三维无限域高频声场模拟的快速奇异边界法研究

Numerical modeling of infinite acoustic fields has extensive and important engineering applications, such as ultrasonic, environmental noise, underwater sonar, and so on. The traditional numerical techniques for these problems encounters bottleneck problems, such as a exponential growth of computing costs with increasing dimensionality and frequency, expensive numerical integration, the fundamental solution in inhomogeneous and heterogeneous media, mesh generation for high-dimensional complex-shaped objects. Consequently, it is computationally a very challenging task to simulate three dimensional (3D) complex problems of high frequency. In addition, the finite element method also demands a troublesome truncated artificial boundary. The singular boundary method (SBM) proposed recently by my group uses the fundamental solution as the interpolation basis function which satisfies the acoustic governing equation and infinite boundary condition. The method requires boundary-only discretization on the surface of complex-shaped three-dimensional computational domains and do not need mesh and numerical integration and is a mathematically simple, easy-to-program, computationally efficient semi-analytical numerical technique. This proposal is aimed at further tackling of critical problems facing this method to simulate large-scale high frequency 3D infinite acoustic fields in complex medium. We will introduce the new algorithms for the fast solution of dense matrix equation to reduce CPU time from O(N^2) in the standard SBM to O(N) of magnitude in the fast SBM and storage requirements from O(N^2) to O(NlogN). And then we will construct the generalized distances underlying the properties of inhomogeneous and heterogeneous media and extend the SBM to acoustic cloak models. The overall purpose of this proposed project is to develop a novel technique which can simulate high frequency infinite acoustic fields in 3D complex medium with numerically stable, highly accurate and efficient, meshless performances.

无限域声场的数值模拟有着广泛而重要的应用。如超声波、环境噪声、声纳等。传统的数值方法在模拟这类问题时遇到计算量随维数频率呈指数增长、数值积分计算量大、复杂介质的基本解构造和复杂域的网格生成等瓶颈问题，难以计算高频三维复杂问题。此外，有限元法还存在构造人工截断边界的问题。我们最近提出的奇异边界法的基本解插值基函数满足声场无穷远边界条件和控制方程，三维复杂计算域仅需表面边界离散，无网格生成和数值积分，是一个数学和编程简单、计算高效的半解析算法。本项目拟进一步解决该方法计算大规模三维复杂无限域高频声场的关键技术难题，结合快速计算稠密矩阵方程的新算法，将计算量和存储量分别减少到O(N)和O(NlogN)量级；构造刻画各向异性和不均匀介质特性的广义距离，应用于数值检验声学斗篷模型。目标是发展数值稳定、高精度、无网格、高效率地计算三维复杂介质无限域高频声场的新型方法。

无限域声场的数值模拟在超声波、环境噪声等领域存在着非常重要的作用。传统的数值方法在模拟此类问题时常遇到计算量随着维数和频率增加、奇异积分计算困难、复杂介质基本解构造困难、以及无限域人工边界选取等难题。本项目采用奇异边界法，一种新型的无网格方法数值研究三维无限域高频声场问题。与边界元法类似，奇异边界法采用控制方程的基本解作为基函数，因此可以将问题降一维求解，并且可以直接适用于无限域声场问题的模拟。为了避免基本解的源点奇异性，奇异边界法引入源点强度因子的概念。.本项目采用奇异边界法数值模拟三维无限域高频声场，主要研究内容包括以下几个方面：（1）研究奇异边界法的算法理论，提出能够稳定求解源点强度因子的新技术，发展三维无限域声场模拟的高精度奇异边界法模型；（2）研究奇异边界法模拟三维无限域高频声场所涉及的矩阵稀疏化技术、预调节技术和自适应技术，基于低频快速多极算法和高频快速多极算法，建立低频快速多极奇异边界法以及高频快速多子极奇异边界法模型；（3）在高性能计算机上布置快速奇异边界法程序，模拟仿真上亿节点的无限域高频声场问题，优化程序和算法。.针对源点强度因子的高精度求解，项目组基于积分平均技术，并与传统数值反插值技术以及解析技术进行比较，提出了一类能够精确求解二维及三维声学问题基本解源点强度因子的新技术，将其应用于二维、三维任意边界条件下的无限域声学问题，进行精度、稳定性、和收敛性分析。.数值模拟三维无限域高频声场问题时，计算时间随着频率增加显著增加。本项目发展了基于低频快速多极子算法以及高频快速多极子算法的快速多极奇异边界法，并引入自适应技术以及矩阵预调节技术，极大降低了计算时间，提高计算效率。.另一方面，本项目将发展的快速多极子奇异边界法布置在高性能计算机上，并对算法和程序进行进一步优化，成功模拟了上亿节点的超大规模无限域高频声场问题，开发并完善了程序软件包，申请了三项软件著作权。