发表机构
University of Edinburgh; IRCAM; CNRS; Sorbonne Université（爱丁堡大学; 法国声学/音乐研究与协作学院; 法国国家科学研究中心; 索邦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对冯·卡门方程，提出一种在空间域计算乘积项、模态域计算导数的伪谱方法，结合标量辅助变量技术实现显式稳定时间积分，降低了模态合成的计算成本并保留其频率控制优势。
AI 中文摘要
模态合成是模拟乐器动力学的广泛应用技术。在线性情形下，模态分解会得到由阻尼受迫简谐振荡器组成的非耦合系统，可通过标准时间步进法高效求解。但扩展至非线性问题颇具挑战，因控制方程中存在模态展开式的乘积项。对于冯·卡门板，模态间的非线性耦合由四阶张量描述，在模态域计算成本过高。本研究提出一种伪谱方法：在空间域网格上计算乘积项，同时在模态域精确计算空间导数；利用模态域与空间域间的离散正弦变换和离散余弦变换，施加板的简支边界条件。最后，证明系统非线性势能的非负性，并采用标量辅助变量技术，实现模态域的显式稳定时间积分。该方法在保留模态合成对模拟频率范围精确控制等优势的同时，降低了计算成本，文中还给出了声音示例。
英文摘要
Modal synthesis is a widely-used technique for simulation of musical instrument dynamics. In the linear case, a modal decomposition leads to an uncoupled system of damped and forced harmonic oscillators which can be efficiently solved by standard time-stepping methods. However, extensions to nonlinear problems are challenging due to the presence of products of modal expansions in the governing equations. In the case of the Föppl-von Kármán plate, the nonlinear coupling between the modes is described by a fourth-order tensor and is prohibitively expensive to evaluate in the modal domain. In this work, we propose a pseudospectral method in which the products are evaluated on a grid in the spatial domain while spatial derivatives are computed exactly in the modal domain. Discrete sine and cosine transforms between the modal and spatial domains are used to impose simply supported boundary conditions for the plate. Finally, we prove non-negativity of the nonlinear potential energy of the system and employ a scalar auxiliary variable technique for explicit and stable time integration in the modal domain. As a result, we reduce the computational cost of modal synthesis while preserving its advantages like a precise control over the simulated frequency range. Sound examples are presented.
CommentsTo be presented at Forum Acusticum 2026, Graz, Austria, September 2026
