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Clustering analysis for elastodynamic homogenization

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Abstract

We propose a reduced-order method for the elastodynamic homogenization of periodic composites. With the help of Bloch-wave expansion and Green’s function, the Lippmann–Schwinger equations relating the dynamic field variable tensors of strain, velocity, stress and linear momentum are obtained. Then the constitutive relation of the averaged dynamic field tensors and the dispersion relation between frequency and wave vector in Willis theory are formulated explicitly. Using the data compression algorithm, k-means clustering, we decompose computational domain into clusters of possibly disjoint cells. The Lippmann–Schwinger equations are then discretized and solved efficiently. Numerical tests for three-dimensional particle-reinforced composites verify the effectiveness and efficiency of the proposed method.

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Acknowledgements

This work is supported partially by NSFC under Grant Numbers 11832001, 11988102, and 11890681.

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Authors and Affiliations

  1. HEDPS and LTCS, College of Engineering, Peking University, Beijing, 100871, China

    Xi Zhu & Shaoqiang Tang

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  1. Xi Zhu
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  2. Shaoqiang Tang
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Correspondence to Shaoqiang Tang.

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Appendix: A Constitutive tensors

Appendix: A Constitutive tensors

In this appendix, denote

$$\begin{aligned} \varvec{f} *\varvec{g}=\int _T \varvec{f}(\varvec{x}-\varvec{y})\varvec{g}(\varvec{y})\textrm{d} \varvec{y}, \end{aligned}$$
(59)

and then Lippmann–Schwinger equations read

$$\begin{aligned} \begin{aligned}&\tilde{\varvec{\varepsilon }}-\tilde{\varvec{E}}=\dfrac{1}{|T|}(\varvec{\varPhi }*\tilde{\varvec{\tau }} +\varvec{\varPsi }*\tilde{\varvec{\pi }}),\\&\tilde{\varvec{v}}-\tilde{\varvec{V}}=\dfrac{1}{|T|}(\varvec{\varTheta }*\tilde{\varvec{\tau }}+\varvec{\varGamma }*\tilde{\varvec{\pi }}). \end{aligned} \end{aligned}$$
(60)

\(\varvec{\varPhi }_{\varvec{\xi }},\varvec{\varPsi }_{\varvec{\xi }},\varvec{\varTheta }_{\varvec{\xi }},\varvec{\varGamma }_{\varvec{\xi }} \in {\mathbb {R}}\) leads to the properties

$$\begin{aligned} \varvec{\varPhi }({\varvec{x}})= & {} \varvec{\varPhi }^*(-{\varvec{x}}),\ \varvec{\varPsi }({\varvec{x}})=\varvec{\varPsi }^*(-{\varvec{x}}), \nonumber \\ \varvec{\varTheta }({\varvec{x}})= & {} \varvec{\varTheta }^*(-{\varvec{x}}),\ \varvec{\varGamma }({\varvec{x}})=\varvec{\varGamma }^*(-{\varvec{x}}). \end{aligned}$$
(61)

\(\varvec{\varPhi }_{\varvec{0}}=\varvec{0},\varvec{\varPsi }_{\varvec{0}}=\varvec{0}, \varvec{\varTheta }_{\varvec{0}}=\varvec{0}, \varvec{\varGamma }_{\varvec{0}}=\varvec{0}\) shows that

$$\begin{aligned} \begin{aligned} \int _T \varvec{\varPhi }(\varvec{x}-\varvec{y})\textrm{d} \varvec{x}&=\varvec{0},\int _T \varvec{\varPsi }(\varvec{x}-\varvec{y})\textrm{d} \varvec{x}=\varvec{0},\\ \int _T \varvec{\varTheta }(\varvec{x}-\varvec{y})\textrm{d} \varvec{x}&=\varvec{0},\int _T \varvec{\varGamma }(\varvec{x}-\varvec{y})\textrm{d} \varvec{x}=\varvec{0},\\ \int _T \varvec{\varPhi }(\varvec{x}-\varvec{y})\textrm{d} \varvec{y}&=\varvec{0},\int _T \varvec{\varPsi }(\varvec{x}-\varvec{y})\textrm{d} \varvec{y}=\varvec{0}, \\ \int _T \varvec{\varTheta }(\varvec{x}-\varvec{y})\textrm{d} \varvec{y}&=\varvec{0},\int _T \varvec{\varGamma }(\varvec{x}-\varvec{y})\textrm{d} \varvec{y}=\varvec{0}. \end{aligned} \end{aligned}$$
(62)

The fact that \(\varvec{\varPhi }_{\varvec{\xi }},\varvec{\varPsi }_{\varvec{\xi }},\varvec{\varTheta }_{\varvec{\xi }},\varvec{\varGamma }_{\varvec{\xi }} \) have symmetry about their subscripts leads to

$$\begin{aligned} \varPhi _{ijkl}= & {} \varPhi _{klij},\quad \varPsi _{ijk}=\varPsi _{jik}=-\varTheta _{kij}=-\varTheta _{kji}, \nonumber \\ \varGamma _{ij}= & {} \varGamma _{ji}. \end{aligned}$$
(63)

For ease of notations, we rewrite (63) as

$$\begin{aligned} \varvec{\varPhi }=\varvec{\varPhi }^T,\ \varvec{\varPsi }^T=-\varvec{\varTheta },\ \varvec{\varTheta }^T=-\varvec{\varPsi },\ \varvec{\varGamma }=\varvec{\varGamma }^T, \end{aligned}$$
(64)

and treat other tensors transpose and complex conjugate transpose (with superscript \(\cdot ^H\)) in the same way.

Denote

$$\begin{aligned} \varvec{f}_{\delta }*\varvec{g}=\int _T \varvec{f}(\varvec{x})\delta (\varvec{x}-\varvec{y}) \varvec{g}(\varvec{y})\textrm{d} \varvec{y}=\varvec{f}(\varvec{x})\varvec{g}(\varvec{x}), \end{aligned}$$
(65)

and rewrite (60) as

$$\begin{aligned}{} & {} \Bigg ( \left[ \begin{array}{cc} \delta &{}0 \\ 0 &{} \delta \end{array} \right] -\dfrac{1}{|T|} \left[ \begin{array}{cc} \varvec{\varPhi }*(\varvec{C}_{\delta }-\varvec{C}^0_{\delta })&{}\varvec{\varPsi }*(\varvec{\rho }_{\delta }-\varvec{\rho }^0_{\delta }) \\ \varvec{\varTheta }*(\varvec{C}_{\delta }-\varvec{C}^0_{\delta }) &{}\varvec{\varGamma }*(\varvec{\rho }_{\delta }-\varvec{\rho }^0_{\delta }) \end{array} \right] \Bigg ) * \nonumber \\{} & {} \quad \left[ \begin{array}{c} \tilde{\varvec{\varepsilon }} \\ \tilde{\varvec{v}} \end{array} \right] = \left[ \begin{array}{c} \tilde{\varvec{E}} \\ \tilde{\varvec{V}} \end{array} \right] . \end{aligned}$$
(66)
$$\begin{aligned} \begin{aligned}\left[ \begin{array}{c} \tilde{\varvec{\Sigma }} \\ \tilde{\varvec{P}} \end{array} \right]&=\dfrac{1}{|T|} \left[ \begin{array}{cc} 1 &{}0 \\ 0 &{} 1 \end{array} \right] * \left[ \begin{array}{c} \tilde{\varvec{\sigma }} \\ \tilde{\varvec{p}} \end{array} \right] \\&= \dfrac{1}{|T|} \left[ \begin{array}{cc} 1 &{}0 \\ 0 &{} 1 \end{array} \right] * \left[ \begin{array}{cc} \varvec{C}_{\delta }&{}0 \\ 0 &{} \varvec{\rho }_{\delta } \end{array} \right] * \left[ \begin{array}{c} \tilde{\varvec{\varepsilon }} \\ \tilde{\varvec{v}} \end{array} \right] \\&=\dfrac{1}{|T|} \left[ \begin{array}{cc} 1 &{}0 \\ 0 &{} 1 \end{array} \right] * \left[ \begin{array}{cc} \varvec{C}_{\delta }&{}0 \\ 0 &{} \varvec{\rho }_{\delta } \end{array} \right] \\&\quad *\Bigg ( \left[ \begin{array}{cc} \delta &{}0 \\ 0 &{} \delta \end{array} \right] -\dfrac{1}{|T|} \left[ \begin{array}{cc} \varvec{\varPhi }*(\varvec{C}_{\delta }-\varvec{C}^0_{\delta })&{}\varvec{\varPsi }*(\varvec{\rho }_{\delta }-\varvec{\rho }^0_{\delta }) \\ \varvec{\varTheta }*(\varvec{C}_{\delta }-\varvec{C}^0_{\delta }) &{}\varvec{\varGamma }*(\varvec{\rho }_{\delta }-\varvec{\rho }^0_{\delta }) \end{array} \right] \Bigg )^{-1} \\&\quad *\left[ \begin{array}{cc} 1 &{}0 \\ 0 &{} 1 \end{array} \right] \left[ \begin{array}{c} \tilde{\varvec{E}} \\ \tilde{\varvec{V}} \end{array} \right] , \end{aligned} \end{aligned}$$
(67)

where \( 1 \) represents constant function with value equals to 1. We remark that operators in (67) are real except \(\varvec{\varPhi },\varvec{\varPsi },\varvec{\varTheta },\varvec{\varGamma }\).

The macroscopic constitutive relation is expressed explicitly as

$$\begin{aligned}{} & {} \left[ \begin{array}{cc} \overline{\tilde{\varvec{C}}} &{} \overline{\tilde{\varvec{S^1}}} \\ \overline{\tilde{\varvec{S^2}}} &{} \overline{\tilde{\varvec{\rho }}} \end{array} \right] \nonumber \\= & {} \dfrac{1}{|T|} \left[ \begin{array}{cc} 1 &{}0 \\ 0 &{} 1 \end{array} \right] * \left[ \begin{array}{cc} \varvec{C}_{\delta }&{}0 \\ 0 &{} \varvec{\rho }_{\delta } \end{array} \right] \nonumber \\{} & {} \quad * \Bigg ( \left[ \begin{array}{cc} \delta &{}0 \\ 0 &{} \delta \end{array} \right] -\dfrac{1}{|T|} \left[ \begin{array}{cc} \varvec{\varPhi }*(\varvec{C}_{\delta }-\varvec{C}^0_{\delta })&{}\varvec{\varPsi }*(\varvec{\rho }_{\delta }-\varvec{\rho }^0_{\delta }) \\ \varvec{\varTheta }*(\varvec{C}_{\delta }-\varvec{C}^0_{\delta }) &{}\varvec{\varGamma }*(\varvec{\rho }_{\delta }-\varvec{\rho }^0_{\delta }) \end{array} \right] \Bigg )^{-1} \nonumber \\{} & {} \quad *\left[ \begin{array}{cc} 1 &{}0 \\ 0 &{} 1 \end{array} \right] . \end{aligned}$$
(68)

Performing power series expansion to the inverse operator, and we obtain the expansion of macroscopic constitutive relation.

Zero-order:

$$\begin{aligned} \begin{aligned}&\left[ \begin{array}{cc} \overline{\tilde{\varvec{C}}}^{(0)} &{} \overline{\tilde{\varvec{S^1}}}^{(0)} \\ \overline{\tilde{\varvec{S^2}}}^{(0)} &{} \overline{\tilde{\varvec{\rho }}}^{(0)} \end{array} \right] \\&\quad =\dfrac{1}{|T|} \left[ \begin{array}{cc} 1 &{}0 \\ 0 &{} 1 \end{array} \right] * \left[ \begin{array}{cc} \varvec{C}_{\delta }&{}0 \\ 0 &{} \varvec{\rho }_{\delta } \end{array} \right] * \left[ \begin{array}{cc} \delta &{}0 \\ 0 &{} \delta \end{array} \right] * \left[ \begin{array}{cc} 1 &{}0 \\ 0 &{} 1 \end{array} \right] \\ {}&\quad = \dfrac{1}{|T|}\left[ \begin{array}{cc} 1 *\varvec{C}_{\delta }* 1 &{}0 \\ 0 &{} 1 *\varvec{\rho }_{\delta }* 1 \end{array} \right] \\ {}&\quad = \dfrac{1}{|T|}\left[ \begin{array}{cc} \int _T \varvec{C}(\varvec{x})\delta (\varvec{x}-\varvec{y})\textrm{d}\varvec{x}\textrm{d}\varvec{y}&{}0 \\ 0 &{} \int _T \varvec{\rho }(\varvec{x})\delta (\varvec{x}-\varvec{y})\textrm{d}\varvec{x}\textrm{d}\varvec{y} \end{array} \right] \\&\quad =\left[ \begin{array}{cc} \frac{1}{|T|}\int _T \varvec{C}(\varvec{x})\textrm{d}\varvec{x}&{}0 \\ 0 &{} \frac{1}{|T|}\int _T \varvec{\rho }(\varvec{x})\textrm{d}\varvec{x} \end{array} \right] . \end{aligned} \end{aligned}$$
(69)

It follows that

$$\begin{aligned} \overline{\tilde{\varvec{C}}}^{(0)}=\overline{\tilde{\varvec{C}}}^{(0)T}=\overline{\tilde{\varvec{C}}}^{(0)H},\overline{\tilde{\varvec{\rho }}}^{(0)}=\overline{\tilde{\varvec{\rho }}}^{(0)T}=\overline{\tilde{\varvec{\rho }}}^{(0)H}. \end{aligned}$$
(70)

First-order:

$$\begin{aligned}&\left[ \begin{array}{cc} \overline{\tilde{\varvec{C}}}^{(1)} &{} \overline{\tilde{\varvec{S^1}}}^{(1)} \\ \overline{\tilde{\varvec{S^2}}}^{(1)} &{} \overline{\tilde{\varvec{\rho }}}^{(1)} \end{array} \right] \\&\quad =\dfrac{1}{|T|} \left[ \begin{array}{cc} 1 &{}0 \\ 0 &{} 1 \end{array} \right] * \left[ \begin{array}{cc} \varvec{C}_{\delta }&{}0 \\ 0 &{} \varvec{\rho }_{\delta } \end{array} \right] \\&\qquad * \dfrac{1}{|T|} \left[ \begin{array}{cc} \varvec{\varPhi }*(\varvec{C}_{\delta }-\varvec{C}^0_{\delta })&{}\varvec{\varPsi }*(\varvec{\rho }_{\delta }-\varvec{\rho }^0_{\delta }) \\ \varvec{\varTheta }*(\varvec{C}_{\delta }-\varvec{C}^0_{\delta }) &{}\varvec{\varGamma }*(\varvec{\rho }_{\delta }-\varvec{\rho }^0_{\delta }) \end{array} \right] * \left[ \begin{array}{cc} 1 &{}0 \\ 0 &{} 1 \end{array} \right] \\ {}&\quad = \dfrac{1}{|T|^2}\left[ \begin{array}{cc} \varvec{C}_{\delta }*\varvec{\varPhi }*(\varvec{C}_{\delta }-\varvec{C}^0_{\delta })&{}\varvec{C}_{\delta }*\varvec{\varPsi }*(\varvec{\rho }_{\delta }-\varvec{\rho }^0_{\delta }) \\ \varvec{\rho }_{\delta }*\varvec{\varTheta }*(\varvec{C}_{\delta }-\varvec{C}^0_{\delta }) &{}\varvec{\rho }_{\delta }*\varvec{\varGamma }*(\varvec{\rho }_{\delta }-\varvec{\rho }^0_{\delta }) \end{array} \right] \\ {}&\quad = \dfrac{1}{|T|^2}\left[ \begin{array}{cc} \int _T \varvec{C}(\varvec{x})\varvec{\varPhi }(\varvec{x}-\varvec{y})(\varvec{C}(\varvec{y})-\varvec{C}^0) \textrm{d}\varvec{x}\textrm{d}\varvec{y} \\ \quad \int _T \varvec{C}(\varvec{x})\varvec{\varPsi }(\varvec{x}-\varvec{y})(\varvec{\rho }(\varvec{y})-\varvec{\rho }^0) \textrm{d}\varvec{x}\textrm{d}\varvec{y} \\ \int _T \varvec{\rho }(\varvec{x})\varvec{\varTheta }(\varvec{x}-\varvec{y})(\varvec{C}(\varvec{y})-\varvec{C}^0) \textrm{d}\varvec{x}\textrm{d}\varvec{y} \\ \quad \int _T \varvec{\rho }(\varvec{x})\varvec{\varGamma }(\varvec{x}-\varvec{y})(\varvec{\rho }(\varvec{y})-\varvec{\rho }^0) \textrm{d}\varvec{x}\textrm{d}\varvec{y} \end{array} \right] \\ {}&\quad =\!=!=\!={(61)(62) } \dfrac{1}{|T|^2}\\&\left[ \begin{array}{cc} \int _T \varvec{C}(\varvec{x})\varvec{\varPhi }(\varvec{x}-\varvec{y})\varvec{C}(\varvec{y}) \textrm{d}\varvec{x}\textrm{d}\varvec{y} \\ \quad \int _T \varvec{C}(\varvec{x})\varvec{\varPsi }(\varvec{x}-\varvec{y})\varvec{\rho }(\varvec{y}) \textrm{d}\varvec{x}\textrm{d}\varvec{y} \\ \int _T \varvec{\rho }(\varvec{x})\varvec{\varTheta }(\varvec{x}-\varvec{y})\varvec{C}(\varvec{y}) \textrm{d}\varvec{x}\textrm{d}\varvec{y} \\ \quad \int _T \varvec{\rho }(\varvec{x})\varvec{\varGamma }(\varvec{x}-\varvec{y})\varvec{\rho }(\varvec{y}) \textrm{d}\varvec{x}\textrm{d}\varvec{y} \end{array} \right] . \end{aligned}$$

It follows that

$$\begin{aligned} \begin{aligned} \overline{\tilde{\varvec{C}}}^{(1)H}&=\Big (\frac{1}{|T|^2} \int _T \varvec{C}(\varvec{x})\varvec{\varPhi }(\varvec{x}-\varvec{y})\varvec{C}(\varvec{y}) \textrm{d}\varvec{x}\textrm{d}\varvec{y}\Big )^H \\&=\frac{1}{|T|^2} \int _T \varvec{C}^T(\varvec{y})\varvec{\varPhi }^H(\varvec{x}-\varvec{y})\varvec{C}^T(\varvec{x}) \textrm{d}\varvec{x}\textrm{d}\varvec{y} \\&=\!=!=\!={(61)(64)} \frac{1}{|T|^2} \int _T \varvec{C}(\varvec{y})\varvec{\varPhi }(\varvec{y}-\varvec{x})\varvec{C}(\varvec{x}) \textrm{d}\varvec{x}\textrm{d}\varvec{y} \\&=\frac{1}{|T|^2} \int _T \varvec{C}(\varvec{x})\varvec{\varPhi }(\varvec{x}-\varvec{y})\varvec{C}(\varvec{y}) \textrm{d}\varvec{x}\textrm{d}\varvec{y} \\&=\overline{\tilde{\varvec{C}}}^{(1)}. \end{aligned} \end{aligned}$$
(71)

Similarly,

$$\begin{aligned} \overline{\tilde{\varvec{\rho }}}^{(1)H}= & {} \overline{\tilde{\varvec{\rho }}}^{(1)}, \end{aligned}$$
(72)
$$\begin{aligned} \overline{\tilde{\varvec{S^1}}}^{(1)H}= & {} \Big (\frac{1}{|T|^2} \int _T \varvec{C}(\varvec{x})\varvec{\varPsi }(\varvec{x}-\varvec{y})\varvec{\rho }(\varvec{y}) \textrm{d}\varvec{x}\textrm{d}\varvec{y}\Big )^H \nonumber \\= & {} \frac{1}{|T|^2} \int _T \varvec{\rho }(\varvec{y})\varvec{\varPsi }^H(\varvec{x}-\varvec{y})\varvec{C}(\varvec{x}) \textrm{d}\varvec{x}\textrm{d}\varvec{y} \nonumber \\{} & {} =\!=!=\!={(64)} \frac{1}{|T|^2} \int _T \varvec{\rho }(\varvec{y})(-\varvec{\varTheta }(\varvec{x}-\varvec{y}))^*\varvec{C}(\varvec{x}) \textrm{d}\varvec{x}\textrm{d}\varvec{y}\nonumber \\{} & {} =\!=!=\!={(61)} -\frac{1}{|T|^2} \int _T \varvec{\rho }(\varvec{y})\varvec{\varTheta }(\varvec{y}-\varvec{x})\varvec{C}(\varvec{x}) \textrm{d}\varvec{x}\textrm{d}\varvec{y}\nonumber \\= & {} -\overline{\tilde{\varvec{S^2}}}^{(1)}. \end{aligned}$$
(73)

Second order:

$$\begin{aligned}&\left[ \begin{array}{cc} \overline{\tilde{\varvec{C}}}^{(2)} &{} \overline{\tilde{\varvec{S^1}}}^{(2)} \\ \overline{\tilde{\varvec{S^2}}}^{(2)} &{} \overline{\tilde{\varvec{\rho }}}^{(2)} \end{array} \right] \\&=\dfrac{1}{|T|^3} \left[ \begin{array}{cc} 1 *\varvec{C}_{\delta }&{}0 \\ 0 &{} 1 *\varvec{\rho }_{\delta } \end{array} \right] \\&\quad * \left[ \begin{array}{cc} \varvec{\varPhi }*(\varvec{C}_{\delta }-\varvec{C}^0_{\delta })&{}\varvec{\varPsi }*(\varvec{\rho }_{\delta }-\varvec{\rho }^0_{\delta }) \\ \varvec{\varTheta }*(\varvec{C}_{\delta }-\varvec{C}^0_{\delta }) &{}\varvec{\varGamma }*(\varvec{\rho }_{\delta }-\varvec{\rho }^0_{\delta }) \end{array} \right] ^2 * \left[ \begin{array}{cc} 1 &{}0 \\ 0 &{} 1 \end{array} \right] \\ {}&=\dfrac{1}{|T|^3} \left[ \begin{array}{cc} 1 *\varvec{C}_{\delta }&{}0 \\ 0 &{} 1 *\varvec{\rho }_{\delta } \end{array} \right] * \\&\quad \left[ \begin{array}{cc} \varvec{\varPhi }*(\varvec{C}_{\delta }-\varvec{C}^0_{\delta })&{}\varvec{\varPsi }*(\varvec{\rho }_{\delta }-\varvec{\rho }^0_{\delta }) \\ \varvec{\varTheta }*(\varvec{C}_{\delta }-\varvec{C}^0_{\delta }) &{}\varvec{\varGamma }*(\varvec{\rho }_{\delta }-\varvec{\rho }^0_{\delta }) \end{array} \right] \\&\quad *\left[ \begin{array}{cc} \varvec{\varPhi }*(\varvec{C}_{\delta }-\varvec{C}^0_{\delta })&{}\varvec{\varPsi }*(\varvec{\rho }_{\delta }-\varvec{\rho }^0_{\delta }) \\ \varvec{\varTheta }*(\varvec{C}_{\delta }-\varvec{C}^0_{\delta }) &{}\varvec{\varGamma }*(\varvec{\rho }_{\delta }-\varvec{\rho }^0_{\delta }) \end{array} \right] * \left[ \begin{array}{cc} 1 &{}0 \\ 0 &{} 1 \end{array} \right] \\ {}&=\!=!=\!={ (62) } \dfrac{1}{|T|^3} \left[ \begin{array}{cc} 1 *\varvec{C}_{\delta }*\varvec{\varPhi }*(\varvec{C}_{\delta }-\varvec{C}^0_{\delta }) &{} 1 *\varvec{C}_{\delta }*\varvec{\varPsi }* (\varvec{\rho }_{\delta }-\varvec{\rho }^0_{\delta }) \\ 1 *\varvec{\rho }_{\delta }*\varvec{\varTheta }*(\varvec{C}_{\delta }-\varvec{C}^0_{\delta }) &{} 1 *\varvec{\rho }_{\delta }*\varvec{\varGamma }*(\varvec{\rho }_{\delta }-\varvec{\rho }^0_{\delta }) \end{array} \right] *\\&\left[ \begin{array}{cc} \varvec{\varPhi }*\varvec{C}_{\delta }* 1 &{}\varvec{\varPsi }*\varvec{\rho }_{\delta }* 1 \\ \varvec{\varTheta }*\varvec{C}_{\delta }* 1 &{}\varvec{\varGamma }*\varvec{\rho }_{\delta }* 1 \end{array} \right] . \end{aligned}$$

It follows that

$$\begin{aligned} \begin{aligned} \overline{\tilde{\varvec{C}}}^{(2)H}&=\Big (\frac{1}{|T|^3} \int _T \varvec{C}(\varvec{x})\varvec{\varPhi }(\varvec{x}-\varvec{y})(\varvec{C}(\varvec{y})-\varvec{C}^0)\\&\quad \times \varvec{\varPhi }(\varvec{y}-\varvec{z})\varvec{C}(\varvec{z}) \textrm{d}\varvec{x}\textrm{d}\varvec{y}\textrm{d}\varvec{z}\Big )^H \\&\quad +\Big (\frac{1}{|T|^3} \int _T \varvec{C}(\varvec{x})\varvec{\varPsi }(\varvec{x}-\varvec{y})(\varvec{\rho }(\varvec{y})-\varvec{\rho }^0)\\&\quad \times \varvec{\varTheta }(\varvec{y}-\varvec{z})\varvec{C}(\varvec{z}) \textrm{d}\varvec{x}\textrm{d}\varvec{y}\textrm{d}\varvec{z}\Big )^H\\&=\!=!=\!={(61)(64)} \frac{1}{|T|^3} \int _T \varvec{C}(\varvec{z})\varvec{\varPhi }(\varvec{z}-\varvec{y})(\varvec{C}(\varvec{y})-\varvec{C}^0)\\&\quad \times \varvec{\varPhi }(\varvec{y}-\varvec{x})\varvec{C}(\varvec{x}) \textrm{d}\varvec{x}\textrm{d}\varvec{y}\textrm{d}\varvec{z} \\&\quad +\int _T \varvec{C}(\varvec{z})\big (-\varvec{\varPsi }(\varvec{z}-\varvec{y})\big )(\varvec{\rho }(\varvec{y})-\varvec{\rho }^0)\\&\quad \times \big (-\varvec{\varTheta }(\varvec{y}-\varvec{x})\big )\varvec{C}(\varvec{x}) \textrm{d}\varvec{x}\textrm{d}\varvec{y}\textrm{d}\varvec{z} =\overline{\tilde{\varvec{C}}}^{(2)}. \end{aligned} \end{aligned}$$
(74)

Similarly,

$$\begin{aligned} \overline{\tilde{\varvec{\rho }}}^{(2)H}= & {} \overline{\tilde{\varvec{\rho }}}^{(2)}, \end{aligned}$$
(75)
$$\begin{aligned} \overline{\tilde{\varvec{S^1}}}^{(2)H}= & {} \Big (\frac{1}{|T|^3} \int _T \varvec{C}(\varvec{x})\varvec{\varPhi }(\varvec{x}-\varvec{y})(\varvec{C}(\varvec{y})-\varvec{C}^0)\nonumber \\{} & {} \times \varvec{\varPsi }(\varvec{y}-\varvec{z})\varvec{\rho }(\varvec{z}) \textrm{d}\varvec{x}\textrm{d}\varvec{y}\textrm{d}\varvec{z}\Big )^H \nonumber \\{} & {} +\Big (\frac{1}{|T|^3} \int _T \varvec{C}(\varvec{x})\varvec{\varPsi }(\varvec{x}-\varvec{y})(\varvec{\rho }(\varvec{y})-\varvec{\rho }^0)\nonumber \\{} & {} \times \varvec{\varGamma }(\varvec{y}-\varvec{z})\varvec{\rho }(\varvec{z}) \textrm{d}\varvec{x}\textrm{d}\varvec{y}\textrm{d}\varvec{z}\Big )^H\nonumber \\{} & {} =\!=!=\!={(61)(64)} \frac{1}{|T|^3} \int _T \varvec{\rho }(\varvec{z})\big (-\varvec{\varTheta }(\varvec{z}-\varvec{y})\big )(\varvec{C}(\varvec{y})-\varvec{C}^0)\nonumber \\{} & {} \times \, \varvec{\varPhi }(\varvec{y}-\varvec{x})\varvec{C}(\varvec{x}) \textrm{d}\varvec{x}\textrm{d}\varvec{y}\textrm{d}\varvec{z} \nonumber \\{} & {} +\int _T \varvec{\rho }(\varvec{z})\varvec{\varGamma }(\varvec{z}-\varvec{y})(\varvec{\rho }(\varvec{y})-\varvec{\rho }^0)\nonumber \\{} & {} \big (-\varvec{\varTheta }(\varvec{y}-\varvec{x})\big )\varvec{C}(\varvec{x}) \textrm{d}\varvec{x}\textrm{d}\varvec{y}\textrm{d}\varvec{z}\nonumber \\{} & {} =-\overline{\tilde{\varvec{S^2}}}^{(2)H}. \end{aligned}$$
(76)

By the same token, higher order terms also satisfy the same equations. Thus we obtain

$$\begin{aligned} \overline{\tilde{\varvec{C}}}=\overline{\tilde{\varvec{C}}}^{H},\overline{\tilde{\varvec{\rho }}}=\overline{\tilde{\varvec{\rho }}}^{H},\overline{\tilde{\varvec{S^1}}}^{H}=-\overline{\tilde{\varvec{S^2}}}^{H}. \end{aligned}$$
(77)

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Zhu, X., Tang, S. Clustering analysis for elastodynamic homogenization. Comput Mech 72, 725–741 (2023). https://doi.org/10.1007/s00466-023-02315-w

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