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| Hauptseite > Publikationsdatenbank > Investigating the Boundary Element Method for the Helmholtz Equation in Fractal Domains |
| Hauptseite > Publikationsdatenbank > Investigating the Boundary Element Method for the Helmholtz Equation in Fractal Domains |
| Master Thesis | FZJ-2026-04311 |
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2026
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Please use a persistent id in citations: doi:10.34734/FZJ-2026-04311
Abstract: Resolving boundary value problems, such as the Helmholtz acoustic scattering problem, on complex, non-smooth geometries introduces numerical and computational challenges. While conventional numerical methods necessitate a full discretization of the domain, the Boundary Element Method (BEM) discretizes only the boundary of the domain. Popular variants of the BEM include the collocation and the Galerkin methods. The collocation method utilizes pointwise enforcements of the boundary condition (strong formulation). Although this approach yields highly accurate results on smooth domains, its precision degrades on non-smooth domains due to the the geometric singularities.To overcome the limitations, the Galerkin method accounts the entire boundary by utilizing a weak formulation. Consequently, the primary objective of this project is to implement a Galerkin solver framework and resolve the Helmholtz acoustic scattering problem up to a fifth-iteration Koch snowflake geometry.This thesis details the implementation, verification, and optimization phases executed to realize a robust Galerkin solver framework utilizing a piecewise polynomial approximation. The inherent numerical challenges arising from kernel singularities during integral evaluation are resolved through a reformulation of the boundary integral expressions combined with a strategic boundary parameterization scheme. The associated computational challenges are mitigated by isolating the optimal discretization parameters that balance numerical accuracy with computational costs, deploying a multi-node parallelization strategy, and systematically evaluating various adaptive numerical quadrature routines. The results demonstrate that pairing optimized parameters with distributed HPC pipelines makes highly complex fractal scattering simulations computationally tractable.
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