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New numerical iterative method for nonlinear wave equations

Cao Na, Chen Shi, Cao Hui, Wang Cheng-Hui, Liu Hang
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• 摘要

提出了一种新的求解非线性波动方程的数值迭代法, 它是一种半解析的方法. 与完全的数值计算方法(如有限元、有限差分法)相比, 这种迭代法的解具有非常清晰的物理含义, 即它的解是各阶谐波的组合. 与微扰法相比, 它能够考虑各阶谐波的相互作用, 且能够满足能量守恒定律. 用它研究了非线性声波在液体中的传播性质, 结果表明, 在微扰法适用的声强范围内迭代法也适用, 在微扰法不适用的一个较宽的声强范围内迭代法依然适用.

Abstract

Nonlinear acoustics is an important branch of acoustics and has important applications in some areas, such as high-intensity focused ultrasound, ultrasonic suspension, acoustic cavitation, acoustic harmonic imaging, and parametric emission array. The solving of nonlinear equations in these fields is very important. Regarding the solution of the wave equation of a nonlinear acoustic system, the methods used at this stage generally include complete numerical calculation method, strict analytical method, and perturbation method. 1) For the complete numerical calculation method, it covers the finite element method and the finite difference method. The physical meaning of the solution obtained by this kind of method is not clear, and it is difficult to reveal the physical nature of nonlinear event. And in many cases it will lead to the numerical divergence problems, and it is not suitable for all nonlinear problems. 2) For the strict analytical method, it can only deal with nonlinear acoustic problems of very few systems, such as the propagation of nonlinear acoustic waves in an ideal fluid. 3) For perturbation method, its advantage is that the method is simple and the physical meaning of the solution is clear, but it is only suitable for dealing with nonlinear effects at low sound intensity. And it takes into consideration only the effect of low-order harmonics on higher-order harmonics, with ignoring its reaction, so it does not satisfy the law of conservation of energy.In this paper, we propose a new, semi-analytical numerical iterative method of solving nonlinear wave equations. It is a form of expanding the sound field into a Fourier series in the frequency domain, realizing the separation of time variables from space coordinates. Then, according to the specific requirements for the calculation accuracy, the high frequency harmonics are cut off to solve the equation. Compared with the results from the complete numerical methods (such as finite element method and finite difference method), the solution from this iterative method has a very clear physical meaning. That is, its solution is a combination of harmonics of all orders. Compared with the perturbation method, it can consider the interaction of various harmonics and can satisfy the law of conservation of energy (provided that the system has no dissipation). It is used to study the propagation properties of nonlinear acoustic waves in liquids. The results show that the iterative method is also applicable in the range of sound intensity where the perturbation method is applicable. In a wide range of sound intensity where the perturbation method is unapplicable, the iterative method is still applicable and satisfies the law of conservation of energy (provided that the system has no dissipation). It is unapplicable only if the sound intensity is extremely loud and strong. And when more high-order harmonics are involved, the calculation time by using the numerical method proposed in this paper does not increase sharply.

作者及机构信息

通信作者: 陈时, chenshi@snnu.edu.cn ; 曹辉, caohui@snnu.edu.cn ;
• 基金项目: 国家自然科学基金(批准号: 11974232, 11374199, 11074159)资助的课题

Authors and contacts

Corresponding author: Chen Shi, chenshi@snnu.edu.cn ; Cao Hui, caohui@snnu.edu.cn ;
• Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11974232, 11374199, 11074159)

施引文献

• 图 1  能量守恒的破坏程度、二次谐波的相对能流和三次谐波的相对能流随入射波能流的变化

Fig. 1.  Damage degree of energy conservation, the relative energy flow of the second harmonic, and the relative energy flow of the third harmonic with incident wave energy flow.

图 2  各阶谐波的相对能流随入射波能流的变化

Fig. 2.  Relation between relative energy of each order of harmonics and incident wave energy.

图 3  入射波能流不同时二阶谐波的相对能流随迭代次数的变化

Fig. 3.  Relative energy flow of the second harmonic varies with the number of iterations under different incident wave energy flow.

图 4  黏度不同的情况下, 各阶谐波的能流随入射波能流的变化(图中曲线的黏度分别是$1 \times {10^{ - 6}}$, $6 \times {10^{ - 1}}$$10 \times {10^{ - 1}} Pa/s, 箭头表示黏度减小的方向) Fig. 4. Relation of relative energy flow of each order of harmonics with incident wave energy flow under different viscosity. Viscosites for different curves are 1 \times {10^{ - 6}}, 6 \times {10^{ - 1}}, 10 \times {10^{ - 1}} Pa/s, respectively. Arrows indicate the direction of decreasing viscosity. 图 5 黏度不同的情况下, 各阶谐波的相对能流随角频率的变化(图中曲线的黏度分别是1 \times {10^{ - 6}}$$1 \times {10^{ - 1}}$ Pa/s, 箭头表示黏度减小的方向)

Fig. 5.  Relative energy flow varies with angular frequency under different visco-sity. Viscosites for different curves are $1 \times {10^{ - 6}}$ and $1 \times {10^{ - 1}}$ Pa/s, respectively. Arrows indicate the direction of decreasing viscosity.