广义有限差分法在含阻抗边界空腔声学分析中的应用

APPLICATION OF GENERALIZED FINITE DIFFERENCE METHOD IN ACOUSTIC ANALYSIS OF CAVITY WITH IMPEDANCE BOUNDARY

声学分析在噪声控制、室内隔音等工程计算中有着重要的作用.由于现实生活中的声学模型往往伴随着吸声材料,因此分析含阻抗边界条件的声学问题显得十分必要.广义有限差分法是一种新型区域型无网格数值离散方法,该方法基于多元函数泰勒级数展开式和加权最小二乘拟合,将控制方程中未知参量的各阶偏导数表示为相邻节点函数值的线性组合.本文首次将广义有限差分法应用于含阻抗边界条件空腔声学问题的分析中,建立了空腔声场问题的广义有限差分法数值离散格式.与传统算法相比,所建立的数值模型具有无需网格剖分和数值积分、计算精度高、适用于大规模声学分析等优点.通过具有解析解的经典算例,研究了总节点数目和局部支撑点数目对数值结果的影响,得到了最大计算频率与节点间距之间关系的经验公式.此外,将广义有限差分法应用于无解析解的二维和三维复杂声学模型,并与COMSOL Multiphysics软件所得的有限元结果进行了比较分析.数值实验表明,该算法是一种高效、精确、稳定、收敛的数值模拟方法,在含阻抗边界空腔声学分析中具有广阔的应用前景.

Acoustic analysis plays significant role in engineering calculations such as noise control and indoor sound insulation.Since the practical acoustic problems usually involve sound-absorbing materials,it is very necessary to analyze acoustic problems with impedance boundary conditions.The generalized finite difference method is a new mesh-less numerical discretization method,this method is based on Taylor series expansion of multivariate function and weighted least square,the partial derivatives of unknown values in the governing equation are expressed as a linear combination of function values at supporting nodes.In this paper,the generalized finite difference method is applied to the analysis of cavity acoustics with impedance boundary conditions firstly,and the corresponding numerical discrete scheme is established.Compared with traditional algorithms,the developed numerical model is a local meshless method with the merits of being mathematically simple,numerically accurate and easy to large-scale acoustic analysis.A benchmark numerical example with analytical solution is examined to verify the influence of the total number of nodes and the number of local supporting nodes on the numerical results,and to obtain an empirical formula of the relationship between maximum computable frequency and node spacing.In addition,the generalized finite difference method is applied to two-dimensional and three-dimensional complex acoustic models without analytical solutions,and is compared with the FEM solutions obtained by COMSOL Multiphysics.Numerical experiments demonstrate that the generalized finite difference method is an efficient,accurate,stable and convergent numerical method,and has broad application prospects in the acoustic analysis of cavities with impedance boundaries.