Pith. sign in

REVIEW 4 major objections 4 minor 28 references

An Implicit Time-Domain Harmonic Balance Method for Radio-Frequency Capacitively Coupled Plasma Simulations

T0 review · 4 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Harmonic balance solves RF plasma periodic state 10x faster than time-marching

desk verdict Solid incremental work that is likely correct, but the headline <0.3% claim is in-sample and overstated; the paper deserves a careful revision, not a desk reject. read the letter →

arxiv 2607.18103 v2 pith:MAELDFUB submitted 2026-07-20 physics.plasm-ph math-phmath.MPphysics.flu-dyn

Yuze Zhu , Yufeng Wei , Yue Zhang , Kun Xu This is my paper · ORCID
classification physics.plasm-phmath-phmath.MPphysics.flu-dyn
keywords harmonicbalancecapacitivelycoupledplasmadrift-diffusion-Poissonelectron-energytransporttime-spectralmethodoperatorsplittingRFsimulationlocal-mean-energyapproximation
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

This paper shows that a time-domain harmonic balance method can directly compute the periodic steady state of radio-frequency capacitively coupled plasmas, skipping the long transient phase that conventional time-marching solvers must integrate. The authors extend harmonic balance beyond simpler plasma models to include full electron-energy transport, using an operator-splitting scheme that keeps all implicit inversions cell-local. On a 1D argon benchmark, retaining eight harmonics resolves the quasi-steady bulk and the sharply nonlinear sheath dynamics, with macroscopic errors under 0.3% versus a highly converged dual-time-stepping reference. The same accuracy is reached in about a tenth of the wall-clock time of a fully converged baseline, even on a single CPU core.

What carries the argument

The central mechanism is the Fourier-collocation time-spectral operator E, a real, dense, skew-symmetric matrix mapping time derivatives at the discrete phase points to a coupling across all collocation points (Eq. 23). Applying E to the drift-diffusion and electron-energy equations converts the unsteady system into a quasi-steady one, and the proposed implicit relaxation factorizes the local Jacobian (Eq. 39) into a spatial factor and a temporal factor, so the dense temporal inversion is only per-cell and per-variable.

What would settle it

Run the same 1D argon benchmark at lower pressure (e.g., 0.1 Torr) or with a metastable population whose lifetime is many RF periods; if the HB solution with NH=8 differs from a fully converged DTS solution by more than 0.3% in the bulk plasma, the strict periodicity assumption is not generally valid.

Watch

Extended reading notes

Core claim

The paper establishes that the time-domain harmonic balance method can be extended to a fully coupled drift-diffusion-Poisson system with electron-energy transport (the local-mean-energy approximation) for RF capacitively coupled plasmas, and that with eight harmonics the periodic solution is strictly consistent with an established time-marching reference. The central technical contribution is converting the periodic-in-time plasma flow problem into a pseudo-steady system sampled at NT=2NH+1 collocation points per RF cycle, then solving it with a spatiotemporal operator splitting: a spatial implicit relaxation sweep followed by a cell-local dense temporal inversion that treats the time-spect

Load-bearing premise

The method assumes the plasma reaches a strictly periodic steady state with period T of the RF drive; if any species or coupled dynamics relax on a longer timescale, the computed periodic solution would not match the time-asymptotic state of a time-marching simulation.

Editorial extensions

If this is right

  • For RF CCP simulations, the periodic state can be obtained without simulating hundreds of RF cycles, making routine parameter sweeps and reactor design iterations much cheaper.
  • Because the method evaluates nonlinear kinetics directly at each phase point, it extends naturally to more complex chemistries and multidimensional geometries without changing the core formulation.
  • The reported errors for ne, εe, and ϕ are all below 0.3% at NH=8, confirming that spectral truncation is not a bottleneck for this problem class.
  • The speedup is achieved on a single core, so the method can be combined with spatial parallelism for further gains.
  • The approach inherits harmonic balance’s clean error control: truncating at NH only changes the spectral error, not the physical time-step error, making it a suitable verification tool for time-marching solvers.

Reading between the lines

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

  • If the method holds in 2D/3D, where the same operator-splitting remains cell-local, the memory advantage over monolithic Jacobian approaches should become even more pronounced.
  • The 10x speedup for a single condition suggests that for optimization studies or design loops that require many successive RF conditions, the effective speedup could compound; a designed experiment varying pressure or voltage would test this.
  • Because the electron temperature is a derived ratio, the near-wall Te error (6.4%) hints that derived quantities in depleted sheaths will always be less accurate than conserved quantities; a post-processing smoothing or a different temperature definition might be worth testing.
  • A natural next test is a discharge with a slowly relaxing neutral metastable population whose period exceeds the RF period, since the method assumes an exactly T-periodic quasi-steady state and could fail or require treating slow manifolds separately.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper presents a time-domain harmonic balance (HB) method for the periodic steady state of a fluid model of radio-frequency capacitively coupled plasmas, specifically a drift-diffusion-Poisson system with electron-energy transport (LMEA). The physical time derivative is replaced by a dense spectral operator at NT = 2NH + 1 temporal collocation points, and the resulting quasi-steady system is solved by an implicit pseudo-time relaxation that combines a spatial block-implicit sweep (decoupled across phases) with a cell-local dense temporal inversion; a semi-implicit Poisson update handles the dielectric-relaxation limit. The method is validated against a 1D argon CCP benchmark, using a dual-time-stepping (DTS) baseline with T/Δt = 200 as reference. The authors select NH = 8 from an FFT of that baseline and report E2 errors below 0.3% for ne, εe, and ϕ, and a 10.26x speedup over the DTS baseline, with the conclusion that eight harmonics 'perfectly resolves' the discharge dynamics.

Significance. If the central claims hold, this is a meaningful contribution: prior HB plasma work (ref. [25]) was limited to the local-field approximation, whereas this paper includes full electron-energy transport and proposes a memory-efficient, cell-local temporal inversion that avoids global Jacobian assembly. The paper contains a careful time-step refinement study showing second-order convergence of the DTS reference and a detailed spatiotemporal comparison against a classical benchmark. However, the headline accuracy claim is weakened by the in-sample selection of NH from the DTS reference, the absence of an error-versus-NH table, and the restriction of all validation to a single 1D operating point. The speedup result, while plausible, is also only demonstrated in 1D on a single processor.

major comments (4)
  1. [§3.3, §3.4, and Table 4] The harmonic count NH = 8 is chosen from an FFT of the DTS baseline (§3.3: 'This preliminary analysis relies on the high-fidelity transient signals extracted directly from the reference DTS baseline'), and the error metrics in Table 4 compare the NH = 8 HB solution to that same baseline. The reported <0.3% errors are therefore in-sample: they show that a truncation tailored to the reference can represent that reference, not that the method reliably predicts the periodic state when the harmonic content is unknown. The sweep NH = 6, 8, 10, 12 in §3.4 is not accompanied by an error table; only convergence histories and selected profiles are shown, so the reader cannot tell whether NH = 8 is a true plateau or a tuned sweet spot. Please provide E2/EMA/E∞ for all four truncation levels for all variables, and ideally an out-of-sample test (e.g., a different V0 or pressure) to support the genera
  2. [Abstract and §3.5, Table 4] The abstract and conclusion state that 'macroscopic relative errors [are] strictly below 0.3%' compared to DTS solutions. Table 4 contradicts this as stated: E∞(ne) = 0.392%, E2(Te) = 1.18%, and E∞(Te) = 6.385%. The body text carefully qualifies the 0.3% statement to the L2 and mean-absolute errors of the directly integrated variables (ne, εe, ϕ), but the abstract and conclusion are unqualified. Revise the headline claim to match the data, e.g., 'L2 and mean-absolute errors below 0.3% for directly integrated variables, with larger localized pointwise errors for derived quantities such as Te near the walls.'
  3. [§3.1, §3.5, Fig. 14] The entire validation is performed in one dimension on a 90-cell mesh. The abstract promises a 'physically rigorous, memory-efficient, and highly accelerated paradigm for practical RF plasma simulations', but no multidimensional test is presented. The proposed cell-local temporal inversion is dimension-independent in principle, but solver robustness, convergence behavior, and the claimed speedup have not been demonstrated in 2D or 3D. Please either add a multidimensional proof-of-concept or explicitly restrict the conclusions and title claims to 1D benchmarks.
  4. [§3.5, Table 4, Eq. (71)] The DTS reference itself carries finite temporal discretization error: Table 3 shows about 0.024% error in spatial averages for T/Δt = 200 versus T/Δt = 400, and local pointwise errors could be larger. The HB errors in Table 4 are measured relative to this reference. In particular, E∞(Te) = 6.385% is attributed to the sensitivity of the ratio εe/ne in depleted sheaths, but that attribution is not verified; part of this deviation could be temporal error of the DTS baseline. To separate HB truncation/aliasing error from DTS discretization error, compare the HB solution against a T/Δt = 400 (or Richardson-extrapolated) DTS reference, at least for Te.
minor comments (4)
  1. [Eq. (66) and §3.1] The notation in Eq. (66) is ambiguous: 'ne,me' appears to mean n_{e,m} times the elementary charge e, but should be written explicitly as such. Also, the dimensionless form of the equations is said to be omitted; providing it, or at least the values of cD, α, and the entries of D_damp, would improve reproducibility.
  2. [§2.4, Eq. (43)–(44), §2.4, Eq. (49)] The numerical diffusion constant c_D, the damping factor α, and the diagonal damping matrix D_damp are user-set parameters but their values are never specified. The paper should list these values and briefly discuss sensitivity to them, since the convergence and stability claims depend on them.
  3. [§3.5, Eq. (71)] The error definition in Eq. (71) samples Ns = 200 phase instances per RF cycle, while the HB solution has only NT = 17 collocation points for NH = 8. The text should state explicitly how the HB solution is evaluated at those 200 samples (e.g., inverse DFT reconstruction from the Fourier coefficients) so that the error metric is unambiguous.
  4. [References] Reference [25] is cited as 'Journal of Computational Physics (2026) 115027' without volume/page details; add the full citation or DOI if available. Also, the caption of Fig. 3 should identify the pseudo-CFL values by line style or color, as the reader cannot distinguish them from the text description alone.

Circularity Check

1 steps flagged · score 4.0 of 10

HB solve is independent, but the headline NH=8 accuracy claim is an in-sample model-selection check: NH is chosen from the DTS reference and then errors are measured against that same reference.

  1. fitted input called prediction [§3.3–§3.5, Table 4]
    "This preliminary analysis relies on the high-fidelity transient signals extracted directly from the reference DTS baseline established in the preceding section. ... To eliminate the minor temperature deviations without incurring the severe computational penalty of higher-order truncations, NH = 8 is selected. ... Table 4: Quantitative spatiotemporal error metrics of the HB solution (NH = 8) evaluated against the fully converged DTS baseline (T/Δt = 200)."

    The harmonic truncation level NH=8 is selected from the spectral content of the DTS reference itself (§3.3 FFT analysis), and the central accuracy claim is then evaluated against that same DTS baseline (§3.5, Table 4). Thus the sub-0.3% E2/EMA agreement is an in-sample consistency check: a representation whose harmonic content was extracted from the reference is shown to reconstruct that reference. It does not independently establish that NH=8 is sufficient for an unseen operating point. The absence of an error-vs-NH table against DTS further prevents verifying that NH=8 is a true convergence plateau. The HB solve itself is a genuine PDE solve and is not fitted to DTS values, so this is partial circularity rather than constructional equivalence.

full rationale

The core numerical derivation is not circular: the HB residuals, the spatiotemporal operator-splitting, the spectral operator E, and the semi-implicit Poisson update are all derived from the governing equations and are not defined in terms of the DTS solution. No physical constants are fitted in the HB solve, and there are no load-bearing self-citations or imported uniqueness theorems. The DTS baseline is used only to set a numerical resolution parameter (NH) and to define error metrics. That said, the paper's flagship claim that NH=8 'perfectly resolves' the periodic state is partially self-referential: the harmonic count is chosen after inspecting the FFT of the DTS reference, and then the comparison is made against that same reference. This is model selection on the test set rather than a fully out-of-sample validation. Separately, the abstract's 'strictly below 0.3%' statement is not supported by Table 4 if one includes maximum pointwise errors (E∞(ne)=0.392%, E∞(Te)=6.385%), but that inconsistency is an accuracy-reporting issue, not a circularity. Overall, the main HB framework has independent content, so the circularity score is moderate rather than high.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new physical entities; it relies on standard plasma fluid modeling assumptions, a central periodicity assumption, and numerical approximations (factorization, damping) whose constants are partly unspecified.

free parameters (5)
  • Number of harmonics NH = 8
    Chosen from the FFT of the DTS reference signal (Sec 3.3); accuracy and cost scale with NH. NH=6 showed small Te artifacts; NH=8 selected as the sweet spot.
  • Pseudo-CFL number = 10000
    Chosen from convergence plateau in Fig. 3; affects speed but not converged solution; used for both DTS and HB.
  • Damping factor α = unspecified
    Used in Eq. (49) to enforce positivity; value not reported, so readers cannot reproduce exactly.
  • Numerical diffusion constant c_D = unspecified
    Appears in the spectral radii in Eq. (44); value not given; affects stability/stiffness.
  • Diagonal damping matrix D_damp = unspecified
    Added to enforce diagonal dominance (Sec 2.4); entries proportional to row sums but scale unspecified.
assumptions (6)
  • domain assumption Inductive effects are negligible; E = -∇ϕ (Eq. 1).
    Requires that the reactor dimensions are much smaller than the electromagnetic wavelength; standard for 13.56 MHz CCP, but an assumption.
  • domain assumption Drift-diffusion flux approximation (Eqs 5-7, 10).
    Assumes highly collisional regime and local mean energy; not valid for low-pressure kinetic regimes.
  • domain assumption The solution reaches a quasi-steady state with period T (Eq. 15).
    Central assumption of harmonic balance; if slow species have longer-timescale drifts, the periodic state may not exist.
  • standard math Nyquist sampling with NT=2NH+1 (Eq. 17) and DFT differentiation matrix.
    Standard spectral differentiation; aliasing errors controlled by NH.
  • ad hoc to paper Approximate factorization neglects the (Δτ)^2 (E⊗Iq)Aii term (Eq. 39).
    Factorization error is second-order in pseudo-time and vanishes at convergence; used to justify the solver splitting.
  • domain assumption Rate coefficients kion and kexc are interpolated from the tabulated data of [18].
    The chemistry model is taken from the cited benchmark; may not transfer to other gases/pressures.

how reviews work

0 comments
Cite this review

Pith. "Pith review of An Implicit Time-Domain Harmonic Balance Method for Radio-Frequency Capacitively Coupled Plasma Simulations." pith.science (2026). https://pith.science/paper/MAELDFUB

@misc{pith2026260718103,
  author       = {Pith},
  title        = {Pith review of: An Implicit Time-Domain Harmonic Balance Method for Radio-Frequency Capacitively Coupled Plasma Simulations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MAELDFUB}},
  note         = {Machine review of arXiv:2607.18103}
}
read the original abstract

Fast and accurate fluid simulation of radio-frequency capacitively coupled plasmas (RF CCPs) is of great importance for the iterative design and parameter optimization of modern plasma reactors. This study presents the first successful extension of the time-domain harmonic balance (HB) method to a fully coupled drift-diffusion-Poisson system with complete electron-energy transport for RF plasma simulations. To resolve the severe numerical stiffness arising from highly nonlinear energy-dependent kinetics and dense phase-coupling, a highly efficient spatiotemporal operator-splitting strategy is employed. By sequentially executing a spatial implicit relaxation and a cell-local temporal inversion, this strategy entirely avoids the memory-intensive assembly of global Jacobians while preserving robust numerical stability. The proposed method is rigorously validated against a standard parallel-plate argon CCP benchmark. Evaluated across all discrete temporal collocation points, the HB solution demonstrates that retaining eight harmonics perfectly resolves both the quasi-steady bulk plasma and the highly nonlinear transient sheath dynamics, yielding macroscopic relative errors strictly below 0.3% compared to conventional dual-time stepping (DTS) solutions. Beyond its high physical fidelity, the time-domain HB method completely bypasses the prohibitive physical transients required by conventional time-marching methods. Evaluated on a purely sequential single-core execution, the HB method delivers a greater than 10-fold speedup over fully converged DTS baselines and remains over 5 times faster than the coarsest time-marching configurations. These results establish the time-domain HB framework as a physically rigorous, memory-efficient, and highly accelerated paradigm for practical RF plasma simulations.

Figures

Figures reproduced from arXiv: 2607.18103 by the authors.

Figure 1
Figure 1. Schematic illustration of the spatio-temporal decoupling strategy in the time [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Schematic diagram of the one-dimensional RF CCP discharge configuration. [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. Convergence of the normalized full BDF residual at a representative physical [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: Electron number density distributions at the beginning of the converged RF [PITH_FULL_IMAGE:figures/full_fig_p025_4.png]
Figure 5
Figure 5. Figure 5: Time-domain signals of elec￾tron number density at different monitor￾ing points [PITH_FULL_IMAGE:figures/full_fig_p026_5.png]
Figure 7
Figure 7. Figure 7: Normalized FFT spectrum of potential at NLE. [PITH_FULL_IMAGE:figures/full_fig_p027_7.png]
Figure 8
Figure 8. Figure 8: Normalized FFT spectrum of potential at BR. [PITH_FULL_IMAGE:figures/full_fig_p027_8.png]
Figure 9
Figure 9. Figure 9: Normalized FFT spectrum of electron number density at NLE. [PITH_FULL_IMAGE:figures/full_fig_p028_9.png]
Figure 11
Figure 11. Figure 11: Truncated reconstruction of NLE electron number density using NH = 5. tion levels (NH = 6, 8, 10, and 12). Preliminary computations revealed that employing lower harmonic counts, such as NH = 4 or 5, inevitably leads to numerical divergence. This instability is primar…
Figure 12
Figure 12. Figure 12: Convergence histories of the spatially averaged electron number density at the [PITH_FULL_IMAGE:figures/full_fig_p029_12.png]
Figure 13
Figure 13. Figure 13: Spatial distributions of macroscopic plasma properties at the beginning of the [PITH_FULL_IMAGE:figures/full_fig_p030_13.png]
Figure 14
Figure 14. Figure 14: Comparison of spatial distributions of (a) electron number density and (b) ion [PITH_FULL_IMAGE:figures/full_fig_p032_14.png]
Figure 15
Figure 15. Figure 15: Spatiotemporal comparison between the DTS reference solution and the HB [PITH_FULL_IMAGE:figures/full_fig_p033_15.png]
Figure 16
Figure 16. Figure 16: Convergence histories of the spatially averaged electron density: a wall-clock [PITH_FULL_IMAGE:figures/full_fig_p036_16.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

28 extracted references

  1. [25]

    D. B. Zulevic, S. Gautam, S. Sriraman, A. Venkattraman, Accelerat- ing low-temperature plasma simulations using the harmonic balance method, Journal of Computational Physics (2026) 115027

    2026

  2. [1]

    M. A. Lieberman, A. J. Lichtenberg, Principles of plasma discharges and materials processing, MRS Bulletin 30 (12) (1994) 899–901

    1994

  3. [2]

    F. F. Chen, J. P. Chang, Lecture notes on principles of plasma process- ing, Springer Science & Business Media, 2003

    2003

  4. [3]

    Chabert, N

    P. Chabert, N. Braithwaite, Physics of radio-frequency plasmas, Cam- bridge University Press, 2011

    2011

  5. [4]

    G. S. Oehrlein, S. Hamaguchi, Foundations of low-temperature plasma enhanced materials synthesis and etching, Plasma Sources Science and Technology 27 (2) (2018) 023001. 38

    2018

  6. [5]

    C. K. Birdsall, Particle-in-cell charged-particle simulations, plus monte carlo collisions with neutral atoms, pic-mcc, IEEE Transactions on plasma science 19 (2) (1991) 65–85

    1991

  7. [6]

    Surendra, D

    M. Surendra, D. B. Graves, Particle simulations of radio-frequency glow discharges, IEEE transactions on plasma science 19 (2) (1991) 144–157

    1991

  8. [7]

    M. J. Kushner, Hybrid modelling of low temperature plasmas for fun- damental investigations and equipment design, Journal of Physics D: Applied Physics 42 (19) (2009) 194013

    2009

Show all 28 references
  1. [8]

    A. P. Matthews, Current advance method and cyclic leapfrog for 2d mul- tispecies hybrid plasma simulations, Journal of Computational Physics 112 (1) (1994) 102–116

    1994

  2. [9]

    Misium, A

    G. Misium, A. Lichtenberg, M. Lieberman, Macroscopic modeling of radio-frequency plasma discharges, Journal of Vacuum Science & Tech- nology A: Vacuum, Surfaces, and Films 7 (3) (1989) 1007–1013

    1989

  3. [10]

    Boeuf, Numerical model of rf glow discharges, Physical review A 36 (6) (1987) 2782

    J.-P. Boeuf, Numerical model of rf glow discharges, Physical review A 36 (6) (1987) 2782

    1987

  4. [11]

    V. A. Godyak, N. Sternberg, Dynamic model of the electrode sheaths in symmetrically driven rf discharges, Physical Review A 42 (4) (1990) 2299

    1990

  5. [12]

    Sharma, Investigation of ion and electron kinetic phenomena in ca- pacitively coupled radio-frequency plasma sheaths: A simulation study, Ph.D

    S. Sharma, Investigation of ion and electron kinetic phenomena in ca- pacitively coupled radio-frequency plasma sheaths: A simulation study, Ph.D. thesis (2013)

    2013

  6. [13]

    G. J. M. Hagelaar, Modeling of microdischarges for display technology (2000)

    2000

  7. [14]

    R. R. Arslanbekov, V. I. Kolobov, Implicit and coupled fluid plasma solver with adaptive cartesian mesh and its applications to non- equilibrium gas discharges, Plasma Sources Science and Technology 30 (4) (2021) 045013

    2021

  8. [15]

    A. D. Gomez, N. Deak, F. Bisetti, Jacobian-free newton–krylov method for the simulation of non-thermal plasma discharges with high-order time integration and physics-based preconditioning, Journal of Compu- tational Physics 480 (2023) 112007. 39

    2023

  9. [16]

    Davoudabadi, J

    M. Davoudabadi, J. S. Shrimpton, F. Mashayek, On accuracy and performance of high-order finite volume methods in local mean en- ergy model of non-thermal plasmas, Journal of Computational Physics 228 (7) (2009) 2468–2479

    2009

  10. [17]

    Trieschmann, L

    J. Trieschmann, L. Vialetto, T. Gergs, Machine learning for advanc- ing low-temperature plasma modeling and simulation, Journal of Mi- cro/Nanopatterning, Materials, and Metrology 22 (4) (2023) 041504– 041504

    2023

  11. [18]

    D. P. Lymberopoulos, D. J. Economou, Fluid simulations of glow dis- charges: Effect of metastable atoms in argon, Journal of applied physics 73 (8) (1993) 3668–3679

    1993

  12. [19]

    K. C. Hall, J. P. Thomas, W. S. Clark, Computation of unsteady nonlin- ear flows in cascades using a harmonic balance technique, AIAA journal 40 (5) (2002) 879–886

    2002

  13. [20]

    K. C. Hall, K. Ekici, J. P. Thomas, E. H. Dowell, Harmonic balance methods applied to computational fluid dynamics problems, Interna- tional Journal of Computational Fluid Dynamics 27 (2) (2013) 52–67

    2013

  14. [21]

    Z. Yan, H. Dai, Q. Wang, S. N. Atluri, Harmonic balancemethods: a review and recent developments (2023)

    2023

  15. [22]

    LaBryer, P

    A. LaBryer, P. Attar, Modeling the nonlinear structural dynamics of a plunging membrane airfoil using a high dimensional harmonic balance approach, in: 50th AIAA/ASME/ASCE/AHS/ASC Structures, Struc- tural Dynamics, and Materials Conference 17th AIAA/ASME/AHS Adaptive Structure...

    2009

  16. [23]

    H. Wu, M. Yang, D. Wang, X. Huang, Efficient forced response mini- mization using a full-viscosity discrete adjoint harmonic balance method, AIAA Journal 62 (10) (2024) 3644–3661

    2024

  17. [24]

    Gopinath, A

    A. Gopinath, A. Jameson, Time spectral method for periodic unsteady computations over two-and three-dimensional bodies, in: 43rd AIAA aerospace sciences meeting and exhibit, 2005, p. 1220. 40

    2005

  18. [26]

    Li, M.-L

    J.-Z. Li, M.-L. Zhao, Y.-R. Zhang, F. Gao, Y.-N. Wang, Fast simu- lation strategy for capacitively-coupled plasmas based on fluid model, Computer Physics Communications 307 (2025) 109392

    2025

  19. [27]

    P. L. Ventzek, T. J. Sommerer, R. J. Hoekstra, M. J. Kushner, Two- dimensional hybrid model of inductively coupled plasma sources for etch- ing, Applied physics letters 63 (5) (1993) 605–607

    1993

  20. [28]

    Sakiyama, D

    Y. Sakiyama, D. B. Graves, Corona-glow transition in the atmospheric pressure rf-excited plasma needle, Journal of Physics D: Applied Physics 39 (16) (2006) 3644–3652. 41

    2006

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.