Appendix A: the range for \(r_a\) and \(\psi\)
This appendix concludes the range for \(r_a\) and \(\psi\) by finding the maximal and minimal values of Eqs. (15) and (16) in the scope \(D = \{\left( \varsigma ,\alpha \right) |\frac{1}{\varsigma _m}\le \varsigma \le \varsigma _m, -\alpha _m\le \alpha \le \alpha _m\}\).
To be more intuitive, Eqs. (15) and (16) are rewritten as
$$\begin{aligned} r_a^2 = \frac{1+\varsigma ^4}{\left( \varsigma ^4-1\right) \cos ^2\alpha +1}-1 \end{aligned}$$
(A1)
and
$$\begin{aligned} \cos ^2\psi = 1-\frac{4}{4+\left( \varsigma ^2-\frac{1}{\varsigma ^2}\right) ^2\sin ^22\alpha }, \end{aligned}$$
(A2)
, respectively.
We consider \(r_a\) first. If \(\alpha _m \ge \frac{\pi }{2}\), it is evident that
$$\begin{aligned}&\frac{1}{\varsigma ^4_m}\le \frac{1}{\varsigma ^4} \le \frac{1+\varsigma ^4}{\left( \varsigma ^4-1\right) \cos ^2\alpha +1}\nonumber \\&-1 \le \varsigma ^4 \le \varsigma _m^4, \quad 1 \le \varsigma \le \varsigma _m \end{aligned}$$
(A3a)
$$\begin{aligned}&\frac{1}{\varsigma ^4_m}\le \varsigma ^4 \le \frac{1+\varsigma ^4}{\left( \varsigma ^4-1\right) \cos ^2\alpha +1}\nonumber \\&-1 \le \frac{1}{\varsigma ^4} \le \varsigma _m^4, \quad \frac{1}{\varsigma _m} \le \varsigma \le 1. \end{aligned}$$
(A3b)
As for the case of \(\alpha _m < \frac{\pi }{2}\), we have
$$\begin{aligned}&\frac{1}{\varsigma ^4_m}\le \frac{1}{\varsigma ^4} \le \frac{1+\varsigma ^4}{\left( \varsigma ^4-1\right) \cos ^2\alpha +1}\nonumber \\&-1 \le \frac{1+\varsigma ^4}{\left( \varsigma ^4-1\right) \cos ^2\alpha _m+1}-1 < \varsigma ^4, \quad 1 \le \varsigma \le \varsigma _m \end{aligned}$$
(A4a)
$$\begin{aligned}&\varsigma ^4 < \frac{1+\varsigma ^4}{\left( \varsigma ^4-1\right) \cos ^2\alpha _m+1}\nonumber \\&-1 \le \frac{1+\varsigma ^4}{\left( \varsigma ^4-1\right) \cos ^2\alpha +1}-1 \le \frac{1}{\varsigma ^4} \le \varsigma _m^4, \quad \frac{1}{\varsigma _m} \le \varsigma \le 1. \end{aligned}$$
(A4b)
Thus, when \(\left( \varsigma ,\alpha \right) \in D\), the minimal value of \(r_a\) is \(\frac{1}{\varsigma ^2_m}\), and the maximum is \(\varsigma _m^2\).
Then, we analyse Eq. (A2). Note that \(\cos ^2\psi\) increases with the growth of \(\sin ^2 2\alpha\) and \(\left( \varsigma ^2-\frac{1}{\varsigma ^2}\right) ^2\). Thus
$$\begin{aligned} 0 \le \cos ^2\psi \le \left\{ \begin{aligned}&1-\frac{4}{4+\left( \varsigma _m^2-\frac{1}{\varsigma _m^2}\right) ^2\sin ^22\alpha _m}=\frac{\left( \varsigma _m^4-1\right) ^2\sin ^2 2\alpha _m}{\left( \varsigma _m^4-1\right) ^2\sin ^2 2\alpha _m+4\varsigma _m^4},\quad 0\le \alpha _m < \frac{\pi }{4,}\\&1-\frac{4}{4+\left( \varsigma _m^2-\frac{1}{\varsigma _m^2}\right) ^2} = \left( \frac{\varsigma ^4_m-1}{\varsigma ^4_m+1}\right) ^2, \quad \alpha _m\ge \frac{\pi }{4} \end{aligned} \right. \end{aligned}$$
(A5)
for each \(\left( \varsigma ,\alpha \right) \in D\). Therefore, the range for \(\psi\) is determined.
Appendix B: derivation of the geometric-feature-based governing equations
This appendix proves that \(\lambda _l\), \(\varsigma _l\), \(\alpha\) , and \(\beta\) satisfy Eq. (21). Firstly, expanding Eqs. (20a, 20b) and substituting
$$\begin{aligned} \lambda = e^{\lambda _l}, \qquad \varsigma = e^{\varsigma _l} \end{aligned}$$
(B6)
into gives
$$\begin{aligned} \begin{aligned}&e^{-\lambda _l}\left( e^{-\varsigma _l}\sin \beta \cos \alpha -e^{\varsigma _l}\cos \beta \sin \alpha \right) \mathchoice{\frac{{\partial }\lambda _l}{{\partial }x_1}}{{\partial }\lambda _l/{\partial }x_1}{{\partial }\lambda _l/{\partial }x_1}{{\partial }\lambda _l/{\partial }x_1}\\&-e^{-\lambda _l}\left( e^{-\varsigma _l}\cos \beta \cos \alpha +e^{\varsigma _l}\sin \beta \sin \alpha \right) \mathchoice{\frac{{\partial }\lambda _l}{{\partial }x_2}}{{\partial }\lambda _l/{\partial }x_2}{{\partial }\lambda _l/{\partial }x_2}{{\partial }\lambda _l/{\partial }x_2}\\&+e^{-\lambda _l}\left( e^{-\varsigma _l}\sin \beta \cos \alpha +e^{\varsigma _l}\cos \beta \sin \alpha \right) \mathchoice{\frac{{\partial }\varsigma _l}{{\partial }x_1}}{{\partial }\varsigma _l/{\partial }x_1}{{\partial }\varsigma _l/{\partial }x_1}{{\partial }\varsigma _l/{\partial }x_1}\\&+e^{-\lambda _l}\left( -e^{-\varsigma _l}\cos \beta \cos \alpha +e^{\varsigma _l}\sin \beta \sin \alpha \right) \mathchoice{\frac{{\partial }\varsigma _l}{{\partial }x_2}}{{\partial }\varsigma _l/{\partial }x_2}{{\partial }\varsigma _l/{\partial }x_2}{{\partial }\varsigma _l/{\partial }x_2}\\&+e^{-\lambda _l}\left( e^{-\varsigma _l}\sin \beta \sin \alpha +e^{\varsigma _l}\cos \beta \cos \alpha \right) \mathchoice{\frac{{\partial }\alpha }{{\partial }x_1}}{{\partial }\alpha /{\partial }x_1}{{\partial }\alpha /{\partial }x_1}{{\partial }\alpha /{\partial }x_1}\\&+e^{-\lambda _l}\left( -e^{-\varsigma _l}\cos \beta \sin \alpha +e^{\varsigma _l}\sin \beta \cos \alpha \right) \mathchoice{\frac{{\partial }\alpha }{{\partial }x_2}}{{\partial }\alpha /{\partial }x_2}{{\partial }\alpha /{\partial }x_2}{{\partial }\alpha /{\partial }x_2}\\&-e^{-\lambda _l}\left( e^{-\varsigma _l}\cos \beta \cos \alpha +e^{\varsigma _l}\sin \beta \sin \alpha \right) \mathchoice{\frac{{\partial }\beta }{{\partial }x_1}}{{\partial }\beta /{\partial }x_1}{{\partial }\beta /{\partial }x_1}{{\partial }\beta /{\partial }x_1}\\&-e^{-\lambda _l}\left( e^{-\varsigma _l}\sin \beta \cos \alpha -e^{\varsigma _l}\cos \beta \sin \alpha \right) \mathchoice{\frac{{\partial }\beta }{{\partial }x_2}}{{\partial }\beta /{\partial }x_2}{{\partial }\beta /{\partial }x_2}{{\partial }\beta /{\partial }x_2}\\&=0, \end{aligned} \end{aligned}$$
(B7a)
$$\begin{aligned} \begin{aligned}&e^{-\lambda _l}\left( e^{-\varsigma _l}\sin \beta \sin \alpha +e^{\varsigma _l}\cos \beta \cos \alpha \right) \mathchoice{\frac{{\partial }\lambda _l}{{\partial }x_1}}{{\partial }\lambda _l/{\partial }x_1}{{\partial }\lambda _l/{\partial }x_1}{{\partial }\lambda _l/{\partial }x_1}\\&-e^{-\lambda _l}\left( e^{-\varsigma _l}\cos \beta \sin \alpha -e^{\varsigma _l}\sin \beta \cos \alpha \right) \mathchoice{\frac{{\partial }\lambda _l}{{\partial }x_2}}{{\partial }\lambda _l/{\partial }x_2}{{\partial }\lambda _l/{\partial }x_2}{{\partial }\lambda _l/{\partial }x_2}\\&+e^{-\lambda _l}\left( e^{-\varsigma _l}\sin \beta \sin \alpha -e^{\varsigma _l}\cos \beta \cos \alpha \right) \mathchoice{\frac{{\partial }\varsigma _l}{{\partial }x_1}}{{\partial }\varsigma _l/{\partial }x_1}{{\partial }\varsigma _l/{\partial }x_1}{{\partial }\varsigma _l/{\partial }x_1}\\&-e^{-\lambda _l}\left( e^{-\varsigma _l}\cos \beta \sin \alpha +e^{\varsigma _l}\sin \beta \cos \alpha \right) \mathchoice{\frac{{\partial }\varsigma _l}{{\partial }x_2}}{{\partial }\varsigma _l/{\partial }x_2}{{\partial }\varsigma _l/{\partial }x_2}{{\partial }\varsigma _l/{\partial }x_2}\\&-e^{-\lambda _l}\left( e^{-\varsigma _l}\sin \beta \cos \alpha -e^{\varsigma _l}\cos \beta \sin \alpha \right) \mathchoice{\frac{{\partial }\alpha }{{\partial }x_1}}{{\partial }\alpha /{\partial }x_1}{{\partial }\alpha /{\partial }x_1}{{\partial }\alpha /{\partial }x_1}\\&+e^{-\lambda _l}\left( e^{-\varsigma _l}\cos \beta \cos \alpha +e^{\varsigma _l}\sin \beta \sin \alpha \right) \mathchoice{\frac{{\partial }\alpha }{{\partial }x_2}}{{\partial }\alpha /{\partial }x_2}{{\partial }\alpha /{\partial }x_2}{{\partial }\alpha /{\partial }x_2}\\&-e^{-\lambda _l}\left( e^{-\varsigma _l}\cos \beta \sin \alpha -e^{\varsigma _l}\sin \beta \cos \alpha \right) \mathchoice{\frac{{\partial }\beta }{{\partial }x_1}}{{\partial }\beta /{\partial }x_1}{{\partial }\beta /{\partial }x_1}{{\partial }\beta /{\partial }x_1}\\&-e^{-\lambda _l}\left( e^{-\varsigma _l}\sin \beta \sin \alpha +e^{\varsigma _l}\cos \beta \cos \alpha \right) \mathchoice{\frac{{\partial }\beta }{{\partial }x_2}}{{\partial }\beta /{\partial }x_2}{{\partial }\beta /{\partial }x_2}{{\partial }\beta /{\partial }x_2}\\&=0. \end{aligned} \end{aligned}$$
(B7b)
Then, re-writing Eqs. (B7a, B7b) in matrix form and extracting the common factor gives
$$\begin{aligned} \begin{aligned}&e^{-\lambda _l} \begin{bmatrix} e^{-\varsigma _l}\cos \alpha &{}-e^{\varsigma _l}\sin \alpha \\ e^{-\varsigma _l}\sin \alpha &{}e^{\varsigma _l}\cos \alpha \end{bmatrix} \left( \begin{bmatrix} \sin \beta &{}-\cos \beta \\ \cos \beta &{}\sin \beta \end{bmatrix} \begin{bmatrix} \mathchoice{\frac{{\partial }\lambda _l}{{\partial }x_1}}{{\partial }\lambda _l/{\partial }x_1}{{\partial }\lambda _l/{\partial }x_1}{{\partial }\lambda _l/{\partial }x_1}\\ \mathchoice{\frac{{\partial }\lambda _l}{{\partial }x_2}}{{\partial }\lambda _l/{\partial }x_2}{{\partial }\lambda _l/{\partial }x_2}{{\partial }\lambda _l/{\partial }x_2} \end{bmatrix}\right. \\&\left. +\begin{bmatrix} \sin \beta &{}-\cos \beta \\ -\cos \beta &{}-\sin \beta \end{bmatrix} \begin{bmatrix} \mathchoice{\frac{{\partial }\varsigma _l}{{\partial }x_1}}{{\partial }\varsigma _l/{\partial }x_1}{{\partial }\varsigma _l/{\partial }x_1}{{\partial }\varsigma _l/{\partial }x_1}\\ \mathchoice{\frac{{\partial }\varsigma _l}{{\partial }x_2}}{{\partial }\varsigma _l/{\partial }x_2}{{\partial }\varsigma _l/{\partial }x_2}{{\partial }\varsigma _l/{\partial }x_2} \end{bmatrix} \right. \\&\left. + \begin{bmatrix} e^{2\varsigma _l} \cos \beta &{}e^{2\varsigma _l} \sin \beta \\ -e^{-2\varsigma _l} \sin \beta &{}e^{-2\varsigma _l} \cos \beta \end{bmatrix} \begin{bmatrix} \mathchoice{\frac{{\partial }\alpha }{{\partial }x_1}}{{\partial }\alpha /{\partial }x_1}{{\partial }\alpha /{\partial }x_1}{{\partial }\alpha /{\partial }x_1}\\ \mathchoice{\frac{{\partial }\alpha }{{\partial }x_2}}{{\partial }\alpha /{\partial }x_2}{{\partial }\alpha /{\partial }x_2}{{\partial }\alpha /{\partial }x_2} \end{bmatrix}\right. \\&\left. +\begin{bmatrix} -\cos \beta &{}-\sin \beta \\ \sin \beta &{}-\cos \beta \end{bmatrix} \begin{bmatrix} \mathchoice{\frac{{\partial }\beta }{{\partial }x_1}}{{\partial }\beta /{\partial }x_1}{{\partial }\beta /{\partial }x_1}{{\partial }\beta /{\partial }x_1}\\ \mathchoice{\frac{{\partial }\beta }{{\partial }x_2}}{{\partial }\beta /{\partial }x_2}{{\partial }\beta /{\partial }x_2}{{\partial }\beta /{\partial }x_2} \end{bmatrix}\right) ={\begin{bmatrix} 0\\ 0 \end{bmatrix}}. \end{aligned} \end{aligned}$$
(B8)
Since \(e^{-\lambda _l} \left[ {\begin{array}{*{20}{c}} e^{-\varsigma _l}\cos \alpha &{} -e^{\varsigma _l}\sin \alpha \\ e^{-\varsigma _l}\sin \alpha &{} e^{\varsigma _l}\cos \alpha \end{array}} \right]\) is invertible, it can be eliminated from Eq. (B8). As a result, Eq. (21) is obtained.
Appendix C: the specific representation of the coefficient matrices
The representations and determinants of the coefficient matrices in Eq. (29) are listed in this appendix.
$$\begin{aligned}&\begin{aligned}&{\textbf {M}}^{\lambda _l\alpha } = -\left[ {\begin{array}{*{20}{c}} \cos \textit{h}2\varsigma _l+\sin \textit{h}2\varsigma _l\cos 2\beta &{} \sin \textit{h} 2\varsigma _l\sin 2\beta \\ \sin \textit{h} 2\varsigma _l\sin 2\beta &{} \cos \textit{h}2\varsigma _l-\sin \textit{h}2\varsigma _l\cos 2\beta \end{array}} \right] ,\\&{\textbf {M}}^{\alpha \lambda _l} = \left[ {\begin{array}{*{20}{c}} \cos \textit{h}2\varsigma _l+\sin \textit{h}2\varsigma _l\cos 2\beta &{} \sin \textit{h} 2\varsigma _l\sin 2\beta \\ \sin \textit{h} 2\varsigma _l\sin 2\beta &{} \cos \textit{h}2\varsigma _l-\sin \textit{h}2\varsigma _l\cos 2\beta \end{array}} \right] , \end{aligned} \end{aligned}$$
(C9a)
$$\begin{aligned}&\det {\textbf {M}}^{\lambda _l\alpha } = \det {\textbf {M}}^{\alpha \lambda _l}= 1, \end{aligned}$$
(C9b)
$$\begin{aligned}&{\textbf {M}}^{\lambda _l\beta } = \left[ {\begin{array}{*{20}{c}} 1&{}0\\ 0&{}1\ \end{array}} \right] ,\qquad {\textbf {M}}^{\beta \lambda _l} = -\left[ {\begin{array}{*{20}{c}} 1&{}0\\ 0&{}1 \end{array}} \right] , \end{aligned}$$
(C9c)
$$\begin{aligned}&\det {\textbf {M}}^{\lambda _l\beta } = \det {\textbf {M}}^{\beta \lambda _l} = 1, \end{aligned}$$
(C9d)
$$\begin{aligned}&{\textbf {M}}^{\varsigma _l\lambda _l} ={\textbf {M}}^{\lambda _l\varsigma _l} = \left[ {\begin{array}{*{20}{c}} -\sin 2\beta &{}\cos 2\beta \\ \cos 2\beta &{}\sin 2\beta \ \end{array}} \right] , \end{aligned}$$
(C9e)
$$\begin{aligned}&\det {\textbf {M}}^{\varsigma _l\lambda _l} = \det {\textbf {M}}^{\lambda _l\varsigma _l}=-1, \end{aligned}$$
(C9f)
$$\begin{aligned}&{\textbf {M}}^{\varsigma _l\beta }={\textbf {M}}^{\beta \varsigma _l}= \left[ {\begin{array}{*{20}{c}} \cos 2\beta &{}\sin 2\beta \\ \sin 2\beta &{}-\cos 2\beta \ \end{array}} \right] , \end{aligned}$$
(C9g)
$$\begin{aligned}&\det {\textbf {M}}^{\varsigma _l\beta }=\det {\textbf {M}}^{\beta \varsigma _l}=-1, \end{aligned}$$
(C9h)
$$\begin{aligned}&\begin{aligned}&{\textbf {M}}^{\alpha \beta }= \left[ {\begin{array}{*{20}{c}} \sin \textit{h}2\varsigma _l\sin 2\beta &{}-\cos \textit{h}2\varsigma _l-\sin \textit{h}2\varsigma _l\cos 2\beta \\ \cos \textit{h}2\varsigma _l-\sin \textit{h}2\varsigma _l\cos 2\beta &{}-\sin \textit{h}2\varsigma _l\sin 2\beta \end{array}} \right] ,\\&{\textbf {M}}^{\beta \alpha }= \left[ {\begin{array}{*{20}{c}} -\sin \textit{h}2\varsigma _l\sin 2\beta &{}-\cos \textit{h}2\varsigma _l+\sin \textit{h}2\varsigma _l\cos 2\beta \\ \cos \textit{h}2\varsigma _l+\sin \textit{h}2\varsigma _l\cos 2\beta &{}\sin \textit{h}2\varsigma _l\sin 2\beta \end{array}} \right] , \end{aligned} \end{aligned}$$
(C9i)
$$\begin{aligned}&\det {\textbf {M}}^{\alpha \beta }=1, \quad \det {\textbf {M}}^{\beta \alpha } = 1, \end{aligned}$$
(C9j)
$$\begin{aligned}&{\textbf {M}}^{\alpha \varsigma _l}={\textbf {M}}^{\varsigma _l\alpha } = -\left[ {\begin{array}{*{20}{c}} \sin \textit{h}2\varsigma _l+\cos \textit{h}2\varsigma _l\cos 2\beta &{}\cos \textit{h}2\varsigma _l\sin 2\beta \\ \cos \textit{h}2\varsigma _l\sin 2\beta &{}\sin \textit{h}2\varsigma _l-\cos \textit{h}2\varsigma _l\cos 2\beta \end{array}} \right] , \end{aligned}$$
(C9k)
$$\begin{aligned}&\det {\textbf {M}}^{\alpha \varsigma _l}=\det {\textbf {M}}^{\varsigma _l\alpha } = -1, \end{aligned}$$
(C9l)
Appendix D: specific expressions of the sensitivity of Mode 1 and Mode 2
This appendix provides the sensitivity expressions of the optimisation formulations Eqs. (45) and (46), respectively.
Here, the geometric quantity fields (i.e. design variables) \(\beta\), \(\varsigma _l\), and \(\alpha\) are digitally represented as
$$\begin{aligned}&\beta = \sum ^{n}_{k=1}N_k\beta ^k,\quad \beta ^k \in {\textbf {Bn}} = \left( \beta ^1,\beta ^2,\ldots ,\beta ^n\right) ^{\top }, \end{aligned}$$
(D10a)
$$\begin{aligned}&\varsigma _l = \sum ^{n}_{k=1}N_k\varsigma _l^k,\quad \varsigma _l^k \in {\textbf {Sn}} = \left( \varsigma _l^1,\varsigma _l^2,\ldots ,\varsigma _l^n\right) ^{\top }, \end{aligned}$$
(D10b)
$$\begin{aligned}&\alpha = \sum ^{n}_{k=1}N_k \alpha ^k, \quad \alpha ^k \in {\textbf {An}} = \left( \alpha ^1,\alpha ^2,\ldots ,\alpha ^n\right) ^{\top }, \end{aligned}$$
(D10c)
where \(N_k\) represents the shape function of elements; \(\varsigma _l^k\), \(\alpha ^k\) and \(\beta ^k\) denote the values of the corresponding quantities on the nodes of elements. As for the \(\lambda _{lb}\) defined on the boundary, we introduce the arc-length coordinate s, and interpolate \(\lambda _{lb}\) along the closed curve of the boundary. In this scenario, \(\lambda _{lb}\) can be written as
$$\begin{aligned} \lambda _{lb}\left( s\right) =\sum ^{m}_{i=1} \omega _i\left( s\right) \lambda _{lb}^{i}, \quad \lambda _{lb}^{i} \in {\textbf {Lb}}=\left( \lambda _{lb}^{1},\lambda _{lb}^{2},\ldots ,\lambda _{lb}^{m}\right) ^{\top }, \end{aligned}$$
(D11)
where \(\lambda _{lb}^{i}\) represents the value of \(\lambda _{lb}\) on the interpolation nodes of \(\Gamma\); \(\omega _i\left( s\right)\) is the corresponding basis function.
For numerically given design variables \({\textbf {D}}=\left( {\textbf {Bn}}^{\top }, {\textbf {Sn}}^{\top }, {\textbf {Lb}}^{\top }, {{\bar{\alpha }}}\right) ^{\top }\), the sensitivity of the optimisation formulation Eq. (45) based on Mode 1 is categorised as
$$\begin{aligned} \mathchoice{\frac{{\partial }{\mathcal {C}}^{\text {H}}}{{\partial }v}}{{\partial }{\mathcal {C}}^{\text {H}}/{\partial }v}{{\partial }{\mathcal {C}}^{\text {H}}/{\partial }v}{{\partial }{\mathcal {C}}^{\text {H}}/{\partial }v}=\left\{ \begin{aligned}&\begin{aligned} -&\int _{\Omega }\mathchoice{\frac{{\partial }{\mathscr {L}}}{{\partial }x_i}}{{\partial }{\mathscr {L}}/{\partial }x_i}{{\partial }{\mathscr {L}}/{\partial }x_i}{{\partial }{\mathscr {L}}/{\partial }x_i}\mathchoice{\frac{{\partial }M^{\lambda _l\alpha }_{ij}}{{\partial }\beta }}{{\partial }M^{\lambda _l\alpha }_{ij}/{\partial }\beta }{{\partial }M^{\lambda _l\alpha }_{ij}/{\partial }\beta }{{\partial }M^{\lambda _l\alpha }_{ij}/{\partial }\beta }\mathchoice{\frac{{\partial }\beta }{{\partial }v}}{{\partial }\beta /{\partial }v}{{\partial }\beta /{\partial }v}{{\partial }\beta /{\partial }v}\mathchoice{\frac{{\partial }\alpha }{{\partial }x_j}}{{\partial }\alpha /{\partial }x_j}{{\partial }\alpha /{\partial }x_j}{{\partial }\alpha /{\partial }x_j}+\mathchoice{\frac{{\partial }{\mathscr {L}}}{{\partial }x_i}}{{\partial }{\mathscr {L}}/{\partial }x_i}{{\partial }{\mathscr {L}}/{\partial }x_i}{{\partial }{\mathscr {L}}/{\partial }x_i} M^{\lambda _l\beta }_{ij}\mathchoice{\frac{{\partial }}{{\partial }x_j}}{{\partial }/{\partial }x_j}{{\partial }/{\partial }x_j}{{\partial }/{\partial }x_j}\left( \mathchoice{\frac{{\partial }\beta }{{\partial }v}}{{\partial }\beta /{\partial }v}{{\partial }\beta /{\partial }v}{{\partial }\beta /{\partial }v}\right) +\mathchoice{\frac{{\partial }{\mathscr {L}}}{{\partial }x_i}}{{\partial }{\mathscr {L}}/{\partial }x_i}{{\partial }{\mathscr {L}}/{\partial }x_i}{{\partial }{\mathscr {L}}/{\partial }x_i}\mathchoice{\frac{{\partial }M^{\lambda _l\varsigma _l}_{ij}}{{\partial }\beta }}{{\partial }M^{\lambda _l\varsigma _l}_{ij}/{\partial }\beta }{{\partial }M^{\lambda _l\varsigma _l}_{ij}/{\partial }\beta }{{\partial }M^{\lambda _l\varsigma _l}_{ij}/{\partial }\beta }\mathchoice{\frac{{\partial }\beta }{{\partial }v}}{{\partial }\beta /{\partial }v}{{\partial }\beta /{\partial }v}{{\partial }\beta /{\partial }v} \mathchoice{\frac{{\partial }\varsigma _l}{{\partial }x_j}}{{\partial }\varsigma _l/{\partial }x_j}{{\partial }\varsigma _l/{\partial }x_j}{{\partial }\varsigma _l/{\partial }x_j}\\&+\mathchoice{\frac{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}}{{\partial }\beta }}{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}/{\partial }\beta }{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}/{\partial }\beta }{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}/{\partial }\beta }\mathchoice{\frac{{\partial }\beta }{{\partial }v}}{{\partial }\beta /{\partial }v}{{\partial }\beta /{\partial }v}{{\partial }\beta /{\partial }v}\mathchoice{\frac{{\partial }u_i^{\text {H}}}{{\partial }x_j}}{{\partial }u_i^{\text {H}}/{\partial }x_j}{{\partial }u_i^{\text {H}}/{\partial }x_j}{{\partial }u_i^{\text {H}}/{\partial }x_j}\cdot \mathchoice{\frac{{\partial }u_k^{\text {H}}}{{\partial }x_l}}{{\partial }u_k^{\text {H}}/{\partial }x_l}{{\partial }u_k^{\text {H}}/{\partial }x_l}{{\partial }u_k^{\text {H}}/{\partial }x_l}\mathop {}\!\textrm{d}{\textbf {x}}, \end{aligned}\quad{} & {} v \in {\textbf {Bn}},\\&-\int _{\Omega }\mathchoice{\frac{{\partial }{\mathscr {L}}}{{\partial }x_i}}{{\partial }{\mathscr {L}}/{\partial }x_i}{{\partial }{\mathscr {L}}/{\partial }x_i}{{\partial }{\mathscr {L}}/{\partial }x_i}\mathchoice{\frac{{\partial }M^{\lambda _l\alpha }_{ij}}{{\partial }\varsigma _l}}{{\partial }M^{\lambda _l\alpha }_{ij}/{\partial }\varsigma _l}{{\partial }M^{\lambda _l\alpha }_{ij}/{\partial }\varsigma _l}{{\partial }M^{\lambda _l\alpha }_{ij}/{\partial }\varsigma _l}\mathchoice{\frac{{\partial }\varsigma _l}{{\partial }v}}{{\partial }\varsigma _l/{\partial }v}{{\partial }\varsigma _l/{\partial }v}{{\partial }\varsigma _l/{\partial }v}\mathchoice{\frac{{\partial }\alpha }{{\partial }x_j}}{{\partial }\alpha /{\partial }x_j}{{\partial }\alpha /{\partial }x_j}{{\partial }\alpha /{\partial }x_j}+\mathchoice{\frac{{\partial }{\mathscr {L}}}{{\partial }x_i}}{{\partial }{\mathscr {L}}/{\partial }x_i}{{\partial }{\mathscr {L}}/{\partial }x_i}{{\partial }{\mathscr {L}}/{\partial }x_i} M^{\lambda _l\varsigma _l}_{ij} \mathchoice{\frac{{\partial }}{{\partial }x_j}}{{\partial }/{\partial }x_j}{{\partial }/{\partial }x_j}{{\partial }/{\partial }x_j}\left( \mathchoice{\frac{{\partial }\varsigma _l}{{\partial }v}}{{\partial }\varsigma _l/{\partial }v}{{\partial }\varsigma _l/{\partial }v}{{\partial }\varsigma _l/{\partial }v}\right) +\mathchoice{\frac{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}}{{\partial }\varsigma _l}}{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}/{\partial }\varsigma _l}{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}/{\partial }\varsigma _l}{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}/{\partial }\varsigma _l}\mathchoice{\frac{{\partial }\varsigma _l}{{\partial }v}}{{\partial }\varsigma _l/{\partial }v}{{\partial }\varsigma _l/{\partial }v}{{\partial }\varsigma _l/{\partial }v}\mathchoice{\frac{{\partial }u_i^{\text {H}}}{{\partial }x_j}}{{\partial }u_i^{\text {H}}/{\partial }x_j}{{\partial }u_i^{\text {H}}/{\partial }x_j}{{\partial }u_i^{\text {H}}/{\partial }x_j}\cdot \mathchoice{\frac{{\partial }u_k^{\text {H}}}{{\partial }x_l}}{{\partial }u_k^{\text {H}}/{\partial }x_l}{{\partial }u_k^{\text {H}}/{\partial }x_l}{{\partial }u_k^{\text {H}}/{\partial }x_l}\mathop {}\!\textrm{d}{\textbf {x}},\quad{} & {} v \in {\textbf {Sn}},\\&-\int _{\Gamma }{\mathscr {L}}\mathchoice{\frac{{\partial }}{{\partial }\varvec{\uptau }}}{{\partial }/{\partial }\varvec{\uptau }}{{\partial }/{\partial }\varvec{\uptau }}{{\partial }/{\partial }\varvec{\uptau }}\left( \mathchoice{\frac{{\partial }\lambda _{lb}}{{\partial }v}}{{\partial }\lambda _{lb}/{\partial }v}{{\partial }\lambda _{lb}/{\partial }v}{{\partial }\lambda _{lb}/{\partial }v}\right) \mathop {}\!\textrm{d}\Gamma ,\quad{} & {} v \in {\textbf {Lb}},\\&-\int _{\Omega }\mathchoice{\frac{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}}{{\partial }\alpha }}{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}/{\partial }\alpha }{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}/{\partial }\alpha }{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}/{\partial }\alpha }\mathchoice{\frac{{\partial }u_i^{\text {H}}}{{\partial }x_j}}{{\partial }u_i^{\text {H}}/{\partial }x_j}{{\partial }u_i^{\text {H}}/{\partial }x_j}{{\partial }u_i^{\text {H}}/{\partial }x_j}\cdot \mathchoice{\frac{{\partial }u_k^{\text {H}}}{{\partial }x_l}}{{\partial }u_k^{\text {H}}/{\partial }x_l}{{\partial }u_k^{\text {H}}/{\partial }x_l}{{\partial }u_k^{\text {H}}/{\partial }x_l}\mathop {}\!\textrm{d}{\textbf {x}},\quad{} & {} v ={{\bar{\alpha }}}, \end{aligned}\right. \end{aligned}$$
(D12)
where \({\mathscr {L}}\) denotes the corresponding Lagrange multiplier, satisfying the following Neumann problem
$$\begin{aligned} \left\{ \begin{aligned}&\mathchoice{\frac{{\partial }}{{\partial }x_j}}{{\partial }/{\partial }x_j}{{\partial }/{\partial }x_j}{{\partial }/{\partial }x_j}\left( M^{\lambda _l\alpha }_{ij}\mathchoice{\frac{{\partial }{\mathscr {L}}}{{\partial }x_i}}{{\partial }{\mathscr {L}}/{\partial }x_i}{{\partial }{\mathscr {L}}/{\partial }x_i}{{\partial }{\mathscr {L}}/{\partial }x_i}\right) -\mathchoice{\frac{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}}{{\partial }\alpha }}{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}/{\partial }\alpha }{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}/{\partial }\alpha }{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}/{\partial }\alpha }\mathchoice{\frac{{\partial }u_i^{\text {H}}}{{\partial }x_j}}{{\partial }u_i^{\text {H}}/{\partial }x_j}{{\partial }u_i^{\text {H}}/{\partial }x_j}{{\partial }u_i^{\text {H}}/{\partial }x_j}\cdot \mathchoice{\frac{{\partial }u_k^{\text {H}}}{{\partial }x_l}}{{\partial }u_k^{\text {H}}/{\partial }x_l}{{\partial }u_k^{\text {H}}/{\partial }x_l}{{\partial }u_k^{\text {H}}/{\partial }x_l}+\frac{1}{\int _\Omega \mathop {}\!\textrm{d}\varvec{\xi }}\int _{\Omega }\mathchoice{\frac{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}}{{\partial }\alpha }}{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}/{\partial }\alpha }{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}/{\partial }\alpha }{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}/{\partial }\alpha } \mathchoice{\frac{{\partial }u_i^{\text {H}}}{{\partial }x_j}}{{\partial }u_i^{\text {H}}/{\partial }x_j}{{\partial }u_i^{\text {H}}/{\partial }x_j}{{\partial }u_i^{\text {H}}/{\partial }x_j}\cdot \mathchoice{\frac{{\partial }u_k^{\text {H}}}{{\partial }x_l}}{{\partial }u_k^{\text {H}}/{\partial }x_l}{{\partial }u_k^{\text {H}}/{\partial }x_l}{{\partial }u_k^{\text {H}}/{\partial }x_l}\mathop {}\!\textrm{d}{\textbf {x}}=0,\\&n_jM^{\lambda _l\alpha }_{ij}\mathchoice{\frac{{\partial }{\mathscr {L}}}{{\partial }x_i}}{{\partial }{\mathscr {L}}/{\partial }x_i}{{\partial }{\mathscr {L}}/{\partial }x_i}{{\partial }{\mathscr {L}}/{\partial }x_i}=0. \end{aligned} \right. \end{aligned}$$
(D13)
The sensitivity of Eq. (46) with respect to \({\textbf {D}} = \left( {\textbf {An}}^{\top },{\textbf {Sn}}^{\top },{\textbf {Lb}}^{\top },{{\bar{\beta }}}\right) ^{\top }\)
is given by
$$\begin{aligned} \mathchoice{\frac{{\partial }{\mathcal {C}}^{\text {H}}}{{\partial }v}}{{\partial }{\mathcal {C}}^{\text {H}}/{\partial }v}{{\partial }{\mathcal {C}}^{\text {H}}/{\partial }v}{{\partial }{\mathcal {C}}^{\text {H}}/{\partial }v}=\left\{ \begin{aligned}&-\int _{\Omega } \mathchoice{\frac{{\partial }{\mathscr {L}}}{{\partial }x_i}}{{\partial }{\mathscr {L}}/{\partial }x_i}{{\partial }{\mathscr {L}}/{\partial }x_i}{{\partial }{\mathscr {L}}/{\partial }x_i}M^{\lambda _l\alpha }_{ij}\mathchoice{\frac{{\partial }}{{\partial }x_j}}{{\partial }/{\partial }x_j}{{\partial }/{\partial }x_j}{{\partial }/{\partial }x_j}\left( \mathchoice{\frac{{\partial }\alpha }{{\partial }v}}{{\partial }\alpha /{\partial }v}{{\partial }\alpha /{\partial }v}{{\partial }\alpha /{\partial }v}\right) +\mathchoice{\frac{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}}{{\partial }\alpha }}{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}/{\partial }\alpha }{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}/{\partial }\alpha }{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}/{\partial }\alpha }\mathchoice{\frac{{\partial }\alpha }{{\partial }v}}{{\partial }\alpha /{\partial }v}{{\partial }\alpha /{\partial }v}{{\partial }\alpha /{\partial }v}\mathchoice{\frac{{\partial }u_i^{\text {H}}}{{\partial }x_j}}{{\partial }u_i^{\text {H}}/{\partial }x_j}{{\partial }u_i^{\text {H}}/{\partial }x_j}{{\partial }u_i^{\text {H}}/{\partial }x_j}\cdot \mathchoice{\frac{{\partial }u_k^{\text {H}}}{{\partial }x_l}}{{\partial }u_k^{\text {H}}/{\partial }x_l}{{\partial }u_k^{\text {H}}/{\partial }x_l}{{\partial }u_k^{\text {H}}/{\partial }x_l}\mathop {}\!\textrm{d}{\textbf {x}}\quad{} & {} v \in {\textbf {An}},\\&-\int _{\Omega }\mathchoice{\frac{{\partial }{\mathscr {L}}}{{\partial }x_i}}{{\partial }{\mathscr {L}}/{\partial }x_i}{{\partial }{\mathscr {L}}/{\partial }x_i}{{\partial }{\mathscr {L}}/{\partial }x_i}\mathchoice{\frac{{\partial }M^{\lambda _l\alpha }_{ij}}{{\partial }\varsigma _l}}{{\partial }M^{\lambda _l\alpha }_{ij}/{\partial }\varsigma _l}{{\partial }M^{\lambda _l\alpha }_{ij}/{\partial }\varsigma _l}{{\partial }M^{\lambda _l\alpha }_{ij}/{\partial }\varsigma _l}\mathchoice{\frac{{\partial }\varsigma _l}{{\partial }v}}{{\partial }\varsigma _l/{\partial }v}{{\partial }\varsigma _l/{\partial }v}{{\partial }\varsigma _l/{\partial }v}\mathchoice{\frac{{\partial }\alpha }{{\partial }x_j}}{{\partial }\alpha /{\partial }x_j}{{\partial }\alpha /{\partial }x_j}{{\partial }\alpha /{\partial }x_j}+\mathchoice{\frac{{\partial }{\mathscr {L}}}{{\partial }x_i}}{{\partial }{\mathscr {L}}/{\partial }x_i}{{\partial }{\mathscr {L}}/{\partial }x_i}{{\partial }{\mathscr {L}}/{\partial }x_i} M^{\lambda _l\varsigma _l}_{ij} \mathchoice{\frac{{\partial }}{{\partial }x_j}}{{\partial }/{\partial }x_j}{{\partial }/{\partial }x_j}{{\partial }/{\partial }x_j}\left( \mathchoice{\frac{{\partial }\varsigma _l}{{\partial }v}}{{\partial }\varsigma _l/{\partial }v}{{\partial }\varsigma _l/{\partial }v}{{\partial }\varsigma _l/{\partial }v}\right) +\mathchoice{\frac{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}}{{\partial }\varsigma _l}}{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}/{\partial }\varsigma _l}{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}/{\partial }\varsigma _l}{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}/{\partial }\varsigma _l}\mathchoice{\frac{{\partial }\varsigma _l}{{\partial }v}}{{\partial }\varsigma _l/{\partial }v}{{\partial }\varsigma _l/{\partial }v}{{\partial }\varsigma _l/{\partial }v}\mathchoice{\frac{{\partial }u_i^{\text {H}}}{{\partial }x_j}}{{\partial }u_i^{\text {H}}/{\partial }x_j}{{\partial }u_i^{\text {H}}/{\partial }x_j}{{\partial }u_i^{\text {H}}/{\partial }x_j}\cdot \mathchoice{\frac{{\partial }u_k^{\text {H}}}{{\partial }x_l}}{{\partial }u_k^{\text {H}}/{\partial }x_l}{{\partial }u_k^{\text {H}}/{\partial }x_l}{{\partial }u_k^{\text {H}}/{\partial }x_l}\mathop {}\!\textrm{d}{\textbf {x}},\quad{} & {} v \in {\textbf {Sn}},\\&-\int _{\Gamma }{\mathscr {L}}\mathchoice{\frac{{\partial }}{{\partial }\varvec{\uptau }}}{{\partial }/{\partial }\varvec{\uptau }}{{\partial }/{\partial }\varvec{\uptau }}{{\partial }/{\partial }\varvec{\uptau }}\left( \mathchoice{\frac{{\partial }\lambda _{lb}}{{\partial }v}}{{\partial }\lambda _{lb}/{\partial }v}{{\partial }\lambda _{lb}/{\partial }v}{{\partial }\lambda _{lb}/{\partial }v}\right) \mathop {}\!\textrm{d}\Gamma ,\quad{} & {} v \in {\textbf {Lb}},\\&-\int _\Omega \mathchoice{\frac{{\partial }{\mathscr {L}}}{{\partial }x_i}}{{\partial }{\mathscr {L}}/{\partial }x_i}{{\partial }{\mathscr {L}}/{\partial }x_i}{{\partial }{\mathscr {L}}/{\partial }x_i}\mathchoice{\frac{{\partial }M^{\lambda _l\varsigma _l}_{ij}}{{\partial }\beta }}{{\partial }M^{\lambda _l\varsigma _l}_{ij}/{\partial }\beta }{{\partial }M^{\lambda _l\varsigma _l}_{ij}/{\partial }\beta }{{\partial }M^{\lambda _l\varsigma _l}_{ij}/{\partial }\beta }\mathchoice{\frac{{\partial }\varsigma _l}{{\partial }x_j}}{{\partial }\varsigma _l/{\partial }x_j}{{\partial }\varsigma _l/{\partial }x_j}{{\partial }\varsigma _l/{\partial }x_j} +\mathchoice{\frac{{\partial }{\mathscr {L}}}{{\partial }x_i}}{{\partial }{\mathscr {L}}/{\partial }x_i}{{\partial }{\mathscr {L}}/{\partial }x_i}{{\partial }{\mathscr {L}}/{\partial }x_i}\mathchoice{\frac{{\partial }M^{\lambda _l\alpha }_{ij}}{{\partial }\beta }}{{\partial }M^{\lambda _l\alpha }_{ij}/{\partial }\beta }{{\partial }M^{\lambda _l\alpha }_{ij}/{\partial }\beta }{{\partial }M^{\lambda _l\alpha }_{ij}/{\partial }\beta }\mathchoice{\frac{{\partial }\alpha }{{\partial }x_j}}{{\partial }\alpha /{\partial }x_j}{{\partial }\alpha /{\partial }x_j}{{\partial }\alpha /{\partial }x_j} +\mathchoice{\frac{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}}{{\partial }\beta }}{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}/{\partial }\beta }{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}/{\partial }\beta }{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}/{\partial }\beta }\mathchoice{\frac{{\partial }u_i^{\text {H}}}{{\partial }x_j}}{{\partial }u_i^{\text {H}}/{\partial }x_j}{{\partial }u_i^{\text {H}}/{\partial }x_j}{{\partial }u_i^{\text {H}}/{\partial }x_j}\cdot \mathchoice{\frac{{\partial }u_k^{\text {H}}}{{\partial }x_l}}{{\partial }u_k^{\text {H}}/{\partial }x_l}{{\partial }u_k^{\text {H}}/{\partial }x_l}{{\partial }u_k^{\text {H}}/{\partial }x_l} \mathop {}\!\textrm{d}{\textbf {x}},\quad{} & {} v ={{\bar{\beta }}}. \end{aligned}\right. \end{aligned}$$
(D14)
Here the Lagrange multiplier \({\mathscr {L}}\) in Eq. (D14) satisfies the following equation
$$\begin{aligned} \left\{ \begin{aligned}&\begin{aligned}&\mathchoice{\frac{{\partial }}{{\partial }x_j}}{{\partial }/{\partial }x_j}{{\partial }/{\partial }x_j}{{\partial }/{\partial }x_j}\left( M^{\lambda _l\beta }_{ij}\mathchoice{\frac{{\partial }{\mathscr {L}}}{{\partial }x_i}}{{\partial }{\mathscr {L}}/{\partial }x_i}{{\partial }{\mathscr {L}}/{\partial }x_i}{{\partial }{\mathscr {L}}/{\partial }x_i}\right) =\mathchoice{\frac{{\partial }{\mathscr {L}}}{{\partial }x_i}}{{\partial }{\mathscr {L}}/{\partial }x_i}{{\partial }{\mathscr {L}}/{\partial }x_i}{{\partial }{\mathscr {L}}/{\partial }x_i}\mathchoice{\frac{{\partial }M^{\lambda _l\varsigma _l}_{ij}}{{\partial }\beta }}{{\partial }M^{\lambda _l\varsigma _l}_{ij}/{\partial }\beta }{{\partial }M^{\lambda _l\varsigma _l}_{ij}/{\partial }\beta }{{\partial }M^{\lambda _l\varsigma _l}_{ij}/{\partial }\beta }\mathchoice{\frac{{\partial }\varsigma _l}{{\partial }x_j}}{{\partial }\varsigma _l/{\partial }x_j}{{\partial }\varsigma _l/{\partial }x_j}{{\partial }\varsigma _l/{\partial }x_j} +\mathchoice{\frac{{\partial }{\mathscr {L}}}{{\partial }x_i}}{{\partial }{\mathscr {L}}/{\partial }x_i}{{\partial }{\mathscr {L}}/{\partial }x_i}{{\partial }{\mathscr {L}}/{\partial }x_i}\mathchoice{\frac{{\partial }M^{\lambda _l\alpha }_{ij}}{{\partial }\beta }}{{\partial }M^{\lambda _l\alpha }_{ij}/{\partial }\beta }{{\partial }M^{\lambda _l\alpha }_{ij}/{\partial }\beta }{{\partial }M^{\lambda _l\alpha }_{ij}/{\partial }\beta }\mathchoice{\frac{{\partial }\alpha }{{\partial }x_j}}{{\partial }\alpha /{\partial }x_j}{{\partial }\alpha /{\partial }x_j}{{\partial }\alpha /{\partial }x_j} +\mathchoice{\frac{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}}{{\partial }\beta }}{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}/{\partial }\beta }{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}/{\partial }\beta }{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}/{\partial }\beta }\mathchoice{\frac{{\partial }u_i^{\text {H}}}{{\partial }x_j}}{{\partial }u_i^{\text {H}}/{\partial }x_j}{{\partial }u_i^{\text {H}}/{\partial }x_j}{{\partial }u_i^{\text {H}}/{\partial }x_j}\cdot \mathchoice{\frac{{\partial }u_k^{\text {H}}}{{\partial }x_l}}{{\partial }u_k^{\text {H}}/{\partial }x_l}{{\partial }u_k^{\text {H}}/{\partial }x_l}{{\partial }u_k^{\text {H}}/{\partial }x_l}\\&-\frac{1}{\int _\Omega \mathop {}\!\textrm{d}\varvec{\xi }}\int _\Omega \mathchoice{\frac{{\partial }{\mathscr {L}}}{{\partial }x_i}}{{\partial }{\mathscr {L}}/{\partial }x_i}{{\partial }{\mathscr {L}}/{\partial }x_i}{{\partial }{\mathscr {L}}/{\partial }x_i}\mathchoice{\frac{{\partial }M^{\lambda _l\varsigma _l}_{ij}}{{\partial }\beta }}{{\partial }M^{\lambda _l\varsigma _l}_{ij}/{\partial }\beta }{{\partial }M^{\lambda _l\varsigma _l}_{ij}/{\partial }\beta }{{\partial }M^{\lambda _l\varsigma _l}_{ij}/{\partial }\beta }\mathchoice{\frac{{\partial }\varsigma _l}{{\partial }x_j}}{{\partial }\varsigma _l/{\partial }x_j}{{\partial }\varsigma _l/{\partial }x_j}{{\partial }\varsigma _l/{\partial }x_j} +\mathchoice{\frac{{\partial }{\mathscr {L}}}{{\partial }x_i}}{{\partial }{\mathscr {L}}/{\partial }x_i}{{\partial }{\mathscr {L}}/{\partial }x_i}{{\partial }{\mathscr {L}}/{\partial }x_i}\mathchoice{\frac{{\partial }M^{\lambda _l\alpha }_{ij}}{{\partial }\beta }}{{\partial }M^{\lambda _l\alpha }_{ij}/{\partial }\beta
}{{\partial }M^{\lambda _l\alpha }_{ij}/{\partial }\beta }{{\partial }M^{\lambda _l\alpha }_{ij}/{\partial }\beta }\mathchoice{\frac{{\partial }\alpha }{{\partial }x_j}}{{\partial }\alpha /{\partial }x_j}{{\partial }\alpha /{\partial }x_j}{{\partial }\alpha /{\partial }x_j} +\mathchoice{\frac{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}}{{\partial }\beta }}{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}/{\partial }\beta }{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}/{\partial }\beta }{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}/{\partial }\beta } \mathchoice{\frac{{\partial }u_i^{\text {H}}}{{\partial }x_j}}{{\partial }u_i^{\text {H}}/{\partial }x_j}{{\partial }u_i^{\text {H}}/{\partial }x_j}{{\partial }u_i^{\text {H}}/{\partial }x_j}\cdot \mathchoice{\frac{{\partial }u_k^{\text {H}}}{{\partial }x_l}}{{\partial }u_k^{\text {H}}/{\partial }x_l}{{\partial }u_k^{\text {H}}/{\partial }x_l}{{\partial }u_k^{\text {H}}/{\partial }x_l} \mathop {}\!\textrm{d}{\textbf {x}}, \end{aligned}\\&n_jM^{\lambda _l\beta }_{ij}\mathchoice{\frac{{\partial }{\mathscr {L}}}{{\partial }x_i}}{{\partial }{\mathscr {L}}/{\partial }x_i}{{\partial }{\mathscr {L}}/{\partial }x_i}{{\partial }{\mathscr {L}}/{\partial }x_i}=0. \end{aligned} \right. \end{aligned}$$
(D15)
Referring to Eq. (40), the derivatives of \({\mathbb {C}}^{\text {H}}\) with respect to \(\beta\), \(\varsigma _l\) and \(\alpha\) in Eqs. (D12)–(D15) can be further expanded as
$$\begin{aligned}&\mathchoice{\frac{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}}{{\partial }\beta }}{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}/{\partial }\beta }{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}/{\partial }\beta }{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}/{\partial }\beta } =\mathchoice{\frac{{\partial }\left( R_{ip}R_{jq}R_{ks}R_{lt}\right) }{{\partial }\beta }}{{\partial }\left( R_{ip}R_{jq}R_{ks}R_{lt}\right) /{\partial }\beta }{{\partial }\left( R_{ip}R_{jq}R_{ks}R_{lt}\right) /{\partial }\beta }{{\partial }\left( R_{ip}R_{jq}R_{ks}R_{lt}\right) /{\partial }\beta }\hat{{\mathbb {C}}}^{\text {H}}_{pqst}, \end{aligned}$$
(D16a)
$$\begin{aligned}&\mathchoice{\frac{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}}{{\partial }\varsigma _l}}{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}/{\partial }\varsigma _l}{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}/{\partial }\varsigma _l}{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}/{\partial }\varsigma _l} = R_{ip}R_{jq}R_{ks}R_{lt}\mathchoice{\frac{{\partial }\hat{{\mathbb {C}}}^{\text {H}}_{pqst}}{{\partial }\varsigma _l}}{{\partial }\hat{{\mathbb {C}}}^{\text {H}}_{pqst}/{\partial }\varsigma _l}{{\partial }\hat{{\mathbb {C}}}^{\text {H}}_{pqst}/{\partial }\varsigma _l}{{\partial }\hat{{\mathbb {C}}}^{\text {H}}_{pqst}/{\partial }\varsigma _l}, \end{aligned}$$
(D16b)
$$\begin{aligned}&\mathchoice{\frac{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}}{{\partial }\alpha }}{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}/{\partial }\alpha }{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}/{\partial }\alpha }{{\partial }{\mathbb {C}}^{\text {H}}_{ijkl}/{\partial }\alpha }=\mathchoice{\frac{{\partial }\left( R_{ip}R_{jq}R_{ks}R_{lt}\right) }{{\partial }\alpha }}{{\partial }\left( R_{ip}R_{jq}R_{ks}R_{lt}\right) /{\partial }\alpha }{{\partial }\left( R_{ip}R_{jq}R_{ks}R_{lt}\right) /{\partial }\alpha }{{\partial }\left( R_{ip}R_{jq}R_{ks}R_{lt}\right) /{\partial }\alpha }\hat{{\mathbb {C}}}^{\text {H}}_{pqst}+ R_{ip}R_{jq}R_{ks}R_{lt}\mathchoice{\frac{{\partial }\hat{{\mathbb {C}}}^{\text {H}}_{pqst}}{{\partial }\alpha }}{{\partial }\hat{{\mathbb {C}}}^{\text {H}}_{pqst}/{\partial }\alpha }{{\partial }\hat{{\mathbb {C}}}^{\text {H}}_{pqst}/{\partial }\alpha }{{\partial }\hat{{\mathbb {C}}}^{\text {H}}_{pqst}/{\partial }\alpha }, \end{aligned}$$
(D16c)
where \(\mathchoice{\frac{{\partial }\hat{{\mathbb {C}}}^{\text {H}}_{pqst}}{{\partial }\varsigma _l}}{{\partial }\hat{{\mathbb {C}}}^{\text {H}}_{pqst}/{\partial }\varsigma _l}{{\partial }\hat{{\mathbb {C}}}^{\text {H}}_{pqst}/{\partial }\varsigma _l}{{\partial }\hat{{\mathbb {C}}}^{\text {H}}_{pqst}/{\partial }\varsigma _l}\) and \(\mathchoice{\frac{{\partial }\hat{{\mathbb {C}}}^{\text {H}}_{pqst}}{{\partial }\alpha }}{{\partial }\hat{{\mathbb {C}}}^{\text {H}}_{pqst}/{\partial }\alpha }{{\partial }\hat{{\mathbb {C}}}^{\text {H}}_{pqst}/{\partial }\alpha }{{\partial }\hat{{\mathbb {C}}}^{\text {H}}_{pqst}/{\partial }\alpha }\) can be directly calculated with central difference scheme, i.e.
$$\begin{aligned}&\mathchoice{\frac{{\partial }\hat{{\mathbb {C}}}^{\text {H}}_{pqst}}{{\partial }\varsigma _l}}{{\partial }\hat{{\mathbb {C}}}^{\text {H}}_{pqst}/{\partial }\varsigma _l}{{\partial }\hat{{\mathbb {C}}}^{\text {H}}_{pqst}/{\partial }\varsigma _l}{{\partial }\hat{{\mathbb {C}}}^{\text {H}}_{pqst}/{\partial }\varsigma _l} = \frac{\hat{{\mathbb {C}}}^{\text {H}}_{pqst}\left( \varsigma _l+\Delta \varsigma _l,\alpha \right) -\hat{{\mathbb {C}}}^{\text {H}}_{pqst}\left( \varsigma _l-\Delta \varsigma _l,\alpha \right) }{2\Delta \varsigma _l}, \end{aligned}$$
(D17a)
$$\begin{aligned}&\mathchoice{\frac{{\partial }\hat{{\mathbb {C}}}^{\text {H}}_{pqst}}{{\partial }\alpha }}{{\partial }\hat{{\mathbb {C}}}^{\text {H}}_{pqst}/{\partial }\alpha }{{\partial }\hat{{\mathbb {C}}}^{\text {H}}_{pqst}/{\partial }\alpha }{{\partial }\hat{{\mathbb {C}}}^{\text {H}}_{pqst}/{\partial }\alpha } = \frac{\hat{{\mathbb {C}}}^{\text {H}}_{pqst}\left( \varsigma _l,\alpha +\Delta \alpha \right) -\hat{{\mathbb {C}}}^{\text {H}}_{pqst}\left( \varsigma _l,\alpha -\Delta \alpha \right) }{2\Delta \alpha }. \end{aligned}$$
(D17b)
