A topology optimisation of composite elastic metamaterial slabs based on the manipulation of far-field behaviours

Abstract

We propose a numerical method for a topology optimisation of composite elastic metamaterial slabs. We aim to realise some anomalous functionalities such as perfect absorption, wave-mode conversion, and negative refraction by designing the shape and topology of (visco-)elastic inclusions. Instead of manipulating effective material constants, we propose to utilise the far-field characteristics of scattered waves. This allows us to achieve novel functionalities for waves in not only low- but also high-frequency ranges. The design sensitivity corresponding to the far-field characteristics is rigorously derived using the adjoint variable method and incorporated into a level-set–based topology optimisation algorithm. The design sensitivity is computed by the boundary element method with periodic Green’s function instead of the standard finite element method to rigorously deal with the radiation of scattered waves without absorbing boundaries. We show some numerical examples to demonstrate the effectiveness of the proposed method.

Appendices

Appendix A: Periodic Green’s function and far-field characteristics

1.1 A.1 Periodic Green’s function

To analyse the periodic scattering problem (9)–(15), we first consider Green’s function $G^{\mathrm {p}}_{ij}$ satisfying:

$$ \begin{array}{@{}rcl@{}} (\lambda+\mu)G^{\mathrm{p}}_{kj,ik}({x},{y})+\mu G^{\mathrm{p}}_{ij,kk}({x},{y}) + \rho\omega^{2} G^{\mathrm{p}}_{ij}({x},{y}) \\ = -\delta({x}-{y})\delta_{ij}, \end{array} $$

(68)

$$ \begin{array}{@{}rcl@{}} G^{\mathrm{p}}_{ij}({x}+L{e}_{1},{y}) = G^{\mathrm{p}}_{ij}({x},{y})\mathrm{e}^{\mathrm{i}\upbeta}, \end{array} $$

(69)

and the radiation condition, where δ_ij is the Kronecker delta, and δ(x) is the Dirac delta function. This Green’s function $G^{\mathrm {p}}_{ij}$ is called periodic Green’s function and known to have the following representation:

$$ \begin{array}{@{}rcl@{}} G^{\mathrm{p}}_{ij}({x},{y}) = \sum\limits_{n=-\infty}^{\infty} G_{ij}({x}-nL{e}_{1},{y})\mathrm{e}^{\mathrm{i} n\upbeta}, \end{array} $$

(70)

where G_ij is the fundamental solution for the two-dimensional elastodynamics, expressed by:

$$ \begin{array}{@{}rcl@{}} &&G_{ij}({x},{y}) = \frac{\mathrm{i}}{4\mu}\left[ H_{0}^{(1)}( k_{\mathrm{T}}|{x}-{y}|)\delta_{ij}\right.\\ &&\left.\qquad+\frac{1}{ k_{\mathrm{T}}^{2}}\frac{1}{\partial y_{i} \partial y_{j}}\left( H_{0}^{(1)}( k_{\mathrm{T}}|{x} - {y}|) - H_{0}^{(1)}( k_{\mathrm{L}}|{x} - {y}|)\right) \right], \end{array} $$

(71)

with the Hankel functions $H^{(1)}_{n}$ of the first kind and order n and wavenumbers:

$$ \begin{array}{@{}rcl@{}} k_{\mathrm{L}} &=& \omega\sqrt{\frac{\rho}{\lambda+2\mu}}, \end{array} $$

(72)

$$ \begin{array}{@{}rcl@{}} k_{\mathrm{T}} &=& \omega\sqrt{\frac{\rho}{\mu}}. \end{array} $$

(73)

The lattice sum (70) would be the simplest expression of $G^{\mathrm {p}}_{ij}$ but has computational limitations. From (71), we see that G_ij asymptotically behaves as G(x,y) = O(|x − y|^− 1/2) when |x − y| tends to the infinity if Im[λ] = Im[μ] = 0. This implies that the convergence speed of the lattice sum (70) is extremely slow; thus, we require another representation of $G^{\mathrm {p}}_{ij}$ whose convergence is guaranteed and rapid.

For now, we assume that x₂ − y₂ ≠ 0 and consider the following Fourier transform of the fundamental solution G_ij(x,y) with respect to x₁:

$$ \begin{array}{@{}rcl@{}} &&\!\!\!\!\!\!\!\!\!\!\!\mathcal{F}_{1}[G_{ij}](\xi,x_{2},{y}) \\ &:=& {\int}_{-\infty}^{\infty} G_{ij}({x},{y})\mathrm{e}^{-\mathrm{i}\xi x_{1}}\mathrm{d} x_{1} \\ &=& \begin{cases} \frac{\mathrm{i}}{2(\lambda+2\mu)\sqrt{ k_{\mathrm{L}}^{2}-\xi^{2}}}d^{\mathrm{L+}}_{i}(\xi) d^{\mathrm{L+}}_{j}(\xi) \\ \times\mathrm{e}^{\mathrm{i} k_{\mathrm{L}}(-p^{\mathrm{L+}}_{1}(\xi) y_{1}+p^{\mathrm{L+}}_{2}(\xi)(x_{2}-y_{2}))} \\ +\frac{\mathrm{i}}{2\mu\sqrt{ k_{\mathrm{T}}^{2}-\xi^{2}}}d^{\mathrm{T+}}_{i}(\xi) d^{\mathrm{T+}}_{j}(\xi) \\ \times\mathrm{e}^{\mathrm{i} k_{\mathrm{T}}(-p^{\mathrm{T+}}_{1}(\xi) y_{1}+p^{\mathrm{T+}}_{2}(\xi)(x_{2}-y_{2}))} & (x_{2}-y_{2}>0) \\ \frac{\mathrm{i}}{2(\lambda+2\mu)\sqrt{ k_{\mathrm{L}}^{2}-\xi^{2}}}d^{\mathrm{L-}}_{i}(\xi) d^{\mathrm{L-}}_{j}(\xi) \\ \times\mathrm{e}^{\mathrm{i} k_{\mathrm{L}}(-p^{\mathrm{L-}}_{1}(\xi) y_{1}+p^{\mathrm{L-}}_{2}(\xi)(x_{2}-y_{2}))} \\ +\frac{\mathrm{i}}{2\mu\sqrt{ k_{\mathrm{T}}^{2}-\xi^{2}}}d^{\mathrm{T-}}_{i}(\xi) d^{\mathrm{T-}}_{j}(\xi) \\ \times\mathrm{e}^{\mathrm{i} k_{\mathrm{T}}(-p^{\mathrm{T-}}_{1}(\xi) y_{1}+p^{\mathrm{T-}}_{2}(\xi)(x_{2}-y_{2}))} & (x_{2}-y_{2}<0) \end{cases} , \end{array} $$

(74)

where p^L±(ξ), d^L±(ξ), p^T±(ξ), and d^T±(ξ) are defined as follows:

$$ \begin{array}{@{}rcl@{}} {p}^{\text{L}\pm}(\xi) &=& {d}^{\text{L}\pm}(\xi) = \frac{1}{ k_{\mathrm{L}}} \begin{pmatrix} \xi \\ \pm\sqrt{ k_{\mathrm{L}}^{2}-\xi^{2}} \end{pmatrix}, \end{array} $$

(75)

$$ \begin{array}{@{}rcl@{}} {p}^{\text{T}\pm}(\xi) &=& \frac{1}{ k_{\mathrm{T}}} \begin{pmatrix} \xi \\ \pm\sqrt{ k_{\mathrm{T}}^{2}-\xi^{2}} \end{pmatrix}, \end{array} $$

(76)

$$ \begin{array}{@{}rcl@{}} {d}^{\text{T}\pm}(\xi) &=& \frac{1}{ k_{\mathrm{T}}} \begin{pmatrix} \pm\sqrt{ k_{\mathrm{T}}^{2}-\xi^{2}} \\ -\xi \end{pmatrix} . \end{array} $$

(77)

Using Poisson’s summation formula, the lattice sum (70) can be converted into the following series:

$$ \begin{array}{@{}rcl@{}} G^{\mathrm{p}}_{ij}({x},{y}) &=& \begin{cases} G^{p+}_{ij}({x},{y}) & (x_{2}-y_{2} > 0) \\ G^{p-}_{ij}({x},{y}) & (x_{2}-y_{2} < 0) \end{cases} , \end{array} $$

(78)

$$ \begin{array}{@{}rcl@{}} G^{\mathrm{p}\pm}_{ij}({x},{y}) &=& \sum\limits_{n=-\infty}^{\infty} G_{ij}({x}-nL{e}_{1}, {y})\mathrm{e}^{\mathrm{i} n\upbeta} \\ &=& \frac{1}{L}\sum\limits_{m=-\infty}^{\infty}\mathrm{e}^{\mathrm{i} \xi_{m} x_{1}}\mathcal{F}_{1}[G_{ij}](\xi_{m},x_{2},{y}) \\ &=&\frac{\mathrm{i}}{2L}\sum\limits_{m=-\infty}^{\infty} \left( F^{\text{L}\pm}_{ij}(\xi_{m}) \mathrm{e}^{\mathrm{i} k_{\mathrm{L}}({x}-{y})\cdot{p}^{\text{L}\pm}_{m}}\right.\\ &&\left.\quad +F^{\text{T}\pm}_{ij}(\xi_{m})\mathrm{e}^{\mathrm{i} k_{\mathrm{T}}({x}-{y}) \cdot{p}^{\text{T}\pm}_{m}} \right), \end{array} $$

(79)

where ${p}^{\text {L}\pm }_{m}= {p}^{\text {L}\pm }(\xi _{m})$, ${d}^{\text {L}\pm }_{m}= {d}^{\text {L}\pm }(\xi _{m})$, ${p}^{\text {T}\pm }_{m}= {p}^{\text {T}\pm }(\xi _{m})$, ${d}^{\text {T}\pm }_{m}= {d}^{\text {T}\pm }(\xi _{m})$, and

$$ \begin{array}{@{}rcl@{}} \xi_{m}&=&(\upbeta+2m\pi)/L , \end{array} $$

(80)

$$ \begin{array}{@{}rcl@{}} F^{\text{L}\pm}_{ij}(\xi_{m}) &=& \frac{1}{(\lambda+2\mu)\sqrt{ k_{\mathrm{L}}^{2}-{\xi_{m}^{2}}}}d^{\text{L}\pm}_{i}(\xi_{m}) d^{\text{L}\pm}_{j}(\xi_{m}) , \end{array} $$

(81)

$$ \begin{array}{@{}rcl@{}} F^{\text{T}\pm}_{ij}(\xi_{m}) &=& \frac{1}{\mu\sqrt{ k_{\mathrm{T}}^{2}-{\xi_{m}^{2}}}}d^{\text{T}\pm}_{i}(\xi_{m}) d^{\text{T}\pm}_{j}(\xi_{m}). \end{array} $$

(82)

The series (79) converges rapidly because of the exponential functions unless |x₂ − y₂| becomes zero; otherwise, the summands become O(|m|^− 1) as $|m|\to \infty$. We can improve this convergence rate by using Kummer’s transformation and obtain:

$$ \begin{array}{@{}rcl@{}} G^{\mathrm{p}}_{ij}({x},{y}) = \sum\limits_{s=1}^{N_{K}} c_{s}\sum\limits_{n=-\infty}^{\infty} \tilde{G}_{ij}({x}-nL{e}_{1},{y};\sqrt{q_{s}})\mathrm{e}^{\mathrm{i} n\upbeta} \\ + \sum\limits_{m=-\infty}^{\infty} \hat{G}_{ij}({x},{y},\xi_{m}), \end{array} $$

(83)

$$ \begin{array}{@{}rcl@{}} \hat{G}_{ij}({x},{y},\xi_{m}) = \begin{cases} \hat{G}^{+}_{ij}({x},{y},\xi_{m}) & (x_{2}-y_{2} \leq 0) \\ \hat{G}^{-}_{ij}({x},{y},\xi_{m}) & (x_{2}-y_{2} \geq 0) \end{cases}, \end{array} $$

(84)

$$ \begin{array}{@{}rcl@{}} \hat{G}^{\pm}_{ij}({x},{y},\xi_{m}) = \frac{\mathrm{i}}{2L} \left[ F^{\text{L}\pm}_{ij}(\xi_{m})\mathrm{e}^{\mathrm{i} k_{\mathrm{L}}{p}^{\text{L}\pm}_{m}\cdot({x}-{y})}\right. \\ + F^{\text{T}\pm}_{ij}(\xi_{m})\mathrm{e}^{\mathrm{i} k_{\mathrm{T}}{p}^{\text{T}\pm}_{m}\cdot({x}-{y})} \\ +\sum\limits_{s=1}^{N_{K}} \left( \tilde{F}^{\text{L}\pm}_{ij}(\xi_{m})\mathrm{e}^{-\sqrt{q_{s}} k_{\mathrm{L}}\tilde{{p}}^{\text{L}\pm}_{m}(\sqrt{q_{s}})\cdot({x}-{y})}\right. \\ \left.\left.+ \tilde{F}^{\text{T}\pm}_{ij}(\xi_{m})\mathrm{e}^{-\sqrt{q_{s}} k_{\mathrm{T}}\tilde{{p}}^{\text{T}\pm}_{m}(\sqrt{q_{s}})\cdot({x}-{y})} \right) \right] , \end{array} $$

(85)

where the vectors and functions with the tilde symbol are defined by replacing (λ, μ) with (−λ/q_s,−μ/q_s) (correspondingly (k_L,k_T) with ($\mathrm {i}\sqrt {q_{s}} k_{\mathrm {L}}, \mathrm {i}\sqrt {q_{s}} k_{\mathrm {T}}$)). We can easily show that the summands of the first series in (83) become at worst (i.e. when |x₂ − y₂| = 0) $O(|m|^{-2N_{K}-1})$ as $|m|\to \infty$ when q_s > 0 and c_s solve the following linear system:

$$ \begin{array}{@{}rcl@{}} \begin{bmatrix} q_{1} & q_{2} & {\cdots} & q_{N_{K}} \\ {q_{1}^{2}} & {q_{2}^{2}} & {\cdots} & q_{N_{K}}^{2} \\ {\vdots} & {\vdots} & {\ddots} & {\vdots} \\ q_{1}^{N_{K}} & q_{2}^{N_{K}} & {\cdots} & q_{N_{K}}^{N_{K}} \end{bmatrix} \begin{pmatrix} c_{1} \\ c_{2} \\ {\vdots} \\ c_{N_{K}} \end{pmatrix} = \begin{pmatrix} 1 \\ -1 \\ {\vdots} \\ (-1)^{{N_{K}}+1} \end{pmatrix} . \end{array} $$

(86)

Thus, we determine c_s by solving (86); q_s > 0 are regarded as parameters. Note that (85) would suffer from a cancellation of siginificant digits in this case and thus require some transformations such as $e^{z}-1=-2\mathrm {i}\mathrm {e}^{z/2}\sin \limits (iz/2)$. On the other hand, the first series in (83) always converges rapidly since $\tilde {G}_{ij}( {x}, {y})=O(\mathrm {e}^{-k| {x}- {y}|})~(k>0)$ as $| {x}- {y}|\to \infty$. Note that the representation (83)–(85) holds even if x₂ − y₂ = 0 though we assumed otherwise. For more details, refer to Matsushima et al. (2018a).

1.2 A.2 Far-field characteristics

Periodic Green’s function expressed by (79) implies that a scattered field can be expanded into a sum of plane P- and S-waves. This can be shown by substituting (79) into the representation formula (Kitahara et al. 1989):

$$ \begin{array}{@{}rcl@{}} u_{i}({x}) = u^{\text{in}}_{i}({x}) + {\int}_{\Gamma} \left( C_{kljm}G^{\mathrm{p}}_{ki,l}({x},{y})n_{m}({y})u_{j}({y})\right. \\ \left.- G^{\mathrm{p}}_{ij}({x},{y})t_{j}({y}) \right) \mathrm{d}{\Gamma}_{y} \quad {x}\in U\setminus\overline{\Omega}, \end{array} $$

(87)

which yields the plane-wave expansion (17) and formulae (21) and (22).

Appendix B: Boundary element method

We describe the numerical solution of the periodic scattering problem (9)–(15). We first convert it into the Burton-Miller-type boundary integral equations (Burton and Miller 1971):

$$ \begin{array}{@{}rcl@{}} \left\{\left[\!\left( \frac{1}{2}\mathcal{I}+\mathcal{D}^{}\!\right)+\alpha\mathcal{N}^{}\right]{u}\right\}_{i} - \left\{\left[\mathcal{S}^{} - \alpha\left( \frac{1}{2}\mathcal{I}-\mathcal{D}^{*}\right)\right]{t}\right\}_{i} \\ = u^{\text{in}}_{i} + \alpha C^{}_{ijkl}u^{\text{in}}_{k,l}n_{j}, \end{array} $$

(88)

$$ \begin{array}{@{}rcl@{}} \left[\left( \frac{1}{2}\mathcal{I}-\mathcal{D}^{\prime}\right){u}\right]_{i} + \left( \mathcal{S}^{\prime}{t}\right)_{i} = 0, \end{array} $$

(89)

where $\mathcal {I}$ is the identity operator, and $\mathcal {S}^{}$, $\mathcal {D}^{}$, $\mathcal {D}^{*}$, and $\mathcal {N}^{}$ are the integral operators defined by:

$$ \begin{array}{@{}rcl@{}} (\mathcal{S}^{}{\phi})_{i}({x}) = {\int}_{\Gamma} G^{\mathrm{p}}_{ij}({x},{y})\phi_{j}({y})\mathrm{d}{\Gamma}_{y}, \end{array} $$

(90)

$$ \begin{array}{@{}rcl@{}} (\mathcal{D}^{}{\phi})_{i}({x}) = -\mathrm{v.p.}\!{\int}_{\Gamma} \!C^{}_{kljm}G^{p}_{ki,l}({x},{y})n_{m}({y})\phi_{j}({y})\mathrm{d}{\Gamma}_{y}, \end{array} $$

(91)

$$ \begin{array}{@{}rcl@{}} (\mathcal{D}^{*}\phi_{i})({x}) = \mathrm{v.p.}{\int}_{\Gamma} C^{}_{kljm}G^{p}_{ki,l}({x},{y})n_{m}({x})\phi_{j}({y})\mathrm{d}{\Gamma}_{y}, \end{array} $$

(92)

$$ \begin{array}{@{}rcl@{}} (\mathcal{N}^{}\phi_{i})({x}) = -\mathrm{p.f.}{\int}_{\Gamma} C^{}_{impq}C^{}_{kljn}G^{p}_{kp,lq}({x},{y})n_{m}({x}) \\ \times n_{n}({y})\phi_{j}({y})\mathrm{d}{\Gamma}_{y}, \end{array} $$

(93)

and $\mathcal {S}^{\prime }$ and $\mathcal {D}^{\prime }$ are defined by replacing (ρ,λ,μ) in $\mathcal {S}$ and $\mathcal {D}$ with $(\rho ^{\prime }, \lambda ^{\prime }, \mu ^{\prime })$, respectively. Furthermore, ‘v.p.’ and ‘p.f.’ stand for Cauchy’s principal value and the finite part of divergent integrals, respectively. The coupling parameter $\alpha \in \mathbb {C}$ is arbitrary if it has a non-zero imaginary part, but the best condition number of the boundary integrals (88) and 89 is often achieved when α = −i/(μk_T) (Matsushima et al. 2018b).

The boundary integral equations (88) and (89) are numerically solved after being discretised by a collocation method with piecewise constant elements, which results in a system of linear equations with a fully populated coefficient matrix. To reduce the computational cost of the linear algebraic operations, we apply the ${\mathscr{H}}$-matrix method (Bebendorf 2008) to the coefficient matrix and solve the linear system by an accelerated LU factorisation.