Study of Numerical Methods for Wave Propagation Phenomena in Unbounded Region and its Applications

本研究的目的是发展波在无界区域传播问题的近似方法，并作为其应用，研究声音产生的数值模拟。我们将问题表述为外部亥姆霍兹方程，并在人工边界上引入人工边界条件，将问题简化为有界区域内的问题。在有限元离散化方法的基础上，提出了求解该问题的数值方法，并对语音生成等应用问题进行了研究。首席调查员Kako的调查结果如下。他找到了声道变形的复特征值的变分公式。特征值与频率响应函数的共振峰有关，共振峰对声音的产生至关重要。然后，他利用变分公式开发了设计声道形状的算法，并通过更多的数值模拟验证了该算法。他还研究了时域有限差分法在声学问题中的应用，并获得了几个基本结果。针对语音问题，吉田开发了利用神经网络获得发音参数到语音传递特征映射的方法。Suito研究了具有空间变化阻抗参数的管状区域中波传播反射最小化的形状优化问题，并获得了不同寻常的数值结果。与波动问题的数值方法相关，Koyama利用虚拟域方法研究了三维亥姆霍兹问题，并推导了近似的先验误差估计，并通过一些数值经验验证了其有效性。Ushijima采用基于基本解的配点法研究了Helmholtz问题，得到了近似解指数收敛的充分条件，并试图通过多精度算术计算验证理论结果。对于包括亥姆霍兹方程在内的应用问题中出现的求解大型线性方程的数值方法，Zhang研究了快速有效的迭代方法。Imamura在实现大型线性系统的并行计算方法时，开发了具有高精算性和稳定性的自动调谐技术。少

The purpose of this research project is to develop the approximation methods for wave propagation problems in unbounded region and as its applications we study the numerical simulation of voice generation. We formulate the problem as the exterior Helmholtz equation, and reduce the problem to the one in a bounded region by introducing the artificial boundary condition on an artificial boundary. We developed the numerical methods for this problem based on the finite element discretization method and study the application problems including the voice generation.The results of the head investigator Kako are the followings. He found out the variational formula of the complex eigenvalues with respect to the deformation of vocal tract. The eigenvalues are related to the formants of frequency response function that is important for voice generation. He then developed the algorithm for designing the shape of vocal tract by use of the variational formula, and validated the algorithm through nume … More rical simulations. He also studied the application of the Finite Difference Time Domain method to the acoustic problem and obtained several basic results.For the voice problem, Yoshida developed the method to obtain the mapping from the articulation parameters to the phonetic transmission characteristics by use of the neural networks. Suito studied the shape optimization problem for minimizing the reflection of the wave propagating in a tubular region with spatially changing impedance parameters and obtained unusual numerical results.Related to the numerical methods for the wave problem, Koyama studied the three dimensional Helmholtz problem by use of the fictitious domain method and derived the a priori error estimates for the approximation, and investigated the validity by some numerical experiences. Ushijima studied the Helmholtz problem by the collocation method based on the fundamental solutions and obtained a sufficient condition for the exponential convergence of the approximate solution and tried to validate of the theoretical results by multi-precision arithmetic computation.As for the numerical methods for solving large linear equations appearing in the application problems including the Helmholtz equation, Zhang studied the fast and efficient iteration methods. Imamura developed the automatic tuning techniques with high actuary and stability in the implementation for the parallel computation methods for the large linear systems. Less