水下大规模结构声固耦合问题的广义有限差分-快速多极奇异边界法研究

Prediction and control of the vibration and noise are greatly significant for coupled acoustic-structural analysis of underwater large-scale structures, which has wide applications in the defense industry and civilian fields such as subsea pipelines, submarines, ships and naval vessels, etc. In the numerical study, traditional finite element and boundary element methods encounter great challenges in high-quality mesh generation, large number of element integrations, and efficient iterative solution of the system equations for resolving the complicated acoustic-structural interaction model. The research goal of this proposal is to develop a novel fast meshless collocation method for coupled acoustic-structural problems of underwater large-scale structures. We address existing challenges by introducing advanced numerical techniques including the generalized finite difference method (GFDM), fast multipole singular boundary method (FMSBM), and matrix preconditioning strategies. This research project consists of three phases: First, we will investigate adaptive selection of nodes, and develop the meshless GFDM model to predict vibration response of large-scale structures. Next, we will derive the wideband FMSBM formulations for exterior acoustic problems based on the partial wave and plane wave expansions of the fundamental solution. Finally, we will combine the GFDM and the wideband FMSBM, and investigate a class of preconditioners to accelerate the convergence of iterative solvers for the coupled system of the acoustic-structural problem. The purpose of this proposal is to develop a highly efficient numerical prediction model with no mesh generation and integration for analysis of coupled acoustic-structural systems of underwater large-scale structures.

在声固耦合分析中，水下大规模结构的振动噪声预报和控制具有重要的研究意义，广泛应用于海底管道、潜艇、舰船等军工和民用领域。传统的有限元和边界元方法在数值模拟复杂结构声固耦合系统时，遇到高质量网格生成、数值积分计算量过高、耦合系统方程迭代求解效率低等困难。本项目基于广义有限差分法、快速多极奇异边界法和矩阵预处理技术，拟发展水下大规模结构声固耦合分析的一种新型快速无网格配点法。首先，研究自适应选点技术，建立大规模结构振动响应分析的无网格广义有限差分法计算模型；其次，基于基本解的分波和平面波展开式，推导无限域声场问题的宽频快速多极奇异边界法公式；最后，基于广义有限差分法和宽频快速多极奇异边界法，建立声固耦合快速无网格配点法，并研究矩阵预处理技术，提高耦合系统方程的迭代求解效率。目标是建立水下大规模结构声固耦合分析的一种无网格、无数值积分的高效仿真模型。

发展水下大规模结构声固耦合问题的高性能数值算法，有利于潜艇、水下自主航行器等设备振动噪声的精准预报和控制。本项目以广义有限差分法和奇异边界法为研究核心，建立了声固耦合分析的一种新型快速无网格配点法。主要的结果如下：1）提出了“二十区域准则”的自适应选点技术，提升了非均匀布点情形下无网格广义有限差分法的精度和稳定性；2）在计算域的边界上引入“辅助节点”，发展了求解薄板弯曲振动问题的改进型广义有限差分法，避免了传统方法离散后形成的超定方程组问题；3）基于低频区域的基本解分波展开式和高频区域的平面波展开式，构造了无限域声场问题的宽频快速多极奇异边界法公式；4）以声压为未知变量，并利用结构面边界条件的连续性，耦合了广义有限差分法和宽频快速多极奇异边界法，建立了声固耦合分析的新型快速无网格配点法。数值结果表明，发展的快速无网格配点法精确、高效且稳定，实现了具有数百万自由度的大规模结构声振特性的快速分析，为水下复杂装备的振动和噪声控制提供了数值依据。