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Dealiasing harmonic balance method for obtaining periodic solutions of an aeroelastic system

机译：用于获得气动弹性系统周期解的去混和谐波平衡方法

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摘要

The harmonic balance method with a dealiasing scheme, referred to as DHB method, is proposed to obtain the semi -analytical periodic solutions of a two dimensional subsonic airfoil. The equations of motion of this autonomous system are a set of integro-differential equations. To solve this problem, the original model is first transformed into a system of nonlinear ordinary differential equations by means of integral transformations, and consequently this system is transformed into nonlinear algebraic equations (NAEs) via the harmonic balance method. Previous study demonstrated that the harmonic balance method may yield mathematically aliased solutions. To remedy this drawback, a dealiasing scheme, based on shifting the high order harmonic coefficients to the correct low order positions and properly scaling the frequency, is proposed to effectively suppress mathematical aliasing. As for solving the resultant NAEs, a series of scalar homotopy methods (SHMs) are introduced. The SHM methods are robust to initial conditions, and do not require the calculation of the inverse of Jacobian matrix. Therefore, the computing cost can be reduced and the inaccuracy arising from inverting the ill-conditioned Jacobian can be eliminated. Finally, numerical examples are carried out to verify the efficiency and accuracy of the present method. (C) 2018 Elsevier Masson SAS. All rights reserved.
机译：为了获得二维亚音速机翼的半解析周期解，提出了一种带有脱氮方案的谐波平衡法，称为DHB法。该自治系统的运动方程是一组积分微分方程。为了解决这个问题，首先通过积分变换将原始模型转换为非线性常微分方程组，然后通过谐波平衡法将该系统转换为非线性代数方程组（NAE）。先前的研究表明，谐波平衡方法可以产生数学上混叠的解决方案。为了弥补这一缺点，提出了一种基于将高次谐波系数移至正确的低阶位置并适当缩放频率的去磁方案，以有效地抑制数学混叠。为了解决所得的NAE，引入了一系列标量同伦方法（SHM）。 SHM方法对初始条件具有鲁棒性，不需要计算雅可比矩阵的逆矩阵。因此，可以降低计算成本，并且可以消除因病态雅可比矩阵求逆而引起的不准确性。最后，通过数值算例验证了本方法的有效性和准确性。 （C）2018 Elsevier Masson SAS。版权所有。

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