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Topology optimization of a flexible multibody system with variable-length bodies described by ALE–ANCF

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Abstract

Recent years have witnessed the application of topology optimization to flexible multibody systems (FMBS) so as to enhance their dynamic performances. In this study, an explicit topology optimization approach is proposed for an FMBS with variable-length bodies via the moving morphable components (MMC). Using the arbitrary Lagrangian–Eulerian (ALE) formulation, the thin plate elements of the absolute nodal coordinate formulation (ANCF) are used to describe the platelike bodies with variable length. For the thin plate element of ALE–ANCF, the elastic force and additional inertial force, as well as their Jacobians, are analytically deduced. In order to account for the variable design domain, the sets of equivalent static loads are reanalyzed by introducing the actual and virtual design domains so as to transform the topology optimization problem of dynamic response into a static one. Finally, the novel MMC-based topology optimization method is employed to solve the corresponding static topology optimization problem by explicitly evolving the shapes and orientations of a set of structural components. The effect of the minimum feature size on the optimization of an FMBS is studied. Three numerical examples are presented to validate the accuracy of the thin plate element of ALE–ANCF and the efficiency of the proposed topology optimization approach, respectively.

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Acknowledgements

This work was supported in part by the National Natural Science Foundation of China under Grants 11290151 and 11472042. It was also supported in part by Postgraduate Research & Practice Innovation Program of Jiangsu Province under Grants KYCX17_0226 and by China Scholarship Council under Grants 201706830011.

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Authors and Affiliations

  1. State Key Laboratory of Mechanics and Control of Mechanical Structures, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    Jialiang Sun & Haiyan Hu

  2. MOE Key Laboratory of Dynamics and Control of Flight Vehicle, School of Aerospace Engineering, Beijing Institute of Technology, Beijing, 100081, China

    Qiang Tian & Haiyan Hu

  3. Department of Mechanical Engineering, Solid Mechanics, Technical University of Denmark, Kongens Lyngby, 2800, Denmark

    Niels L. Pedersen

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  1. Jialiang Sun
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Correspondence to Haiyan Hu.

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Appendix

Appendix

First of all, the strain tensor \({\varvec{\upvarepsilon }}\) as shown in Eq. (7) can be rewritten as

$$\begin{aligned} {\varvec{\upvarepsilon }}=\left[ {{\begin{array}{l} {{\left( {\mathbf{r}_{,m}^{\mathrm{T}} \mathbf{r}_{,m} -1} \right) }/2} \\ {{\left( {\mathbf{r}_{,n}^{\mathrm{T}} \mathbf{r}_{,n} -1} \right) }/2} \\ {\mathbf{r}_{,m}^{\mathrm{T}} \mathbf{r}_{,n} } \\ \end{array} }} \right] =\left[ {{\begin{array}{l} {{\left( {\left( \mathbf{A} \right) _{ij} q_i q_j -1} \right) }/2} \\ {{\left( {\left( \mathbf{B} \right) _{ij} q_i q_j -1} \right) }/2} \\ {\left( \mathbf{C} \right) _{ij} q_i q_j } \\ \end{array} }} \right] , \nonumber \\ \end{aligned}$$
(A.1)

where the subscripts \(i,j=1,2,\ldots ,36\), and \(\mathbf{A}=\mathbf{N}_{e,m}^{\mathrm{T}} \mathbf{N}_{e,m} \), \(\mathbf{B}=\mathbf{N}_{e,n}^{\mathrm{T}} \mathbf{N}_{e,n} \), \(\mathbf{C}=\mathbf{N}_{e,m}^{\mathrm{T}} \mathbf{N}_{e,n} \).

Then, the invariant matrices \(\left( {\mathbf{K}_1 } \right) _{ikab} \), \(\left( {\mathbf{K}_2 } \right) _{ikab} \), \(\left( {\mathbf{K}_3 } \right) _{ikab} \), \(\left( {\mathbf{K}_4 } \right) _{ijkab} \), \(\left( {\mathbf{K}_5 } \right) _{ijkab} \), \(\left( {\mathbf{K}_6 } \right) _{ijkab} \), \(\left( {\mathbf{G}_1 } \right) _{ik} \), \(\left( {\mathbf{G}_2 } \right) _{ik} \), \(\left( {\mathbf{G}_3 } \right) _{ijk} \) and \(\left( {\mathbf{G}_4 } \right) _{ijk} \) in Eq. (11) can be computed as follows

$$\begin{aligned}&\left( {\mathbf{K}_1 } \right) _{ikab} =\int _{V_0 } {\mathbf{E}_{11}^\varepsilon \left( {\left( \mathbf{A} \right) _{ik} \left( \mathbf{A} \right) _{ab} +\left( \mathbf{B} \right) _{ik} \left( \mathbf{B} \right) _{ab} } \right) \hbox {d}V} ,\;\;\;\;\;~~\nonumber \\&\left( {\mathbf{K}_2 } \right) _{ikab} =\int _{V_0 } {\mathbf{E}_{12}^\varepsilon \left( {\left( \mathbf{B} \right) _{ik} \left( \mathbf{A} \right) _{ab} +\left( \mathbf{A} \right) _{ik} \left( \mathbf{B} \right) _{ab} } \right) \hbox {d}V} ,\;\;\;\;~~~~\nonumber \\&\left( {\mathbf{K}_3 } \right) _{ikab} =\int _{V_0 } {2\mathbf{E}_{33}^\varepsilon \left( {\left( \mathbf{C} \right) _{ik} \left( \mathbf{C} \right) _{ab} +\left( \mathbf{C} \right) _{ki} \left( \mathbf{C} \right) _{ab} } \right) \hbox {d}V} ,\;\;\;~~~~\nonumber \\&\left( {\mathbf{K}_4 } \right) _{ijkab} =\int _{V_0 } {\mathbf{E}_{11}^\varepsilon \left( {\left( \mathbf{A} \right) _{ij,q_k } \left( \mathbf{A} \right) _{ab} +\left( \mathbf{B} \right) _{ij,q_k } \left( \mathbf{B} \right) _{ab} } \right) \hbox {d}V} ,\nonumber \\&\left( {\mathbf{K}_5 } \right) _{ijkab} =\int _{V_0 } {\mathbf{E}_{12}^\varepsilon \left( {\left( \mathbf{B} \right) _{ij,q_k } \left( \mathbf{A} \right) _{ab} +\left( \mathbf{A} \right) _{ij,q_k } \left( \mathbf{B} \right) _{ab} } \right) \hbox {d}V} , \nonumber \\&\left( {\mathbf{K}_6 } \right) _{ijkab} =\int _{V_0 } {4\mathbf{E}_{33}^\varepsilon \left( \mathbf{C} \right) _{ij,q_k } \left( \mathbf{C} \right) _{ab} \hbox {d}V} ,\nonumber \\&\left( {\mathbf{G}_1 } \right) _{ik} =\int _{V_0 } {\mathbf{E}_{11}^\varepsilon \left( {\left( \mathbf{A} \right) _{ik} +\left( \mathbf{B} \right) _{ik} } \right) \hbox {d}V},\nonumber \\&\left( {\mathbf{G}_2 } \right) _{ik} =\int _{V_0 } {\mathbf{E}_{12}^\varepsilon \left( {\left( \mathbf{A} \right) _{ik} +\left( \mathbf{B} \right) _{ik} } \right) \hbox {d}V},\nonumber \\&\left( {\mathbf{G}_3 } \right) _{ijk} =\int _{V_0 } {\mathbf{E}_{11}^\varepsilon \left( {\left( \mathbf{A} \right) _{ij,q_k } +\left( \mathbf{B} \right) _{ij,q_k } } \right) \hbox {d}V},\nonumber \\&\left( {\mathbf{G}_4 } \right) _{ijk} =\int _{V_0 } {\mathbf{E}_{12}^\varepsilon \left( {\left( \mathbf{A} \right) _{ij,q_k } +\left( \mathbf{B} \right) _{ij,q_k } } \right) \hbox {d}V}, \end{aligned}$$
(A.2)

where the subscripts \(i,\;j,\;a,\;b=1,\;2,\;\ldots ,\;36\) and \(\mathbf{E}_{11}^\varepsilon \), \(\mathbf{E}_{12}^\varepsilon \), \(\mathbf{E}_{33}^\varepsilon \) are the entries of the elastic coefficient matrix \(\mathbf{E}^{\varepsilon }\) in Eq. (8). From Eq. (A.2), it can be observed that \(\mathbf{K}_1 \) and \(\mathbf{K}_2 \) are symmetric about i, k and a, b, respectively. \(\mathbf{K}_4 \) and \(\mathbf{K}_5 \) are symmetric about i, j and a, b, respectively. \(\mathbf{K}_3 \), \(\mathbf{G}_1 \) and \(\mathbf{G}_2 \) are symmetric about i, k. \(\mathbf{K}_6 \), \(\mathbf{G}_3 \) and \(\mathbf{G}_4 \) are symmetrical about i, j.

In Eqs. (13) and (15), the expressions of \(\left( {\mathbf{K}_1 } \right) _{ikab,q_l } \), \(\left( {\mathbf{K}_2 } \right) _{ikab,q_l } \), \(\left( {\mathbf{K}_3 } \right) _{ikab,q_l } \), \(\left( {\mathbf{K}_4 } \right) _{ijkab,q_l } \), \(\left( {\mathbf{K}_5 } \right) _{ijkab,q_l } \), \(\left( {\mathbf{K}_6 } \right) _{ijkab,q_l } \), \(\left( {\mathbf{G}_1 } \right) _{ik,q_l } \), \(\left( {\mathbf{G}_2 } \right) _{ik,q_l } \), \(\left( {\mathbf{G}_3 } \right) _{ijk,q_l } \) and \(\left( {\mathbf{G}_4 } \right) _{ijk,q_l } \) are listed as follows

$$\begin{aligned}&\left( {\mathbf{K}_1 } \right) _{ikab,q_l } =\int _{V_0 } {\mathbf{E}_{11}^\varepsilon \left( {\left( \mathbf{A} \right) _{ik,q_l } \left( \mathbf{A} \right) _{ab} +\left( \mathbf{A} \right) _{ik} \left( \mathbf{A} \right) _{ab,q_l } } \right) \hbox {d}V} \nonumber \\&\quad +\int _{V_0 } {\mathbf{E}_{11}^\varepsilon \left( {\left( \mathbf{B} \right) _{ik,q_l } \left( \mathbf{B} \right) _{ab} +\left( \mathbf{B} \right) _{ik} \left( \mathbf{B} \right) _{ab,q_l } } \right) \hbox {d}V,} \;\;\;\;~~~\;~~ \nonumber \\&\left( {\mathbf{K}_2 } \right) _{ikab,q_l } =\int _{V_0 } {\mathbf{E}_{12}^\varepsilon \left( {\left( \mathbf{B} \right) _{ik,q_l } \left( \mathbf{A} \right) _{ab} +\left( \mathbf{B} \right) _{ik} \left( \mathbf{A} \right) _{ab,q_l } } \right) \hbox {d}V} \nonumber \\&\quad +\int _{V_0 } {\mathbf{E}_{12}^\varepsilon \left( {\left( \mathbf{A} \right) _{ik,q_l } \left( \mathbf{B} \right) _{ab} +\left( \mathbf{A} \right) _{ik} \left( \mathbf{B} \right) _{ab,q_l } } \right) \hbox {d}V,} \;\;\;\;~~~~ \nonumber \\&\left( {\mathbf{K}_3 } \right) _{ikab,q_l } =\int _{V_0 } {2\mathbf{E}_{33}^\varepsilon \left( {\left( \mathbf{C} \right) _{ik,q_l } \left( \mathbf{C} \right) _{ab} +\left( \mathbf{C} \right) _{ik} \left( \mathbf{C} \right) _{ab,q_l } } \right) \hbox {d}V} \nonumber \\&\quad +\int _{V_0 } {2\mathbf{E}_{33}^\varepsilon \left( {\left( \mathbf{C} \right) _{ki,q_l } \left( \mathbf{C} \right) _{ab} +\left( \mathbf{C} \right) _{ki} \left( \mathbf{C} \right) _{ab,q_l } } \right) \hbox {d}V,} \;\;\;~~~~~ \nonumber \\&\left( {\mathbf{K}_4 } \right) _{ijkab,q_l } =\int _{V_0 } {\mathbf{E}_{11}^\varepsilon \left( {\left( \mathbf{A} \right) _{ij,q_k q_l } \left( \mathbf{A} \right) _{ab} +\left( \mathbf{A} \right) _{ij,q_k } \left( \mathbf{A} \right) _{ab,q_l } } \right) \hbox {d}V} \nonumber \\&\quad +\int _{V_0 } {\mathbf{E}_{11}^\varepsilon \left( {\left( \mathbf{B} \right) _{ij,q_k q_l } \left( \mathbf{B} \right) _{ab} +\left( \mathbf{B} \right) _{ij,q_k } \left( \mathbf{B} \right) _{ab,q_l } } \right) \hbox {d}V} ,\;~~ \nonumber \\&\left( {\mathbf{K}_5 } \right) _{ijkab,q_l } =\int _{V_0 } {\mathbf{E}_{12}^\varepsilon \left( {\left( \mathbf{B} \right) _{ij,q_k q_l } \left( \mathbf{A} \right) _{ab} +\left( \mathbf{B} \right) _{ij,q_k } \left( \mathbf{A} \right) _{ab,q_l } } \right) \hbox {d}V} \nonumber \\&\quad +\int _{V_0 } {\mathbf{E}_{12}^\varepsilon \left( {\left( \mathbf{A} \right) _{ij,q_k q_l } \left( \mathbf{B} \right) _{ab} +\left( \mathbf{A} \right) _{ij,q_k } \left( \mathbf{B} \right) _{ab,q_l } } \right) \hbox {d}V} ,~~ \nonumber \\&\left( {\mathbf{K}_6 } \right) _{ijkab,q_l } =\int _{V_0 } {4\mathbf{E}_{33}^\varepsilon \left( {\left( \mathbf{C} \right) _{ij,q_k q_l } \left( \mathbf{C} \right) _{ab} +\left( \mathbf{C} \right) _{ij,q_k } \left( \mathbf{C} \right) _{ab,q_l } } \right) \hbox {d}V} ,\nonumber \\&\left( {\mathbf{G}_1 } \right) _{ik,q_l } =\int _{V_0 } {\mathbf{E}_{11}^\varepsilon \left( {\left( \mathbf{A} \right) _{ik,q_l } +\left( \mathbf{B} \right) _{ik,q_l } } \right) \hbox {d}V}, \nonumber \\&\left( {\mathbf{G}_2 } \right) _{ik,q_l } =\int _{V_0 } {\mathbf{E}_{12}^\varepsilon \left( {\left( \mathbf{A} \right) _{ik,q_l } +\left( \mathbf{B} \right) _{ik,q_l } } \right) \hbox {d}V} ,\nonumber \\&\left( {\mathbf{G}_3 } \right) _{ijk,q_l } =\int _{V_0 } {\mathbf{E}_{11}^\varepsilon \left( {\left( \mathbf{A} \right) _{ij,q_k q_l } +\left( \mathbf{B} \right) _{ij,q_k q_l } } \right) \hbox {d}V} ,\nonumber \\&\left( {\mathbf{G}_4 } \right) _{ijk,q_l } =\int _{V_0 } {\mathbf{E}_{12}^\varepsilon \left( {\left( \mathbf{A} \right) _{ij,q_k q_l } +\left( \mathbf{B} \right) _{ij,q_k q_l } } \right) \hbox {d}V} , \end{aligned}$$
(A.3)

where the subscripts \(i,\;j,\;a,\;b=1,\;2,\;\ldots ,\;36\).

Besides, the expressions of \(\left( {\mathbf{H}_1 } \right) _{ka} \), \(\left( {\mathbf{H}_2 } \right) _{ka} , {\ldots }, \left( {\mathbf{H}_{14} } \right) _{ka} \) and \(\left( {\mathbf{H}_{15} } \right) _{kba} \), \(\left( {\mathbf{H}_{16} } \right) _{kba} , {\ldots }, \left( {\mathbf{H}_{28} } \right) _{kba} \) in Eqs. (17) and (18) are listed as follows

$$\begin{aligned} \left( {\mathbf{H}_1 } \right) _{ka}= & {} 2\rho \int _{V_0 } {\left( {\mathbf{N}_e } \right) _{jk} \left( {\mathbf{N}_{e,m_1 } } \right) _{ja} \hbox {d}V} ,~~~~\;~~\nonumber \\ \left( {\mathbf{H}_2 } \right) _{ka}= & {} 2\rho \int _{V_0 } {\left( {\mathbf{N}_e } \right) _{jk} \left( {\mathbf{N}_{e,n_1 } } \right) _{ja} \hbox {d}V} ,~~~~\;~~\nonumber \\ \left( {\mathbf{H}_3 } \right) _{ka}= & {} 2\rho \int _{V_0 } {\left( {\mathbf{N}_e } \right) _{jk} \left( {\mathbf{N}_{e,m_2 } } \right) _{ja} \hbox {d}V} ,~~~\;~~\nonumber \\ \left( {\mathbf{H}_4 } \right) _{ka}= & {} 2\rho \int _{V_0 } {\left( {\mathbf{N}_e } \right) _{jk} \left( {\mathbf{N}_{e,n_2 } } \right) _{ja} \hbox {d}V} ,~~~~\;~~\nonumber \\ \left( {\mathbf{H}_5 } \right) _{ka}= & {} \rho \int _{V_0 } {\left( {\mathbf{N}_e } \right) _{jk} \left( {\mathbf{N}_{e,m_1 m_1 } } \right) _{ja} \hbox {d}V} ~,~~~~~~~~\nonumber \\ \left( {\mathbf{H}_6 } \right) _{ka}= & {} \rho \int _{V_0 } {\left( {\mathbf{N}_e } \right) _{jk} \left( {\mathbf{N}_{e,n_1 n_1 } } \right) _{ja} \hbox {d}V} ,~~~~~~~~~~~\nonumber \\ \left( {\mathbf{H}_7 } \right) _{ka}= & {} \rho \int _{V_0 } {\left( {\mathbf{N}_e } \right) _{jk} \left( {\mathbf{N}_{e,m_2 m_2 } } \right) _{ja} \hbox {d}V} ,~~~~~~~~\nonumber \\ \left( {\mathbf{H}_8 } \right) _{ka}= & {} \rho \int _{V_0 } {\left( {\mathbf{N}_e } \right) _{jk} \left( {\mathbf{N}_{e,n_2 n_2 } } \right) _{ja} \hbox {d}V} ,~~~~~~~~~~\nonumber \\ \left( {\mathbf{H}_9 } \right) _{ka}= & {} 2\rho \int _{V_0 } {\left( {\mathbf{N}_e } \right) _{jk} \left( {\mathbf{N}_{e,m_1 n_1 } } \right) _{ja} \hbox {d}V} ,~~~~\nonumber \\ \left( {\mathbf{H}_{10} } \right) _{ka}= & {} 2\rho \int _{V_0 } {\left( {\mathbf{N}_e } \right) _{jk} \left( {\mathbf{N}_{e,m_1 m_2 } } \right) _{ja} \hbox {d}V} ,\nonumber \\ \left( {\mathbf{H}_{11} } \right) _{ka}= & {} 2\rho \int _{V_0 } {\left( {\mathbf{N}_e } \right) _{jk} \left( {\mathbf{N}_{e,m_1 n_2 } } \right) _{ja} \hbox {d}V} ,~~\nonumber \\ \left( {\mathbf{H}_{12} } \right) _{ka}= & {} 2\rho \int _{V_0 } {\left( {\mathbf{N}_e } \right) _{jk} \left( {\mathbf{N}_{e,n_1 m_2 } } \right) _{ja} \hbox {d}V} ,~\nonumber \\ \left( {\mathbf{H}_{13} } \right) _{ka}= & {} 2\rho \int _{V_0 } {\left( {\mathbf{N}_e } \right) _{jk} \left( {\mathbf{N}_{e,n_1 n_2 } } \right) _{ja} \hbox {d}V} ,~~\nonumber \\ \left( {\mathbf{H}_{14} } \right) _{ka}= & {} 2\rho \int _{V_0 } {\left( {\mathbf{N}_e } \right) _{jk} \left( {\mathbf{N}_{e,m_2 n_2 } } \right) _{ja} \hbox {d}V} , \end{aligned}$$
(A.4)

and

$$\begin{aligned} \left( {\mathbf{H}_{15} } \right) _{kba}= & {} 2\rho \int _{V_0 } {\left( {\mathbf{N}_{e,q_k } } \right) _{jb} \left( {\mathbf{N}_{e,m_1 } } \right) _{ja} \hbox {d}V} ,~~~~~~~~\nonumber \\ \left( {\mathbf{H}_{16} } \right) _{kba}= & {} 2\rho \int _{V_0 } {\left( {\mathbf{N}_{e,q_k } } \right) _{jb} \left( {\mathbf{N}_{e,n_1 } } \right) _{ja} \hbox {d}V} ,~~~~~~~~~\nonumber \\ \left( {\mathbf{H}_{17} } \right) _{kba}= & {} 2\rho \int _{V_0 } {\left( {\mathbf{N}_{e,q_k } } \right) _{jb} \left( {\mathbf{N}_{e,m_2 } } \right) _{ja} \hbox {d}V} ,~~~~~~~~\nonumber \\ \left( {\mathbf{H}_{18} } \right) _{kba}= & {} 2\rho \int _{V_0 } {\left( {\mathbf{N}_{e,q_k } } \right) _{jb} \left( {\mathbf{N}_{e,n_2 } } \right) _{ja} \hbox {d}V} ,~~~~~~~~~\nonumber \\ \left( {\mathbf{H}_{19} } \right) _{kba}= & {} \rho \int _{V_0 } {\left( {\mathbf{N}_{e,q_k } } \right) _{jb} \left( {\mathbf{N}_{e,m_1 m_1 } } \right) _{ja} \hbox {d}V} ,\;~~~~\nonumber \\ \left( {\mathbf{H}_{20} } \right) _{kba}= & {} \rho \int _{V_0 } {\left( {\mathbf{N}_{e,q_k } } \right) _{jb} \left( {\mathbf{N}_{e,n_1 n_1 } } \right) _{ja} \hbox {d}V} ,\;~~~~~~\nonumber \\ \left( {\mathbf{H}_{21} } \right) _{kba}= & {} \rho \int _{V_0 } {\left( {\mathbf{N}_{e,q_k } } \right) _{jb} \left( {\mathbf{N}_{e,m_2 m_2 } } \right) _{ja} \hbox {d}V} ,~\;~~\nonumber \\ \left( {\mathbf{H}_{22} } \right) _{kba}= & {} \rho \int _{V_0 } {\left( {\mathbf{N}_{e,q_k } } \right) _{jb} \left( {\mathbf{N}_{e,n_2 n_2 } } \right) _{ja} \hbox {d}V} ,~~~~~~~~\nonumber \\ \left( {\mathbf{H}_{23} } \right) _{kba}= & {} 2\rho \int _{V_0 } {\left( {\mathbf{N}_{e,q_k } } \right) _{jb} \left( {\mathbf{N}_{e,m_1 n_1 } } \right) _{ja} \hbox {d}V} ,~~\nonumber \\ \left( {\mathbf{H}_{24} } \right) _{kba}= & {} 2\rho \int _{V_0 } {\left( {\mathbf{N}_{e,q_k } } \right) _{jb} \left( {\mathbf{N}_{e,m_1 m_2 } } \right) _{ja} \hbox {d}V} ,\nonumber \\ \left( {\mathbf{H}_{25} } \right) _{kba}= & {} 2\rho \int _{V_0 } {\left( {\mathbf{N}_{e,q_k } } \right) _{jb} \left( {\mathbf{N}_{e,m_1 n_2 } } \right) _{ja} \hbox {d}V~} ,\nonumber \\ \left( {\mathbf{H}_{26} } \right) _{kba}= & {} 2\rho \int _{V_0 } {\left( {\mathbf{N}_{e,q_k } } \right) _{jb} \left( {\mathbf{N}_{e,n_1 m_2 } } \right) _{ja} \hbox {d}V} ,~\nonumber \\ \left( {\mathbf{H}_{27} } \right) _{kba}= & {} 2\rho \int _{V_0 } {\left( {\mathbf{N}_{e,q_k } } \right) _{jb} \left( {\mathbf{N}_{e,n_1 n_2 } } \right) _{ja} \hbox {d}V} ,~~\nonumber \\ \left( {\mathbf{H}_{28} } \right) _{kba}= & {} 2\rho \int _{V_0 } {\left( {\mathbf{N}_{e,q_k } } \right) _{jb} \left( {\mathbf{N}_{e,m_2 n_2 } } \right) _{ja} \hbox {d}V} . \end{aligned}$$
(A.5)

In Eqs. (A.4) and (A.5), the subscripts \(a,\;b=1,\;2,\;\ldots ,\;36\) and \(j=1,2,3\).

Likewise, the expressions of \(\left( {\mathbf{H}_1 } \right) _{ka,q_l } \), \(\left( {\mathbf{H}_2 } \right) _{ka,q_l } \), ..., \(\left( {\mathbf{H}_{14} } \right) _{ka,q_l } \) and \(\left( {\mathbf{H}_{15} } \right) _{kba,q_l } \), \(\left( {\mathbf{H}_{16} } \right) _{kba,q_l } \), ..., \(\left( {\mathbf{H}_{28} } \right) _{kba,q_l } \) in Eqs. (19)–(22) are listed as follows

$$\begin{aligned} \left( {\mathbf{H}_1 } \right) _{ka,q_l }= & {} 2\rho \int _{V_0 } \left[ \left( {\mathbf{N}_{e,q_l } } \right) _{jk} \left( {\mathbf{N}_{e,m_1 } } \right) _{ja} \right. \nonumber \\&\left. {+} \left( {\mathbf{N}_e } \right) _{jk} \left( {\mathbf{N}_{e,m_1 q_l } } \right) _{ja} \right] \hbox {d}V ,\;\;~~~~~~~~~~~~ \nonumber \\ \left( {\mathbf{H}_2 } \right) _{ka,q_l }= & {} 2\rho \int _{V_0 } \left[ \left( {\mathbf{N}_{e,q_l } } \right) _{jk} \left( {\mathbf{N}_{e,n_1 } } \right) _{ja} \right. \nonumber \\&\left. {+} \left( {\mathbf{N}_e } \right) _{jk} \left( {\mathbf{N}_{e,n_1 q_l } } \right) _{ja} \right] \hbox {d}V ,\;\;~~~~~~~~~~~~~ \nonumber \\ \left( {\mathbf{H}_3 } \right) _{ka,q_l }= & {} 2\rho \int _{V_0 } \left[ \left( {\mathbf{N}_{e,q_l } } \right) _{jk} \left( {\mathbf{N}_{e,m_2 } } \right) _{ja} \right. \nonumber \\&\left. {+} \left( {\mathbf{N}_e } \right) _{jk} \left( {\mathbf{N}_{e,m_2 q_l } } \right) _{ja} \right] \hbox {d}V ,\;\;~~~~~~~~~~ \nonumber \\ \left( {\mathbf{H}_4 } \right) _{ka,q_l }= & {} 2\rho \int _{V_0 } \left[ \left( {\mathbf{N}_{e,q_l } } \right) _{jk} \left( {\mathbf{N}_{e,n_2 } } \right) _{ja} \right. \nonumber \\&\left. {+} \left( {\mathbf{N}_e } \right) _{jk} \left( {\mathbf{N}_{e,n_2 q_l } } \right) _{ja} \right] \hbox {d}V ,\;\;~~~~~~~~~~~ \nonumber \\ \left( {\mathbf{H}_5 } \right) _{ka,q_l }= & {} \rho \int _{V_0 } \left[ \left( {\mathbf{N}_{e,q_l } } \right) _{jk} \left( {\mathbf{N}_{e,m_1 m_1 } } \right) _{ja} \right. \nonumber \\&\left. {+} \left( {\mathbf{N}_e } \right) _{jk} \left( {\mathbf{N}_{e,m_1 m_1 q_l } } \right) _{ja} \right] \hbox {d}V ,\;\;~~~ \nonumber \\ \left( {\mathbf{H}_6 } \right) _{ka,q_l }= & {} \rho \int _{V_0 } \left[ \left( {\mathbf{N}_{e,q_l } } \right) _{jk} \left( {\mathbf{N}_{e,n_1 n_1 } } \right) _{ja} \right. \nonumber \\&\left. {+} \left( {\mathbf{N}_e } \right) _{jk} \left( {\mathbf{N}_{e,n_1 n_1 q_l } } \right) _{ja} \right] \hbox {d}V ,\;\;~~~~~~~~ \nonumber \\ \left( {\mathbf{H}_7 } \right) _{ka,q_l }= & {} \rho \int _{V_0 } \left[ \left( {\mathbf{N}_{e,q_l } } \right) _{jk} \left( {\mathbf{N}_{e,m_2 m_2 } } \right) _{ja} \right. \nonumber \\&\left. {+} \left( {\mathbf{N}_e } \right) _{jk} \left( {\mathbf{N}_{e,m_2 m_2 q_l } } \right) _{ja} \right] \hbox {d}V ,\;~~~~\nonumber \\ \left( {\mathbf{H}_8 } \right) _{ka,q_l }= & {} \rho \int _{V_0 } \left[ \left( {\mathbf{N}_{e,q_l } } \right) _{jk} \left( {\mathbf{N}_{e,n_2 n_2 } } \right) _{ja} \right. \nonumber \\&\left. {+} \left( {\mathbf{N}_e } \right) _{jk} \left( {\mathbf{N}_{e,n_2 n_2 q_l } } \right) _{ja} \right] \hbox {d}V ,\;\;~~~~ \nonumber \\ \left( {\mathbf{H}_9 } \right) _{ka,q_l }= & {} 2\rho \int _{V_0 } \left[ \left( {\mathbf{N}_{e,q_l } } \right) _{jk} \left( {\mathbf{N}_{e,m_1 n_1 } } \right) _{ja} \right. \nonumber \\&\left. {+} \left( {\mathbf{N}_e } \right) _{jk} \left( {\mathbf{N}_{e,m_1 n_1 q_l } } \right) _{ja} \right] \hbox {d}V ,\;~~~ \nonumber \\ \left( {\mathbf{H}_{10} } \right) _{ka,q_l }= & {} 2\rho \int _{V_0 } \left[ \left( {\mathbf{N}_{e,q_l } } \right) _{jk} \left( {\mathbf{N}_{e,m_1 m_2 } } \right) _{ja} \right. \nonumber \\&\left. {+} \left( {\mathbf{N}_e } \right) _{jk} \left( {\mathbf{N}_{e,m_1 m_2 q_l } } \right) _{ja} \right] \hbox {d}V , \nonumber \\ \left( {\mathbf{H}_{11} } \right) _{ka,q_l }= & {} 2\rho \int _{V_0 } \left[ \left( {\mathbf{N}_{e,q_l } } \right) _{jk} \left( {\mathbf{N}_{e,m_1 n_2 } } \right) _{ja} \right. \nonumber \\&\left. {+} \left( {\mathbf{N}_e } \right) _{jk} \left( {\mathbf{N}_{e,m_1 n_2 q_l } } \right) _{ja} \right] \hbox {d}V ,~~ \nonumber \\ \left( {\mathbf{H}_{12} } \right) _{ka,q_l }= & {} 2\rho \int _{V_0 } \left[ \left( {\mathbf{N}_{e,q_l } } \right) _{jk} \left( {\mathbf{N}_{e,n_1 m_2 } } \right) _{ja} \right. \nonumber \\&\left. {+} \left( {\mathbf{N}_e } \right) _{jk} \left( {\mathbf{N}_{e,n_1 m_2 q_l } } \right) _{ja} \right] \hbox {d}V ,~~ \nonumber \\ \left( {\mathbf{H}_{13} } \right) _{ka,q_l }= & {} 2\rho \int _{V_0 } \left[ \left( {\mathbf{N}_{e,q_l } } \right) _{jk} \left( {\mathbf{N}_{e,n_1 n_2 } } \right) _{ja} \right. \nonumber \\&\left. {+} \left( {\mathbf{N}_e } \right) _{jk} \left( {\mathbf{N}_{e,n_1 n_2 q_l } } \right) _{ja} \right] \hbox {d}V ,~~~~ \nonumber \\ \left( {\mathbf{H}_{14} } \right) _{ka,q_l }= & {} 2\rho \int _{V_0 } \left[ \left( {\mathbf{N}_{e,q_l } } \right) _{jk} \left( {\mathbf{N}_{e,m_2 n_2 } } \right) _{ja} \right. \nonumber \\&\left. {+} \left( {\mathbf{N}_e } \right) _{jk} \left( {\mathbf{N}_{e,m_2 n_2 q_l } } \right) _{ja} \right] \hbox {d}V , \end{aligned}$$
(A.6)

and

$$\begin{aligned} \left( {\mathbf{H}_{15} } \right) _{kba,q_l }= & {} 2\rho \int _{V_0 } \left[ \left( {\mathbf{N}_{e,q_k q_l } } \right) _{jb} \left( {\mathbf{N}_{e,m_1 } } \right) _{ja} \right. \nonumber \\&\left. {+}\left( {\mathbf{N}_{e,q_k } } \right) _{jb} \left( {\mathbf{N}_{e,m_1 q_l } } \right) _{ja} \right] \hbox {d}V ,\;\;\;~~~~~~ \nonumber \\ \left( {\mathbf{H}_{16} } \right) _{kba,q_l }= & {} 2\rho \int _{V_0 } \left[ \left( {\mathbf{N}_{e,q_k q_l } } \right) _{jb} \left( {\mathbf{N}_{e,n_1 } } \right) _{ja} \right. \nonumber \\&\left. {+}\left( {\mathbf{N}_{e,q_k } } \right) _{jb} \left( {\mathbf{N}_{e,n_1 q_l } } \right) _{ja} \right] \hbox {d}V ,\;\;\;\;~~~~~ \nonumber \\ \left( {\mathbf{H}_{17} } \right) _{kba,q_l }= & {} 2\rho \int _{V_0 } \left[ \left( {\mathbf{N}_{e,q_k q_l } } \right) _{jb} \left( {\mathbf{N}_{e,m_2 } } \right) _{ja} \right. \nonumber \\&\left. {+}\left( {\mathbf{N}_{e,q_k } } \right) _{jb} \left( {\mathbf{N}_{e,m_2 q_l } } \right) _{ja} \right] \hbox {d}V ,\;\;~~~~~~~~ \nonumber \\ \left( {\mathbf{H}_{18} } \right) _{kba,q_l }= & {} 2\rho \int _{V_0 } \left[ \left( {\mathbf{N}_{e,q_k q_l } } \right) _{jb} \left( {\mathbf{N}_{e,n_2 } } \right) _{ja} \right. \nonumber \\&\left. {+}\left( {\mathbf{N}_{e,q_k } } \right) _{jb} \left( {\mathbf{N}_{e,n_2 q_l } } \right) _{ja} \right] \hbox {d}V ,\;\;\;~~~~~~~ \nonumber \\ \left( {\mathbf{H}_{19} } \right) _{kba,q_l }= & {} \rho \int _{V_0 } \left[ \left( {\mathbf{N}_{e,q_k q_l } } \right) _{jb} \left( {\mathbf{N}_{e,m_1 m_1 } } \right) _{ja} \right. \nonumber \\&\left. {+}\left( {\mathbf{N}_{e,q_k } } \right) _{jb} \left( {\mathbf{N}_{e,m_1 m_1 q_l } } \right) _{ja} \right] \hbox {d}V ,\;~~~~~ \nonumber \\ \left( {\mathbf{H}_{20} } \right) _{kba,q_l }= & {} \rho \int _{V_0 } \left[ \left( {\mathbf{N}_{e,q_k q_l } } \right) _{jb} \left( {\mathbf{N}_{e,n_1 n_1 } } \right) _{ja} \right. \nonumber \\&\left. {+}\left( {\mathbf{N}_{e,q_k } } \right) _{jb} \left( {\mathbf{N}_{e,n_1 n_1 q_l } } \right) _{ja} \right] \hbox {d}V ,\nonumber \\ \left( {\mathbf{H}_{21} } \right) _{kba,q_l }= & {} \rho \int _{V_0 } \left[ \left( {\mathbf{N}_{e,q_k q_l } } \right) _{jb} \left( {\mathbf{N}_{e,m_2 m_2 } } \right) _{ja} \right. \nonumber \\&\left. {+}\left( {\mathbf{N}_{e,q_k } } \right) _{jb} \left( {\mathbf{N}_{e,m_2 m_2 q_l } } \right) _{ja} \right] \hbox {d}V ,\;\nonumber \\ \left( {\mathbf{H}_{22} } \right) _{kba,q_l }= & {} \rho \int _{V_0 } \left[ \left( {\mathbf{N}_{e,q_k q_l } } \right) _{jb} \left( {\mathbf{N}_{e,n_2 n_2 } } \right) _{ja} \right. \nonumber \\&\left. {+}\left( {\mathbf{N}_{e,q_k } } \right) _{jb} \left( {\mathbf{N}_{e,n_2 n_2 q_l } } \right) _{ja} \right] \hbox {d}V ,\;\;~~~\nonumber \\ \left( {\mathbf{H}_{23} } \right) _{kba,q_l }= & {} 2\rho \int _{V_0 } \left[ \left( {\mathbf{N}_{e,q_k q_l } } \right) _{jb} \left( {\mathbf{N}_{e,m_1 n_1 } } \right) _{ja} \right. \nonumber \\&\left. {+}\left( {\mathbf{N}_{e,q_k } } \right) _{jb} \left( {\mathbf{N}_{e,m_1 n_1 q_l } } \right) _{ja} \right] \hbox {d}V ,~~~~~ \nonumber \\ \left( {\mathbf{H}_{24} } \right) _{kba,q_l }= & {} 2\rho \int _{V_0 } \left[ \left( {\mathbf{N}_{e,q_k q_l } } \right) _{jb} \left( {\mathbf{N}_{e,m_1 m_2 } } \right) _{ja} \right. \nonumber \\&\left. {+}\left( {\mathbf{N}_{e,q_k } } \right) _{jb} \left( {\mathbf{N}_{e,m_1 m_2 q_l } } \right) _{ja} \right] \hbox {d}V , \nonumber \\ \left( {\mathbf{H}_{25} } \right) _{kba,q_l }= & {} 2\rho \int _{V_0 } \left[ \left( {\mathbf{N}_{e,q_k q_l } } \right) _{jb} \left( {\mathbf{N}_{e,m_1 n_2 } } \right) _{ja} \right. \nonumber \\&\left. {+}\left( {\mathbf{N}_{e,q_k } } \right) _{jb} \left( {\mathbf{N}_{e,m_1 n_2 q_l } } \right) _{ja} \right] \hbox {d}V ,\nonumber \\ \left( {\mathbf{H}_{26} } \right) _{kba,q_l }= & {} 2\rho \int _{V_0 } \left[ \left( {\mathbf{N}_{e,q_k q_l } } \right) _{jb} \left( {\mathbf{N}_{e,n_1 m_2 } } \right) _{ja} \right. \nonumber \\&\left. {+}\left( {\mathbf{N}_{e,q_k } } \right) _{jb} \left( {\mathbf{N}_{e,n_1 m_2 q_l } } \right) _{ja} \right] \hbox {d}V ,\nonumber \\ \left( {\mathbf{H}_{27} } \right) _{kba,q_l }= & {} 2\rho \int _{V_0 } \left[ \left( {\mathbf{N}_{e,q_k q_l } } \right) _{jb} \left( {\mathbf{N}_{e,n_1 n_2 } } \right) _{ja} \right. \nonumber \\&\left. {+}\left( {\mathbf{N}_{e,q_k } } \right) _{jb} \left( {\mathbf{N}_{e,n_1 n_2 q_l } } \right) _{ja} \right] \hbox {d}V ,\nonumber \\ \left( {\mathbf{H}_{28} } \right) _{kba,q_l }= & {} 2\rho \int _{V_0 } \left[ \left( {\mathbf{N}_{e,q_k q_l } } \right) _{jb} \left( {\mathbf{N}_{e,m_2 n_2 } } \right) _{ja} \right. \nonumber \\&\left. {+}\left( {\mathbf{N}_{e,q_k } } \right) _{jb} \left( {\mathbf{N}_{e,m_2 n_2 q_l } } \right) _{ja} \right] \hbox {d}V . \end{aligned}$$
(A.7)

In Eqs. (A.6) and (A.7), the subscripts \(a,\;b=1,\;2,\;\ldots ,\;36\) and \(j=1,2,3\).

The matrices in Eqs. (A.2)–(A.7) are all invariant and can be computed and stored with sparse matrix technique in the preprocessing procedure to greatly improve the computation efficiency.

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Sun, J., Tian, Q., Hu, H. et al. Topology optimization of a flexible multibody system with variable-length bodies described by ALE–ANCF. Nonlinear Dyn 93, 413–441 (2018). https://doi.org/10.1007/s11071-018-4201-6

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