Over the last 15 years, literature on nondestructive testing has shown that the generation of higher harmonics and nonlinear mixing of waves could be used to obtain the nonlinearity parameters of an elastic medium and thereby gather information about its state, e.g., aging and fatigue. To design ultrasound measurement setups based on these phenomena, efficient numerical modeling tools are needed. In this paper, the iterative nonlinear contrast source method for numerical modeling of nonlinear acoustic waves is extended to the one-dimensional elastic case. In particular, nonlinear mixing of two collinear bulk waves (one compressional, one shear) in a homogeneous, isotropic medium is considered, taking into account its third-order elastic constants ( A,  B,  and  C). The obtained results for nonlinear propagation are in good agreement with a benchmark solution based on the modified Burgers equation. The results for the resonant waves that are caused by the one-way and two-way mixing of primary waves are in quantitative agreement with the results in the literature [Chen, Tang, Zhao, Jacobs, and Qu, J. Acoust. Soc. Am. 136(5), 2389-2404 (2014)]. The contrast source approach allows the identification of the propagating and evanescent components of the scattered wavefield in the wavenumber-frequency domain, which provides physical insight into the mixing process and explains the propagation direction of the resonant wave.@en

In nondestructive testing, nonlinear wave mixing could be used to obtain the nonlinearity parameters of an elastic medium and thereby get information about its state, e.g., aging and fatigue. To better understand the mixing mechanisms and optimize the design of measurement setups, a physics-oriented tool for the simulation of nonlinear elastic wave propagation would be valuable. In this presentation, we extend the iterative nonlinear contrast source method (INCS) to study the nonlinear mixing of two plane, collinear bulk waves (one compressional, one shear) in a homogeneous, isotropic, elastic medium with two independent coefficients of nonlinearity (βL and βT). The method successfully captured the resonant wave generated due to the mixing (one-way and two-way) of primary waves of different frequencies. The obtained results for the resonant wave were in good agreement with the results reported in the literature. In addition, the contrast source allowed the propagating and evanescent components of the scattered wave field to be studied in the wavenumber-frequency domain, which provides physical insight into the mixing process and explains the propagation direction of the scattered wave. Thus, the INCS method seems to be a useful tool to investigate and predict wave mixing in nonlinear elastic media.@en

There are many applications in which the properties of nonlinear waves play a role. For example, in medical diagnostic imaging, the higher harmonics in nonlinear ultrasound are used to reduce clutter from nearby artifacts and to improve resolution, and in non-destructive testing nonlinear mixing products from crossing ultrasound beams are used to assess the state of materials. Simulation of nonlinear waves is important for optimizing equipment and quantitatively assessing phenomena. Unfortunately, analytical approaches cannot deal with complex realistic situations, while traditional Finite Element and Finite Difference methods require extremely large grids to capture waves with many higher harmonics in large three-dimensional domains. The Iterative Nonlinear Contrast Source method was developed to overcome these issues. It is based on an integral equation involving the analytic Green’s function of the linearized medium and a distributed contrast source representing nonlinearity. The integral equation is solved iteratively, and the computations are based on Fast Fourier Transforms that only require two grid points of the shortest wavelength. In this presentation, the basic principles behind the method will be explained and its extension to lossy and inhomogeneous media will be shown. Moreover, a novel extension will be presented that enables the computation of nonlinear elastic waves.@en