| Acoustic problems,especially underwater acoustic scattering problems,are a fundamental branch of naval architecture and ocean engineering,for example,they have extensive applications in marine fishery,marine resource exploitation,marine transportation,military industrials and so on.Until now,in computational acoustics finite element method is still one of frequently-used numerical methods,which is widely integrated in many mainstream commercial software packages for simulating acoustic phenomena.However,numerical solutions obtained from the classical finite element method are significantly degraded for moderate and high computational frequencies,yielding expensive computational efforts in order to obtain accurate results.So it is of great importance to find the improved numerical methods.In this work,on basis of G space theory and the generalized gradient smoothing technique in terms of element edges,a group of the novel smoothed point interpolation meshfree acoustical models(ES-PIM/RPIM)are constructed using point interpolation shape functions.InG~1 space,the discontinuous assumed functions are admissible as long as the assumed functions are square integrable within the entire problem domain,which hence resolves the incompatible issue associated withH~1 space.Then,by employing the triangular background mesh a series of edge-based smoothing domains are formed,in which the generalized gradient smoothing technique is utilized to smooth the original gradient field.As a result,the generalized smoothed Galerkin weak forms of the wave equation and the Helmholtz equation are further established based on the standard Galerkin weak forms.Compared with the standard Galerkin weak form,the continuity requirement on shape functions is further reduced by the generalized smoothed Galerkin weak form,and the smoothed stiffness of the discrete model is much“softer”than the original stiffness.By virtue of the generalized gradient smoothing technique and the triangular-mesh-based node selection schemes,the stiffness of the smoothed point interpolation acoustical models can be“adjustable”.In the manuscript,we have thoroughly analyzed the dispersion error effect from frequency-domain acoustic computations,and the numerical results accord well with the predictions of the dispersion error analysis,demonstrating that ES-PIM-Tr3 and ES-RPIM-Tr6 acoustical models are the two best models here.Among all the considered smoothed point interpolation acoustical models,the smoothed stiffness of ES-PIM-Tr3 and ES-RPIM-Tr6 is closer to the exact stiffness,resulting in the numerical wave velocity closer to the exact wave velocity.Employing the meshfree perfectly matched layers based on the complex coordinate transformation with the smoothing domain related to the edges,the smoothed point interpolation-perfectly matched layer acoustical models are established for frequency-domain acoustic scattering problems.The undetermined parameters of perfectly matched layers are reasonably chosen through the theoretical analysis and a number of numerical examples in order to ensure that the outgoing waves can be totally absorbed.Numerical results show that the smoothed point interpolation-perfectly matched layer acoustical models can well deal with frequency-domain acoustic scattering problems.For transient acoustics,employing the results of the dispersion error analysis,we have investigated the total wave velocity errors of several acoustical models using the Newmark trapezoidal rule and the Bathe time integration method,in terms of the relationship between the dispersion errors from the spatial discretization and the total wave velocity errors from the temporal discretization.Numerical examples show that the total wave velocity errors of several acoustical models can accurately predict the accuracy of the numerical solutions of the transient acoustic wave propagation,and that the Bathe time integration method can more effectively depress the spurious high-frequency fluctuations from the temporal discretization.Employing the total wave velocity error results,the smoothed point interpolation acoustical models are utilized to analyze several transient acoustic scattering problems in comparison to the linear and quadratic finite element acoustical models,manifesting that the smoothed point interpolation acoustical models can properly tackle transient acoustic scattering problems.In general,not only the computational accuracy of ES-PIM-Tr3 and ES-RPIM-Tr6 is quite satisfying,but they can also be better than the classical finite element method in the aspects of the convergence rate,the computational cost and the reliability in using irregular node distributions.In addition,it is observed that ES-PIM-Tr3 and ES-RPIM-Tr6 acoustical models can possess the accuracy comparable to the quadratic finite element acoustical model,and behave much better than the linear finite element acoustical model in underwater acoustic scattering. |
PDF Full Text Request