A convolution quadrature method for acoustic time domain boundary element method
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摘要:
核函数中保留Dirac函数的原型, 形成关于时间的卷积积分, 是声学时域边界元法中一种稳定、有效的时间数值积分计算方法(CQ-BEM)。然而, 传统CQ-BEM中卷积积分系数的获取有计算量大、耗时长, 且对不同单元需要重新计算的问题, 极大地降低了CQ-BEM法计算时域声场的效率。针对传统CQ-BEM积分系数计算效率低的问题, 本文利用多项式展开定理给出了待求函数泰勒系数的解析表达与数值计算方法, 建立了不同单元间待求系数的转换理论, 可以在一次循环迭代内完成不同单元的积分系数的计算, 大幅降低了计算量, 提高了CQ-BEM方法的声场计算效率。脉动球源数值算例结果表明, 在相同要求下, 本文方法计算时间较传统方法减少50%以上, 相对误差小5个数量级以上, 且计算时间随单元数的增长率仅为传统方法的2.34%。因此, 本文提出的系数计算方法能够有效提高CQ-BEM方法的时域声场计算效率, 拓展了CQ-BEM在大型机电设备时域声场模拟的计算规模。
Abstract:The convolution quadrature based boundary element method (CQ-BEM) can preserve the kernel Dirac function when dealing with integral in time domain, forming a time convolution, which is a stable and effective numerical method in the time domain acoustic boundary element method. However, the traditional way to acquire the convolution integral coefficients in CQ-BEM requires large computation, long calculating time, and recalculation for different elements, which greatly reduces the efficiency of calculating the sound field in time domain. In order to solve the problem, this paper proposes the analytical expression and the numerical method of the coefficients based on polynomial theorem. The conversion theory of the coefficients between different elements is established so that the coefficients of different elements can be acquired in a single calculation, which greatly reduces the calculation amount. The sound field calculation efficiency of CQ-BEM method is then improved. In the numerical example of pulsating spherical source, the calculation time, relative error and growth rate of calculation time with the number of elements of traditional method and proposed method are compared. The results show that, under the same requirements, the calculation time of this method is more than 50% less than the traditional method, the relative error is more than 5 orders of magnitude less, and the growth rate of calculation time with the number of elements is only 2.34% of the traditional method, which can effectively improve the computational efficiency of CQ-BEM method in time domain sound field.
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表 1 截断项数 − 阶数(L−n)二次曲线拟合参数
相对误差 $a$ $b$ $c$ ${10^{ - 2}}$ −3.24408 × 10−3 2.75610 × 10−1 5.60391 × 100 ${10^{ - 4}}$ −3.39617 × 10−3 2.98290 × 10−1 8.42215 × 100 ${10^{ - 6}}$ −3.45414 × 10−3 3.11518 × 10−1 1.08118 × 101 ${10^{ - 8}}$ −3.47816 × 10−3 3.20749 × 10−1 1.29764 × 101 ${10^{ - 10}}$ −3.48680 × 10−3 3.27784 × 10−1 1.49946 × 101 
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表 2 卷积系数
${\alpha _n}$ 关于重复计算次数x的线性拟合参数${k_1}$ ${b_1}$ 传统方法 1.73953×10−3 6.70258×10−3 本文方法 3.97174×10−5 1.95126×10−1
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表 3 计算时间−迭代步数一次曲线拟合参数
${k_2}$ ${b_2}$ 传统方法 5.20367×10−1 2.40009×103 本文方法 5.60708×10−1 4.86759×102
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