UT-BATTELLE 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

UT-BATTELLE 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 Source Image

UT-BATTELLE 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

UT-BATTELLE 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 First weld Second weld First weld

UT-BATTELLE Higher harmonics important Fundamental 2nd Fundamental 3rd 2nd3rd Huge difference in the harmonics between different directions

UT-BATTELLE 3 rd harmonic cause of poor quality of second weld? Fundamental2 nd harmonic3 rd harmonic Second weld First weld