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Finite-element solution of Maxwell's equations with Helmholtz forms

K D Paulsen1

  • 1Thayer School of Engineering, Dartmouth College, Hanover, New Hampshire 03755.

Journal of the Optical Society of America. A, Optics, Image Science, and Vision
|April 1, 1994
PubMed
Summary

This study presents a computational method using the Helmholtz formulation for solving Maxwell's equations for complex 3D objects. The approach offers spurious-mode resistance and efficient solutions for electromagnetic scattering and absorption problems.

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Area of Science:

  • Computational electromagnetics
  • Numerical methods for wave propagation

Background:

  • Solving scattering and absorption problems for 3D penetrable bodies is computationally challenging.
  • The finite-element method is a suitable numerical technique for geometrically and electrically complex objects.

Purpose of the Study:

  • To present an overview of the Helmholtz formulation for solving Maxwell's equations.
  • To highlight the spurious-mode-resistant properties of the Helmholtz formulation.
  • To detail efficient solution procedures and mesh generation for 3D finite-element analysis.

Main Methods:

  • Utilizing Helmholtz weak forms for electromagnetic wave problems.
  • Developing efficient and reliable solution procedures for the resulting algebraic systems.
  • Implementing unstructured mesh generation techniques for complex geometries.

Related Experiment Videos

  • Applying the finite-element method to solve Maxwell's equations in a workstation environment.

    • Main Results:

    • Demonstrated the spurious-mode-resistant properties of the Helmholtz formulation.
    • Showcased efficient and reliable solution procedures for algebraic systems.
    • Presented an approach for unstructured mesh generation suitable for complex 3D objects.
    • Provided examples of practical 3D calculations achievable with readily available computing power.

    Conclusions:

    • The Helmholtz formulation, coupled with efficient solution procedures and mesh generation, provides a robust methodology for 3D finite-element solutions of Maxwell's equations.
    • This approach enables practical 3D electromagnetic scattering and absorption calculations on standard workstations.
    • The method exhibits favorable properties, including resistance to spurious modes, making it reliable for complex problems.