In this project, we consider the shape optimization of a dielectric scatterer aiming at efficient directional routing of light. In the studied setting, light interacts with a penetrable scatterer with dimension comparable to the wavelength of an incoming planar wave. The design objective is to maximize the scattering efficiency inside a target angle window. For this, a Helmholtz problem with a piecewise constant refractive index medium models the wave propagation, and an accurate Dirichlet-to-Neumann map models an exterior domain. The strategy consists of using a high-order finite element (FE) discretization combined with gradient-based numerical optimization. The latter consists of a quasi-Newton (BFGS) with backtracking line search. A discrete adjoint method is used to compute the sensitivities with respect to the design variables. Particularly, for the FE representation of the curved shape, we use a bilinear transfinite interpolation formula, which admits explicit differentiation with respect to the design variables. We exploit this fact and show in detail how sensitivities are obtained in the discrete setting. We test our strategy for a variety of target angles, different wave frequencies, and refractive indexes. In all cases, we efficiently reach designs featuring high scattering efficiencies that satisfy the required criteria.

中文翻译：

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电介质纳米棒强定向散射的形状优化

在这个项目中，我们考虑了电介质散射体的形状优化，旨在实现光的有效定向路由。在所研究的环境中，光与尺寸与入射平面波的波长相当的可穿透散射体相互作用。设计目标是最大化目标角度窗口内的散射效率。为此，具有分段恒定折射率介质的亥姆霍兹问题对波的传播进行建模，而精确的 Dirichlet-to-Neumann 映射则对外部域进行建模。该策略包括使用高阶有限元 (FE) 离散化与基于梯度的数值优化相结合。后者由具有回溯线搜索的准牛顿 (BFGS) 组成。离散伴随方法用于计算关于设计变量的敏感性。特别，对于弯曲形状的有限元表示，我们使用双线性超限插值公式，该公式允许关于设计变量的显式微分。我们利用这一事实并详细展示了如何在离散设置中获得灵敏度。我们针对各种目标角度、不同的波频率和折射率测试我们的策略。在所有情况下，我们都能有效地实现满足所需标准的高散射效率设计。和折射率。在所有情况下，我们都能有效地实现满足所需标准的高散射效率设计。和折射率。在所有情况下，我们都能有效地实现满足所需标准的高散射效率设计。