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Fiber bundle topology optimization of hierarchical microtextures for wetting behavior in Cassie-Baxter mode

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Abstract

This paper presents the topology optimization of hierarchical microtextures for wetting behavior in the Cassie-Baxter mode, considering a structural unit of the hierarchical microtexture composed of base and secondary structures. The geometrical configuration of the considered structural unit can be described as a fiber bundle composed of an external surface of the base structure and the pattern of the secondary structures. Thus, two design variables are defined, one for the external surface of the base structure, and the other for the pattern of the secondary structures. The Young-Laplace equation, including a term depending on the mean curvature of the external surface, is used to describe the liquid/vapor interface imposed with a surface tension in the Cassie-Baxter mode. To overcome the difficulty of numerically computing the second-order derivative of the external surface, two partial differential equation filters are sequentially applied to the design variable of the base structure to ensure the numerical accuracy and feasibility of using an efficient linear-element-based finite element method to solve the Young-Laplace equation. To improve the performance of the hierarchical microtextures, the volume of the liquid bulges suspended at the liquid/vapor interface in the Cassie-Baxter mode, before the transition into the Wenzel mode, is minimized to optimize the match between the external surface of the base structure and the pattern of the secondary structures. In the topology optimization process, penalization of the material density of the surface tension is achieved by an artificial Marangoni phenomenon. In numerical examples, solid surfaces are tiled into textures with axial symmetry, radial symmetry, chirality, and quasiperiodicity; and structural units are derived consisting of base structures with peak shapes and dense secondary structures surrounding the crests of the peaks. The optimized performance of the derived structural units has been confirmed by comparisons.

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Acknowledgments

The authors are grateful to Prof. O. Sigmund for inviting to submit this work to the Special Issue of Structural and Multidisciplinary Optimization, to the reviewers for their kind attention and very valuable suggestions, and to the audience at WCSMO13 for constructive comments on this work presented there. The authors are also grateful to K. Svanberg of KTH for supplying the MMA codes.

Funding

Y. Deng acknowledges a Humboldt Research Fellowship for Experienced Researchers (Humboldt-ID: 1197305), the support from the National Natural Science Foundation of China (No. 51875545), the Youth Innovation Promotion Association of the Chinese Academy of Sciences (No. 2018253); Z. Liu acknowledges the support from the National Natural Science Foundation of China (No. 51675506); W. Zhang and J. Zhu acknowledge the support from the National Key Research and Development Program (2017YFB1102800) and the National Natural Science Foundation of China (11722219, 11620101002); J.G. Korvink acknowledges the support from an EU2020 FET grant (TiSuMR, 737043), the DFG under grant KO 1883/20-1 Metacoils, in the framework of the German Excellence Intitiative under grant EXC 2082 “3D Matter Made to Order,” and the VirtMat initiative “Virtual Materials Design.”

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Correspondence to Yongbo Deng, Weihong Zhang, Zhenyu Liu, Jihong Zhu or Jan G. Korvink.

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Replication of results

Details on the numerical implementation for the replication of the results have been provided in Section 2.5, with the pseudocode in Table 1 and the optimization parameters in Table 2. The size of the meshes used to discretize the dimensionless design domains in Section 3 has been illustrated in Fig. 7. If the information provided in the paper is not sufficient, we sincerely welcome scientists or interested parties to contact us for further explanations.

Appendix

Appendix

In Appendixes AA, the variational formulations of the related PDEs and the adjoint equations are detailed, based on the Kurash-Kuhn-Tucker (KKT) condition of the PDE constrained optimization problem and Lagrangian multiplier-based adjoint method (Hinze et al. 2009). In Appendix A, Pareto fronts for the duty ratio of the fiber bundles for the hierarchical microtextures are provided.

1.1 Appendix 1: Variational formulations of PDEs

Based on the Galerkin variational principle, the variational formulations of the related PDEs can be derived as

  • Variational formulation of the PDE filter in (14):find \( z_{f} \in {\mathscr{H}}\left ({\Omega }\right )\) with zf = 0 on Ω, satisfying

    $$ \begin{array}{@{}rcl@{}} &&{\int}_{\Omega} {r_{m}^{2}} \nabla z_{f} \cdot \nabla \tilde{z}_{f} + z_{f} \tilde{z}_{f} - z_{m} \tilde{z}_{f} \mathrm{d}{\Omega} = 0,\\ && \forall \tilde{z}_{f}\in\mathcal{H}\left( {\Omega}\right); \end{array} $$
    (51)
  • Variational formulation of the PDE filter in (15):find \( z_{s} \in {\mathscr{H}}\left ({\Omega }\right )\) with zs = 0 on Ω, satisfying

    $$ \begin{array}{@{}rcl@{}} &&{\int}_{\Omega} {r_{m}^{2}} \nabla z_{s} \cdot \nabla \tilde{z}_{s} + z_{s} \tilde{z}_{s} - b_{z} z_{f} \tilde{z}_{s} \mathrm{d}{\Omega} = 0,\\ &&\forall \tilde{z}_{s} \in \mathcal{H}\left( {\Omega}\right); \end{array} $$
    (52)
  • Variational formulation of the surface-PDE filter in (17):find \(\gamma _{f} \in {\mathscr{H}}\left ({\Sigma }\right )\) with γf = 0 at Σ, satisfying

    $$ \begin{array}{@{}rcl@{}} &&{\int}_{\Sigma} {r_{f}^{2}} \nabla_{s} \gamma_{f} \cdot \nabla_{s} \tilde{\gamma}_{f} + \gamma_{f} \tilde{\gamma}_{f} - \gamma \tilde{\gamma}_{f} \mathrm{d}{\Sigma} = 0,\\ &&\forall \tilde{\gamma}_{f}\in\mathcal{H}\left( {\Sigma}\right); \end{array} $$
    (53)
  • Variational formulation of the dimensionless Young-Laplace (24):find \(\bar {d} \in {\mathscr{H}}\left ({\Sigma }\right )\) with \(\bar {d} =0\) at Σ and \(\kappa \in {\mathscr{H}}^{-1}\left ({\Omega }\right )\), satisfying

    $$ \begin{array}{@{}rcl@{}} && {\int}_{\Sigma} - \bar{\sigma} { \nabla_{s} \bar{d} \cdot \nabla_{s} \tilde{\bar{d}} \over \sqrt { 1/{d_{0}^{2}} + \left| \nabla_{s} \bar{d} \right|^{2}}} - \left( 1 - p_{\kappa} \kappa\right) \tilde{\bar{d}} \mathrm{d}{\Sigma}\\ &&+ {\int}_{\Omega} \left[ \kappa - {\bar{\sigma}_{l} \over {r_{m}^{2}}} { z_{s} - b_{z} z_{f} \over \left( \sqrt{ 1 + \left| \nabla z_{s} \right|^{2} } \right)^{3} } \right] \tilde{\kappa} \mathrm{d}{\Omega} = 0,\\ && \forall \tilde{\bar{d}}\in\mathcal{H}\left( {\Sigma}\right)~\text{and}~\forall \tilde{\kappa} \in \overset{\circ}{\mathcal{H}}\left( {\Omega}\right) \end{array} $$
    (54)

where \(\tilde {z}_{f}\), \(\tilde {z}_{s}\), \(\tilde {\gamma }_{f}\), \(\tilde {\bar {d}}\), and \(\tilde {\kappa }\) are the test functions of zf, zs, γf, \(\bar {d}\), and κ, respectively; \({\mathscr{H}}\left ({\Sigma }\right )\) and \({\mathscr{H}}\left ({\Omega }\right )\) are the first order Sobolev spaces on Σ and Ω, respectively; \(\overset {\circ }{{\mathscr{H}}}\left ({\Omega }\right )\) is the closure of \(C_{0}^{\infty }\left ({\Omega }\right )\); \({\mathscr{H}}^{-1}\left ({\Omega }\right )\) is the dual space of \(\overset {\circ }{{\mathscr{H}}}\left ({\Omega }\right )\).

1.2 Appendix 2: Adjoint analysis for \(J \left |{\Sigma }\right |^{2}\) in design objective

Based on the variational formulations of the PDEs in (5154) and Lagrangian multiplier-based adjoint method, the augmented Lagrangian for \(J \left |{\Sigma }\right |^{2} = {\int \limits }_{\Sigma } \bar {d}^{2} \mathrm {d}{\Sigma }\) can be formulated as

$$ \begin{array}{@{}rcl@{}} \widehat{J \left|{\Sigma}\right|^{2}} &= & {\int}_{\Sigma} \bar{d}^{2} - \bar{\sigma} { \nabla_{s} \bar{d} \cdot \nabla_{s} \bar{d}_{a} \over \sqrt { 1/{d_{0}^{2}} + \left| \nabla_{s} \bar{d} \right|^{2}}} - \left( 1 - p_{\kappa} \kappa\right)\bar{d}_{a}\\ && + {r_{f}^{2}} \nabla_{s} \gamma_{f} \cdot \nabla_{s} \gamma_{fa} + \gamma_{f} \gamma_{fa} - \gamma \gamma_{fa} \mathrm{d}{\Sigma} \\ && + {\int}_{\Omega} \left[ \kappa - {\bar{\sigma}_{l} \over {r_{m}^{2}}} { z_{s} - b_{z} z_{f} \over \left( \sqrt{ 1 + \left| \nabla z_{s} \right|^{2} } \right)^{3} } \right] \kappa_{a} + {r_{m}^{2}} \nabla z_{f} \\ &&\cdot \nabla z_{fa} + z_{f} z_{fa} - z_{m} z_{fa} + {r_{m}^{2}} \nabla z_{s} \cdot \nabla z_{sa} \\ &&+ z_{s} z_{sa} - b_{z} z_{f} z_{sa} \mathrm{d}{\Omega} \\ &= & {\int}_{\Omega} \left[ \bar{d}^{2} - \bar{\sigma} { \nabla_{s} \bar{d} \cdot \nabla_{s} \bar{d}_{a} \over \sqrt { 1/{d_{0}^{2}} + \left| \nabla_{s} \bar{d} \right|^{2}}} - \left( 1 - p_{\kappa} \kappa \right)\bar{d}_{a} \right.\\ &&\qquad\left. + {r_{f}^{2}} \nabla_{s} \gamma_{f} \cdot \nabla_{s} \gamma_{fa} + \gamma_{f} \gamma_{fa} - \gamma \gamma_{fa} \vphantom{{ \nabla_{s} \bar{d} \cdot \nabla_{s} \bar{d}_{a} \over \sqrt { 1/{d_{0}^{2}} + \left| \nabla_{s} \bar{d} \right|^{2}}}}\right]\\ &&\times \sqrt{1 + \left| \nabla z_{s} \right|^{2} } + \left[ \kappa - {\bar{\sigma}_{l} \over {r_{m}^{2}}} { z_{s} - b_{z} z_{f} \over \left( \sqrt{ 1 + \left| \nabla z_{s} \right|^{2} } \right)^{3} } \right]\\ && \times \kappa_{a} + {r_{m}^{2}} \nabla z_{f} \cdot \nabla z_{fa} + z_{f} z_{fa} - z_{m} z_{fa} \\ && + {r_{m}^{2}} \nabla z_{s} \cdot \nabla z_{sa} + z_{s} z_{sa} - b_{z} z_{f} z_{sa} \mathrm{d}{\Omega}, \end{array} $$
(55)

where \(\bar {d}_{a}~=~0\) and γfa = 0 are satisfied at Σ; zfa = 0 and zsa = 0 are satisfied on Ω. The first-order variational of the augmented Lagrangian \(\hat {J}\) is

$$ \begin{array}{@{}rcl@{}} \delta \widehat{J \left|{\Sigma}\right|^{2}} &= & {\int}_{\Omega} \left[ 2\bar{d} \delta\bar{d} - \bar{\sigma} { \nabla_{s} \delta \bar{d} \cdot \nabla_{s} \bar{d}_{a} \over \sqrt { 1/{d_{0}^{2}} + \left| \nabla_{s} \bar{d} \right|^{2}}} + \bar{\sigma} { \left( \nabla_{s} \bar{d} \cdot \nabla_{s} \bar{d}_{a} \right) \left( \nabla_{s} \bar{d} \cdot \nabla_{s} \delta \bar{d} \right) \over \left( \sqrt { 1/{d_{0}^{2}} + \left| \nabla_{s} \bar{d} \right|^{2}}\right)^{3} } \right] \sqrt{1+ \left| \nabla z_{s} \right|^{2} } + p_{\kappa} \bar{d}_{a} \sqrt{1+ \left| \nabla z_{s} \right|^{2} } \delta\kappa \\ && +\kappa_{a} \delta\kappa + \left( {r_{f}^{2}} \nabla_{s} \delta \gamma_{f} \cdot \nabla_{s} \gamma_{fa} + \delta \gamma_{f} \gamma_{fa} - { \partial \bar{\sigma} \over \partial \gamma_{p} } { \partial \gamma_{p} \over \partial \gamma_{f} } { \nabla_{s} \bar{d} \cdot \nabla_{s} \bar{d}_{a} \over \sqrt { 1/{d_{0}^{2}} + \left| \nabla_{s} \bar{d} \right|^{2}}} \delta \gamma_{f} + {\partial p_{\kappa} \over \partial \gamma_{p}} {\partial \gamma_{p} \over \partial \gamma_{f}} \kappa \bar{d}_{a} \delta \gamma_{f} \right) \sqrt{1+ \left| \nabla z_{s} \right|^{2} } \\ && - \gamma_{fa} \sqrt{1+ \left| \nabla z_{s} \right|^{2} } \delta \gamma + {r_{m}^{2}} \nabla \delta z_{f} \cdot \nabla z_{fa} + \delta z_{f} z_{fa} + {r_{m}^{2}} \nabla \delta z_{s} \cdot \nabla z_{sa} + \delta z_{s} z_{sa} - b_{z} \delta z_{f} z_{sa} \\ && + {r_{f}^{2}} \left[ \nabla_{s}^{\left( \delta z_{s}\right)} \gamma_{f} \cdot \nabla_{s} \gamma_{fa} + \nabla_{s} \gamma_{f} \cdot \nabla_{s}^{\left( \delta z_{s}\right)} \gamma_{fa} \right] \sqrt{1+ \left| \nabla z_{s} \right|^{2} } \\ && - \bar{\sigma} \left[ { \nabla_{s}^{\left( \delta z_{s}\right)} \bar{d} \cdot \nabla_{s} \bar{d}_{a} + \nabla_{s} \bar{d} \cdot \nabla_{s}^{\left( \delta z_{s}\right)} \bar{d}_{a} \over \sqrt { 1/{d_{0}^{2}} + \left| \nabla_{s} \bar{d} \right|^{2}}} - { \left( \nabla_{s} \bar{d} \cdot \nabla_{s} \bar{d}_{a} \right) \left( \nabla_{s}^{\left( \delta z_{s}\right)} \bar{d} \cdot \nabla_{s} \bar{d} \right) \over \left( \sqrt { 1/{d_{0}^{2}} + \left| \nabla_{s} \bar{d} \right|^{2}}\right)^{3} } \right] \sqrt{1+ \left| \nabla z_{s} \right|^{2} } \\ && + \left[ \bar{d}^{2} - \bar{\sigma} { \nabla_{s} \bar{d} \cdot \nabla_{s} \bar{d}_{a} \over \sqrt { 1/{d_{0}^{2}} + \left| \nabla_{s} \bar{d} \right|^{2}}} - \left( 1 - p_{\kappa} \kappa \right) \bar{d}_{a} + {r_{f}^{2}} \nabla_{s} \gamma_{f} \cdot \nabla_{s} \gamma_{fa} + \gamma_{f} \gamma_{fa} - \gamma \gamma_{fa} \right] { \nabla z_{s} \cdot \nabla \delta z_{s} \over \sqrt{1+ \left| \nabla z_{s} \right|^{2} } } \\ && - {\bar{\sigma}_{l} \over {r_{m}^{2}}} \left[ { \delta z_{s} - b_{z} \delta z_{f} \over \left( \sqrt{ 1 + \left| \nabla z_{s} \right|^{2} }\right)^{3} } - 3 { \left( z_{s} - b_{z} z_{f} \right) \left( \nabla z_{s} \cdot \nabla \delta z_{s} \right) \over \left( \sqrt{ 1 + \left| \nabla z_{s} \right|^{2} }\right)^{5} } \right] \kappa_{a} - z_{fa} \delta z_{m} \mathrm{d}{\Omega}, \end{array} $$
(56)

where ∇s has the transformed form in (19), and its first order variational to zs has the form as that in (21) with \(\tilde {z}_{s}\) replaced to be δzs; δzs, δzf, δzm, δγf, and δγ are the first-order variational of zs, zf, zm, γf, and γ, respectively. According to the KKT condition of the PDE constrained optimization problem (Hinze et al. 2009), the first-order variational of the augmented Lagrangian to the variables \(\bar {d}\), \(\bar {\gamma }_{f}\), zf, and zs can be set to be zero as follows:

$$ \begin{array}{@{}rcl@{}} && {\int}_{\Omega} \left[ 2\bar{d} \delta\bar{d} - \bar{\sigma} { \nabla_{s} \bar{d}_{a} \cdot \nabla_{s} \delta \bar{d} \over \sqrt { 1/{d_{0}^{2}} + \left| \nabla_{s} \bar{d} \right|^{2}}} + \bar{\sigma} { \left( \nabla_{s} \bar{d} \cdot \nabla_{s} \bar{d}_{a} \right) \left( \nabla_{s} \bar{d} \cdot \nabla_{s} \delta \bar{d} \right) \over \left( \sqrt { 1/{d_{0}^{2}} + \left| \nabla_{s} \bar{d} \right|^{2}}\right)^{3} } \right] \sqrt{1+ \left| \nabla z_{s} \right|^{2} } \mathrm{d}{\Omega} = 0 \\ &\Rightarrow & {\int}_{\Sigma} 2\bar{d} \delta\bar{d} - \bar{\sigma} { \nabla_{s} \bar{d}_{a} \cdot \nabla_{s} \delta \bar{d} \over \sqrt { 1/{d_{0}^{2}} + \left| \nabla_{s} \bar{d} \right|^{2}}} + \bar{\sigma} { \left( \nabla_{s} \bar{d} \cdot \nabla_{s} \bar{d}_{a} \right) \left( \nabla_{s} \bar{d} \cdot \nabla_{s} \delta \bar{d} \right) \over \left( \sqrt { 1/{d_{0}^{2}} + \left| \nabla_{s} \bar{d} \right|^{2}}\right)^{3} } \mathrm{d}{\Sigma} = 0; \end{array} $$
(57)
$$ {\int}_{\Omega} \left( p_{\kappa} \bar{d}_{a} \sqrt{1+ \left| \nabla z_{s} \right|^{2} } + \kappa_{a} \right) \delta\kappa \mathrm{d}{\Omega} = 0; $$
(58)
$$ \begin{array}{@{}rcl@{}} && {\int}_{\Omega} \left( {r_{f}^{2}} \nabla_{s} \delta \gamma_{f} \cdot \nabla_{s} \gamma_{fa} + \delta \gamma_{f} \gamma_{fa} - { \partial \bar{\sigma} \over \partial \gamma_{p} } { \partial \gamma_{p} \over \partial \gamma_{f} } { \nabla_{s} \bar{d} \cdot \nabla_{s} \bar{d}_{a} \over \sqrt { 1/{d_{0}^{2}} + \left| \nabla_{s} \bar{d} \right|^{2}}} \delta \gamma_{f} + {\partial p_{\kappa} \over \partial \gamma_{p}} {\partial \gamma_{p} \over \partial \gamma_{f}} \kappa \bar{d}_{a} \delta \gamma_{f} \right) \sqrt{1+ \left| \nabla z_{s} \right|^{2} } \mathrm{d}{\Omega} = 0 \\ && \Rightarrow {\int}_{\Sigma} {r_{f}^{2}} \nabla_{s} \delta \gamma_{f} \cdot \nabla_{s} \gamma_{fa} + \delta \gamma_{f} \gamma_{fa} - { \partial \bar{\sigma} \over \partial \gamma_{p} } { \partial \gamma_{p} \over \partial \gamma_{f} } { \nabla_{s} \bar{d} \cdot \nabla_{s} \bar{d}_{a} \over \sqrt { 1/{d_{0}^{2}} + \left| \nabla_{s} \bar{d} \right|^{2}}} \delta \gamma_{f} + {\partial p_{\kappa} \over \partial \gamma_{p}} {\partial \gamma_{p} \over \partial \gamma_{f}} \kappa \bar{d}_{a} \delta \gamma_{f} \mathrm{d}{\Sigma} = 0; \end{array} $$
(59)
$$ \begin{array}{@{}rcl@{}} && {\int}_{\Omega} {r_{m}^{2}} \nabla \delta z_{s} \cdot \nabla z_{sa} + \delta z_{s} z_{sa} + {r_{f}^{2}} \left( \nabla_{s}^{\left( \delta z_{s}\right)} \gamma_{f} \cdot \nabla_{s} \gamma_{fa} + \nabla_{s} \gamma_{f} \cdot \nabla_{s}^{\left( \delta z_{s}\right)} \gamma_{fa} \right) \sqrt{1+ \left| \nabla z_{s} \right|^{2} } \\ && - \bar{\sigma} \left[ { \nabla_{s}^{\left( \delta z_{s}\right)} \bar{d} \cdot \nabla_{s} \bar{d}_{a} + \nabla_{s} \bar{d} \cdot \nabla_{s}^{\left( \delta z_{s}\right)} \bar{d}_{a} \over \sqrt { 1/{d_{0}^{2}} + \left| \nabla_{s} \bar{d} \right|^{2}}} - { \left( \nabla_{s} \bar{d} \cdot \nabla_{s} \bar{d}_{a} \right) \left( \nabla_{s}^{\left( \delta z_{s}\right)} \bar{d} \cdot \nabla_{s} \bar{d} \right) \over \left( \sqrt { 1/{d_{0}^{2}} + \left| \nabla_{s} \bar{d} \right|^{2}}\right)^{3} } \right] \sqrt{1+ \left| \nabla z_{s} \right|^{2} } \\ && + \left[ \bar{d}^{2} - \bar{\sigma} { \nabla_{s} \bar{d} \cdot \nabla_{s} \bar{d}_{a} \over \sqrt { 1/{d_{0}^{2}} + \left| \nabla_{s} \bar{d} \right|^{2}}} - \left( 1-p_{\kappa} \kappa\right) \bar{d}_{a} + {r_{f}^{2}} \nabla_{s} \gamma_{f} \cdot \nabla_{s} \gamma_{fa} + \gamma_{f} \gamma_{fa} - \gamma \gamma_{fa} \right] { \nabla z_{s} \cdot \nabla \delta z_{s} \over \sqrt{1+ \left| \nabla z_{s} \right|^{2} } } \\ && - {\bar{\sigma}_{l} \over {r_{m}^{2}}} \left[ { \delta z_{s} \over \left( \sqrt{ 1 + \left| \nabla z_{s} \right|^{2} }\right)^{3} } - 3 { \left( z_{s} - b_{z} z_{f} \right) \left( \nabla z_{s} \cdot \nabla \delta z_{s} \right) \over \left( \sqrt{ 1 + \left| \nabla z_{s} \right|^{2} }\right)^{5} } \right] \kappa_{a} \mathrm{d}{\Omega} = 0; \end{array} $$
(60)
$$ \begin{array}{@{}rcl@{}} &&{\int}_{\Omega} {r_{m}^{2}} \nabla \delta z_{f} \cdot \nabla z_{fa} + \delta z_{f} z_{fa} - b_{z} \delta z_{f} z_{sa}\\ &&+ {\bar{\sigma}_{l} \over {r_{m}^{2}}} { b_{z} \kappa_{a} \over \left( \sqrt{ 1 + \left| \nabla z_{s} \right|^{2} }\right)^{3} } \delta z_{f} \mathrm{d}{\Omega} = 0. \end{array} $$
(61)

Further, the adjoint sensitivity of \(J \left |{\Sigma }\right |^{2}\) is derived from \(\delta \widehat {J \left |{\Sigma }\right |^{2}}\):

$$ \begin{array}{@{}rcl@{}} \delta \left( J \left|{\Sigma}\right|^{2}\right) &=& - {\int}_{\Omega} \gamma_{fa} \sqrt{1+ \left| \nabla z_{s} \right|^{2} } \delta \gamma + z_{fa} \delta z_{m} \mathrm{d}{\Omega} \\ &=& - {\int}_{\Sigma} \gamma_{fa} \delta \gamma \mathrm{d}{\Sigma} - {\int}_{\Omega} z_{fa} \delta z_{m} \mathrm{d}{\Omega}. \end{array} $$
(62)

Without losing the arbitrariness of \(\delta \bar {d}\), δκ, δγf, δzf, and δzs, one can set \(\delta \bar {d}=\tilde {\bar {d}}_{a}\) with \(\forall \tilde {\bar {d}}_{a}\in {\mathscr{H}}\left ({\Sigma }\right )\), \(\delta \kappa =\tilde {\kappa }_{a}\) with \(\forall \tilde {\kappa }_{a}\in {\mathscr{H}}^{-1}\left ({\Omega }\right )\), \(\delta \gamma _{f}=\tilde {\gamma }_{fa}\) with \(\forall \tilde {\gamma }_{fa}\in {\mathscr{H}}\left ({\Sigma }\right )\), \(\delta z_{f}=\tilde {z}_{fa}\) with \(\forall \tilde {z}_{fa}\in {\mathscr{H}}\left ({\Omega }\right )\), and \(\delta z_{s} =\tilde {z}_{sa}\) with \(\forall \tilde {z}_{sa}\in {\mathscr{H}}\left ({\Omega }\right )\), to derive the variational formulations of the adjoint system in (2932).

1.3 Appendix 3: Adjoint analysis for manifold area \(\left |{\Sigma }\right |\) in design objective and duty-ratio constraint

Based on the variational formulations of the PDEs in (51) and (52) and the Lagrangian multiplier-based adjoint method, the augmented Lagrangian for \(\left | {\Sigma } \right | = {\int \limits }_{\Sigma } 1 \mathrm {d}{\Sigma } = {\int \limits }_{\Omega } \sqrt {1+ \left | \nabla z_{s} \right |^{2}} d{\Omega } \) can be formulated as

$$ \begin{array}{@{}rcl@{}} \widehat{\left| {\Sigma} \right|} &= & {\int}_{\Sigma} 1 \mathrm{d}{\Sigma} + {\int}_{\Omega} {r_{m}^{2}} \nabla z_{f} \cdot \nabla z_{fa} + z_{f} z_{fa} - z_{m} z_{fa}\\ && + {r_{m}^{2}} \nabla z_{s} \cdot \nabla z_{sa} + z_{s} z_{sa} - b_{z} z_{f} z_{sa} \mathrm{d}{\Omega} \\ &= & {\int}_{\Omega} \sqrt{1+ \left| \nabla z_{s} \right|^{2}} + {r_{m}^{2}} \nabla z_{f} \cdot \nabla z_{fa} + z_{f} z_{fa} - z_{m} z_{fa}\\ && + {r_{m}^{2}} \nabla z_{s} \cdot \nabla z_{sa} + z_{s} z_{sa} - b_{z} z_{f} z_{sa} \mathrm{d}{\Omega}, \end{array} $$
(63)

where zfa = 0 and zsa = 0 are satisfied on Ω. The first-order variational of the augmented Lagrangian \(\widehat {\left | {\Sigma } \right |}\) is

$$ \begin{array}{@{}rcl@{}} \delta \widehat{\left| {\Sigma} \right|} \!&=&\! {\int}_{\Omega} { \nabla z_{f} \cdot \nabla \delta z_{f} \over \sqrt{1+ \left| \nabla z_{f} \right|^{2}}} + {r_{m}^{2}} \nabla \delta z_{f} \cdot \nabla z_{fa} + \delta z_{f} z_{fa}\\ &&\!- \delta z_{m} z_{fa} \!+ {r_{m}^{2}} \nabla \delta z_{s} \cdot \nabla z_{sa} \!+ \delta z_{s} z_{sa} - b_{z} \delta z_{f} z_{sa} \mathrm{d}{\Omega}.\\ \end{array} $$
(64)

According to the KKT condition of the PDE constrained optimization problem, the first-order variational of the augmented Lagrangian to the variables zf and z can be set to be zero:

$$ {\int}_{\Omega} { \nabla z_{s} \cdot \nabla \delta z_{s} \over \sqrt{1+ \left| \nabla z_{s} \right|^{2}}} + {r_{m}^{2}} \nabla \delta z_{s} \cdot \nabla z_{sa} + \delta z_{s} z_{sa} \mathrm{d}{\Omega} = 0; $$
(65)
$$ {\int}_{\Omega} {r_{m}^{2}} \nabla \delta z_{f} \cdot \nabla z_{fa} + \delta z_{f} z_{fa} - b_{z} z_{sa} \delta z_{f} \mathrm{d}{\Omega} = 0. $$
(66)

Further, the adjoint sensitivity of \(\left |{\Sigma }\right |\) is derived from \(\delta \widehat {\left |{\Sigma }\right |}\):

$$ \delta \left|{\Sigma}\right| = - {\int}_{\Omega} z_{fa} \delta z_{m} \mathrm{d}{\Omega}. $$
(67)

Without losing the arbitrariness of δzs and δzf, one can set \(\delta z_{s}=\tilde {z}_{sa}\) with \(\forall \tilde {z}_{sa}\in {\mathscr{H}}\left ({\Omega }\right )\) and \(\delta z_{f}=\tilde {z}_{fa}\) with \(\forall \tilde {z}_{fa}\in {\mathscr{H}}\left ({\Omega }\right )\), to derive the variational formulations of the adjoint system in (35) and (36).

1.4 Appendix 4: Adjoint analysis for \(f_{d} \left | {\Sigma } \right |\) in duty-ratio constraint

Based on the variational formulations of the PDEs in (5153) and Lagrangian multiplier-based adjoint method, the augmented Lagrangian for \(f_{d} \left | {\Sigma } \right | = {\int \limits }_{\Sigma } 1- \gamma _{p} \mathrm {d}{\Sigma }\) can be formulated as

$$ \begin{array}{@{}rcl@{}} \widehat{f_{d} \left| {\Sigma} \right|} \!&=&\! {\int}_{\Sigma} 1- \gamma_{p} + {r_{f}^{2}} \nabla_{s} \gamma_{f} \!\cdot\! \nabla_{s} \gamma_{fa} + \gamma_{f} \gamma_{fa} - \gamma \gamma_{fa} \mathrm{d}{\Sigma} \\ && \!+ {\int}_{\Omega} {r_{m}^{2}} \nabla z_{f} \cdot \nabla z_{fa} + z_{f} z_{fa} - z_{m} z_{fa} \mathrm{d}{\Omega} \\ && \!+ {\int}_{\Omega} {r_{m}^{2}} \nabla z_{s} \cdot \nabla z_{sa} + z_{s} z_{sa} - b_{z} z_{f} z_{sa} \mathrm{d}{\Omega} \\ \!&=&\! {\int}_{\Omega} \left( 1 - \gamma_{p} + {r_{f}^{2}} \nabla_{s} \gamma_{f} \cdot \nabla_{s} \gamma_{fa} + \gamma_{f} \gamma_{fa} \!- \gamma \gamma_{fa} \right)\\ && \times \sqrt{1+ \left| \nabla z_{s} \right|^{2} } + {r_{m}^{2}} \nabla z_{f} \cdot \nabla z_{fa} + z_{f} z_{fa} - z_{m} z_{fa}\\ && + {r_{m}^{2}} \nabla z_{s} \cdot \nabla z_{sa} + z_{s} z_{sa} - b_{z} z_{f} z_{sa} \mathrm{d}{\Omega}, \end{array} $$
(68)

where γfa = 0 is satisfied at Σ; zfa = 0 is satisfied on Ω. The first-order variational of the augmented Lagrangian \(\widehat {f_{d} \left | {\Sigma } \right |}\) is

$$ \begin{array}{@{}rcl@{}} \delta \widehat{f_{d} \left| {\Sigma} \right|} &= & {\int}_{\Omega} \left( - {\partial\gamma_{p}\over\partial\gamma_{f}}\delta\gamma_{f} + {r_{f}^{2}} \nabla_{s} \delta \gamma_{f} \cdot \nabla_{s} \gamma_{fa} + \delta \gamma_{f} \gamma_{fa}\right.\\ &&\qquad \left. - \delta \gamma \gamma_{fa} \vphantom{{\partial\gamma_{p}\over\partial\gamma_{f}}}\right) \sqrt{1+ \left| \nabla z_{s} \right|^{2} } \\ && + {r_{f}^{2}} \left( \nabla_{s}^{\left( \delta z_{s} \right)} \gamma_{f} \cdot \nabla_{s} \gamma_{fa} + \nabla_{s} \gamma_{f} \cdot \nabla_{s}^{\left( \delta z_{s} \right)} \gamma_{fa} \right) \\ &&\times \sqrt{1+ \left| \nabla z_{s} \right|^{2} } \\ && + \left( 1- \gamma_{p} + {r_{f}^{2}} \nabla_{s} \gamma_{f} \cdot \nabla_{s} \gamma_{fa} + \gamma_{f} \gamma_{fa} - \gamma \gamma_{fa} \right)\\ && \times { \nabla z_{s} \cdot \nabla \delta z_{s} \over \sqrt{1+ \left| \nabla z_{s} \right|^{2} }} + {r_{m}^{2}} \nabla \delta z_{f} \cdot \nabla z_{fa} + \delta z_{f} z_{fa} \\ &&- \delta z_{m} z_{fa} + {r_{m}^{2}} \nabla \delta z_{s} \cdot \nabla z_{sa} + \delta z_{s} z_{sa} \\ &&- b_{z} \delta z_{f} z_{sa} \mathrm{d}{\Omega}. \end{array} $$
(69)

According to the KKT condition of the PDE constrained optimization problem, the first-order variational of the augmented Lagrangian to the variables γf, zf, and zs can be set to be zero:

$$ \begin{array}{@{}rcl@{}} && {\int}_{\Omega} \left( - {\partial\gamma_{p}\over\partial\gamma_{f}}\delta\gamma_{f} + {r_{f}^{2}} \nabla_{s} \delta \gamma_{f} \cdot \nabla_{s} \gamma_{fa} + \delta \gamma_{f} \gamma_{fa} \right)\\ &&\times \sqrt{1+ \left| \nabla z_{s} \right|^{2} } \mathrm{d}{\Omega} = 0 \\ \Rightarrow && {\int}_{\Sigma} - {\partial\gamma_{p}\over\partial\gamma_{f}}\delta\gamma_{f} + {r_{f}^{2}} \nabla_{s} \delta \gamma_{f} \cdot \nabla_{s} \gamma_{fa} + \delta \gamma_{f} \gamma_{fa} \mathrm{d}{\Sigma} = 0;\\ \end{array} $$
(70)
$$ \begin{array}{@{}rcl@{}} && {\int}_{\Omega} {r_{f}^{2}} \left( \nabla_{s}^{\left( \delta z_{s} \right)} \gamma_{f} \!\cdot\! \nabla_{s} \gamma_{fa} + \nabla_{s} \gamma_{f} \cdot \nabla_{s}^{\left( \delta z_{s} \right)} \gamma_{fa} \right) \sqrt{1+ \left| \nabla z_{s} \right|^{2} } \\ && + \left( 1- \gamma_{p} + {r_{f}^{2}} \nabla_{s} \gamma_{f} \cdot \nabla_{s} \gamma_{fa} + \gamma_{f} \gamma_{fa} -\gamma \gamma_{fa} \right)\\ &&\times { \nabla z_{s} \cdot \nabla \delta z_{s} \over \sqrt{1+ \left| \nabla z_{s} \right|^{2} }} + {r_{m}^{2}} \nabla \delta z_{s} \cdot \nabla z_{sa} + \delta z_{s} z_{sa} \mathrm{d}{\Omega} = 0; \end{array} $$
(71)
$$ {\int}_{\Omega} {r_{m}^{2}} \nabla \delta z_{f} \cdot \nabla z_{fa} + \delta z_{f} z_{fa} - b_{z} z_{sa} \delta z_{f} \mathrm{d}{\Omega} = 0. $$
(72)

Further, the adjoint sensitivity of \(f_{d} \left | {\Sigma } \right |\) is derived from \(\delta \widehat {f_{d} \left | {\Sigma } \right |}\):

$$ \begin{array}{@{}rcl@{}} \delta \left( f_{d} \left| {\Sigma} \right| \right) &=& - {\int}_{\Omega} \gamma_{fa} \sqrt{1+ \left| \nabla z_{s} \right|^{2} } \delta \gamma + z_{fa} \delta z_{m} \mathrm{d}{\Omega} \\ &=& - {\int}_{\Sigma} \gamma_{fa} \delta \gamma \mathrm{d}{\Sigma} - {\int}_{\Omega} z_{fa} \delta z_{m} \mathrm{d}{\Omega}. \end{array} $$
(73)

Without losing the arbitrariness of δγf, δzf, and δzs, one can set \(\delta \gamma _{f}=\tilde {\gamma }_{fa}\) with \(\forall \tilde {\gamma }_{fa}\in {\mathscr{H}}\left ({\Sigma }\right )\), \(\delta z_{f}=\tilde {z}_{fa}\) with \(\forall \tilde {z}_{fa}\in {\mathscr{H}}\left ({\Omega }\right )\), and \(\delta z_{s}=\tilde {z}_{sa}\) with \(\forall \tilde {z}_{sa}\in {\mathscr{H}}\left ({\Omega }\right )\), to derive the variational formulations of the adjoint system in (39) and (40).

Fig. 22
figure 22

Plots for the converged values of the performance measurement for the derived fiber bundles with different duty ratios, corresponding to the three regular-polygon tilings in Figs. 8a1, 9a1 and 10a1, respectively

1.5 Appendix 5: Adjoint analysis for v in volume-fraction constraint

Based on the variational formulations of the PDEs in (51) and (52) and the Lagrangian multiplier-based adjoint method, the augmented Lagrangian for \(v = {\int \limits }_{\Omega } z_{s} \mathrm {d}{\Sigma } / \left (b_{z} \left |{\Omega }\right | \right )\) can be formulated as

$$ \begin{array}{@{}rcl@{}} \hat{v} &=& {\int}_{\Omega} {1\over b_{z} \left|{\Omega}\right|} z_{s} + {r_{m}^{2}} \nabla z_{f} \cdot \nabla z_{fa} + z_{f} z_{fa} - z_{m} z_{fa}\\ &&+ {r_{m}^{2}} \nabla z_{s} \cdot \nabla z_{sa} + z_{s} z_{sa} - b_{z} z_{f} z_{sa} \mathrm{d}{\Omega}. \end{array} $$
(74)

where zfa = 0 is satisfied on Ω. The first-order variational of the augmented Lagrangian \(\widehat {v}\) is

$$ \begin{array}{@{}rcl@{}} \delta \hat{v} &=& {\int}_{\Omega} {1\over b_{z} \left|{\Omega}\right|} \delta z_{s} + {r_{m}^{2}} \nabla \delta z_{f} \cdot \nabla z_{fa} + \delta z_{f} z_{fa} - \delta z_{m} z_{fa} \\ &&+ {r_{m}^{2}} \nabla \delta z_{s} \cdot \nabla z_{sa} + \delta z_{s} z_{sa} - b_{z} \delta z_{f} z_{sa} \mathrm{d}{\Omega}. \end{array} $$
(75)

According to the KKT condition of the PDE constrained optimization problem, the first-order variational of the augmented Lagrangian to the variable zs and zf can be set to be zero:

$$ {\int}_{\Omega} {1\over b_{z} \left|{\Omega}\right|} \delta z_{s} + {r_{m}^{2}} \nabla \delta z_{s} \cdot \nabla z_{sa} + \delta z_{s} z_{sa} \mathrm{d}{\Omega} = 0; $$
(76)
$$ {\int}_{\Omega} {r_{m}^{2}} \nabla \delta z_{f} \cdot \nabla z_{fa} + \delta z_{f} z_{fa} - b_{z} \delta z_{f} z_{sa} \mathrm{d}{\Omega} = 0. $$
(77)

Further, the adjoint sensitivity of v is derived from \(\delta \hat {v}\):

$$ \delta v = - {\int}_{\Omega} z_{fa} \delta z_{m} \mathrm{d}{\Omega}. $$
(78)

Without losing the arbitrariness of δzs and δzf, one can set \(\delta z_{s}=\tilde {z}_{sa}\) with \(\forall \tilde {z}_{sa}\in {\mathscr{H}}\left ({\Omega }\right )\) and \(\delta z_{f}=\tilde {z}_{fa}\) with \(\forall \tilde {z}_{fa}\in {\mathscr{H}}\left ({\Omega }\right )\), to derive the variational formulations of the adjoint system in (49) and (50).

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Deng, Y., Zhang, W., Liu, Z. et al. Fiber bundle topology optimization of hierarchical microtextures for wetting behavior in Cassie-Baxter mode. Struct Multidisc Optim 61, 2523–2556 (2020). https://doi.org/10.1007/s00158-020-02558-8

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