The acoustic radiation of general structures with Neumann's boundary condition using the variational boundary element method (VBEM) is considered. The classical numerical implementation of the VBEM suffers from the computation cost associated with double surface integrals. To circumvent this limitation a novel acceleration method is proposed. It is based on the expansion of the cross influence matrices in terms of multipoles using the expansion of the Green's function in terms of spherical Bessel functions. Since the resulting multipoles are not dependent on the elements locations, the technique results in large computation time savings for homogeneous meshes. The theory behind the approach, its convergence, and its numerical implementation in a VBEM code will be presented. It will be shown that by accounting for the monopole, dipole, and quadrupole terms in the multipoles expansion, the classical convergence criteria usually used for quadratic element hold. Numerical applications to acoustic radiation from plates, open cylinders, and closed boxes will be presented to demonstrate the accuracy and efficiency of the method.