T)T ) \nabla \tilde{T} \cdot \nabla \delta T \\ & +& \frac{\text{d} \gamma (T)}{\text{d} T} \tilde{T} \nabla V \cdot \nabla V +\frac{\text{d}}{\text{d} T} (\alpha(T)\gamma(T) ) \tilde{T} \nabla T \cdot \nabla V \\ & +& \alpha(T)\gamma(T) \tilde{T} \nabla \delta T \cdot \nabla V - \frac{\text{d} \gamma (T)}{\text{d} T} \nabla \tilde{V} \cdot \nabla V \\ & -&\frac{\text{d} }{\text{d} T}(\alpha(T)\gamma(T) ) \nabla \tilde{V} \cdot \nabla T\\ & -& \alpha(T)\gamma(T) \nabla \tilde{V} \cdot \nabla \delta T ) \text{d} {\Omega}= 0, \end{array} $$

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$$\begin{array}{@{}rcl@{}} {\int}_{{\Omega}_{p}} (&-& \alpha(T)\gamma(T)T \nabla \tilde{T} \cdot \nabla \delta V + 2\gamma (T) \tilde{T} \nabla \delta V \cdot \nabla V \\ & +& \gamma (T) \tilde{T} \nabla V \cdot \nabla \delta V+ \alpha(T)\gamma(T)\tilde{T} \nabla T \cdot \nabla \delta V\\ & -&\gamma(T) \nabla \tilde{V} \cdot \nabla \delta V ) \text{d} {\Omega} = 0, \end{array} $$

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where $\tilde { {u}}$, $\tilde {T}$, and $\tilde {V}$ are the adjoint variables that respectively satisfy the above three equations. Consequently, the design sensitivity, $\bar {F}^{\prime }$, is obtained as

$$\begin{array}{@{}rcl@{}} \langle \bar{F}^{\prime}, \delta \tilde{\chi}_{\phi}\rangle &=& {\int}_{D} (\epsilon({u}):\mathbf{E} : \epsilon (\tilde{{u}}) - \alpha_{T} T \textbf{I} : \mathbf{E} : \epsilon(\tilde{{u}})\\ &&+ \kappa \nabla T \cdot \nabla \tilde{T}+ \lambda )\delta \tilde{\chi}_{\phi} \text{d} {\Omega} , \end{array} $$

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where

$$\begin{array}{@{}rcl@{}} \bar{F}^{\prime}& = &\epsilon({u}):\mathbf{E} : \epsilon (\tilde{{u}}) - \alpha_{T} T \textbf{I} : \mathbf{E} : \epsilon(\tilde{{u}})\\ &&+ \kappa \nabla T \cdot \nabla \tilde{T}+ \lambda. \end{array} $$

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