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REVIEW 3 major objections 6 minor 39 references

A 3-D Hybrid Numerical Method for Simulating Electromagnetic Fields in Structures with Multiple Inhomogeneous Layered Media

T0 review · 3 major / 6 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read The paper claims that a 3-D hybrid method—numerical mode-matching for inhomogeneous layered media coupled to a tree-cotree mixed finite element solver for arbitrary scattering regions—can reproduce full-wave S-parameters to within about 1.5

desk verdict A genuinely new hybrid NMM/MFEM formulation with credible COMSOL checks, but the mode-truncation convergence evidence is too thin to lock in the reported accuracy and speedup numbers. read the letter →

arxiv 2608.03639 v1 pith:OVSHIQMO submitted 2026-08-04 physics.comp-ph cs.NAmath.NA

Jie Liu , Taihe Li , Ke Chen , Mingwei Zhuang , Qing Huo Liu This is my paper · ORCID
classification physics.comp-phcs.NAmath.NA MSC 78A4578M1065N3065N25 PACS 41.20.Jb02.70.Dh
keywords electromagneticscatteringinhomogeneouslayeredmedianumericalmodematchingmixedfiniteelementmethodtree-cotreedecompositionwaveguideeigenmodesS-parameters3-Dhybrid
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to establish that a carefully coupled hybrid of the numerical mode-matching method (NMM) and a tree-cotree mixed finite element method (MFEM) can simulate 3-D electromagnetic scattering when non-layered scatterers are embedded in multiple inhomogeneous layered waveguides, with accuracy close to a full 3-D FEM reference and much lower cost. The authors report 2.00–5.46× speedups, 1.95–12.87× fewer unknowns, and 1.64–15.78× memory savings across five test structures, with S-parameter relative errors of 0.17–1.48%. If true, this gives circuit, antenna, and geophysical modelers a practical alternative for structures where pure full-domain methods are expensive and pure mode-matching cannot represent arbitrary 3-D scatterers.

What carries the argument

The load-bearing object is the coupled variational problem (18)–(21): the bilinear form for the 3-D region is augmented by port boundary terms that carry incident and scattered transverse magnetic fields from each layered waveguide, and the interface equations enforce tangential E and H continuity exactly. Two supporting mechanisms carry the argument: (i) the 2.5-D mixed FEM eigenvalue solver for each waveguide layer, which suppresses spurious modes by incorporating Gauss's law, and (ii) the tree-cotree decomposition of the 3-D electric field, which writes the tree-edge part as gradients of nodal functions so the material-weighted divergence-free condition holds without extra unknowns. The r

What would settle it

Run the multilayered checkerboard example of Section 3.1 with the retained mode count increased from 100 to 500 and record |S11| and |S21| at each stop. If the S-parameters drift by more than the reported 1.48% error budget, or fail to converge monotonically, the modal-truncation premise would be shown to be load-bearing and unsafe at the stated cost. A stronger test: build a structure with strong transverse inhomogeneity and an electrically long layered section, and compare the hybrid method against a converged full 3-D FEM over a frequency band where evanescent modes matter.

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Extended reading notes

Core claim

The central claim is that the 3-D scattering problem splits cleanly: layered waveguides are solved semi-analytically as superpositions of waveguide eigenmodes, and the arbitrary 3-D region is discretized with a tree-cotree MFEM, with the two descriptions tied together by exact tangential field continuity at shared interfaces. The NMM part uses a 2.5-D MFEM to compute spurious-free eigenmodes of arbitrarily shaped inhomogeneous anisotropic waveguides, then builds global reflection and transmission matrices recursively without relying on mode orthogonality; the MFEM part enforces the divergence-free condition on the electric field through tree-cotree splitting rather than Lagrange multipliers

Load-bearing premise

The whole economy of the method rests on the assumption that a modest finite set of waveguide eigenmodes, selected empirically in each case, captures the transverse field structure well enough that truncation error stays below the reported S-parameter accuracy; the paper gives no a priori bound on the required number of modes.

Editorial extensions

If this is right

  • Layered regions no longer need to be homogeneous: transverse inhomogeneity, anisotropy, loss, and nonreciprocal permeability are handled by the 2.5-D MFEM eigenmode solver.
  • Interfaces are matched through exact tangential field boundary conditions, avoiding approximate numerical fluxes and the costly generalized admittance matrix inversion used in some prior hybrid schemes.
  • The dimensionality reduction becomes stronger relative to full 3-D FEM when the layered media are electrically thick or have complex cross-sections.
  • Across plane-wave, TEM, TE10, and TE11 excitations, the reported relative errors stay within 0.17%–1.48% of a commercial reference solver while DOF and memory reduce by factors of about 2–13 and 1.6–16, respectively.
  • The coupled system has size (Nc+Nn+m), with the mode count m typically much smaller than the 3-D unknowns, so cost is dominated by the scattering region alone.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same interface-matching variational framework could plausibly be extended to time-domain or nonlinear material models, since it does not depend on mode orthogonality; a natural next test is a high-power or biased-ferrite device where permeability depends on field strength.
  • Because mode counts are chosen empirically, an adaptive mode-selection criterion based on estimated port-to-scatterer coupling could remove trial-and-error and make the method more robust for users encountering unfamiliar geometries.
  • If a layered region becomes electrically very long or highly lossy, the recursive reflection matrices may dominate cost differently; a testable variant would switch such regions to a pure 3-D FEM treatment within the same interface framework.
  • The absence of generalized-admittance matrix inversion suggests the method may scale more gracefully to many ports than earlier hybrid schemes; a direct scaling study with P = 2, 4, 8, 16 ports would test this.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper presents a 3-D hybrid numerical method (HNM) that couples the numerical mode-matching (NMM) method for inhomogeneous layered media with a tree-cotree-based mixed finite element method (MFEM) for an arbitrary 3-D scattering region. The NMM expresses fields in each layer as a superposition of 2-D waveguide eigenmodes, with reflection/transmission handled recursively; the MFEM discretizes the scattering region while enforcing tangential continuity and a discrete divergence-free condition. The method is validated against COMSOL on five examples (multi-layered media with a scatterer, bend waveguide, multiple scatterers, millimeter-wave circulator, three-port divider) with S-parameter relative errors from 0.17% to 1.48% and reported speedups of 2.00x to 5.46x. The central claim is that HNM provides an efficient and accurate alternative for structures containing non-layered regions embedded in layered media.

Significance. If the result holds, the paper makes a useful contribution by combining the dimensionality reduction of NMM with the geometric flexibility of MFEM, while addressing known shortcomings of prior hybrid MM/FEM formulations: spurious modes, reliance on orthogonal TE/TM modes, and costly GSM matrix inversion. The validation against COMSOL across five diverse examples, including anisotropic, lossy, dispersive, and nonreciprocal media, is a strength, and the reported consistency of S-parameter errors supports the credibility of the implementation. However, the accuracy and efficiency claims depend critically on the number of retained waveguide eigenmodes, and the evidence for mode-truncation convergence is incomplete; this is a load-bearing gap that must be addressed before the conclusions can be fully accepted.

major comments (3)
  1. [Sec. 3, Table 2] The mode-truncation convergence is not established. The text states that Table 2 confirms the selected mode numbers are sufficient for convergence, but the data in Table 2 show the opposite for Example A: increasing modes from 50 to 100 changes the |S11| error from 1.22% to 0.77% and the |S21| error from 1.82% to 1.48%. The solution is still moving appreciably at the highest tested mode count, and no run at 150, 200, or more modes is reported. Since the reported accuracy and speedup claims (e.g., 2.09x at m=100 for Example A) are computed at this mode count, the validation does not separate mode-truncation error from the claimed method error. Please add a systematic convergence study in m for at least Example A (and ideally for the other examples with non-homogeneous layers), and report whether the final selected m places the truncation error well below the COMSOL comparison error.
  2. [Sec. 2.1, Eqs. (4)-(9); Sec. 3] The paper provides no a priori error bound or rigorous convergence criterion for the number of eigenmodes m; the statement that the required number 'depends on transverse inhomogeneity, frequency, and port-scattering-region coupling' is only heuristic. The selected mode counts vary widely across examples (4, 10, 50, 80, 100), so the choice is not governed by a uniform rule. Given that the central efficiency/accuracy claim depends on m being small enough for speedup and large enough for accuracy, please either provide a formal completeness/error estimate for the mode expansion or a reproducible empirical protocol (e.g., convergence threshold on S-parameters) that can be applied without ad hoc per-example tuning.
  3. [Sec. 3, Eq. (28)] The definition of S-parameters in Eq. (28) is in dB, but Table 2 reports 'relative errors' as percentages. It is unclear whether the reported errors are relative differences in linear magnitude, in dB, or in power. This matters for interpreting the accuracy claim (for example, 1.48% on a dB scale implies a very different absolute error than 1.48% on a linear magnitude). Please state the error metric explicitly and consistently.
minor comments (6)
  1. [Throughout] There are numerous typographical and formatting errors, including 'efficient' (abstract), 'T able' (Tables 1, 4, 5), 'egde-DoFs' (Sec. 2.3), 'P¨¦rez' (reference 9), and inconsistent use of '¡ª' in references. A careful proofread is needed.
  2. [Sec. 3.3] The sentence 'A TE10 mode excitation at Port 1 sweeps15 ∼ 21GHz over 201 Tables 5 shows...' is ungrammatical and should be rewritten (e.g., '...over 201 frequency points. Table 5 shows...').
  3. [Sec. 3.5, Table 7] The caption of Table 7 reads 'Computational Costs for Simplified Circulator Model,' but the example is a three-port divider, not a circulator. The caption should be corrected.
  4. [Sec. 3.4] The material labeled 'M0' is used in the first sentence of the material-properties paragraph but is not defined or introduced. Please clarify what M0 is (presumably air). Also, the phrase 'with a relative error of 134.71%' for Ph21 and Ph12 is confusing: is this the difference between the two phases (a physical nonreciprocity effect) or an error between HNM and COMSOL? As written, it appears to be the former and should be labeled as such, not as an 'error.'
  5. [Sec. 2.1] Equations (1)-(13) are taken largely from prior works [21,31] without derivation. While citations are appropriate, the manuscript should state explicitly which components are new (the hybrid coupling, Eqs. (16)-(27)) and which are reproduced from earlier papers, to help readers assess the novelty and reproduce the implementation.
  6. [Sec. 3.1 after Table 3] The conclusion states that HNM 'can also handle multiple scattering regions and layered media,' but none of the five examples contains multiple scattering regions. Either provide such an example or soften the claim to 'can handle multiple layered media and a scattering region' as demonstrated.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the hybrid method's central claims are validated against an external benchmark, and the cited prior work is used as building blocks, not as the target result.

full rationale

The paper's derivation chain is self-contained in the sense that its claimed new contribution—the variational coupling of the 3-D numerical mode-matching method with a tree-cotree-based mixed finite element method—is derived in the manuscript itself (Sec. 2.2, Eqs. (16)–(21), and the discretized system (26)–(27)). The NMM recursion (Eqs. (4)–(9)) and the 2.5-D MFEM eigenmode solver are imported from prior works [21] and [31], both of which include overlapping authors. This is genuine self-citation, but it is not circular: those citations supply established building blocks, and the paper does not use them to assert the validity of the new hybrid formulation. The new hybrid formulation is instead tested against COMSOL, an independent external solver, with S-parameter relative errors and computational-cost comparisons. No fitted parameter from the COMSOL data is renamed as a prediction, and no quantity in the derivation is defined in terms of the validation data. The mode-truncation study (Table 2) is a heuristic convergence check and no rigorous a priori bound is provided; the errors do still decrease between the two tested mode counts in some examples. That is a correctness/convergence gap, not a circularity, because the reported S-parameters are computed from the same HNM equations and compared with an independent solver. Therefore the central claim—that HNM provides an efficient alternative for non-layered regions embedded in layered media—does not reduce to its inputs and is not circular.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central claim rests on two pillars from prior work by the same group: the 2.5-D MFEM eigenmode solver [21] and the recursive reflection/transmission formulation [31], plus the standard tree-cotree FEM theory [32,37]. The only hand-chosen numerical parameters are the per-example mode counts, selected by convergence tests without a rigorous rule.

free parameters (1)
  • Number of retained eigenmodes m = Example A: 100; B: 4; C: 80; D: 20; E: 80 (from Table 2 and text)
    Chosen by convergence testing rather than derived; the paper gives no a priori rule for m, and the reported accuracy and efficiency depend on these hand-picked values.
assumptions (4)
  • domain assumption Fields in each layer of the inhomogeneous layered media can be represented as a finite superposition of 2-D waveguide eigenmodes computed by the 2.5-D MFEM (Sec. 2.1, Eqs. (1)-(9)).
    Quoted from [21,31]; the completeness of eigenmodes for inhomogeneous/anisotropic waveguides is assumed, and mode truncation is not rigorously justified.
  • domain assumption The recursive reflection/transmission and global reflection matrices satisfy the stated recursion (Eqs. (6)-(7), from [31]) and converge.
    No derivation or convergence analysis is given here; the method depends on these recursion formulas.
  • domain assumption Tree-cotree splitting of edge-element spaces enforces both tangential continuity and material-weighted divergence-free condition (Sec. 2.3, Eq. (25)).
    This is a standard finite element theory result (cited [32,37]); the paper relies on it to justify no extra unknowns.
  • domain assumption The boundary of the 3-D scattering region outside the ports S_L is PEC/PMC so the s0 boundary term vanishes (Appendix 5.1).
    Used in the variational form (20); in examples outer boundaries are PEC/PMC/Bloch.

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Cite this review

Pith. "Pith review of A 3-D Hybrid Numerical Method for Simulating Electromagnetic Fields in Structures with Multiple Inhomogeneous Layered Media." pith.science (2026). https://pith.science/paper/OVSHIQMO

@misc{pith2026260803639,
  author       = {Pith},
  title        = {Pith review of: A 3-D Hybrid Numerical Method for Simulating Electromagnetic Fields in Structures with Multiple Inhomogeneous Layered Media},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OVSHIQMO}},
  note         = {Machine review of arXiv:2608.03639}
}
read the original abstract

A three-dimensional (3-D) hybrid numerical method (HNM) is presented for electromagnetic scattering in structures with multiple inhomogeneous layered media coupled to an arbitrary 3-D non-layered scattering region. It integrates the 3-D numerical mode-matching (NMM) method with a tree-cotree-based mixed finite element method (MFEM). In the NMM, the fields in the 3-D layered media are reduced to a superposition of 2-D waveguide eigenmodes, while the MFEM discretizes the 3-D scattering region. The HNM thus inherits the dimensionality-reduction advantages of both conventional hybrid MM/FEM and pure NMM. Numerical experiments show that the HNM provides an effcient alternative for scattering problems involving non-layered regions embedded in layered media.

Figures

Figures reproduced from arXiv: 2608.03639 by the authors.

Figure 1
Figure 1. Sketch of EM scattering by a non-layered 3-D scattering region Ω with the multiple scatterers. The region Ω connects to multiple layered waveguides LMp (p = 1, 2, . . . P) via interface Sp at w = w0. Each LMp is divided into Lp layers along the w-direction at interfaces w = wα (α = 0, 1, . . . , Lp), with the p-th port at the end of the Lp-th layer. Layer counts may differ among LMp, and each layer is filled with in… view at source ↗
Figure 2
Figure 2. Flowchart of the 3-D hybrid numerical method (HNM). 2.1. NMM Method for the Inhomogeneous Layered Media A key step in the NMM method is to obtain the waveguide eigenmodes for each layer within the inhomogeneous layered media LMp. Thus, we start by solving the eigenmodes in an inhomogeneous [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Schematic of multi-layered inhomogeneous media with a 3-D scattering region containing a metallic sphere. The top, bottom, front, and rear boundaries are Bloch periodic. A plane wave at 30◦ to the z-axis illuminates the structure from Port 1. The scattering region is discretized with a 3-D mesh for the MFEM, while the layered-media EM fields are described via the NMM with second-order basis functions in both HNM and… view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: S-parameters at the ports of multi-layered inhomogeneous media as shown in [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: Geometry of bend waveguide model. (a) The model consists of a 90◦ curved copper conductor and two Teflon hollow cubic supports. Air-filled regions surround the copper conductor except for the Teflon supports. All outer boundaries are PEC except at the two ports. (b) Me…
Figure 6
Figure 6. Figure 6: S-parameters at the ports of bend waveguide model. The same frequency sampling (99 frequency points) is adopted for both HNM and COMSOL. (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 7
Figure 7. Figure 7: Geometry of the multiple scatterers model. (a) The model consists of three layered media LMi (i = 1, 2, 3) and a scattering region. LM2 and LM3 are obtained by rotating LM1 by 90◦ and 180◦ around the z−axis, respectively. Their constituent blocks are filled with materi…
Figure 8
Figure 8. Figure 8: S-parameters at the ports of the multiple-scatterers model in [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]
Figure 9
Figure 9. Figure 9: Geometry of the millimeter-wave circulator model. (a) Sectional view. (b) The circulator model consists of three layered media LMi (i = 1, 2, 3) and a scattering region, with three ports at the extremities of LMi . All other outer boundaries are PEC. The scattering reg…
Figure 10
Figure 10. Figure 10: S-parameters and their phases for the circulator model in [PITH_FULL_IMAGE:figures/full_fig_p014_10.png]
Figure 11
Figure 11. Figure 11: Geometry of the simplified divider model. (a) Sectional view. (b) The simplified divider model comprises a scattering region and three layered media LMi (i = 1, 2, 3). The extremities of LMi are set as the ports and the outer boundaries are covered with the PEC. The f…
Figure 12
Figure 12. Figure 12: S-parameters and their phases at the ports of the divider model in [PITH_FULL_IMAGE:figures/full_fig_p016_12.png]

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Reference graph

Works this paper leans on

39 extracted references · 39 canonical work pages

  1. [1]

    Numerical methods in electromagnetic scattering theoy,

    F. M. Kahnert, “Numerical methods in electromagnetic scattering theoy,” J. Quant. Spectrosc. Radiat. Transf., vol. 79-80, pp. 775-824, Jun.-Sep. 2003

    work page 2003

  2. [2]

    M. S. Zhdanov, Geophysical Inverse Theory and Regularization Problems . New York: NY, USA: Elsevier, 2002

    work page 2002

  3. [3]

    Imaging of biomedical data using a multiplicative regularized contrast source inversion method,

    A. Abubakar, P. M. van den Berg, and J. J. Mallorqui, “Imaging of biomedical data using a multiplicative regularized contrast source inversion method,”IEEE Trans. Microw. Theory Techn., vol. 50, no. 7, pp. 1761-1771, Jul. 2002

    work page 2002

  4. [4]

    Efficient use of closed-form Green’s functions for the analysis of planar geometries with vertical connections,

    N. Kinayman and M. I. Aksun, “Efficient use of closed-form Green’s functions for the analysis of planar geometries with vertical connections,”IEEE Trans. Microw. Theory Techn. , vol. 45, no. 5, pp. 593-603, May 1997. 20 Jie Liu, T aihe Li, Ke Chen, Mingwei Zhuang and Qing Huo Liu

    work page 1997

  5. [5]

    Fast inhomogeneous plane wave algorithm for scattering from objects above the multilayered medium,

    B. Hu and W. C. Chew, “Fast inhomogeneous plane wave algorithm for scattering from objects above the multilayered medium,”IEEE Trans. Geosci. Remote Sens., vol. 39, no. 5, pp. 1028-1038, May 2001

    work page 2001

  6. [6]

    A novel method to analyze electromagnetic scattering of complex objects,

    K. Umashankar and A. Taflove, “A novel method to analyze electromagnetic scattering of complex objects,” IEEE Trans. Electromagn. Compat., vol. 24, no. 4, pp. 397–405, Nov. 1982

    work page 1982

  7. [7]

    The finite-difference time-domain (FDTD) and the finite-volume time- domain (FVTD) methods in solving Maxwell’s equations,

    K. S. Yee and J. S. Chen, “The finite-difference time-domain (FDTD) and the finite-volume time- domain (FVTD) methods in solving Maxwell’s equations,”IEEE Trans. Antennas Propag., vol. 45, no. 3, pp. 354-363, Mar. 1997

    work page 1997

  8. [8]

    Moment method with isoparametric elements for three-dimensional anisotropic scatterers,

    R. D. Graglia, P. L. E. Uslenghi, and R. S. Zich, “Moment method with isoparametric elements for three-dimensional anisotropic scatterers,”Proc. IEEE, vol. 77, no. 5, pp. 750-760, May 1989

    work page 1989

Show all 39 references
  1. [9]

    Windowed green function MoM for second-kind surface integral equation formulations of layered media electromagnetic scattering problems,

    R. Arrieta and C. P¨¦rez-Arancibia, “Windowed green function MoM for second-kind surface integral equation formulations of layered media electromagnetic scattering problems,”IEEE Trans. Antennas Propag., vol. 70, no. 12, pp. 11978-11989, Dec. 2022

    2022

  2. [10]

    Electromagnetic modeling of inhomogeneous and anisotropic structures by volume integral equation methods,

    Y. S. Cao, P. Li, L.-J. Jiang, and A. E. Ruehli, “Electromagnetic modeling of inhomogeneous and anisotropic structures by volume integral equation methods,”IEEE Trans. Antennas Propag., vol. 65, no. 2, pp. 948-953, Feb. 2017

    2017

  3. [11]

    Jin, The Finite Element Method in Electromagnetics

    J. Jin, The Finite Element Method in Electromagnetics . New York: John Wiley & Sons Inc., 1993

    1993

  4. [12]

    A finite element method for approximating electromagnetic scattering from a conducting object,

    A. Kirsch and P. Monk, “A finite element method for approximating electromagnetic scattering from a conducting object,”Numer. Math., vol. 92, no. 3, pp. 501-534, Sep. 2002

    2002

  5. [13]

    Adjoint-based accelerated adaptive refinement in frequency domain 3-D finite element method scattering problems,

    J. J. Harmon, C. Key, D. Estep, T. Butler, and B. M. Notaroˇs, “Adjoint-based accelerated adaptive refinement in frequency domain 3-D finite element method scattering problems,” IEEE Trans. Antennas Propag., vol. 69, no. 2, pp. 940-949, Feb. 2021

    2021

  6. [14]

    Efficient modal analysis of waveguide filters including the orthogonal mode coupling elements by an MM/FE method,

    R. Beyer and F. Arndt, “Efficient modal analysis of waveguide filters including the orthogonal mode coupling elements by an MM/FE method,”IEEE Microwave and Guided Wave Lett. , vol. 5, no. 1, pp. 9-10, Jan. 1995

    1995

  7. [15]

    Automated design of waveguide components using hybrid mode matching/numerical EM building blocks in optimization-oriented CAD frame-works¡ªstate-of-the-art and recent advances,

    F. Arndt, R. Beyer, J. M. Reiter, T. Sieverding, and T. Wolf, “Automated design of waveguide components using hybrid mode matching/numerical EM building blocks in optimization-oriented CAD frame-works¡ªstate-of-the-art and recent advances,” IEEE Trans. Microw. Theory Techn. , ...

    1997

  8. [16]

    Fast CAD and optimization of waveguide components and aperture antennas by hybrid MM/FE/MoM/FD methods¡ªstate-of-the-art and recent advances,

    F. Arndt, J. Brandt, V. Catina, J. Ritter, I. Rullhusen, J. Dauelsberg, U. Hilgefort, and W. Wessel, “Fast CAD and optimization of waveguide components and aperture antennas by hybrid MM/FE/MoM/FD methods¡ªstate-of-the-art and recent advances,”IEEE Trans. Microw. Theory Techn....

    2004

  9. [17]

    Transmission matrix representation of finline discontinuities,

    A. S. Omar and K. Schonemann, “Transmission matrix representation of finline discontinuities,” IEEE Trans. Microw. Theory Techn. , vol. 33, no. 9, pp. 765-770, Sep. 1985

    1985

  10. [18]

    A hybrid mode matching/FEM technique withedgeelementsforsolvingwaveguidesdiscontinuityproblems,

    D. Arena, M. Ludovico, G. Manara, and A. Monorchio, “A hybrid mode matching/FEM technique withedgeelementsforsolvingwaveguidesdiscontinuityproblems,” in Proc. IEEE Antennas Propag. Soc. Int. Symp. , vol. 4, Jul. 2000, pp. 2028-2031

    2000

  11. [19]

    Wideband design of a short-slot 2-plane coupler by the mode matching/FEM hybrid analysis considering the structural symmetry,

    M. Wakasa, D.-H. Kim, T. Tomura, and J. Hirokawa, “Wideband design of a short-slot 2-plane coupler by the mode matching/FEM hybrid analysis considering the structural symmetry,”IEICE Trans. Commun., vol. E102.B, no. 5, pp. 1019¨C1026, May 2019

    2019

  12. [20]

    Analysis of passive microwave circuits by using a hybrid 2-D and 3-D finite-element mode-matching method,

    J. Rubio, J. Arroyo and J. Zapata, “Analysis of passive microwave circuits by using a hybrid 2-D and 3-D finite-element mode-matching method,”IEEE Trans. Microw. Theory Techn., vol. 47, no. 9, pp. 1-4, Sep. 1999

    1999

  13. [21]

    Microscopic modeling of metasurfaces by the mixed finite element numerical mode-matching method,

    J. Liu, N. Liu, and Q. H. Liu, “Microscopic modeling of metasurfaces by the mixed finite element numerical mode-matching method,”IEEE Trans. Microw. Theory Techn. , vol. 68, no. 2, pp. 469– 478, Feb. 2020

    2020

  14. [22]

    New mixed SETD and FETD methods to overcome the low-frequency breakdown problems by tree-cotree splitting,

    K. Chen, J. Liu, M. Zhuang, Q. Sun, and Q. H. Liu, “New mixed SETD and FETD methods to overcome the low-frequency breakdown problems by tree-cotree splitting,”IEEE Trans. Microw. Theory Techn., vol. 68, no. 8, pp. 3219–3228, Aug. 2020

    2020

  15. [23]

    Method to calculate the reflection and transmission of guided A 3-D HNM for Simulating EM Fields in Structures with Multiple Inhomogeneous Layered Media 21 waves,

    M. A. A. Pudensi, L. G. Ferreira, “Method to calculate the reflection and transmission of guided A 3-D HNM for Simulating EM Fields in Structures with Multiple Inhomogeneous Layered Media 21 waves, ”Journal of the Optical Society of America , vol. 72, no. 1, pp. 126-130, Jan. 1982

    1982

  16. [24]

    Analysis of circular and noncircular waveguides and striplines with multilayered uniaxial anisotropic medium,

    V. Kamra and A. Dreher, “Analysis of circular and noncircular waveguides and striplines with multilayered uniaxial anisotropic medium,”IEEE Trans. Microw. Theory Techn. , vol. 67, no. 2, pp. 548–591, Feb. 2019

    2019

  17. [25]

    Efficient computation of electromagnetic wave in anisotropic orthogonal- plano-cylindrically layered media using the improved numerical mode matching (NMM) method,

    J. W. Dai and Q. H. Liu, “Efficient computation of electromagnetic wave in anisotropic orthogonal- plano-cylindrically layered media using the improved numerical mode matching (NMM) method,” IEEE Trans. Antennas Propag., vol. 63, no. 8, pp. 3569–3578, Aug. 2015

    2015

  18. [26]

    A numerical mode matching (NMM) method for optical fibers with Kerr nonlinearity,

    X. L. Wu, J. Dai, J. Liu, J. H. Chen, and Q. H. Liu, “A numerical mode matching (NMM) method for optical fibers with Kerr nonlinearity,”IEEE Trans. Antennas Propag. , vol. 71, no. 6, pp. 5166-5177, June 2023

    2023

  19. [27]

    Spectral numerical mode matching method for metasurfaces,

    J. Liu, G. X. Cai, J. Yao, N. Liu, and Q. H. Liu, “Spectral numerical mode matching method for metasurfaces,” IEEE Trans. Microw. Theory Techn. , vol. 67, no. 7, pp. 2629-2639, July 2019

    2019

  20. [28]

    Spectral numerical mode-matching method for 3-D layered multiregion structures,

    J. Liu, N. Liu, and Q. H. Liu, “Spectral numerical mode-matching method for 3-D layered multiregion structures,” IEEE Trans. Antennas Propag., vol. 68, no. 2, pp. 986-996, Feb. 2020

    2020

  21. [29]

    Mixed finite element numerical mode matching method for designing infrared broadband polarization-independent metamaterial absorbers,

    C. F. Tong, J. Liu, and Q. H. Liu, “Mixed finite element numerical mode matching method for designing infrared broadband polarization-independent metamaterial absorbers,”Opt. Express, vol. 30, no. 25, pp. 45031-45042, Dec. 2022

    2022

  22. [30]

    3-D numerical mode matching method for off- centered electromagnetic well logging tools in noncircular vertical borehole and invasion zones in multilayered media,

    J. Liu, J. L. Zhuo, W. Jiang, and Q. H. Liu, “3-D numerical mode matching method for off- centered electromagnetic well logging tools in noncircular vertical borehole and invasion zones in multilayered media,”IEEE Trans. Geosci. Remote Sens. , vol. 60, Jan. 2022, Art. no. 4504313

    2022

  23. [31]

    3-D NMM method for fully anisotropic and nonreciprocal media,

    J. Liu, C. F. Tong, W. Jiang, and Q. H. Liu, “3-D NMM method for fully anisotropic and nonreciprocal media,” IEEE Trans. Microw. Theory Techn. , vol. 70, no. 7, pp. 3428–3441, July 2022

    2022

  24. [32]

    Removal of spurious DC modes in edge element solutions for modeling three-dimensional resonators,

    N. V. Venkatarayalu and J.-F. Lee, “Removal of spurious DC modes in edge element solutions for modeling three-dimensional resonators,” IEEE Trans. Microw. Theory Techn. , vol. 54, no. 7, pp. 3019–3025, July 2006

    2006

  25. [33]

    Mixed spectral-element methods for 3-D Maxwell¡¯s eigenvalue problems with Bloch periodic and open resonators,

    S. J. Wang, J. Liu, M. W. Zhuang, K. Chen, and Q. H. Liu. “Mixed spectral-element methods for 3-D Maxwell¡¯s eigenvalue problems with Bloch periodic and open resonators,”IEEE Trans. Microw. Theory Techn., vol. 69, no. 3, 1547-1558, March 2021

    2021

  26. [34]

    Mixed and penalty formulations for finite element analysis of an eigenvalue problem in electromagnetism,

    K. Fumio, “Mixed and penalty formulations for finite element analysis of an eigenvalue problem in electromagnetism,” Comput. Methods Appl. Mech. Eng. , vol. 64, nos. 1-3, pp. 509-521, Oct. 1987

    1987

  27. [35]

    Hiptmair, Finite Elements in Computational Electromagnetism, Cambridge, U.K.: Cambridge Univ

    R. Hiptmair, Finite Elements in Computational Electromagnetism, Cambridge, U.K.: Cambridge Univ. Press, 2002

    2002

  28. [36]

    Monk, Finite Element Methods for Maxwell¡¯s Equations

    P. Monk, Finite Element Methods for Maxwell¡¯s Equations . Oxford, U.K.: Oxford Univ. Press, 2003

    2003

  29. [37]

    A spurious-free domain decomposition method for 3-D Maxwell¡¯s eigenvalue problems,

    S. J. Wang, J. Liu, K. Chen, and Q. H. Liu, “A spurious-free domain decomposition method for 3-D Maxwell¡¯s eigenvalue problems,”IEEE Trans. Microw. Theory Techn., vol. 71, no. 2, pp. 548-560, Feb. 2023

    2023

  30. [38]

    Solving unsymmetric sparse systems of linear equations with PARDISO,

    O. Schenk and K. Gärtner, “Solving unsymmetric sparse systems of linear equations with PARDISO,” Future Gener. Comput. Syst. , vol. 20, pp. 475-487, 2004

    2004

  31. [39]

    A fast, higher order three-dimensional finite- element analysis of microwave waveguide devices,

    J. Liu, J.-M. Jin, E. K. N. Yung, and R. S. Chen, “A fast, higher order three-dimensional finite- element analysis of microwave waveguide devices,”Microw. Opt. Technol. Lett., vol. 32, no. 5, pp. 344-352, Jan. 2002

    2002

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.