2025-10-02 –, Coffee room
Language: English
The Helmholtz equation is essential to modeling wave scattering problems in periodic structures. Green’s functions, which solve linear partial differential equations with Dirac delta function sources δ(x), provide a foundation for recasting these problems into boundary integral equations. While closed-form expressions exist for many settings, the quasi-periodic Green’s function for the Helmholtz equation poses a unique challenge: it is defined as an infinite series that converges slowly as x2 → 0 in 2D (or x3 → 0 in 3D).
In this talk, we introduce QPGreen.jl, which implements an FFT-based algorithm to efficiently evaluate these Green’s functions. It combines spectral truncation and interpolation to overcome the limitations of direct series summation. Leveraging Julia’s composability, we combine QPGreen.jl with Inti.jl—a boundary integral equation solver—and demonstrate applications to wave scattering problems with quasi-periodic incident fields.
The integral equation method is a popular method for simulating acoustic wave scattering in periodic structures, requiring efficient evaluation of quasi-periodic Green’s functions.
For the Helmholtz equation, the series representation of quasi-periodic Green’s functions converges too slowly to be practical in boundary integral equation methods. To address this, we introduce QPGreen.jl (https://github.com/gregoirepourtier/QPGreen.jl), which implements an FFT-based algorithm based on [1]. The method proceeds in two stages:
- Preparation step: precomputes various Fourier coefficients and evaluates the Green’s function at grid points. This step is performed once per set of quasi-periodicities and wavenumbers.
- Evaluation step: efficiently interpolates values between grid points to compute values at arbitrary points between precomputed grid nodes, delivering efficient and accurate pointwise evaluations.
This approach overcome the limitations of direct series summation while maintaining high accuracy, making it ideal when a large number of evaluations is required.
By leveraging Julia’s composability and package extension capabilities, we combine QPGreen.jl with Inti.jl (https://github.com/IntegralEquations/Inti.jl) - a boundary integral equation solver - to efficiently solve 2D and 3D scattering problems involving periodic gratings.
[1] B. Zhang and R. Zhang. An FFT-based algorithm for efficient computation of Green’s functions for the Helmholtz and Maxwell’s equations in periodic domains. SIAM Journal on Scientific Computing 40, B915 – B941 (2018).
I am a PhD student in the research group for analysis and applications at TU Berlin. My research focuses on the numerical solution of scattering problems, primarily using boundary integral equation methods.