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 problemΓ0 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