Concurrent shape optimization of a multiscale structure for controlling macrostructural stiffness

Abstract

We propose a novel shape optimization method for designing a multiscale structure with the desired stiffness. The shapes of the macro- and microstructures are concurrently optimized. The squared error norm between actual and target displacements of the macrostructure is minimized as an objective function. The design variables are the shape variation fields of the outer and interface shapes of the macrostructure and the shapes of holes in the microstructures. Subdomains with independent periodic microstructures are arbitrarily defined in the macrostructure in advance. Homogenized elastic tensors are calculated and applied to the correspondent subdomains. The shape gradient functions are theoretically derived with respect to each shape variation of the macro- and microstructures, and applied to the H¹ gradient method to determine the optimum shapes. The proposed method is applied to several numerical examples, including Poisson’s ratio design and deformation control designs of an L-shaped bracket and a both ends fixed beam with holes. The results of the design examples confirm that the desired stiff or compliant deformation can be achieved while obtaining clear and smooth boundaries. The influence on the final results of the initial shape of the unit cell, the connectivity of adjacent microstructures, and interface optimization is also discussed.

Appendix

1.1 List of symbols

$\left( {\, \cdot \,} \right)^{M}$	Variables for the macrostructure
$\left( {\, \cdot \,} \right)_{M}$	Function defined for the macrostructure
${\varvec{V}}_{{}}^{\left( I \right)}$	Shape variation vector (or field) of unit cell I
${\varvec{V}}_{{}}^{M\left( I \right)}$	Shape variation vector distributed on the outer boundary of subdomain I in the macrostructure
${\varvec{V}}_{{}}^{{B\left( {KL} \right)}}$	Shape variation vector distributed on the interface boundary between subdomains K and L in the macrostructure
$\varvec{u}^{M}$	Actual displacement vector of the macrostructure
${\mathbf{w}}^{M}$	Target displacement vector of the macrostructure
$F_{M}^{(I)} (\, \cdot ,\, \cdot \,)$	Squared error of subdomain I in the macrostructure
$a_{M\left( I \right)} ( \cdot \,,\, \cdot )$	Bilinear form for internal virtual work of subdomain I in the macrostructure
$h_{M\left( I \right)} ( \cdot , \cdot )$	Bilinear form for virtual work by the Cauchy stress vector on the interface Boundary of subdomain I in the macrostructure
$l_{M\left( I \right)} ( \cdot )$	Linear form for external virtual work of subdomain I in the macrostructure
$a_{\left( I \right)} ( \cdot \,,\, \cdot )$	Bilinear form for internal virtual work of unit cell I
$\Omega^{M}$	Domain of the macrostructure
$\Gamma^{M}$	Boundary of macrostructure domain $\Omega^{M}$
$\Gamma^{D}$	Boundary for defining the target displacement in the macrostructure
${\varvec{u}}^{M\left( I \right)}$	Displacement vector in subdomain I in the macrostructure
$\Omega^{M\left( I \right)}$	Subdomain I in the macrostructure
$\Gamma^{M\left( I \right)}$	Boundary of subdomain I in the macrostructure
${\varvec{E}}_{{}}^{M\left( I \right)}$	Elastic tensor of subdomain I in the macrostructure.
$\Gamma^{{M\left( {I,J} \right)}}$	Interface boundary between subdomains I and J in the macrostructure
$\Omega^{\left( I \right)}$	Domain of unit cell I
${{\varvec{\chi}}}^{\left( I \right)kl}$	Characteristic displacement vector of microstructure I for unit initial strain component kl
${\varvec{I}}^{\left( I \right)kl}$	Unit initial strain tensor with the component kl of unit cell I
$\overline{{\left( {\, \cdot \,} \right)}}$	Virtual displacement or adjoint displacement or Lagrange variable
N	Number of domain divisions
$H_{0}^{1}$	Sobolev space of order 1
$U_{Y}^{\left( I \right)}$	Y-periodic allowable displacement space of unit cell I
$U^{M}$	Allowable displacement space of the macrostructure
${\varvec{n}}_{{}}^{M\left( K \right)}$	Outward unit normal vector of subdomain K
${\varvec{n}}_{{}}^{\left( K \right)}$	Outward unit normal vector of microstructure K.
${\varvec{P}}$	Surface force vector on the macrostructure.
$\left| Y \right|$	Periodic unit area of the microstructure.
$\left( {\, \cdot \,} \right)^{\prime }$	Shape derivative.
$\mathop {\left( {\, \cdot \,} \right)}\limits$	Material derivative.
${\varvec{G}}^{M\left( I \right)}$ ($= G^{M\left( I \right)} {\varvec{n}}^{{{(}I{)}}}$)	Shape gradient function for subdomain I in the macrostructure.
$G^{M\left( I \right)}$	Shape gradient density function for subdomain I in the macrostructure.
$G^{{B\left( {KL} \right)}}$	Shape gradient density function for the interface boundary between subdomains K and L.
$G^{\left( I \right)}$	Shape gradient density function for microstructure I
$\hat{G}^{\left( I \right)}$	Mean shape gradient density function in microstructure I
EN	Number of elements in each subdomain
$A_{el}^{M\left( I \right)}$	Area of element el in subdomain I in the macrostructure
$\Delta s^{M}$	Small positive coefficient for macrostructure
$\Delta s$	Small positive coefficient for microstructure
$\overline{\varvec{v}}^{M}$	Virtual displacement vector of the macrostructure
$\overline{\varvec{v}}$	Virtual displacement vector of the microstructure
$C_{M}$	Kinematically admissible function space for the macrostructure
$C_{\Theta }$	Kinematically admissible function space for the unit cell

1.2 Derivation of Eq. (13)

The calculation process for deriving Eq. (15) from Eq. (14) is described here.

Equation (30) shows the material derivative of the first term on the right side of Eqs. (14), and (31), (32), (33), and (34) are the second, third, fourth, and fifth terms, respectively.

$$\dot{L} = 2\sum\limits_{I = 1}^{3} {F_{M\left( I \right)} \left( {\varvec{u}^{M\left( I \right)} - {\varvec{w}}^{M\left( I \right)} ,\varvec{u}^{M\left( I \right)\prime } } \right)}$$

(30)

$$- \sum\limits_{I = 1}^{3} {\left\{ {a_{M\left( I \right)} \left( {{\varvec{u}}^{M\left( I \right)\prime } ,{\overline{\varvec{u}}}^{M\left( I \right)} } \right) + a_{M\left( I \right)} \left( {{\varvec{u}}^{M\left( I \right)} ,{\overline{\varvec{u}}}^{M\left( I \right)\prime } } \right)} \right\}} \, - \sum\limits_{I = 1}^{3} {\int_{{\Gamma^{M\left( I \right)} }} {E_{ijkl}^{M\left( I \right)} u_{k,l}^{M\left( I \right)} \overline{u}_{i,j}^{M\left( I \right)} n_{s}^{M\left( I \right)} V_{s}^{M\left( I \right)} {\text{d}}\Gamma^{M\left( I \right)} } } - \sum\limits_{I = 1}^{3} {\int_{{\Omega^{M\left( I \right)} }} {\left[ {\frac{1}{\left| Y \right|}\int_{{\Omega^{\left( I \right)} }} {\left( { - E_{ijmn}^{{}} \frac{{\partial \chi_{m}^{\left( I \right)kl\prime } }}{{\partial y_{n} }}} \right)} {\text{d}}\Omega^{\left( I \right)} } \right.} } \left. { + \frac{1}{\left| Y \right|}\int_{{\Gamma^{\left( I \right)} }} {\left( {E_{ijkl}^{{}} - E_{ijmn}^{{}} \frac{{\partial \chi_{m}^{\left( I \right)kl} }}{{\partial y_{n} }}} \right)n_{s}^{\left( I \right)} V_{s}^{\left( I \right)} {\text{d}}\Gamma^{\left( I \right)} } } \right]u_{k,l}^{M\left( I \right)} \overline{u}_{i,j}^{M\left( I \right)} {\text{d}}\Omega^{M\left( I \right)}$$

(31)

$$\begin{gathered} \,\,\,\,\,\,\,\,\, + \sum\limits_{I = 1}^{3} {\left\{ {h_{M\left( I \right)} \left( {{\varvec{u}}^{M\left( I \right)\prime } ,{\overline{\varvec{u}}}^{M\left( I \right)} } \right) + h_{M\left( I \right)} \left( {{\varvec{u}}^{M\left( I \right)} ,{\overline{\varvec{u}}}^{M\left( I \right)\prime } } \right)} \right\}} \hfill \\ \,\,\,\,\,\,\,\,\, + \int_{{\Gamma^{{M\left( {1,2} \right)}} }} {\left[ {\left( {E_{ijkl}^{M\left( 1 \right)} u_{k,l}^{M\left( 1 \right)} n_{j}^{M\left( 1 \right)} \overline{u}_{i}^{M\left( 1 \right)} } \right),_{q} n_{q}^{M} + E_{ijkl}^{M\left( 1 \right)} u_{k,l}^{M\left( 1 \right)} n_{j}^{M\left( 1 \right)} \overline{u}_{i}^{M\left( 1 \right)} \kappa^{M\left( 1 \right)} } \right]n_{s}^{M\left( 1 \right)} V_{s}^{M\left( 1 \right)} } {\text{d}}\Gamma^{{M\left( {1,2} \right)}} \hfill \\ \,\,\,\,\,\,\,\,\, + \int_{{\Gamma^{{M\left( {1,3} \right)}} }} {\left[ {\left( {E_{ijkl}^{M\left( 1 \right)} u_{k,l}^{M\left( 1 \right)} n_{j}^{M\left( 1 \right)} \overline{u}_{i}^{M\left( 1 \right)} } \right),_{q} n_{q}^{M} + E_{ijkl}^{M\left( 1 \right)} u_{k,l}^{M\left( 1 \right)} n_{j}^{M\left( 1 \right)} \overline{u}_{i}^{M\left( 1 \right)} \kappa^{M\left( 1 \right)} } \right]n_{s}^{M\left( 1 \right)} V_{s}^{M\left( 1 \right)} } {\text{d}}\Gamma^{{M\left( {1,3} \right)}} \hfill \\ \,\,\,\,\,\,\,\,\, + \int_{{\Gamma^{{M\left( {2,1} \right)}} }} {\left[ {\left( {E_{ijkl}^{M\left( 2 \right)} u_{k,l}^{M\left( 2 \right)} n_{j}^{M\left( 2 \right)} \overline{u}_{i}^{M\left( 2 \right)} } \right),_{q} n_{q}^{M} + E_{ijkl}^{M\left( 2 \right)} u_{k,l}^{M\left( 2 \right)} n_{j}^{M\left( 2 \right)} \overline{u}_{i}^{M\left( 2 \right)} \kappa^{M\left( 2 \right)} } \right]n_{s}^{M\left( 2 \right)} V_{s}^{M\left( 2 \right)} } {\text{d}}\Gamma^{{M\left( {2,1} \right)}} \hfill \\ \,\,\,\,\,\,\,\,\, + \int_{{\Gamma^{{M\left( {2,3} \right)}} }} {\left[ {\left( {E_{ijkl}^{M\left( 2 \right)} u_{k,l}^{M\left( 2 \right)} n_{j}^{M\left( 2 \right)} \overline{u}_{i}^{M\left( 2 \right)} } \right),_{q} n_{q}^{M} + E_{ijkl}^{M\left( 2 \right)} u_{k,l}^{M\left( 2 \right)} n_{j}^{M\left( 2 \right)} \overline{u}_{i}^{M\left( 2 \right)} \kappa^{M\left( 2 \right)} } \right]n_{s}^{M\left( 2 \right)} V_{s}^{M\left( 2 \right)} } {\text{d}}\Gamma^{{M\left( {2,3} \right)}} \hfill \\ \,\,\,\,\,\,\,\,\, + \int_{{\Gamma^{{M\left( {3,1} \right)}} }} {\left[ {\left( {E_{ijkl}^{M\left( 3 \right)} u_{k,l}^{M\left( 3 \right)} n_{j}^{M\left( 3 \right)} \overline{u}_{i}^{M\left( 3 \right)} } \right),_{q} n_{q}^{M} + E_{ijkl}^{M\left( 3 \right)} u_{k,l}^{M\left( 3 \right)} n_{j}^{\left( 3 \right)} \overline{u}_{i}^{M\left( 3 \right)} \kappa^{M\left( 3 \right)} } \right]n_{s}^{M\left( 3 \right)} V_{s}^{M\left( 3 \right)} } {\text{d}}\Gamma^{{M\left( {3,1} \right)}} \hfill \\ \,\,\,\,\,\,\,\,\, + \int_{{\Gamma^{{M\left( {3,2} \right)}} }} {\left[ {\left( {E_{ijkl}^{M\left( 3 \right)} u_{k,l}^{M\left( 3 \right)} n_{j}^{M\left( 3 \right)} \overline{u}_{i}^{M\left( 3 \right)} } \right),_{q} n_{q}^{M} + E_{ijkl}^{M\left( 3 \right)} u_{k,l}^{M\left( 3 \right)} n_{j}^{\left( 3 \right)} \overline{u}_{i}^{M\left( 3 \right)} \kappa^{M\left( 3 \right)} } \right]n_{s}^{M\left( 3 \right)} V_{s}^{M\left( 3 \right)} } {\text{d}}\Gamma^{{M\left( {3,2} \right)}} \hfill \\ \end{gathered}$$

(32)

$$\,\,\,\,\,\,\,\,\, + \sum\limits_{I = 1}^{3} {\left\{ {l_{M\left( I \right)} \left( {{\overline{\varvec{u}}}^{M\left( I \right)\prime } } \right)} \right\}}$$

(33)

$$\begin{gathered} \,\,\,\,\,\,\,\,\, - \sum\limits_{I = 1}^{3} {\int_{{\Omega^{M\left( I \right)} }} {\sum\limits_{k = 1}^{2} {\sum\limits_{l = 1}^{2} {\left\{ {a_{\left( I \right)} \left( {{{\varvec{\chi}}}^{\left( I \right)kl\prime } ,{\overline{\varvec{\chi }}}^{\left( I \right)kl} } \right) + a_{\left( I \right)} \left( {{{\varvec{\chi}}}^{\left( I \right)kl} - {\varvec{I}}^{kl} ,{\overline{\varvec{\chi }}}^{\left( I \right)kl\prime } } \right)} \right\}} } {\text{d}}\Omega^{M\left( I \right)} } } \hfill \\ \,\,\,\,\,\,\,\,\, - \sum\limits_{I = 1}^{3} {\int_{{\Omega^{M\left( I \right)} }} {\int_{{\Gamma^{\left( I \right)} }} {\sum\limits_{k,l = 1,2} {\left( {E_{ijmn} \chi_{m,n}^{\left( I \right)kl} \overline{\chi }_{i,j}^{\left( I \right)kl} - E_{ijmn} I_{m,n}^{\left( I \right)kl} \overline{\chi }_{i,j}^{\left( I \right)kl} } \right)} n_{s} V_{s} {\text{d}}\Gamma^{\left( I \right)} } } {\text{d}}\Omega^{M\left( I \right)} } \hfill \\ \end{gathered}$$

(34)

where, $\kappa^{M\left( I \right)}$ is the curvature of the interface boundary on subdomain I. The relationships between the normal vectors (Eq. 35), the curvatures (Eq. 36), the displacements (Eq. 37), the domain variations (Eq. 38), and the equilibrium forces based on the Cauchy stress vector (Eq. 39) at the interface boundary $\Gamma^{M(I,J)}$ are used to derive Eq. (30)–(34) (Shi and Shimoda 2015).

$${\varvec{n}}^{M\left( I \right)} = - {\varvec{n}}^{M\left( J \right)} ,\left( {I = 1,2,3} \right),\,\left( {J = 1,2,3} \right),\,\left( {I \ne J} \right)$$

(35)

$${{\varvec{\kappa}}}^{M\left( I \right)} = - {{\varvec{\kappa}}}^{M\left( J \right)} ,\left( {I = 1,2,3} \right),\,\left( {J = 1,2,3} \right),\,\left( {I \ne J} \right)$$

(36)

$${\varvec{u}}^{M\left( I \right)} = {\varvec{u}}^{M\left( J \right)} ,\left( {I = 1,2,3} \right),\,\left( {J = 1,2,3} \right),\left( {I \ne J} \right)$$

(37)

$${\varvec{V}}_{{}}^{M\left( I \right)} = {\varvec{V}}_{{}}^{M\left( J \right)} ,\left( {I = 1,2,3} \right),\,\left( {J = 1,2,3} \right)$$

(38)

$$E_{ijkl}^{M\left( I \right)} u_{k,l}^{M\left( I \right)} n_{j}^{M\left( I \right)} = - E_{ijkl}^{M\left( J \right)} u_{k,l}^{M\left( J \right)} n_{j}^{M\left( J \right)} ,\left( {I = 1,2,3} \right),\,\left( {J = 1,2,3} \right),\,\left( {I \ne J} \right)$$

(39)

When the following Eqs. (40)–(43) are satisfied, Eq. (30)–(34) is re-expressed as Eq. (44). Here, Eq. (40) is the state equation for ${{\varvec{\chi}}}_{{}}^{\left( I \right)kl}$ in the unit cell and is the same as Eq. (6). Eq. (41) is the adjoint equation for ${\overline{\varvec{\chi }}}_{{}}^{\left( I \right)kl}$. Eq. (42) is the state equation for $\varvec{u}^{M\left( I \right)}$ in the macrostructure and is the same as Eq. (3). Equation (43) is the adjoint equation for $\overline{\varvec{u}}^{M\left( I \right)}$.

$$a_{\left( I \right)} ({{\varvec{\chi}}}^{\left( I \right)kl} - {\varvec{I}}^{\left( I \right)kl} ,{\overline{\varvec{\chi }}}^{\left( I \right)kl\prime } ) = 0, \, \,\,\forall {\overline{\varvec{\chi }}}^{\left( I \right)kl\prime } \in U_{Y}^{\left( I \right)} , \, (k,l = 1,2),\left( {I = 1,2,3} \right)$$

(40)

$$a_{\left( I \right)} ({{\varvec{\chi}}}^{\left( I \right)kl\prime } ,{\overline{\varvec{\chi }}}^{\left( I \right)kl} ) = \frac{1}{\left| Y \right|}\left[ {\int_{{\Omega^{\left( I \right)} }} {(E_{ijmn}^{{}} \chi_{m,n}^{\left( I \right)kl\prime } ){\text{d}}\Omega^{\left( I \right)} } } \right]u_{k,l}^{M\left( I \right)} u_{i,j}^{M\left( I \right)} , \, \forall {{\varvec{\chi}}}^{\left( I \right)kl\prime } \in U_{Y}^{\left( I \right)} , \, (k,l = 1,2),\left( {I = 1,2,3} \right)$$

(41)

$$\sum\limits_{I = 1}^{3} {l_{M\left( I \right)} \left( {{\overline{\varvec{u}}}^{M\left( I \right)\prime } } \right)} = \sum\limits_{I = 1}^{3} {a_{M\left( I \right)} \left( {\varvec{u}^{M\left( I \right)} ,\overline{\varvec{u}}^{M\left( I \right)\prime } } \right)} - \sum\limits_{I = 1}^{3} {h_{M\left( I \right)} \left( {\varvec{u}^{M\left( I \right)} ,\overline{\varvec{u}}^{M\left( I \right)\prime } } \right)} , \, \forall {\overline{\varvec{u}}}^{M\left( I \right)\prime } \in U$$

(42)

$$2\sum\limits_{I = 1}^{3} {F_{M\left( I \right)} \left( {\varvec{u}^{M\left( I \right)} - {\varvec{w}}^{M\left( I \right)} ,\varvec{u}^{M\left( I \right)\prime } } \right)} = \sum\limits_{I = 1}^{3} {a_{M\left( I \right)} \left( {\varvec{u}^{M\left( I \right)\prime } ,\overline{\varvec{u}}^{M\left( I \right)} } \right)} - \sum\limits_{I = 1}^{3} {h_{M\left( I \right)} \left( {\varvec{u}^{M\left( I \right)\prime } ,\overline{\varvec{u}}^{M\left( I \right)} } \right)} ,\forall {\varvec{u}}^{M\left( I \right)\prime } \in U$$

(43)

Finally, $\dot{L}$ is expressed as Eq. (15), and the shape gradient functions are derived as expressed in Eqs. (16)–(18).

$$\dot{L} = \sum\limits_{I = 1}^{3} {\left\{ { - \int_{{\Gamma^{M\left( I \right)} }} {E_{ijkl}^{M\left( I \right)} u_{k,l}^{M\left( I \right)} \overline{u}_{i,j}^{M\left( I \right)} n_{s}^{M\left( I \right)} V_{s}^{M\left( I \right)} {\text{d}}\Gamma^{M\left( I \right)} } } \right\}}$$

(44)

$$\begin{gathered} \,\,\,\,\,\,\, + \int_{{\Gamma^{{M\left( {1,2} \right)}} }} {\left\{ { - \left( {E_{ijkl}^{M\left( 1 \right)} u_{k,l}^{M\left( 1 \right)} \overline{u}_{i,j}^{M\left( 1 \right)} - E_{ijkl}^{M\left( 2 \right)} u_{k,l}^{M\left( 2 \right)} \overline{u}_{i,j}^{M\left( 2 \right)} } \right) + E_{ijkl}^{M\left( 1 \right)} u_{k,l}^{M\left( 1 \right)} n_{j}^{M\left( 1 \right)} \left( {\overline{u}_{i,q}^{M\left( 1 \right)} - \overline{u}_{i,q}^{M\left( 2 \right)} } \right)n_{q}^{M\left( 1 \right)} } \right\}n_{s}^{M\left( 1 \right)} V_{s}^{M\left( 1 \right)} {\text{d}}\Gamma^{{M\left( {1,2} \right)}} } \hfill \\ \,\,\,\,\,\,\, + \int_{{\Gamma^{{M\left( {2,3} \right)}} }} {\left\{ { - \left( {E_{ijkl}^{M\left( 2 \right)} u_{k,l}^{M\left( 2 \right)} \overline{u}_{i,j}^{M\left( 2 \right)} - E_{ijkl}^{M\left( 3 \right)} u_{k,l}^{M\left( 3 \right)} \overline{u}_{i,j}^{M\left( 3 \right)} } \right) + E_{ijkl}^{M\left( 2 \right)} u_{k,l}^{M\left( 2 \right)} n_{j}^{M\left( 2 \right)} \left( {\overline{u}_{i,q}^{M\left( 2 \right)} - \overline{u}_{i,q}^{M\left( 3 \right)} } \right)n_{q}^{M\left( 2 \right)} } \right\}n_{s}^{M\left( 2 \right)} V_{s}^{M\left( 2 \right)} {\text{d}}\Gamma^{{M\left( {2,3} \right)}} } \hfill \\ \,\,\,\,\,\,\, + \int_{{\Gamma^{{M\left( {3,1} \right)}} }} {\left\{ { - \left( {E_{ijkl}^{M\left( 3 \right)} u_{k,l}^{M\left( 3 \right)} \overline{u}_{i,j}^{M\left( 3 \right)} - E_{ijkl}^{M\left( 1 \right)} u_{k,l}^{M\left( 1 \right)} \overline{u}_{i,j}^{M\left( 1 \right)} } \right) + E_{ijkl}^{M\left( 3 \right)} u_{k,l}^{M\left( 3 \right)} n_{j}^{M\left( 3 \right)} \left( {\overline{u}_{i,q}^{M\left( 3 \right)} - \overline{u}_{i,q}^{M\left( 1 \right)} } \right)n_{q}^{M\left( 3 \right)} } \right\}n_{s}^{M\left( 3 \right)} V_{s}^{M\left( 3 \right)} {\text{d}}\Gamma^{{M\left( {3,1} \right)}} } \hfill \\ \end{gathered}$$

(45)

$$\begin{gathered} \,\,\,\,\,\,\, + \sum\limits_{I = 1}^{3} {\left\{ { - \int_{{\Omega^{M\left( I \right)} }} {\left[ {\frac{1}{\left| Y \right|}\int_{{\Gamma^{\left( I \right)} }} {\left( {E_{ijkl}^{{}} - E_{ijmn}^{{}} \frac{{\partial \chi_{m}^{\left( I \right)kl} }}{{\partial y_{n} }}} \right)u_{k,l}^{M\left( 1 \right)} \overline{u}_{i,j}^{M\left( 1 \right)} n_{s}^{\left( I \right)} V_{s}^{\left( I \right)} {\text{d}}\Gamma^{\left( I \right)} } } \right]} {\text{d}}\Omega^{M\left( I \right)} } \right\}} \hfill \\ \,\,\,\,\,\,\, - \sum\limits_{I = 1}^{3} {\int_{{\Omega^{M\left( I \right)} }} {\left( {\int_{{\Gamma^{\left( I \right)} }} {\sum\limits_{k = 1}^{2} {\sum\limits_{l = 1}^{2} {\left( {E_{ijmn}^{{}} \chi_{m,n}^{\left( I \right)kl} \overline{\chi }_{i,j}^{\left( I \right)kl} - E_{ijmn}^{{}} I_{m,n}^{kl} \overline{\chi }_{i,j}^{\left( I \right)kl} } \right)} } n_{s}^{\left( I \right)} V_{s}^{\left( I \right)} {\text{d}}\Gamma^{\left( I \right)} } } \right)} {\text{d}}\Omega^{M\left( I \right)} } \hfill \\ \end{gathered}$$

(46)

$$\begin{gathered} \, = \sum\limits_{I = 1}^{3} {\int_{{\Gamma^{M\left( I \right)} }} {G^{M\left( I \right)} {\varvec{V}}_{{}}^{M\left( I \right)} \cdot {\varvec{n}}^{{M{(}I{)}}} {\text{d}}\Gamma^{M\left( I \right)} } } + \int_{{\Gamma^{{M\left( {1,2} \right)}} }} {G^{{B\left( {12} \right)}} {\varvec{V}}_{{}}^{M\left( 1 \right)} \cdot {\varvec{n}}^{{M{(1)}}} {\text{d}}\Gamma^{{M\left( {1,2} \right)}} } \hfill \\ \, + \int_{{\Gamma^{{M\left( {2,3} \right)}} }} {G^{{B\left( {23} \right)}} {\varvec{V}}_{{}}^{M\left( 2 \right)} \cdot {\varvec{n}}^{{M{(2)}}} {\text{d}}\Gamma^{{M\left( {2,3} \right)}} } + \int_{{\Gamma^{{M\left( {3,1} \right)}} }} {G^{{B\left( {31} \right)}} {\varvec{V}}_{{}}^{M\left( 3 \right)} \cdot {\varvec{n}}^{{M{(3)}}} {\text{d}}\Gamma^{{M\left( {3,1} \right)}} } \hfill \\ \, + \sum\limits_{I = 1}^{3} {\int_{{\Omega^{M\left( I \right)} }} {\left( {\int_{{\Gamma^{\left( I \right)} }} {G_{\left( I \right)} {\varvec{V}}_{{}}^{\left( I \right)} \cdot {\varvec{n}}^{{{(}I{)}}} {\text{d}}\Gamma^{\left( I \right)} } } \right)} {\text{d}}\Omega^{M\left( I \right)} } \hfill \\ \end{gathered}$$

(15)