Exact conic section arc elements in 2D and 2.5D FEM using a coordinate transformation

Abstract

This study introduces the conic section arc elements in 2D and 2.5D finite element method (FEM ). Elements are obtained by deforming an edge in a standard triangular element through a coordinate transformation. This allows to completely eliminate the geometrical error in structures composed of circular, elliptical, hyperbolic and parabolic arcs. The element order is defined independently of the geometrical description of the boundary, allowing the use of simple meshers. Previously available FEM codes can be straightforwardly modified to introduce the conic section boundaries.This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.

Considerations on boundary conditions in transformation optics for complete field computation in generic waveguides

“…Figure 3 compares the eigenvalues for a waveguide with $<math altimg="urn:x-wiley:08952477:media:mop33493:mop33493-math-0110" wiley:location="equation/mop33493-math-0110.png" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mi>R</mi><mo>\unicode{x0003D}</mo><mn>10</mn><mtext>mm</mtext></mrow></mrow></math>$, $<math wiley:location="equation/mop33493-math-0259.png" altimg="urn:x-wiley:08952477:media:mop33493:mop33493-math-0259" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mi>r</mi><mi>b</mi></msub><mo>\unicode{x0003D}</mo><mn>1</mn><mtext>mm</mtext></mrow></mrow></math>$ and varying $<math altimg="urn:x-wiley:08952477:media:mop33493:mop33493-math-0111" wiley:location="equation/mop33493-math-0111.png" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mi>d</mi><mo>\unicode{x02215}</mo><mi>R</mi></mrow></mrow></math>$ (from $<math wiley:location="equation/mop33493-math-0271.png" altimg="urn:x-wiley:08952477:media:mop33493:mop33493-math-0271" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mn>0</mn></mrow></mrow></math>$, circular waveguide, to $<math wiley:location="equation/mop33493-math-0272.png" altimg="urn:x-wiley:08952477:media:mop33493:mop33493-math-0272" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mn>1.5</mn></mrow></mrow></math>$). TE and TM modes are computed with $<math altimg="urn:x-wiley:08952477:media:mop33493:mop33493-math-0112" wiley:location="equation/mop33493-math-0112.png" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mi>M</mi><mo>\unicode{x0003D}</mo><mi>N</mi><mo>\unicode{x0003D}</mo><mn>20</mn></mrow></mrow></math>$ and compared with a two‐dimensional FEM solution 12 . Good agreement is apparent, even for geometries rather different from a circumference ($<math altimg="urn:x-wiley:08952477:media:mop33493:mop33493-math-0113" wiley:location="equation/mop33493-math-0113.png" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mi>d</mi><mo>\unicode{x02215}</mo><mi>R</mi><mo>\unicode{x0003D}</mo><mn>1.5</mn></mrow></mrow></math>$).…”

“…hence involving in the TD, not only the radial but also the azimuthal derivative. While (12) reduces to the usual formula…”

“…It eventuates that the boundary conditions (12) in the TD are satisfied if and only if the boundary conditions in the OD are satisfied.…”

“…The longitudinal magnetic field component is more complex since straightforward expansion in circular waveguide TE modes would not satisfy the boundary condition (12). Bases different from those originally exploited in Gentili et al 5 are necessary: | 375 order m. The functions r ρ l ϕ ( ), ( )…”

“…, and v ϕ ( ) mn represent a degree of freedom allowing to have every single term of the expansion to satisfy the boundary conditions (12). Only f mn is analyzed in the following, and the calculations for g mn are analogous.…”