New method for modeling acoustic waves in plates

1. New method for modeling acoustic waves in plates • A powerful boundary element method is developed for plate geometry • The new method achieves higher numerical accuracy with far fewer number of mesh points than traditional finite difference solutions • Because only the boundary conditions are discretized, the method is orders of magnitude faster than traditional methods • The new method can be easily applied to plates of different shapes and geometry

2. Boundary Element Method • A closed form solution for the wave equation over the entire plate in operator form • Implemented numerically through matrix algebra • Boundary conditions discretized to provide a set of linear equations used to determine the free parameters in the solution • Source points of a driving field modeled using Green’s functions • Convergence of Green’s functions accelerated with image source points outside the boundaries Image Source

3. Verification of the method on a circular plate • Extensional modes of a circular plate are solved analytically • Numerical solutions are compared to the analytic solutions • All solutions are matched to four significant figures using only 20 boundary points Energy density for first few extensional modes of a circular plate

4. Modeling of ultrasonic welding • Experiments show significant variation in weld quality vs direction and order of welds • Initial calculations at 20kHz reveal minimal difference in distribution of energy density Second weld First weld First weld

5. Higher harmonics important Fundamental 2nd 3rd Fundamental 2nd 3rd Huge difference in the harmonics between different directions

6. 3rd harmonic cause of poor quality of second weld? Second weld First weld Fundamental 2nd harmonic 3rd harmonic