Université Pierre et Marie Curie - Paris 6, Institut Jean le Rond d'Alembert (UMR UPMC/CNRS 7190), 4 place Jussieu, 75252 Paris Cedex 05, France.
J Acoust Soc Am. 2011 Jul;130(1):20-32. doi: 10.1121/1.3583549.
Numerical simulation of nonlinear acoustics and shock waves in a weakly heterogeneous and lossless medium is considered. The wave equation is formulated so as to separate homogeneous diffraction, heterogeneous effects, and nonlinearities. A numerical method called heterogeneous one-way approximation for resolution of diffraction (HOWARD) is developed, that solves the homogeneous part of the equation in the spectral domain (both in time and space) through a one-way approximation neglecting backscattering. A second-order parabolic approximation is performed but only on the small, heterogeneous part. So the resulting equation is more precise than the usual standard or wide-angle parabolic approximation. It has the same dispersion equation as the exact wave equation for all forward propagating waves, including evanescent waves. Finally, nonlinear terms are treated through an analytical, shock-fitting method. Several validation tests are performed through comparisons with analytical solutions in the linear case and outputs of the standard or wide-angle parabolic approximation in the nonlinear case. Numerical convergence tests and physical analysis are finally performed in the fully heterogeneous and nonlinear case of shock wave focusing through an acoustical lens.
考虑了在弱非均匀和无损耗介质中非线性声学和冲击波的数值模拟。波动方程的构建方式可以将均匀衍射、非均匀效应和非线性分开。开发了一种称为用于解决衍射的非均匀单通近似（HOWARD）的数值方法，该方法通过忽略反向散射的单通近似在谱域（时间和空间）中求解方程的均匀部分。进行了二阶抛物近似，但仅在小的非均匀部分进行。因此，与通常的标准或广角抛物近似相比，得到的方程更精确。它与所有正向传播波（包括渐逝波）的精确波动方程具有相同的频散方程。最后，通过分析、冲击拟合方法处理非线性项。通过与线性情况下的解析解以及非线性情况下的标准或广角抛物近似的输出进行比较，进行了几个验证测试。最后，在通过声学透镜聚焦冲击波的完全非均匀和非线性情况下进行了数值收敛测试和物理分析。
