计算小尺度封闭空间内声传递函数的自适应有限元法

A New Adaptive Finite Element Method (AFEM) for Calculating Sound Transfer Function in Small Enclosure

如何求解声传递函数成为实现小尺度封闭空间可听化的关键问题。基于 Helmholtz方程的有限元法能够真实反映出声场内的波动现象 ,成为求解该问题的有效方法。文章基于 SPR法对求解声传递函数的有限元法进行误差估计 ;在此基础上结合 Helmholtz方程求解的误差理论 ,提出了求解该问题的自适应有限元法 ,使用该方法能够预测出在各求解频率段内有限元网格的划分情况 ,从而能够满足预先给定的误差要求 ;算例结果表明本文提出的方法是有效可行的。

Due to several reasons, evaluating the precision of finite element method (FEM) based on Helmholtz equation in calculating sound transfer function (STF) in small enclosure is impossible. We propose a new AFEM that appears to be quite promising for obtaining such evaluation. Our AFEM still, of cause, depends on basic error theory analysis of Helmholtz equation but it is also based on posterior error estimation, which is obtained with the help of superconvergent patch recovery (SPR) technique already existing in theory of elasticity. Section 1 reviews the acoustic FEM model in Ref. 6 by the first author et al that is relevant to the understanding of this paper. Section 2 defines the error of STF very carefully so as to make clear all that should be clearly understood about it. Section 3 summarizes briefly basic error theory analysis of Helmholtz equation. Section 4 reviews SPR technique as given in Refs. 4 and 5 and uses it to obtain the important eq.(18) for calculating superconvergent velocity. Section 5 describes the new AFEM we propose. Section 6 uses our new AFEM to calculate STF in one rectangular enclosure with confidence that it will give results of good precision. Section 6 also compares our AFEM results with the known quite precise results obtained by mode superposition. The comparison shows preliminarily that our new AFEM is effective and feasible in simple-shape enclosure. Our new AFEM appears to be promising for calculating STF with good precision in complex-shape enclosure. Experimental work remains to be done about the effectiveness and feasibility of our new AFEM for calculating STF in complex-shape enclosure.