Adaptive harmonic balance method for nonlinear time-periodic flows

Maple, Raymond C.; King, Paul I.; Orkwis, Paul D.; Wolff, J. Mitch

doi:10.1016/j.jcp.2003.08.013

Cited by 34 publications

(20 citation statements)

References 30 publications

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“…The harmonic balance method employed 24 harmonics which produce 49 (=2 × 24 + 1) time instances, and the present method of the Chebyshev pseudospectral approach uses 48 Chebyshev points in the time interval for consistent comparison. The solutions of the inviscid Burgers' equation tend to have discontinuities with higher frequency values as shown in Fig.8, and the number of the discontinuous peaks is increasing proportional to The results from present method are in very good agreements with those from the harmonic balance method and Maple's results 30 in the location and number of the discontinuities (Fig.8).…”

Section: Numerical Case Studiessupporting

confidence: 64%

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A Mapped Chebyshev Pseudospectral Method for Unsteady Flow Analysis

Choi

McClure

et al. 2015

22nd AIAA Computational Fluid Dynamics Conference

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A mapped Chebyshev pseudospectral method is developed as an accurate and yet efficient approach to solve unsteady flows. Preserving the conservation laws, the method discretizes a spatial derivative term implicitly while a time derivative term is treated explicitly using the mapped Chebyshev collocation operator. As standard Chebyshev points make the corresponding spectral derivative matrix ill-conditioned due to uneven distribution of the Chebyshev points clustered heavily towards the ends of the interval, an inverse sine mapping function is applied to the Chebyshev collocation operator so that the point distribution becomes more uniform in a time domains and mitigates numerical instabilities correspondingly. Computations of unsteady flows, including onedimensional Burgers' equation and two-dimensional airfoil flows under oscillation and plunging motions, are carried out. Numerical results of the present study are compared with those of the conventional time-marching solution method and the harmonic balance method for each application, and demonstrate high accuracy at much reduced computational cost. The method shows great potential to solve general unsteady flows involving both periodic and non-periodic motions. Nomenclature A = amplitude of oscillation a = Chebyshev coefficient b = Chebyshev coefficient for first differential D = matrix in harmonic balance equation D ch = Chebyshev collocation matrix D T = Chebyshev coefficient matrix D T = Chebyshev coefficient matrix for first differential F = Fourier transform matix G, H = fluxes g = a mapping funtion g' = derivative of a mapping function k c = reduced frequency M = matrix in frequency domain equation M ch = number of reconstruction points by Chebyshev M ∞ = freestream Mach number N hs = number of harmonics p, q = constant of polynomial mapping function Q = conservative variables R = residual T = Chebyshev polynomial T max = time interval t = time V, W = flux Jacobian matrixs in mapped Chebyshev method y = variable of a mapping function α, β = constant of inverse sine mapping function δ, ε = constant of hyperbolic sine mapping function φ = frequency ρ, σ = constant of tangent mapping function

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Section: Numerical Case Studiessupporting

confidence: 64%

“…The results from the harmonic balance method are compared with the current results 27,30 . The artificial damping term is added to reduce the oscillation of the solution of the inviscid Burgers' equation.…”

Section: Numerical Case Studiesmentioning

confidence: 99%

A Mapped Chebyshev Pseudospectral Method for Unsteady Flow Analysis

Choi

McClure

et al. 2015

22nd AIAA Computational Fluid Dynamics Conference

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“…Maple et al proposed an adaptive harmonic balance method for nonlinear time-periodic flows which results in a significant reduction in computational costs [13].…”

Section: Introductionmentioning

confidence: 99%

Optimization of nonlinear structural resonance using the incremental harmonic balance method

Dou

Jensen

2015

Journal of Sound and Vibration

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We present an optimization procedure for tailoring the nonlinear structural resonant response with time-harmonic loads. A nonlinear finite element method is used for modeling beam structures with a geometric nonlinearity and the incremental harmonic balance method is applied for accurate nonlinear vibration analysis. An optimization procedure based on a gradient-based algorithm is developed and we use the adjoint method for efficient computation of design sensitivities. We consider several examples in which we find optimized beam width distributions that minimize or maximize fundamental or super-harmonic resonant responses.

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“…An adaptive version of the method was developed by Maple [50], in which the number of harmonics vary throughout the domain proportionally to the local level of unsteadiness of the flow. As flows often contain both regions of low and high unsteadiness, this advancement allows for greater computational savings in regions where only a few harmonics are necessary.…”

Section: Frequency-domain Solution Methodsmentioning

confidence: 99%