提出一种高效稳定的非线性板振动模拟方法，解决传统模态合成计算昂贵问题。
Explicit and Stable Pseudospectral Time-Domain Method for the Föppl-von Kármán Equations
- 在模态域计算导数，在空间域求解非线性项，结合快速正弦/余弦变换处理边界条件
- 证明非线性势能恒正，采用标量辅助变量法实现模态域显式稳定时间积分
- 显著降低计算开销，适合音乐乐器动态模拟中对频响精度要求高的场景
模态合成是模拟乐器动力学的常用技术。线性情况下，模态分解可得到一组解耦的阻尼受迫谐振子，可用标准时间步进法高效求解。然而，扩展到非线性问题时，控制方程中出现模态展开的乘积，带来挑战。以Föppl-von Kármán板为例，模态间的非线性耦合由四阶张量描述，直接在模态域求解成本过高。本文提出一种伪谱方法：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。利用离散正弦与余弦变换实现简支边界条件。最后，证明系统非线性势能始终非负，并采用标量辅助变量技术，在模态域实现显式且稳定的时域积分。该方法在保持模态合成优势（如精确频带控制）的同时，大幅降低计算成本。提供了声音示例。
原文摘要 · Abstract (English)
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.
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