Abstract:
This report details a novel methodology for robust computational electromagnetics called Discrete Differential Forms (DDF), developed to overcome the numerical instabilities and limitations of previous simulation methods. Motivated by the need for a provably stable and higher-order accurate method for solving problems on 3D unstructured grids, this approach is elegantly cast in the language of differential geometry. The key insight is that different physical quantities (e.g., electric potential, magnetic flux) have different mathematical properties and must be discretized using different types of finite element basis functions, known as p-forms (0-forms, 1-forms, etc.). This ensures the underlying physics is correctly represented, yielding a stable and conservative solution. A major outcome of this research is FEMSTER, an object-oriented C++ class library that encapsulates these higher-order DDF basis functions and operators. The methodology is validated against several canonical problems, including the Poisson and vector Helmholtz equations, demonstrating optimal, higher-order rates of convergence. The paper concludes by showcasing the versatility of the DDF approach on practical applications such as calculating resistance in integrated circuits, computing resonant modes in accelerator induction cells, and performing full-wave simulations of optical devices like holey fibers and photonic bandgap waveguides.
