Multiphysics design of programmable shape-memory alloy-based smart structures via topology optimization

Abstract

We present a novel multiphysics and multimaterial computational design framework for shape-memory alloy-based smart structures. The proposed framework uses topology optimization to optimally distribute multiple material candidates within the design domain, and leverages a nonlinear phenomenological constitutive model for shape-memory alloys (SMAs), along with a coupled transient heat conduction model. In most practical scenarios, SMAs are activated by a nonuniform temperature field or a nonuniform stress field. This framework accurately captures the coupling between the phase transformation process and the evolution of the local temperature field. Thus, the resulting design framework is able to optimally tailor the two-way shape-memory effect and the superelasticity response of SMAs more precisely than previous algorithms that have relied on the assumption of a uniform temperature distribution. We present several case studies, including the design of a self-actuated bending beam and a gripper mechanism. The results show that the proposed framework can successfully produce SMA-based designs that exhibit targeted displacement trajectories and output forces. In addition, we present an example in which we enforce material-specific thermal constraints in a multimaterial design to enhance its thermal performance. In conclusion, the proposed framework provides a systematic computational approach to consider the nonlinear thermomechanical response of SMAs, thereby providing enhanced programmability of the SMA-based structure.

Appendices

Appendix A: Analytical derivation of adjoint vectors

In this appendix, we present the derivation of an analytical formulation for computing the tangent matrices described in Eq. 35. The motivation behind deriving these formulas is to avoid directly factorizing the ill-conditioned matrix ${\partial {\varvec{H}}_n}/{\partial \varvec{\nu }_n}$. For conciseness, we focus on the case where the behaviors of the SMA in the current time step and next step are inelastic. To start, we first represent the inverse of ${\partial {\varvec{H}}_{\mathfrak {G},n}}/{\partial \varvec{\nu }_{\mathfrak {G},n}}$ using the Schur complement (Zhang 2006).

$$\begin{aligned} \left( {\frac{\partial {\varvec{H}}_{n}}{\partial \varvec{\nu }_{\mathfrak {G},n}}}\right) ^{-1} = \left[ \begin{array}{cc} \mathbb {(A+BC)}^{-1} &{} \mathbb {(A+BC)}^{-1}\mathbb {B} \\ \mathbb {C}\mathbb {(A+BC)}^{-1} &{} -{\varvec{I}} + \mathbb {C(A+BC)}^{-1}\mathbb {B} \\ \end{array}\right] \end{aligned}$$

(A.1)

with

$$\begin{aligned} \frac{\partial {\varvec{H}}_{n}}{\partial \varvec{\nu }_{\mathfrak {G},n}} = \left[ \begin{array}{c|c} \begin{array}{ccc} \partial _{\xi } \Phi _{n}&{} {\varvec{0}}_{1\times 6} &{} 0_{1 \times 1} \\ \varvec{\Lambda }_{n} &{} -{\varvec{I}}_{6 \times 6} &{} {\varvec{0}}_{6 \times 1} \\ \Delta S &{} {\varvec{0}}_{1 \times 6} &{} -I_{1 \times 1} \end{array} &{} \begin{array}{c} \partial _{\varvec{\sigma }_{n}} \Phi _{n}^\mathrm{T} \\ \partial _{\varvec{\sigma }} \varvec{\Lambda }_{n}:\Delta {\xi }_n \\ {\varvec{0}}_{1 \times 6} \end{array} \\ \hline \begin{array}{ccc} {\varvec{0}}_{6 \times 1} &{} -\boxed {{\varvec{S}}_{n}^{-1}} &{} -\boxed {{S}_{n}^{-1}}:\varvec{\sigma }_{n} \end{array}&-{\varvec{I}}_{6 \times 6} \end{array}\right] = \begin{bmatrix} \mathbb {A} &{} \mathbb {B} \\ \mathbb {C} &{} \mathbb {D} \end{bmatrix} \end{aligned}$$

(A.2)

Note that the analytical solution in which we directly factorize ${\partial {\varvec{H}}_n}/{\partial \varvec{\nu }_n}$ is not recommended for calculating the sensitivities. This analytical solution will not change the ill-conditioned characteristics of the matrix, since $(\mathbb {A+BC})$ is nearly singular due to the fact that the large value $S^{-1}$ still appears on the lower triangle. Similarly, ${\partial {\varvec{H}}_{n+1}}/{\partial \varvec{\nu }_n}$ can be defined in block matrix form as

$$\begin{aligned} \frac{\partial {\varvec{H}}_{n+1}}{\partial \varvec{\nu }_{\mathfrak {G},n}} = \left[ \begin{array}{c|c} \begin{array}{ccc} 0_{1 \times 1} &{} {\varvec{0}}_{1 \times 6} &{} 0_{1 \times 1} \\ -\varvec{\Lambda }_{n+1} &{} {\varvec{I}}_{6 \times 6} &{} {\varvec{0}}_{6 \times 1} \\ -\Delta S &{} {\varvec{0}}_{1 \times 6} &{} I_{1 \times 1} \end{array} &{} \begin{array}{c} {\varvec{0}}_{1 \times 6} \\ {\varvec{0}}_{6 \times 6} \\ {\varvec{0}}_{1 \times 6} \end{array} \\ \hline \begin{array}{ccc} {\varvec{0}}_{6 \times 1} &{} {\varvec{0}}_{6 \times 6} &{} {\varvec{0}}_{6 \times 1} \end{array}&{\varvec{0}}_{6 \times 6} \end{array}\right] = \begin{bmatrix} \widetilde{\mathbb {A}} &{} \widetilde{\mathbb {B}} \\ \widetilde{\mathbb {C}} &{} \widetilde{\mathbb {D}} \end{bmatrix} \end{aligned}$$

(A.3)

The five aforementioned matrices then can be represented in a new form shown below.

$$\begin{aligned}&- \bigwedge \limits _\mathrm{el}\sum \limits _{\mathfrak {G}}\frac{\partial {\varvec{R}}_n}{\partial \varvec{\nu }_{\mathfrak {G},n}}\left( \frac{\partial {\varvec{H}}_n}{\partial \varvec{\nu }_{\mathfrak {G},n}}\right) ^{-1}\frac{\partial {\varvec{H}}_{\mathfrak {G},n}}{\partial {\varvec{u}}_{\mathrm{el},n}} \nonumber \\&\quad = \bigwedge \limits _\mathrm{el}\sum \limits _{\mathfrak {G}}w{\varvec{B}}_{\mathfrak {G}}^\mathrm{T}[{\varvec{I}} - \mathbb {C(A+BC)}^{-1}\mathbb {B}]_{\mathfrak {G}}{\varvec{S}}_{\mathfrak {G},n}^{-1}{\varvec{B}} _{\mathfrak {G}}\mathrm{det}{\varvec{J}}_{\mathfrak {G}} \nonumber \\&\quad \frac{\partial {\varvec{R}}_{\mathrm{el},n}}{\partial \varvec{\nu }_{\mathfrak {G},n}}\left( \frac{\partial {\varvec{H}}_{\mathfrak {G},n}}{\partial \varvec{\nu }_{\mathfrak {G},n}}\right) ^{-1} \nonumber \\&\quad = w{\varvec{B}}^\mathrm{T}\left[ \begin{array}{cc} \mathbb {C(A+BC)}^{-1}&-{\varvec{I}} + \mathbb {C(A+BC)}^{-1}\mathbb {B} \end{array}\right] \mathrm{det}{\varvec{J}} \nonumber \\&\quad \left( \frac{\partial {\varvec{H}}_{\mathfrak {G},n}}{\partial \varvec{\nu }_{\mathfrak {G},n}}\right) ^{-1}\frac{\partial {\varvec{H}}_{\mathfrak {G},n}}{\partial {\varvec{u}}_{\mathrm{el},n}} = \left[ \begin{array}{c} \mathbb {(A+BC)}^{-1}\mathbb {B} \\ -I + \mathbb {C(A+BC)}^{-1}\mathbb {B} \end{array}\right] {\varvec{B}}\\&\quad \frac{\partial {\varvec{H}}_{\mathfrak {G},n+1}}{\partial \varvec{\nu }_{\mathfrak {G},n}}\left( \frac{\partial {\varvec{H}}_{\mathfrak {G},n}}{\partial \varvec{\nu }_{\mathfrak {G},n}}\right) ^{-1} \nonumber \\&\quad = \left[ \begin{array}{cc} \widetilde{\mathbb {A}}\mathbb {(A+BC)}^{-1} &{} \widetilde{\mathbb {A}}\mathbb {(A+BC)}^{-1}\mathbb {B}\\ (\widetilde{\mathbb {C}}+\mathbb {C})\mathbb {(A+BC)}^{-1} &{} -{\varvec{I}} + (\widetilde{\mathbb {C}}+\mathbb {C})\mathbb {(A+BC)}^{-1}\mathbb {B} \end{array}\right] \nonumber \\&\quad \frac{\partial {\varvec{H}}_{\mathfrak {G},n+1}}{\partial \varvec{\nu }_{\mathfrak {G},n}}\left( \frac{\partial {\varvec{H}}_{\mathfrak {G},n}}{\partial \varvec{\nu }_{\mathfrak {G},{\mathfrak {G},n}}}\right) ^{-1}\frac{\partial {\varvec{H}}_n}{\partial {\varvec{u}}_{\mathrm{el},n}} \nonumber \\&\quad = \left[ \begin{array}{c} \widetilde{\mathbb {A}}\mathbb {(A+BC)}^{-1}\mathbb {B} -{\varvec{I}} \end{array}\right] {\varvec{S}}_{n}^{-1}{\varvec{B}} \nonumber \end{aligned}$$

(A.4)

It can be observed that the key to obtaining an accurate solution of the tangent matrices lies in solving the inverse of $\mathbb {(A+BC)}$. Here we use the Schur formulation again to calculate the inverse of the matrix.

$$\begin{aligned} (\mathbb {A+BC})^{-1} = \begin{bmatrix} \mathbb {(A'-B'}\mathbb {D'}^{-1}\mathbb {C')}^{-1} &{} -\mathbb {(A'-B'}\mathbb {D'}^{-1}\mathbb {C')}^{-1}\mathbb {B'}\mathbb {D'}^{-1} \\ -\mathbb {D'}^{-1}\mathbb {C'}\mathbb {(A'-B'}\mathbb {D'}^{-1}\mathbb {C')}^{-1} &{} \mathbb {D'}^{-1} + \mathbb {D'}^{-1}\mathbb {C'(A'-B'}\mathbb {D'}^{-1}\mathbb {C')}^{-1}\mathbb {B'}\mathbb {D'}^{-1} \\ \end{bmatrix} \end{aligned}$$

(A.5)

with

$$\begin{aligned} \mathbb {A+BC} = \left[ \begin{array}{c|c} \partial _{\xi } \Phi _{n} &{} \begin{array}{cc} -\partial _{\varvec{\sigma }} \Phi _{n}:{{\varvec{S}}}_{n}^{-1} &{} -\partial _{\varvec{\sigma }} \Phi _{n}:{S}_{n}^{-1}:\varvec{\sigma }_{n} \\ \end{array} \\ \hline \begin{array}{c} \varvec{\Lambda }_{n} \\ \Delta S \\ \end{array} &{} \begin{array}{cc} -({{\varvec{S}}}_{n}+\partial _{\varvec{\sigma }} \varvec{\Lambda }_{n}:\Delta {\xi }_n):{{\varvec{S}}}_{n}^{-1}&{} -\partial _{\varvec{\sigma }} \varvec{\Lambda }_{n}:\Delta {\xi }_n:{S}_{n}^{-1}:\varvec{\sigma }_{n} \\ {\varvec{0}}_{1\times 6} &{} -I_{1 \times 1} \end{array} \end{array}\right] = \begin{bmatrix} \mathbb {A'} &{} \mathbb {B'} \\ \mathbb {C'} &{} \mathbb {D'} \end{bmatrix} \end{aligned}$$

(A.6)

Here, $\mathbb {D'}^{-1}$ and $\mathbb {(A'-B'}\mathbb {D'}^{-1}\mathbb {C')}^{-1}$ need to be solved. Note that it can be easily proven that $\partial _{\varvec{\sigma }} \varvec{\Lambda }$ belongs to the null space of $\varvec{\sigma }$, i.e., $\partial _{\varvec{\sigma }} \varvec{\Lambda }$:$\varvec{\sigma }$ = 0, hence $\partial _{\varvec{\sigma }} \varvec{\Lambda }$ is not invertible. However, $({{\varvec{S}}}_{n}+\partial _{\varvec{\sigma }} \varvec{\Lambda }_{n}:\Delta {\xi }_n)$ is nonsingular. Defining $\varvec{\zeta } = {{\varvec{S}}}_{n}+\partial _{\varvec{\sigma }} \varvec{\Lambda }_{n}:\Delta {\xi }_n$, $\mathbb {D'}^{-1}$ can be solved using the Schur form as

$$\begin{aligned} \mathbb {D'}^{-1} = \left[ \begin{array}{cc} -{{\varvec{S}}}_{n}:{\varvec{\zeta }}_{n}^{-1} &{} {{\varvec{S}}}_{n}:{\varvec{\zeta }}_{n}^{-1}:\partial _{\varvec{\sigma }} \varvec{\Lambda }_{n}:\Delta {\xi }_n:{S}_{n}^{-1}:\varvec{\sigma }_{n} \\ {\varvec{0}}_{1\times 6} &{} -I_{1 \times 1} \end{array}\right] \end{aligned}$$

(A.7)

With the above information, one can obtain that

$$\begin{aligned} \begin{aligned}&\mathbb {B'D'}^{-1} = \left[ \begin{array}{cc} \partial _{\varvec{\sigma }} \Phi _{n}:{\varvec{\zeta }}_{n}^{-1} &{} \partial _{\varvec{\sigma }} \Phi _{n}:{\varvec{\zeta }}_{n}^{-1}:{{\varvec{S}}}_{n}^{-1}:{S}_{n}^{-1}:\varvec{\sigma }_{n} \\ \end{array} \right] \\&\mathbb {D'}^{-1}\mathbb {C'} = \left[ \begin{array}{c} -{{\varvec{S}}}_{n}:{\varvec{\zeta }}_{n}^{-1}:(\varvec{\Lambda }_n-\partial _{\varvec{\sigma }} \varvec{\Lambda }_{n}:\Delta {\xi }_n:{S}_{n}^{-1}:\varvec{\sigma }_{n}:\Delta {S}) \\ -\Delta {S} \end{array} \right] = \left[ \begin{array}{c} -\widetilde{\varvec{\Lambda }}_n\\ -\Delta {S} \end{array} \right] \end{aligned} \end{aligned}$$

(A.8)

Then $\mathbb {(A'-B'}\mathbb {D'}^{-1}\mathbb {C')}^{-1}$ is a scalar and can be calculated as $Q=-\partial _{\varvec{\sigma }} \Phi _{n}:{\varvec{\zeta }}_{n}^{-1}:\partial _{\varvec{\sigma }} \Phi _{n}+\partial _{\xi } \Phi _{n}$. Hence the issue of the ill-conditioned matrix has been solved, and we are able to accurately evaluate the five matrices as follows.

$$\begin{aligned}&- \bigwedge \limits _\mathrm{el}\sum \limits _{\mathfrak {G}}\frac{\partial {\varvec{R}}_n}{\partial \varvec{\nu }_{\mathfrak {G},n}}\left( \frac{\partial {\varvec{H}}_{\mathfrak {G},n}}{\partial \varvec{\nu }_{\mathfrak {G},n}}\right) ^{-1}\frac{\partial {\varvec{H}}_{\mathfrak {G},n}}{\partial {\varvec{u}}_{\mathrm{el},n}} \nonumber \\&\quad = \bigwedge \limits _\mathrm{el}\sum \limits _{\mathfrak {G}}w{\varvec{B}}_{\mathfrak {G}}^\mathrm{T}({\varvec{\zeta }}_{\mathfrak {G},n}^{-1} - \frac{{\varvec{\zeta }}_{\mathfrak {G},n}^{-1}:\partial _{\varvec{\sigma }} \Phi _{\mathfrak {G},n}\otimes {\varvec{\zeta }}_{\mathfrak {G},n}^{-1}:\partial _{\varvec{\sigma }} \Phi _{\mathfrak {G},n}}{\partial _{\varvec{\sigma }} \Phi _{\mathfrak {G},n}:{\varvec{\zeta }}_{\mathfrak {G},n}^{-1}:\partial _{\varvec{\sigma }} \Phi _{\mathfrak {G},n}-\partial _{\xi } \Phi _{\mathfrak {G},n}}){\varvec{B}}_{\mathfrak {G}}\mathrm{det}{\varvec{J}}_{\mathfrak {G}} \nonumber \\&\quad = \bigwedge \limits _\mathrm{el}\sum \limits _{\mathfrak {G}} w{\varvec{B}}_\mathrm{G}^\mathrm{T}\varvec{\mathfrak {L}}_{\mathfrak {G},n}{\varvec{B}}_{\mathfrak {G}}\mathrm{det}{\varvec{J}}_{\mathfrak {G}} \nonumber \\&\quad \frac{\partial {\varvec{R}}_{\mathrm{el},n}}{\partial \varvec{\nu }_{\mathfrak {G},n}}\left( \frac{\partial {\varvec{H}}_{\mathfrak {G},n}}{\partial \varvec{\nu }_{\mathfrak {G},n}}\right) ^{-1} \nonumber \\&\quad = w{\varvec{B}}^\mathrm{T}\left[ \begin{array}{cccc} -\frac{{\varvec{\zeta }}_{n}^{-1}:{\partial _{\varvec{\sigma }} \Phi _n}}{Q}&\varvec{\mathfrak {L}}_{n}&\varvec{\mathfrak {L}}_{n}:\varvec{\mathfrak {C}}^{-1}:\varvec{\sigma }_{n}&-\varvec{\mathfrak {L}}_{n}:{{\varvec{S}}}_{n} \end{array}\right] \mathrm{det}{\varvec{J}} \nonumber \\&\quad \left( \frac{\partial {\varvec{H}}_{\mathfrak {G},n}}{\partial \varvec{\nu }_{\mathfrak {G},n}}\right) ^{-1}\frac{\partial {\varvec{H}}_{\mathfrak {G},n}}{\partial {\varvec{u}}_{\mathrm{el},n}} \nonumber \\&\quad = \left[ \begin{array}{c} \frac{{\partial _{\sigma } \Phi _n}^\mathrm{T}:{\varvec{\zeta }}_{n}^{-1}}{Q}\\ \frac{\widetilde{\varvec{\Lambda }}_{n}:{\partial _{\varvec{\sigma }} \Phi _n}^\mathrm{T}}{Q}:{\varvec{\zeta }}_{n}^{-1}+{{\varvec{S}}}_{n}:{\varvec{\zeta }}_{n}^{-1}-{\varvec{I}}_{6\times 6}\\ \frac{\Delta S:{\partial _{\varvec{\sigma }} \Phi _n}^\mathrm{T}}{Q}:{\varvec{\zeta }}_{n}^{-1}\\ -\varvec{\mathfrak {L}}_{n} \end{array}\right] {\varvec{B}} \nonumber \\&\quad \frac{\partial {\varvec{H}}_{\mathfrak {G},n+1}}{\partial \varvec{\nu }_{\mathfrak {G},n}}\left( \frac{\partial {\varvec{H}}_{\mathfrak {G},n}}{\partial \varvec{\nu }_{\mathfrak {G},n}}\right) ^{-1}&= \left[ \begin{array}{cc} \mathfrak {A}_{8 \times 8} &{} \mathfrak {B}_{8 \times 6}\\ {\varvec{0}}_{6 \times 8} &{} {\varvec{0}}_{6 \times 6} \end{array}\right] \nonumber \\&\quad \frac{\partial {\varvec{H}}_{\mathfrak {G},n+1}}{\partial \varvec{\nu }_{\mathfrak {G},n}}\left( \frac{\partial {\varvec{H}}_{\mathfrak {G},n}}{\partial \varvec{\nu }_{\mathfrak {G},n}}\right) ^{-1}\frac{\partial {\varvec{H}}_{\mathfrak {G},n}}{\partial {\varvec{u}}_{\mathrm{el},n}} \nonumber \\&\quad = \left[ \begin{array}{c} {\varvec{0}}_{1 \times 6}\\ \frac{\widetilde{\varvec{\Lambda }}_{n}-\varvec{\Lambda }_{n+1}}{Q}:{\partial _{\varvec{\sigma }} \Phi _n}^\mathrm{T}:{\varvec{\zeta }}_{n}^{-1}+{{\varvec{S}}}_{n}:{\varvec{\zeta }}_{n}^{-1}-{\varvec{I}}_{6\times 6}\\ {\varvec{0}}_{7 \times 6}\\ \end{array}\right] {\varvec{B}} \end{aligned}$$

(A.9)

where

$$\begin{aligned}&\mathfrak {A} = \left[ \begin{array}{cc} 0_{1 \times 1} &{} {\varvec{0}}_{1 \times 7} \\ \begin{array}{c} \frac{\widetilde{\varvec{\Lambda }}_n - \varvec{\Lambda }_{n+1}}{Q} \\ 0_{1 \times 1} \end{array}&\mathfrak {a}_{7\times 7} \end{array}\right] \nonumber \\&\mathfrak {a} =\left[ \begin{array}{cc} (-{{\varvec{S}}}_{n}-\frac{\widetilde{\varvec{\Lambda }}_n - \varvec{\Lambda }_{n+1}}{Q}:{\partial _{\varvec{\sigma }} \Phi _n}^\mathrm{T}):{\varvec{\zeta }}_{n}^{-1} &{} (-\frac{\widetilde{\varvec{\Lambda }}_n - \varvec{\Lambda }_{n+1}}{Q}:{\partial _{\varvec{\sigma }} \Phi _n}^\mathrm{T}:{\varvec{\zeta }}_{n}^{-1}:{{\varvec{S}}}_{n}+{{\varvec{S}}}_{n}:{\varvec{\zeta }}_{n}^{-1}:\partial _{\varvec{\sigma }} \varvec{\Lambda }_{n}:\Delta {\xi }_n):{S_{n}}^{-1}:\varvec{\sigma }_{n} \\ {\varvec{0}}_{1 \times 6} &{} -I_{1 \times 1} \end{array}\right] \nonumber \\&\mathfrak {B} = \left[ \begin{array}{c} {\varvec{0}}_{1 \times 6} \\ \frac{\widetilde{\varvec{\Lambda }}_n - \varvec{\Lambda }_{n+1}}{Q}:{\partial _{\varvec{\sigma }} \Phi _n}^\mathrm{T}:{\varvec{\zeta }}_{n}^{-1}+{{\varvec{S}}}_{n}:{\varvec{\zeta }}_{n}^{-1}-{\varvec{I}}_{6\times 6}\\ {\varvec{0}}_{1 \times 6} \\ \end{array}\right] :{{\varvec{S}}}_{n} \\&\widetilde{\varvec{\Lambda }}_{n} = {{\varvec{S}}}_{n}:{{\varvec{\zeta }}_{n}^{-1}}:({\varvec{\Lambda }}_{n} - {\partial _{\varvec{\sigma }}{\varvec{\Lambda }}_{n}}:\Delta \xi _{n}:{S_{n}^{-1}}:{\varvec{\sigma }}_{n}:\Delta S) \nonumber \end{aligned}$$

(A.10)

When $\partial _{\varvec{\sigma }}\varvec{\Lambda }=0$ and $\varvec{\zeta } = {\varvec{S}}$, the above analytical solutions degenerate as follows

$$\begin{aligned}&- \bigwedge \limits _\mathrm{el}\sum \limits _{\mathfrak {G}}\frac{\partial {\varvec{R}}_n}{\partial \varvec{\nu }_{\mathfrak {G},n}}\left( \frac{\partial {\varvec{H}}_{\mathfrak {G},n}}{\partial \varvec{\nu }_{\mathfrak {G},n}}\right) ^{-1}\frac{\partial {\varvec{H}}_{\mathfrak {G},n}}{\partial {\varvec{u}}_{\mathrm{el},n}} \nonumber \\&\quad = \bigwedge \limits _\mathrm{el}\sum \limits _{\mathfrak {G}}w{\varvec{B}}_{\mathfrak {G}}^\mathrm{T}({{\varvec{S}}}_{\mathfrak {G},n}^{-1} - \frac{{{\varvec{S}}}_{\mathfrak {G},n}^{-1}:\partial _{\varvec{\sigma }} \Phi _{\mathfrak {G},n}\otimes {{\varvec{S}}}_{\mathfrak {G},n}^{-1}:\partial _{\varvec{\sigma }} \Phi _{\mathfrak {G},n}}{\partial _{\varvec{\sigma }} \Phi _{\mathfrak {G},n}:{{\varvec{S}}}_{\mathfrak {G},n}^{-1}:\partial _{\varvec{\sigma }} \Phi _{\mathfrak {G},n}-\partial _{\xi } \Phi _{\mathfrak {G},n}}){\varvec{B}}_{\mathfrak {G}}\mathrm{det}{\varvec{J}}_{\mathfrak {G}} \nonumber \\&\quad = \bigwedge \limits _\mathrm{el}\sum \limits _{\mathfrak {G}} w{\varvec{B}}_{\mathfrak {G}}^\mathrm{T}\varvec{\mathfrak {L}}_{\mathfrak {G},n}{\varvec{B}}_{\mathfrak {G}}\mathrm{det}{\varvec{J}}_{\mathfrak {G}} \nonumber \\&\quad \frac{\partial {\varvec{R}}_{\mathrm{el},n}}{\partial \varvec{\nu }_{\mathfrak {G},n}}\left( \frac{\partial {\varvec{H}}_{\mathfrak {G},n}}{\partial \varvec{\nu }_{\mathfrak {G},n}}\right) ^{-1} = w{\varvec{B}}^\mathrm{T}\left[ \begin{array}{cccc} \frac{{{{\varvec{S}}}_{n}^{-1}:\partial _{\sigma } \Phi _n}}{Q}&\varvec{\mathfrak {L}}_{n}&\varvec{\mathfrak {L}}_{n}:\varvec{\mathfrak {C}}^{-1}:\varvec{\sigma }_{n} -\varvec{\mathfrak {L}}_{n}:{{\varvec{S}}}_{n} \end{array}\right] \mathrm{det}{\varvec{J}} \nonumber \\&\quad \left( \frac{\partial {\varvec{H}}_{\mathfrak {G},n}}{\partial \varvec{\nu }_{\mathfrak {G},n}}\right) ^{-1}\frac{\partial {\varvec{H}}_{\mathfrak {G},n}}{\partial {\varvec{u}}_{\mathrm{el},n}} = \left[ \begin{array}{c} \frac{{\partial _{\varvec{\sigma }} \Phi _n}^\mathrm{T}:{{\varvec{S}}}_{n}^{-1}}{Q}\\ \frac{\left[ \begin{array}{c} \varvec{\Lambda }_{n} \\ \Delta S \end{array}\right] :{\partial _{\varvec{\sigma }} \Phi _n}^\mathrm{T}:{{\varvec{S}}}_{n}^{-1}}{Q}\\ -\varvec{\mathfrak {L}}_{n} \end{array}\right] {\varvec{B}}\nonumber \\&\quad \frac{\partial {\varvec{H}}_{\mathfrak {G},n+1}}{\partial \varvec{\nu }_{\mathfrak {G},n}}\left( \frac{\partial {\varvec{H}}_{\mathfrak {G},n}}{\partial \varvec{\nu }_{\mathfrak {G},n}}\right) ^{-1} = \left[ \begin{array}{cc} \mathfrak {A}_{8 \times 8} &{} \mathfrak {B}_{8 \times 6}\\ {\varvec{0}}_{6 \times 8} &{} {\varvec{0}}_{6 \times 6} \end{array}\right] \nonumber \\&\quad \frac{\partial {\varvec{H}}_{\mathfrak {G},n+1}}{\partial \varvec{\nu }_{\mathfrak {G},n}}\left( \frac{\partial {\varvec{H}}_{\mathfrak {G},n}}{\partial \varvec{\nu }_{\mathfrak {G},n}}\right) ^{-1}\frac{\partial {\varvec{H}}_{\mathfrak {G},n}}{\partial {\varvec{u}}_{\mathrm{el},n}} = \left[ \begin{array}{c} {\varvec{0}}_{1 \times 6}\\ \left( \frac{\varvec{\Lambda }_{n}-\varvec{\Lambda }_{n+1}}{Q}:{\partial _{\varvec{\sigma }} \Phi _n}^\mathrm{T}:{{\varvec{S}}}_{n}^{-1}\right) \\ {{\varvec{0}}}_{7 \times 6} \end{array}\right] {\varvec{B}} \end{aligned}$$

(A.11)

where

$$\begin{aligned} \begin{aligned} \mathfrak {A}&= \left[ \begin{array}{cc} 0_{1 \times 1} &{} {\varvec{0}}_{1 \times 7} \\ \begin{array}{c} \frac{\varvec{\Lambda }_n - \varvec{\Lambda }_{n+1}}{Q} \\ 0_{1 \times 1} \end{array} &{} -{\varvec{I}}_{7 \times 7}- \frac{\varvec{\Lambda }_n - \varvec{\Lambda }_{n+1}}{Q}:{\partial _{\varvec{\sigma }} \Phi _n}^\mathrm{T}: \left[ \begin{array}{cc} {{\varvec{S}}}_{n}^{-1} &{} S_{n}^{-1}:\varvec{\sigma }_{n} \\ {\varvec{0}}_{1 \times 6} &{} 0_{1 \times 1} \end{array}\right] \end{array}\right] \\ \mathfrak {B}&= \left[ \begin{array}{c} {\varvec{0}}_{1 \times 6} \\ \frac{\varvec{\Lambda }_n - \varvec{\Lambda }_{n+1}}{Q}:{\partial _{\varvec{\sigma }} \Phi _n}^\mathrm{T}\\ {\varvec{0}}_{1 \times 6} \\ \end{array}\right] \\ \end{aligned} \end{aligned}$$

(A.12)

Note that the first equation in A.9 and A.11 can be recognized as the formulas for calculating the tangent stiffness matrix, shown in our work (Kang et al. 2022). Therefore, in addition to improving the accuracy of the sensitivity evaluation, we are able to conserve computational resources by re-using the tangent stiffness matrix already computed during the finite element analysis.

Appendix B: Comparison between analytical and finite difference sensitivity results

This section validates the adjoint sensitivity formulation by comparing the analytical results with finite differences. Figures 23 and 24 show the geometry and boundary conditions of the test case used to calculate the sensitivities in Fig. 6. Note that in the following tables from Tables 4, 5, 6, 7, 8 and 9, t refers to the time step; T and $\xi$ refer to the temperature and martensite volume fraction evaluated at the node and Gauss point that are closest to point A, which is associated with the function of interest $f_{int}$. ANA and FD refer to the analytical and finite difference sensitivity values, respectively.

1.1 Appendix B.1: Simulating two-way shape-memory effects

In the first case study we evaluate the sensitivity of the TWSME of a bi-material beam containing SMA and BC, whose properties are shown in Tables 2 and 3. The material volume fraction r (cf. Eq. 23) is uniformly 0.9 across the structure. The boundary conditions are shown below,

Fig. 23

Boundary conditions for topology optimization of an active bending beam

where a constant uniform surface traction ${\varvec{f}}= 9\times 10^7\mathrm{N/m}$ is applied when cooling the beam. The initial and reference temperature of the beam is set to be 250K. In addition, temperature ${\bar{T}}$ (cf. Fig. 23b) is fixed at 250 K during the cooling process. A heat flux increment $\mathrm{d}q$ = $1 \times 10^2$ w/m per time step, and a heat sink increment $\mathrm{d}Q$ = $-5 \times 10^3$ w/m$^2$ per time step are applied to cool the beam. The function of interest $f_{int}$ for sensitivity analysis (cf. Eq. 31) is the vertical displacement at point A of the structure, i.e., $f_{int} = d_{y}^{A}$. We calculate the sensitivity of the displacement function under different mesh sizes and at different transformation stages. The numerical results are shown in Tables 4, 5, 6 and 6.

Table 4 Sensitivity analysis when simulating TWSME, mesh size = 4

Table 5 Sensitivity analysis when simulating TWSME, mesh size = 40

Table 6 Sensitivity analysis when simulating TWSME, mesh size = 250

1.2 Appendix B.2: Simulating both two-way shape-memory effects and superelasticity

The second case study looks at sensitivity analysis of a single-material SMA structure undergoing both TWSME and superelasticity. The material volume fraction r (cf. Eq. 23) is uniformly 0.5 across the structure. The boundary conditions are shown in Fig. 24. An input force increment $\mathrm{d}{\varvec{F}}_{in} = 1\times 10^6$N per time step is applied to the structure to trigger superelasticity. Meanwhile, the structure is cooled with a heat flux increment $\mathrm{d}q$ = $1 \times 10^2$ w/m per time step, and a heat sink increment $\mathrm{d}Q$ = $-5 \times 10^3$ w/m$^2$ per time step. The initial and reference temperature, as well as the fixed temperature ${\bar{T}}$ for the thermal conduction problem (cf. Fig. 24b) are set to be 230 K. The function of interest $f_{int}$ for sensitivity analysis (cf. Eq. 31) is the mechanical advantage at point A, i.e., $f_{int} = -{\varvec{F}}_{out}/{\varvec{F}}_{in}$. We calculate the sensitivity of the structure, under different mesh sizes and at different transformation stages. The numerical results are shown in Tables 7 to 9

Fig. 24

Boundary conditions for topology optimization of a force inverter

Table 7 Sensitivity analysis when simulating both TWSME and superelasticity, mesh size = 8

Table 8 Sensitivity analysis when simulating both TWSME and superelasticity, mesh size = 100

Table 9 Sensitivity analysis when simulating both TWSME and superelasticity, mesh size = 400