Efficient fictitious eigenfrequency issue-free full-wave simulation of 3D ultrasonic NDE using CFIE-based Nyström method accelerated by kernel independent algorithm.

Conventional boundary integral equations (CBIE) based ultrasonic non-destructive evaluation (UNDE) models admit multiple solutions at some frequencies, which is known as the fictitious eigenfrequency issue. At the fictitious eigenfrequencies, numerical methods used for simulating the UNDE model, such as the Nystrom method (NM), for example, result in singular matrix equations, which cannot be solved. To avoid the fictitious eigenfrequency issue in CBIE for 3D UNDE problem, this paper uses the combined field integral equation (CFIE) formulation and proposes a high-order Nyström method (NM) accelerated by kernel independent algorithm for solving it. By linearly combining the CBIE with the hyper-singular boundary integral equation (HBIE), the proposed CFIE framework effectively suppresses fictitious eigenfrequencies and enhances numerical stability. Within this framework, high-order surface basis functions are employed to discretize unknown displacement and traction fields, while Gaussian interpolation nodes simplify the treatment of singular integrals. The octree structure divides the 3D object under test into near-field, far-field, and diagonal blocks: the near-field and diagonal blocks are directly filled using the conventional locally-corrected NM, whereas the far-field blocks are compressed via the adaptive cross approximation (ACA) algorithm to reduce computational complexity from O(N2) to O(NlogN), where N represents the number of unknowns. In UNDE, to locate and reconstruct the geometric or material information of the crack, it is required to calculate the responses from multiple excitations by scanning the ultrasonic transducer. To enhance the computational efficiency of the CFIE-based high-order NM solver for multiple excitations, the fast excitation sweep (FES) algorithm is proposed, which applies the ACA algorithm to compress the sparse characteristics of multiple right-hand sides (multi-RHSs) matrix to accelerate iterative solving of the matrix system for multi-angle and multi-position scannings. To compute the time domain signals using the inverse Fourier transform, the model needs to be simulated at multiple frequency points according to the Nyquist sampling theorem. To alleviate the computational burden, the model is simulated only at a few frequency points and the Kriging interpolation technique is used to calculate the solution at intermediate frequencies. We show that Kriging interpolation requires fewer sampling points in the frequency domain and shows improved performance over cubic spline technique. Experimental validation on benchmark cases, including unit sphere scattering, spherical cavities in fused-quartz specimen, pillbox cavities within a titanium block, and flat-bottom hole defect in steel specimen, shows excellent agreement between numerical results achieved by the proposed model and measured data. The proposed solver exhibits good computational efficiency while maintaining robustness in solving 3D UNDE problems, thanks to the integration of multiple improvements in the form of CFIE formulation, high-order NM, ACA acceleration, and FES.