Nonlinear ultrasonic guided waves are of great importance in structural health monitoring, since they have both the large-area inspection capability of guided waves and the desirable sensitivity to incipient defects from nonlinear ultrasonic features. Efficient numerical methods play a key role in investigating the nonlinear features of guided waves. In this work, a time domain spectral finite element method (TDSFEM) combined with the bi-potential contact theory is developed to capture the nonlinear interactions between guided waves and breathing cracks. The bi-potential approach has better accuracy than the conventional penalty method, since there is no any user-defined parameter for computation in the former method while the latter one requires a user-experience-based penalty factor. In addition, a semi-explicit algorithm is proposed to integrate the wave equation, which can take full advantage of the diagonal mass matrix of TDSFEM while not sacrificing numerical stability. Numerical verifications against ANSYS show that the computational efficiency of the bi-potential method based TDSFEM is much higher than that of the FEM via the penalty method. Numerical case studies are then performed to investigate the nonlinear interactions between Lamb waves and breathing cracks. Results show that the proposed method can successfully capture the classical nonlinear features, such as higher harmonic generation and zero frequency (DC) response. Moreover, the influence of the length and gap of a breathing crack on the contact acoustic nonlinearity (CAN) is investigated. As an efficient numerical approach, the bi-potential method-based TDSFEM may be a good tool for investigating the CAN.

非线性超声导波在结构健康监测中具有重要意义，因为它们既具有导波的大面积检测能力，又对非线性超声特征的初始缺陷具有理想的敏感性。有效的数值方法在研究导波的非线性特征方面起着关键作用。在这项工作中，开发了一种结合双势接触理论的时域谱有限元方法 (TDSFEM) 来捕获导波和呼吸裂纹之间的非线性相互作用。双势方法比传统的惩罚方法具有更好的准确性，因为前一种方法没有任何用户定义的计算参数，而后一种方法需要基于用户体验的惩罚因子。此外，提出了一种对波动方程进行积分的半显式算法，在不牺牲数值稳定性的情况下，充分利用了TDSFEM的对角质量矩阵。ANSYS 数值验证表明，基于双势法的 TDSFEM 的计算效率远高于通过惩罚法的 FEM。然后进行数值案例研究以研究兰姆波和呼吸裂缝之间的非线性相互作用。结果表明，所提出的方法可以成功捕获经典的非线性特征，例如高次谐波产生和零频率（DC）响应。此外，研究了呼吸裂纹的长度和间隙对接触声学非线性 (CAN) 的影响。作为一种有效的数值方法，