Shape and topology optimization of acoustic lens system using phase field method

Abstract

A layout optimization method for a two-dimensional acoustic lens system used in underwater imaging is presented. To this end, a shape and topology optimization is formulated for the design problem of a lens system for the first time. The layout of a lens system to be optimized includes the number of lenses, shape of lens surfaces, distances between lenses, and lens materials. A phase field function is employed to implicitly parameterize the boundaries of the lenses, which move according to design sensitivities during optimization. Multiple lenses with different materials are optimized using a single phase field function. Because the ratio of the acoustic wavelength with respect to lens dimensions is large, diffraction effects should be taken into account. Accordingly, the performance of a lens system should be analyzed using wave acoustics and not the ray tracing method. The optimization problem is formulated to remove the aberrations of coma and field curvature. The validity of the proposed optimization method is demonstrated by solving benchmark design problems including a lens system with a large field of view.

Appendices

Appendix 1: Finite element formulation

The weak formulation for the governing equation in Equation (8) is

$$ \underset{D}{\int}\left[\nabla q\cdot \frac{1}{\rho}\nabla p-\frac{\omega^2}{\kappa} qp\right] d\Omega =0, $$

(29)

where q is a test function. Using integration by parts and applying boundary conditions in Equation (9) and (10), Equation (29) can be rewritten as

$$ \begin{array}{c}\hfill \frac{1}{\rho}\underset{D}{\int}\nabla q\cdot \nabla pd\Omega -\frac{\omega^2}{\kappa}\underset{D}{\int } q pd\Omega =\frac{1}{\rho}\underset{\Gamma_1}{\int } q\left(2{ikp}_{\mathrm{in}}- ikp\right) d\Gamma, \hfill \\ {}\hfill +\frac{1}{\rho}\underset{\Gamma_2}{\int } q\left(- ikp\right) d\Gamma +\frac{1}{\rho}\underset{\Gamma_3}{\int } q\left(- ikp\right) d\Gamma +\frac{1}{\rho}\underset{\Gamma_4}{\int } q\left(- ikp\right) d\Gamma .\hfill \end{array}\kern5.5em $$

(30)

For the finite element discretization, the field variables are approximated as

$$ \begin{array}{ccc}\hfill p={\mathbf{N}}^T\mathbf{p}\hfill & \hfill \mathrm{and}\hfill & \hfill q={\mathbf{N}}^T\mathbf{q},\hfill \end{array} $$

(31)

where N ^T is the shape function matrix, and p and q are nodal vectors associated with p and q, respectively. Substituting Equation (31) in Equation (30) gives the matrix form of the equation,

$$ \mathbf{SP}=\mathbf{F}, $$

(32)

where

$$ \mathbf{S}=\mathbf{K}-\mathbf{M}+\mathbf{B}, $$

(33)

$$ \mathbf{K}=\underset{D}{\int}\nabla {\mathbf{N}}^T\nabla \mathbf{N} d\Omega, $$

(34)

$$ \mathbf{M}=\frac{\omega^2}{\kappa}\underset{D}{\int }{\mathbf{N}}^T\mathbf{N} d\Omega, $$

(35)

$$ \mathbf{B}=\frac{ik}{\rho}\left(\underset{\Gamma_1}{\int }{\mathbf{N}}^T\mathbf{N} d\Gamma +\underset{\Gamma_2}{\int }{\mathbf{N}}^T\mathbf{N} d\Gamma +\underset{\Gamma_3}{\int }{\mathbf{N}}^T\mathbf{N} d\Gamma +\underset{\Gamma_4}{\int }{\mathbf{N}}^T\mathbf{N} d\Gamma \right), $$

(36)

$$ \mathbf{F}=\frac{2{ikp}_{\mathrm{in}}}{\rho}\underset{\Gamma_1}{\int }{\mathbf{N}}^T d\Gamma . $$

(37)

Appendix 2: Adjoint variable method for sensitivity analysis

Using Equation (32) and the conjugate equation of Equation (32), the objective in Equation (19) can be written as

$$ \begin{array}{l}\overline{\Psi}= f\left(\phi \right)+{\boldsymbol{\uplambda}}_1^T\left({\mathbf{S}\mathbf{p}}_R+ i{\mathbf{S}\mathbf{p}}_I-\mathbf{F}\right)+{\boldsymbol{\uplambda}}_2^T\left(\overline{\mathbf{S}}{\mathbf{p}}_R- i\overline{\mathbf{S}}{\mathbf{p}}_I-\overline{\mathbf{F}}\right)\\ {}\kern1em +\sum_{j=1}^M\left\{-{\ell}_j{C}_j\left(\phi \right)+\frac{1}{2}{\sigma}_j{C}_j^2\left(\phi \right)\right\}+{P}_{\mathrm{dist}},\end{array} $$

(38)

where $\overline{\mathbf{S}}$ and $\overline{\mathbf{F}}$ are conjugates of S and F, respectively, and λ ₁ and λ ₂ are adjoint variable vectors. The sensitivity of Equation (38) is

$$ \begin{array}{l}\frac{d\overline{\Psi}}{d{\phi}_i}=\frac{\partial f}{\partial {\phi}_i}+{\left(\frac{\partial f}{\partial {\mathbf{p}}_R}\right)}^T\frac{\partial {\mathbf{p}}_R}{\partial {\phi}_i}+{\left(\frac{\partial f}{\partial {\mathbf{p}}_I}\right)}^T\frac{\partial {\mathbf{p}}_I}{\partial {\phi}_i}+{\boldsymbol{\uplambda}}_1^T\left[\left(\frac{\partial \mathbf{S}}{\partial {\phi}_i}{\mathbf{p}}_R+\mathbf{S}\frac{\partial {\mathbf{p}}_R}{\partial {\phi}_i}\right)+ i\left(\frac{\partial \mathbf{S}}{\partial {\phi}_i}{\mathbf{p}}_I+\mathbf{S}\frac{\partial {\mathbf{p}}_I}{\partial {\phi}_i}\right)-\frac{\partial \mathbf{F}}{\partial {\phi}_i}\right]\\ {}\kern1em +{\boldsymbol{\uplambda}}_2^T\left[\left(\frac{\partial \overline{\mathbf{S}}}{\partial {\phi}_i}{\mathbf{p}}_R+\overline{\mathbf{S}}\frac{\partial {\mathbf{p}}_R}{\partial {\phi}_i}\right)+ i\left(\frac{\partial \overline{\mathbf{S}}}{\partial {\phi}_i}{\mathbf{p}}_I-\overline{\mathbf{S}}\frac{\partial {\mathbf{p}}_I}{\partial {\phi}_i}\right)-\frac{\partial \overline{\mathbf{F}}}{\partial {\phi}_i}\right]\\ {}\kern1em +\sum_{j=1}^M\left\{-{\ell}_j+{\sigma}_j{C}_j\left(\phi \right)\right\}\frac{{ d C}_j\left(\phi \right)}{d{\phi}_i}+\frac{{ d P}_{\mathrm{dist}}}{d{\phi}_i},\end{array} $$

(39)

where $\frac{\partial {\boldsymbol{\uplambda}}_1^T}{\partial {\phi}_i}\left({\mathbf{Sp}}_R+ i{\mathbf{Sp}}_I-\mathbf{F}\right)=0$ and $\frac{\partial {\boldsymbol{\uplambda}}_2^T}{\partial {\phi}_i}\left(\overline{\mathbf{S}}{\mathbf{p}}_R- i\overline{\mathbf{S}}{\mathbf{p}}_I-\overline{\mathbf{F}}\right)=0$ are used. Reorganizing Equation (39) gives

$$ \begin{array}{l}\frac{d\overline{\Psi}}{d{\phi}_i}=\frac{\partial f}{\partial {\phi}_i}+{\left(\frac{\partial {\mathbf{p}}_R}{\partial {\phi}_i}\right)}^T\left(\frac{\partial f}{\partial {\mathbf{p}}_R}+{\mathbf{S}}^T{\boldsymbol{\uplambda}}_1+{\overline{\mathbf{S}}}^T{\boldsymbol{\uplambda}}_2\right)+{\left(\frac{\partial {\mathbf{p}}_I}{\partial {\phi}_i}\right)}^T\left(\frac{\partial f}{\partial {\mathbf{p}}_I}+ i{\mathbf{S}}^T{\boldsymbol{\uplambda}}_1- i{\overline{\mathbf{S}}}^T{\boldsymbol{\uplambda}}_2\right)\\ {}\kern1em +{\boldsymbol{\uplambda}}_1^T\left(\frac{\partial \mathbf{S}}{\partial {\phi}_i}\mathbf{p}-\frac{\partial \mathbf{F}}{\partial {\phi}_i}\right)+{\boldsymbol{\uplambda}}_2^T\left(\frac{\partial \overline{\mathbf{S}}}{\partial {\phi}_i}\overline{\mathbf{p}}-\frac{\partial \overline{\mathbf{F}}}{\partial {\phi}_i}\right)\\ {}\kern1em +\sum_{j=1}^M\left\{-{\ell}_j+{\sigma}_j{C}_j\left(\phi \right)\right\}\frac{{ d C}_j\left(\phi \right)}{d{\phi}_i}+\frac{{ d P}_{\mathrm{dist}}}{d{\phi}_i}.\end{array} $$

(40)

Because λ ₁ and λ ₂ can be set arbitrarily, we set ${\boldsymbol{\uplambda}}_1={\overline{\boldsymbol{\uplambda}}}_2=\boldsymbol{\uplambda}$ and choose λ to satisfy

$$ {\mathbf{S}}^T\boldsymbol{\uplambda} +{\overline{\mathbf{S}}}^T\overline{\boldsymbol{\uplambda}}=-\frac{\partial f}{\partial {\mathbf{p}}_R}, $$

(41)

$$ i{\mathbf{S}}^T\boldsymbol{\uplambda} - i{\overline{\mathbf{S}}}^T\overline{\boldsymbol{\uplambda}}=-\frac{\partial f}{\partial {\mathbf{p}}_I}. $$

(42)

Multiplying i to Equation (42) and subtracting it from Equation (41) results in

$$ {\mathbf{S}}^T\boldsymbol{\uplambda} =-\frac{1}{2}\left(\frac{\partial f}{\partial {\mathbf{p}}_R}- i\frac{\partial f}{\partial {\mathbf{p}}_I}\right). $$

(43)

Using λ by Equation (43), Equation (13) can be rewritten as

$$ \frac{d\overline{\Psi}}{d{\phi}_i}=\frac{\partial f}{\partial {\phi}_i}+2\mathrm{Re}\left({\boldsymbol{\uplambda}}^T\frac{\partial \mathbf{S}}{\partial {\phi}_i}\mathbf{p}-\frac{\partial \mathbf{F}}{\partial {\phi}_i}\right)+\sum_{j=1}^M\left\{-{\ell}_j+{\sigma}_j{C}_j\left(\phi \right)\right\}\frac{{ d C}_j\left(\phi \right)}{d{\phi}_i}+\frac{{ d P}_{\mathrm{dist}}}{d{\phi}_i}. $$

(44)