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Sahar Sargheini, Alberto Paganini, Ralf Hiptmair, Christian Hafner

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1 Sahar Sargheini, Alberto Paganini, Ralf Hiptmair, Christian Hafner
Shape calculus in nano-optics Sahar Sargheini, Alberto Paganini, Ralf Hiptmair, Christian Hafner

2 Outline Introduction PDE Constraint Shape calculus
Electromagnetic wave scattering problem Numerical results Conclusion SAM-Seminar of Applied Mathematics / IFH - Electromagnetic Field Theory Group

3 Introduction What is shape calculus? Nano-Optics
Study the effect of geometry perturbation Nano-Optics SAM-Seminar of Applied Mathematics / IFH - Electromagnetic Field Theory Group

4 Introduction Nanoantenna SNOM Production based variation:
sensitivity analysis by deriving shape gradient SNOM Reconstructing shape and electric properties is inverse problem. reformulate inverse problem into a PDE constraint optimization problem (using descent approach along shape gradients). SAM-Seminar of Applied Mathematics / IFH - Electromagnetic Field Theory Group

5 PDE Constraint Shape Calculus
Outline Introduction PDE Constraint Shape Calculus Electromagnetic wave scattering problem Numerical results Conclusion SAM-Seminar of Applied Mathematics / IFH - Electromagnetic Field Theory Group

6 PDE Constraint Shape optimization
Problem: find the optimal admissible geometry Solution procedure options: Use parametric model (a few design variables) Consider boundaries as manifolds (infinite-dimensional minimization problems) SAM-Seminar of Applied Mathematics / IFH - Electromagnetic Field Theory Group

7 PDE Constraint Shape optimization
Ts is the flow of a sufficiently smooth (parameter dependent) vectorfield V Vector field: Eulerian derivative of J in the direction V at t=0 is defined by Material derivative: Shape derivative in the direction of V is: SAM-Seminar of Applied Mathematics / IFH - Electromagnetic Field Theory Group

8 Electromagnetic wave scattering problem
Outline Introduction PDE Constraint Shape optimization Electromagnetic wave scattering problem Numerical results Conclusion SAM-Seminar of Applied Mathematics / IFH - Electromagnetic Field Theory Group

9 Electromagnetic wave scattering problem
Γ0 D Γ1 on Γ Ω Objective function: SAM-Seminar of Applied Mathematics / IFH - Electromagnetic Field Theory Group

10 Electromagnetic wave scattering problem
Shape derivative: Adjoint Equation: SAM-Seminar of Applied Mathematics / IFH - Electromagnetic Field Theory Group

11 Electromagnetic wave scattering problem
Steps to be done in each iteration Solve state problem Solve adjoint problem Compute shape gradient Move boundary nodes Smooth the mesh FEM Method LehrFEM Library based on Matlab SAM-Seminar of Applied Mathematics / IFH - Electromagnetic Field Theory Group

12 Electromagnetic wave scattering problem
Compute shape gradient But we would like: Smoothing the mesh Laplace Smoothing SAM-Seminar of Applied Mathematics / IFH - Electromagnetic Field Theory Group

13 Numerical results Outline Introduction
PDE Constraint Shape optimization Electromagnetic wave scattering problem Numerical results Conclusion SAM-Seminar of Applied Mathematics / IFH - Electromagnetic Field Theory Group

14 Numerical results (Example1)
Meshing of reference structure Meshing of first Iteration structure SAM-Seminar of Applied Mathematics / IFH - Electromagnetic Field Theory Group

15 Numerical results(Example1)
SAM-Seminar of Applied Mathematics / IFH - Electromagnetic Field Theory Group

16 Numerical results(Example1)
Reference structure solution Final iteration structure solution SAM-Seminar of Applied Mathematics / IFH - Electromagnetic Field Theory Group

17 Numerical results(Example2)
Meshing of first Iteration structure Meshing of reference structure SAM-Seminar of Applied Mathematics / IFH - Electromagnetic Field Theory Group

18 Numerical results(Example2)
SAM-Seminar of Applied Mathematics / IFH - Electromagnetic Field Theory Group

19 Numerical results(Example2)
Final iteration structure solution Reference structure solution SAM-Seminar of Applied Mathematics / IFH - Electromagnetic Field Theory Group

20 Conclusion Shape gradients can be used to find the optimal shape. Method of mapping provides an analytical representation of the shape gradients. Using FEM as a solver we don’t have access to shape gradients on boundary nodes directly and some approximations are necessary. Despite this drawback, we experienced good convergence in our simulations. Work in Progress: improve gradient recovery, use of second order information SAM-Seminar of Applied Mathematics / IFH - Electromagnetic Field Theory Group

21 Thank you for your attention
Any Question? SAM-Seminar of Applied Mathematics / IFH - Electromagnetic Field Theory Group


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