## Appendix A The Global Phase Relation between a and \(\mathbf{d}\)  
For a typical member \((j)\)\([(j)=(1),(2),\cdots,(m)]\), the compatibility conditions between the generalized displacements/forces of the member in dual local coordinates \((x,y,z)^{|K}\) and \((x,y,z)^{|J}\)\((x^{|K}+x^{|J}=l_{(j)})\) are written as:  
\[\vartheta^{|K}(x^{|K})=-\vartheta^{|J}(l^{|K}-x^{|K}),\,\phi^{|K}(x^{|K})= \phi^{|K}(l^{|K}-x^{|K}); \tag{1}\]  
\[\hat{\mathcal{Q}}^{|K}(x^{|K})=\hat{\mathcal{Q}}^{|K}(l^{|K}-x^{|K}),\,\,\hat {\mathcal{M}}^{|K}(x^{|K})=-\hat{\mathcal{M}}^{|K|}(l^{|K}-x^{|K}).\]  
Substituting the solutions to the generalized displacements (forces) as provided in Equation (3) [Equation (5)] into the former (latter) two formulas of Equation (1) and noticing the functions \(\mathbf{e}^{\mathrm{i}k,x}\) and \(\mathbf{e}^{-\mathrm{i}k,x}\) are independent of each other, one obtains the local phase relation of member \((j)\):  
\[\mathbf{a}_{(j)}=\mathbf{P}_{(j)}(\omega)\widetilde{\mathbf{d}}_{(j)}= \mathbf{P}_{(j)}(\omega)\mathbf{U}_{(j)}\mathbf{d}_{(j)}, \tag{2}\]  
where \(\mathbf{a}_{(j)}=[a_{2}^{|K},a_{3}^{|K},a_{2}^{|K|},a_{3}^{|K|}]^{\mathrm{T}}\) and \(\mathbf{d}_{(j)}=[d_{2}^{|K},d_{3}^{|K},d_{2}^{|K|},a_{3}^{|K|}]^{\mathrm{T}}\) are the arriving and departing wave vectors of member \((j)\), respectively; \(\widetilde{\mathbf{d}}_{(j)}=[d_{2}^{|K},d_{3}^{|K|},d_{2}^{|K|},a_{3}^{|K|}]^{\mathrm{T}}\) has the same components but arranged in different sequence as \(\mathbf{d}_{(j)}\). They are related by \(\widetilde{\mathbf{d}}_{(j)}=\mathbf{U}_{(j)}\mathbf{d}_{(j)}\), where \(\mathbf{U}_{(j)}=[\mathbf{0},\mathbf{I}_{2};\mathbf{I}_{2},\mathbf{0}]\) is an orthogonal permutation matrix of member \((j)\) with \(\mathbf{I}_{2}\) the identity matrix of second order. In Equation (2), \(\mathbf{P}_{(j)}=<-\mathbf{e}^{-\mathrm{i}k_{2(j)}l_{(j)}},-\mathbf{e}^{- \mathrm{i}k_{3(j)}l_{(j)}},-\mathbf{e}^{-\mathrm{i}k_{3(j)}l_{(j)}}>\) is the phase matrix of member \((j)\) with \(<\cdot>\) here and after denoting the (block) diagonal matrix. Note that Equation (2) can also be written out directly according to the wave propagation concept and the sign convention of the wave amplitudes, as indicated in Figure 2d. The resulting equations are \(a_{r}^{|K}=-\mathbf{e}^{-\mathrm{i}k_{r(j)}l_{(j)}}a_{r}^{|K|},a_{r}^{|K|}=- \mathbf{e}^{-\mathrm{i}k_{r(j)}l_{(j)}}a_{r}^{|K|}\). It should be noted that the chosen wavenumbers \(k_{r(j)}\) in Section 2.1 can always guarantee \(\mathbf{Re}[-\mathrm{i}k_{r(j)}l_{(j)}]\leq 0\). Thus, no exponentially growing function appears in Equation (2), which results in the numerically well condition of our matrix formulation.  
Grouping the local phase relations of all members in sequence of member label \((j)=(1),(2),\cdots,(m)\), we have the global phase relation as follows:  
\[\mathbf{a}=\mathbf{P}(\omega)\widetilde{\mathbf{d}}=\mathbf{P}(\omega) \mathbf{U}\mathbf{d}, \tag{3}\]  
where \(\mathbf{a}=[(\mathbf{a}_{(1)})^{\mathrm{T}},(\mathbf{a}_{(2)})^{\mathrm{T}}, \cdots,(\mathbf{a}_{(j)})^{\mathrm{T}},\cdots,(\mathbf{a}_{(m)})^{\mathrm{T}} ]^{\mathrm{T}}\) and \(\mathbf{d}=[(\mathbf{d}_{(1)})^{\mathrm{T}},(\mathbf{d}_{(2)})^{\mathrm{T}}, \cdots,(\mathbf{d}_{(j)})^{\mathrm{T}},\cdots,(\mathbf{d}_{(m)})^{\mathrm{T}} ]^{\mathrm{T}}\) are the global arriving and departing wave vectors that are essentially identical to those provided in Equation (13), and \(\widetilde{\mathbf{d}}=[(\widetilde{\mathbf{d}}_{(1)})^{\mathrm{T}},( \widetilde{\mathbf{d}}_{(2)})^{\mathrm{T}},\cdots,(\widetilde{\mathbf{d}}_{(j )})^{\mathrm{T}},\cdots,(\widetilde{\mathbf{d}}_{(m)})^{\mathrm{T}}]^{\mathrm{T}}\) is a variant of \(\mathbf{d}\) that relating to \(\mathbf{d}\) by \(\widetilde{\mathbf{d}}=\mathbf{U}\mathbf{d}.\mathbf{U}=<\mathbf{U}_{(1)}, \mathbf{U}_{(2)},\cdots,\mathbf{U}_{(j)},\cdots,\mathbf{U}_{(m)}>\) and \(\mathbf{P}=<\mathbf{P}_{(1)},\mathbf{P}_{(2)},\cdots,\mathbf{P}_{(j)},\cdots, \mathbf{P}_{(m)}>\) are the \(4\)\(m\)\(\times\)\(4\)\(m\) global permutation and phase matrices.