A Galerkin Method for Linear PDE Systems in Circular Geometries with Structural Acoustic Problems

A Galerkin method for systems of PDEs in circular geometries is presented with motivating problems being drawn from structural, acoustic, and structural acoustic applications. Depending upon the application under consideration, piecewise splines or Legendre polynomials are used when approximating the system dynamics with modifications included to incorporate the analytic solution decay near the coordinate singularity. This provides an efficient method which retains its accuracy throughout the circular domain without degradation at the singularity. Because the problems under consideration are linear or weakly nonlinear with constant or piecewise constant coefficients, transform methods for the problems are not investigated. While the specific method is developed for the 2-D wave equation on a circular domain and the equation of transverse motion for a thin circular plate, examples demonstrating the extension of the techniques to a fully coupled structural acoustic system are used to illustrate the flexibility of the method when approximating the dynamics of more complex systems.

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