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Contribution in topological optimization and application to nanophotonics

Abstract : This thesis focuses on the mathematical field of shape optimization and explores two topics:one concerns the systematic determination of the design of nanophotonic components and the other one the optimal shape and location of boundary conditions defining partial differential equations (PDE).• In the mathematical setting of the three-dimensional, time-harmonic Maxwell equations, we proposea shape and topology optimization algorithm combining Hadamard’s boundary variation methodwith a level set representation of shapes and their evolution. A particular attention is devotedto the robustness of the optimized devices with respect to small uncertainties over the physical orgeometrical data of the problem. In this respect, we rely on a simple multi-objective formulation todeal with the two main sources of uncertainties plaguing nanophotonic devices, namely uncertaintiesover the incoming wavelength, and geometric uncertainties entailed by the lithography and etchingfabrication process. Several numerical examples are presented and discussed to assess the efficiencyof our methodology.• The second application concern the optimization of the shape of the regions assigned to different typesof boundary conditions in the definition of a “physical” PDE. This problem proves to be difficult inthe case of a Dirichlet-Neumann transition since it requires a precise study of the singular nature ofPDE solutions at the transition between two regions supporting these boundary conditions. On theone hand a full mathematical study is carried out on this theoretical problem and on the other handa numerical method based on a regularization of the boundary conditions is proposed to optimizethese regions. Various numerical examples are eventually presented in order to appraise the efficiencyof the proposed process.
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Nicolas Lebbe. Contribution in topological optimization and application to nanophotonics. Analysis of PDEs [math.AP]. Université Grenoble Alpes, 2019. English. ⟨NNT : 2019GREAM047⟩. ⟨tel-02518286⟩

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