Efficient Implementation of Non-Uniform Refinement Levels in a Geometric Multigrid Finite Element Method for Electromagnetic Waves

This paper develops a topology optimization procedure for three-dimensional electromagnetic waves with an edge element-based finite-element method. In contrast to the two-dimensional case, three-dimensional electromagnetic waves must include an additional divergence-free condition for the field variables. The edge element-based finite-element method is used to both discretize the wave equations and enforce the divergence-free condition. For wave propagation described in terms of the magnetic field in the widely used class of non-magnetic materials, the divergence-free condition is imposed on the magnetic field. This naturally leads to a nodal topology optimization method. When wave propagation is described using the electric field, the divergence-free condition must be imposed on the electric displacement. In this case, the material in the design domain is assumed to be piecewise homogeneous to impose the divergence-free condition on the electric field. This results in an element-wise topology optimization algorithm. The topology optimization problems are regularized using a Helmholtz filter and a threshold projection method and are analysed using a continuous adjoint method. In order to ensure the applicability of the filter in the element-wise topology optimization version, a regularization method is presented to project the nodal into an element-wise physical density variable.

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A finite element method for the analysis of radiation and scattering of electromagnetic waves on complex environments

Computer Methods in Applied Mechanics and Engineering ◽

10.1016/j.cma.2004.05.025 ◽

2005 ◽

Vol 194 (2-5) ◽

pp. 637-655 ◽

Cited By ~ 25

Author(s):

Luis E. García-Castillo ◽

Ignacio Gómez-Revuelto ◽

Francisco Sáez de Adana ◽

Magdalena Salazar-Palma

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Electromagnetic Waves ◽

Complex Environments ◽

Element Method

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Anisotropic Modeling with Geometric Multigrid Preconditioned Finite Element Method

Geophysics ◽

10.1190/geo2021-0592.1 ◽

2022 ◽

pp. 1-21

Author(s):

Qingtao Sun ◽

Runren Zhang ◽

Ke Chen ◽

Naixing Feng ◽

Yunyun Hu

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Large Scale ◽

Data Interpretation ◽

Scale Modeling ◽

Arbitrary Anisotropy ◽

Large Scale Modeling ◽

Element Method ◽

Geometric Multigrid ◽

Electromagnetic Surveys

Formation anisotropy in complicated geophysical environments can have a significant impact on data interpretation of electromagnetic surveys. To facilitate full 3D modeling of arbitrary anisotropy, we have adopted an h-version geometric multigrid preconditioned finite-element method based on vector basis functions. By using a structured mesh, instead of an unstructured one, our method can conveniently construct the restriction and prolongation operators for multigrid implementation, and then recursively coarsen the grid with the F-cycle coarsening scheme. The geometric multigrid method is used as a preconditioner for the biconjugate-gradient stabilized method to efficiently solve the linear system resulting from the finite-element method. Our method avoids the need of interpolation for arbitrary anisotropy modeling as in Yee’s grid-based finite-difference method, and it is also more capable of large-scale modeling with respect to the p-version geometric multigrid preconditioned finite-element method. A numerical example in geophysical well logging is included to demonstrate its numerical performance. Our h-version geometric multigrid preconditioned finite-element method is expected to help formation anisotropy characterization with electromagnetic surveys in complicated geophysical environments.

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