Boundary Element Analysis of Acoustic Wave Problems in Functionally Graded Materials

Abstract

A combined Laplace Transform (LT) and domain-boundary element method (DBEM) was utilized to obtain numerical solutions for a spatio-temporal coefficient Helmholtz-type equation with arbitrary initial conditions and source terms, addressing acoustic wave problems in functionally graded materials (FGMs). The procedure begins by transforming the equation of the time-space variable coefficients into one of the time-variable coefficients. The equation was then converted into an integral equation using the Gaussian divergence theorem. The time variable of the integral equation is reduced by utilizing LT and its convolution theorem to obtain a domain-boundary integral equation. Numerical solutions within the LT framework were then obtained using the standard domain-boundary element method. These numerical solutions were inverted using the Stehfest method to obtain solutions for the original time variable. Several problems involving trigonometric, exponential, and quadratic spatial gradation function coefficients were solved. The numerical validation demonstrates exceptional accuracy with mean absolute relative errors as low as E = 0.001138 for quadratic gradation materials (Ns=10, Nc=32), E = 0.003411 for exponential gradation materials (Ns=8, Nc=32), and E = 0.004279 for trigonometric gradation materials (Ns=8, Nc=16). Computational efficiency analysis reveals optimal performance parameters, with relative CPU times ranging from 0.302452s to 1.93548s depending on the gradation type and discretization parameters. The validity of the analysis used to derive the domain boundary integral equations was verified, and accurate DBEM solutions were obtained. This study successfully extends the LT-DBEM approach to handle arbitrary source terms and initial conditions in variable coefficient problems, providing a robust and computationally efficient method for analyzing acoustic wave propagation in diverse functionally graded material configurations with superior accuracy compared to traditional approaches.