ACA-accelerated Nyström method with fast excitation sweep algorithm for efficient modeling of 3D ultrasonic NDE problems using CBIE

In this paper, an efficient pure algebraic and kernel independent algorithm, the adaptive cross approximation (ACA) algorithm, is developed for modeling three-dimensional (3D) ultrasonic non-destructive evaluation (UNDE) problems by accelerating conventional boundary integral equations (CBIE) which is discretized using the high order Nyström method (NM). The octree structure is constructed to subdivide the 3D object under test into small sub-blocks, which are then classified into diagonal, near and far blocks determined by the distances among them. Unlike boundary element method (BEM), NM selects the interpolation nodes based on Gaussian quadrature, which substantially simplifies the process of dealing with the singularities and near-singularities, and removes the necessities of performing explicit integration in the calculation of matrix elements for far block interactions. The ACA algorithm is then applied to accelerate solving the matrix discretized by NM, reducing the computational complexity to O(NlogN). It also works as the fast excitation sweep algorithm for sweeping across a large number of incident fields, reducing solution time for multiple right-hand-side (RHS) vectors by up to 99.7 %, which is especially useful for dealing with problems such as uncertainty propagation wherein a large number of input excitations need to be simulated. Finally, the scattering amplitudes obtained in frequency domain by solving the discretized equation with multiple RHSs are converted into voltage signals in time domain using inverse Fourier transform and Auld's reciprocity formulas. The accuracy and efficiency of the proposed method are demonstrated using immersion testing benchmark problems.