Towards an Optimization of the Spectral Collocation Method with a New Balancing Algorithm for Plotting Dispersion Curves of Composites with Large Numbers of Layers

This article proposes a new algorithm for computing dispersion curves of ultrasonic guided waves in multi-layered composites with large number of layers. The algorithm is based on balancing the eigenvalue problem of the Spectral Collocation Method (SCM) formulation. The SCM has proven effective in analyzing single-layer and simple waveguides. However, it struggles with large multi-layer structures due to numerical instability caused by irregular and sparse matrices in the eigenvalue problem. The proposed algorithm for balancing has significantly reduced both the conditioning measure and the matrix norm in the matrix system. The optimization of spectral formulation enables accurate calculation of dispersion curves and characterization of displacement/stress profiles using Matlab software. This precise characterization and mode separation are essential for selecting ultrasonic sensors for damage detection.  A comparison was made with Dispersion Calculator (DC) software. The algorithm was first validated using a hybrid multi-layered composite [CFRP-Al-CFRP-Al], which was successful. The validated algorithm was then quantitatively evaluated using a cross-play laminate T۸۰۰M۹۱۳ [۰/۹۰/۰/۹۰] in terms of three parameters: Number of collocation points, wave propagation direction and thickness. The algorithm is then applied to a large number of layers using three configurations: Symmetric layup T۸۰M۹۱۳[۰/۹۰]۱۰s, hybrid layup [CFRP-Al]۵۰ and challenging layup T۸۰۰M۹۱۳[۰/۹۰]۱۰۰. The study found that the balancing algorithm, when combined with the SCM, is effective for structures with a large number of layers. Finally, optimizing the SCM will enhance its competitiveness as an effective tool in ultrasonic non-destructive testing for the studied structures of great industrial interest.

Keywords:

Dispersion curves , Spectral collocation method , Balancing eigenvalue problem. Guided waves , Multi-layered composites