A COMSOL Multiphysics Methodology for Array-Based Noncollinear Ultrasonic Wave Mixing
Noncollinear ultrasonic wave mixing is a powerful nondestructive evaluation (NDE) technique for localizing microscale material damage, as the amplitude of the generated mixed wave is a direct measure of localized material nonlinearity. Recent experimental work combined this technique with phased array technology, using transducers on angled waveguides to steer incident shear-wave beams that mix into a longitudinal wave at a targeted interior location [1]. COMSOL's Nonlinear Structural Materials Module captures large-strain, geometric, and hyperelastic nonlinearity, but not the third-order-elastic-constant (acoustic) material nonlinearity that generates the second harmonics and mixed waves used in NDE. This work presents a COMSOL methodology for array-based noncollinear wave mixing, including mode conversion, beam steering, and damage localization. The required material nonlinearity is introduced through user-defined physics: the elastodynamic constitutive relation is augmented with quadratic, third-order-elastic-constant terms, entered as equation-based modeling expressions in the General Form PDE interface and equivalently through an external material called by the COMSOL Java API. Built on the Structural Mechanics and Acoustics Modules with a Time Dependent study, the model uses perfectly matched layers, mode-conversion wedge geometry, and parametric sweeps over array delay laws, so a single model file scales seamlessly from one element to an arbitrary phased array. Validation is embedded directly in the workflow. Heat-treated aluminum—a homogeneous, well-documented reference with literature-based third-order elastic constants—gives the recovered nonlinearity a known target. A three-dimensional model first generates array-driven Rayleigh surface waves, where second harmonic generation provides a clean check that the implemented nonlinearity and the refined mesh (resolving higher-frequency harmonics and mixed waves) reproduce the expected quadratic amplitude growth before any mixing physics is introduced. With the material model anchored, the two-dimensional plane-strain case follows: longitudinal excitation is mode-converted across a wedge by Snell's law into in-plane shear beams that intersect at the resonance angle to generate a longitudinal mixed wave, mirroring the experimental geometry [1]. Per-element time delays implement the delay law to steer the mixing zone along an arc, and four-way polarity flipping isolates the mixed component from the simulated receiver signals, reproducing the full experimental signal chain and the observed proportionality between mixed-wave amplitude and the product of incident amplitudes. Because the model is parametric and equation-based, it functions as a digital testbed for inspection design—enabling rapid evaluation of higher transducer counts, element spacings, wedge angles, and steering laws, including larger arrays and frequency regimes not yet commercially available. Validating in homogeneous aluminum first deliberately isolates the nonlinear mixing mechanism from scattering and attenuation, establishing a verified foundation for the principal target application: heterogeneous, viscoelastic media such as the concrete of [1], where aggregate scattering and high-frequency attenuation are introduced incrementally to reproduce the field-relevant inspection regime.
