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

2nd 3rd

Huge difference in the harmonics between different directions

UT-BATTELLE

3rd harmonic cause of poor quality of second weld?

Fundamental 2nd harmonic 3rd harmonic

Second weld

First weld