The direct interpolation boundary element technique applied to three-dimensional scalar free vibration problems

Continuing the previous research, in this work the Direct Interpolation Technique with Radial Basis Functions of the Boundary Element Method is applied to three-dimensional free vibration problems, governed by the Helmholtz Equation. Due to the primitive radial interpolation function, the domain integral regarding the inertia of the system is transformed into a boundary integral, generating a dynamic model capable of identifying the natural frequencies spectrum when is written as an eigenvalue problem. By solving two test problems, one can confirm the good performance of the model for solving three-dimensional problems, which present a significantly greater effort concerning the numerical implementation and computational processing comparatively the two-dimensional analysis. Triangular isoparametric elements with double nodes in the corners were used in the classical discrete model of the BEM, while some of the main traditional radial basis functions were tested in the interpolation procedure. Natural frequencies are obtained by solution of the eigenvalue problem, which allow an easier evaluation of the quality of numerical results. Analytical results and Finite Element Method results were used to generate the reference solutions for a suitable evaluation of accuracy.