Dual boundary element method for solving the two-dimensional Helmholtz equation for the damped wave equation

Area of Science:

• Computational Mathematics
• Acoustics and Wave Propagation
• Numerical Analysis

Background:

• The Helmholtz equation models wave phenomena, often involving complex wave numbers due to damping.
• Singular and hypersingular integrals pose challenges in boundary integral formulations.
• Understanding resonance phenomena in damped systems is crucial for various applications.

Purpose of the Study:

• To derive a dual boundary integral formulation for the two-dimensional Helmholtz equation with complex wave numbers.
• To develop a regularization technique for handling singular and hypersingular integrals.
• To investigate the influence of damping on resonance phenomena and validate the formulation's accuracy and applicability.

Main Methods:

• Derivation of the dual boundary integral formulation for the Helmholtz equation with complex wave numbers.
• Application of the addition theorem to expand kernel functions into real-variable series.
• Regularization of singular and hypersingular integrals into summations of regular integrals.
• Computation of regular integrals using Gaussian quadrature.
• Validation using benchmark cases with exact solutions and analysis of irregular geometries.

Main Results:

• A novel dual boundary integral formulation for the damped Helmholtz equation is presented.
• Singular and hypersingular integrals are successfully transformed into regular integrals via a series expansion and regularization technique.
• The influence of damping on eigenvalues and resonances in interior and exterior Helmholtz problems is examined.
• The proposed method demonstrates good convergence and accuracy for benchmark cases and complex geometries.

Conclusions:

• The developed dual boundary integral formulation effectively addresses the challenges posed by complex wave numbers and singular integrals in the Helmholtz equation.
• The method provides a robust approach for analyzing resonance phenomena in damped acoustic systems, even with irregular boundaries.
• The formulation shows significant potential for applications in acoustics, electromagnetics, and other fields involving wave propagation with damping.