Acoustic eigenproblems of elliptical cylindrical cavities with multiple elliptical cylinders by using the collocation multipole method

• Acoustic eigenproblems of elliptical cylindrical cavities are solved by using a 3-D semi-analytical approach.
• A 3-D semi-analytical formulation based on a multipole expansion and directional derivative is proposed.
• The normal derivative of acoustic pressure with respect to non-local elliptical coordinates is derived.
• Some results may be used as benchmark solutions for future researches.
• The proposed method has good accuracy and a fast convergence rate.

Acoustic eigenproblems of 3-D elliptical cylindrical cavities having multiple elliptical cylinders are solved by means of a 3-D semi-analytical formulation based on a multipole expansion, directional derivative, collocation technique and singular value decomposition (SVD). The multipole expansion for the acoustic pressure is formulated in terms of angular and radial Mathieu functions due to elliptical boundaries considered here. The boundary conditions are satisfied by uniformly collocating points along the boundaries. When considering sound-hard or Neumann boundary conditions, the normal derivative of acoustic pressure with respect to non-local elliptic coordinates is derived using the directional derivative for multiply-connected domain problems. By truncating the multipole expansion, a finite linear algebraic system is acquired. The direct searching approach is applied to identify the natural frequencies by using the SVD. Several numerical examples are examined, including those of an elliptical cylindrical cavity, a confocal elliptical annulus cylindrical cavity and an elliptical cylindrical cavity with two elliptical cylinders. Convergence and comparison studies are done to demonstrate the validation and accuracy of the present method. No spurious eigensolutions are found in the proposed formulation. Excellent accuracy and fast rate of convergence are the main features of the present method thanks to its semi-analytical character.