EI / SCOPUS / CSCD 收录

中文核心期刊

时域声学问题的有限线法分析

Finite line method for acoustic problems in time domain

  • 摘要: 采用一种新型数值算法, 即有限线法, 求解时域声学问题。有限线法是一种配点型的强形式算法, 在该方法中, 计算域被离散为一系列配置点, 每个配置点处形成一组穿过该点的直线或曲线, 即交叉线系。每条线的形函数使用拉格朗日插值多项式构造, 通过采用沿弧长求方向导数的方式创建任意物理量对总体坐标的一阶偏导数, 再通过递推技术, 由一阶偏导数公式建立二阶偏导数计算式。导出的偏导数可直接代入到声学问题的控制微分方程及相应的边界条件之中, 从而建立离散的系统方程组。有限线法作为一种强形式的数值方法, 不需要用变分原理或能量原理来建立计算格式, 整个求解过程不需要进行积分。此外, 有限线法在计算过程中生成的系数矩阵往往是高度稀疏的, 这使得其具有很高的计算效率。引入了Newmark差分技术对时间域进行离散, 求解包含时间项的方程组; 对于用狄拉克函数表示的点声源, 提出了一种声源密度函数来近似点声源的作用, 使其能够运用到有限线法中。给出四个数值算例, 验证了该方法的正确性与应用潜力。

     

    Abstract: A new numerical method called the finite line method (FLM), is used to solve time-domain acoustic problems in this paper. The FLM is a strong-form numerical method. In the method, the computational domain is discretized into a series of collocation points, and a set of lines or curves passing through each collocation point is formed, which is called line-set. The shape function of each line is constructed using a Lagrange interpolation polynomial, and the first-order partial derivative of any physical quantity with respect to the global coordinates is created by calculating the directional derivative along the arc length of each line. The second-order partial derivative is then calculated using a recursive technique. The derived partial derivatives can be directly substituted into the governing differential equations and the corresponding boundary conditions of the acoustic problem, so as to establish the discrete system equations. As a strong-form numerical method, the FLM does not need to use the variational principle or energy principle to establish the calculation scheme, and the whole solution process does not require integration. In addition, the coefficient matrix generated by the FLM is highly sparse, making it computationally efficient. This paper introduces the Newmark difference technique to discretize the time domain and solve the system of equations containing time terms. For a point sound source represented by Dirac function, a source density function is proposed to approximate the effect of the point sound source, so that it can be applied to the FLM. Four numerical examples are given to verify the correctness and application potential of the proposed method.

     

/

返回文章
返回