Acoustic scattering of a pair of rigid spheroids based on the T-matrix method

In this study, the T-matrix method combined with the addition theorems of spherical basis functions is applied to semi-analytically compute the underwater far-field acoustic scattering of a pair of rigid spheroids with arbitrary incident angles. The involvement of the addition theorems renders the multiple scattering fields of each spheroid to be translated into an identical origin. The accuracy and convergence property of the proposed method are verified and validated. The interference of specular reflection wave and Franz wave can be spotted from the oscillations of the form function. Furthermore, the propagation paths of specular reflection and Franz waves are quantitatively analyzed in the time domain with conclusions that the Franz waves reach the observation point subsequent to specular reflection waves and the time interval between these two wave series is equal to the time cost of the Franz waves traveling along the sphere surfaces. Finally, the effects of separation distances, aspect ratios (the ratio of the polar radius to equatorial radius), non-dimensional frequencies, and incidence angles of the plane wave on the far-field acoustic scattering of a pair of rigid spheroids are studied by the T-matrix method

Transient Wave Propagation Dynamics with Edge-Based Smoothed Finite Element Method and Bathe Time Integration Technique

In this work, the edge-based smoothed finite element method (ES-FEM) is incorporated with the Bathe time integration scheme to solve the transient wave propagation problems. The edge-based gradient smoothing technique (GST) can properly soften the “overly soft” system matrices from the standard finite element approach; then, the spatial numerical dispersion error of the calculated solutions for wave problems can be significantly reduced. To effectively solve the transient wave propagation problems, the Bathe time integration scheme is employed to perform the involved time integration. Due to the appropriate “numerical dissipation effects” from the Bathe time integration method, the spurious oscillations in the relatively large wave numbers (high frequencies) can be effectively suppressed; then, the temporal numerical dispersion error in the calculated solutions can also be notably controlled. A number of supporting numerical examples are considered to examine the capabilities of the present approach. The numerical results show that ES-FEM works very well with the Bathe time integration technique, and much more numerical solutions can be reached for solving transient wave propagation problems compared to the standard FEM

Free and Forced Vibration Analysis of Two-Dimensional Linear Elastic Solids Using the Finite Element Methods Enriched by Interpolation Cover Functions

In this paper, a novel enriched three-node triangular element with the augmented interpolation cover functions is proposed based on the original linear triangular element for two-dimensional solids. In this enriched triangular element, the augmented interpolation cover functions are employed to enrich the original standard linear shape functions over element patches. As a result, the original linear approximation space can be effectively enriched without adding extra nodes. To eliminate the linear dependence issue of the present method, an effective scheme is used to make the system matrices of the numerical model completely positive-definite. Through several typical numerical examples, the abilities of the present enriched three node triangular element in forced and free vibration analysis of two-dimensional solids are studied. The results show that, compared with the original linear triangular element, the present element can not only provide more accurate numerical results, but also have higher computational efficiency and convergence rate

An Enriched Finite Element Method with Appropriate Interpolation Cover Functions for Transient Wave Propagation Dynamic Problems

A novel enriched finite element method (EFEM) was employed to analyze the transient wave propagation problems. In the present method, the traditional finite element approximation was enriched by employing the appropriate interpolation covers. We mathematically and numerically showed that the present EFEM possessed the important monotonic convergence property with the decrease of the used time steps for transient wave propagation problems when the unconditional stable Newmark time integration scheme was used for time integration. This attractive property markedly distinguishes the present EFEM from the traditional FEM for transient wave propagation problems. Two typical numerical examples were given to demonstrate the capabilities of the present method

An Enriched Finite Element Method with Appropriate Interpolation Cover Functions for Transient Wave Propagation Dynamic Problems

A novel enriched finite element method (EFEM) was employed to analyze the transient wave propagation problems. In the present method, the traditional finite element approximation was enriched by employing the appropriate interpolation covers. We mathematically and numerically showed that the present EFEM possessed the important monotonic convergence property with the decrease of the used time steps for transient wave propagation problems when the unconditional stable Newmark time integration scheme was used for time integration. This attractive property markedly distinguishes the present EFEM from the traditional FEM for transient wave propagation problems. Two typical numerical examples were given to demonstrate the capabilities of the present method

A Modified Radial Point Interpolation Method (M-RPIM) for Free Vibration Analysis of Two-Dimensional Solids

The classical radial point interpolation method (RPIM) is a powerful meshfree numerical technique for engineering computation. In the original RPIM, the moving support domain for the quadrature point is usually employed for the field function approximation, but the local supports of the nodal shape functions are always not in alignment with the integration cells constructed for numerical integration. This misalignment can result in additional numerical integration error and lead to a loss in computation accuracy. In this work, a modified RPIM (M-RPIM) is proposed to address this issue. In the present M-RPIM, the misalignment between the constructed integration cells and the nodal shape function supports is successfully overcome by using a fixed support domain that can be easily constructed by the geometrical center of the integration cell. Several numerical examples of free vibration analysis are conducted to evaluate the abilities of the present M-RPIM and it is found that the computation accuracy of the original RPIM can be markedly improved by the present M-RPIM

The Meshfree Radial Point Interpolation Method (RPIM) for Wave Propagation Dynamics in Non-Homogeneous Media

This work presents a novel simulation approach to couple the meshfree radial point interpolation method (RPIM) with the implicit direct time integration method for the transient analysis of wave propagation dynamics in non-homogeneous media. In this approach, the RPIM is adopted for the discretization of the overall space domain, while the discretization of the time domain is completed by employing the efficient Bathe time stepping scheme. The dispersion analysis demonstrates that, in wave analysis, the amount of numerical dispersion error resulting from the discretization in the space domain can be suppressed at a very low level when the employed nodal support domain of the interpolation function is adequately large. Meanwhile, it is also mathematically shown that the amount of numerical error resulting from the time domain discretization is actually a monotonically decreasing function of the non-dimensional time domain discretization interval. Consequently, the present simulation approach is capable of effectively handling the transient analysis of wave propagation dynamics in non-homogeneous media, and the disparate waves with different speeds can be solved concurrently with very high computation accuracy. This numerical feature makes the present simulation approach more suitable for complicated wave analysis than the traditional finite element approach because the waves with disparate speeds always cannot be concurrently solved accurately. Several numerical tests are given to check the performance of the present simulation approach for the analysis of wave propagation dynamics in non-homogeneous media

A Coupled Overlapping Finite Element Method for Analyzing Underwater Acoustic Scattering Problems

It is found that the classic finite element method (FEM) requires much time for adequate meshes to acquire satisfactory numerical solutions, and is restricted to acoustic problems with low and middle frequencies. In this work, a coupled overlapping finite element method (OFEM) is employed by combining the overlapping finite element and the modified Dirichlet-to-Neumann (mDtN) boundary condition to solve underwater acoustic scattering problems. The main difference between the OFEM and the FEM lies in the construction of the local field approximation. In the OFEM, virtual nodes are utilized to form the partition of unity functions while no degree of freedom is assigned to these virtual nodes, which suppresses the linear dependence issue in other generalized finite element methods. Moreover, the user-defined enrichment functions can be flexibly utilized in the local field, and thus the numerical dispersions can be significantly mitigated. To truncate the infinite problem domain and satisfy the Sommerfeld radiation condition, an artificial boundary is constructed by incorporating the mDtN technique. Several numerical examples are studied and it is shown that the proposed method can greatly diminish the numerical error and is insensitive to distorted meshes, indicating that the proposed method is promising in predicting underwater acoustic scattering

The Extrinsic Enriched Finite Element Method with Appropriate Enrichment Functions for the Helmholtz Equation

The traditional finite element method (FEM) could only provide acceptable numerical solutions for the Helmholtz equation in the relatively small wave number range due to numerical dispersion errors. For the relatively large wave numbers, the corresponding FE solutions are never adequately reliable. With the aim to enhance the numerical performance of the FEM in tackling the Helmholtz equation, in this work an extrinsic enriched FEM (EFEM) is proposed to reduce the inherent numerical dispersion errors in the standard FEM solutions. In this extrinsic EFEM, the standard linear approximation space in the linear FEM is enriched extrinsically by using the polynomial and trigonometric functions. The construction of this enriched approximation space is realized based on the partition of unity concept and the highly oscillating features of the Helmholtz equation in relatively large wave numbers can be effectively captured by the employed specially-designed enrichment functions. A number of typical numerical examples are considered to examine the ability of this extrinsic EFEM to control the dispersion error for solving Helmholtz problems. From the obtained numerical results, it is found that this extrinsic EFEM behaves much better than the standard FEM in suppressing the numerical dispersion effects and could provide much more accurate numerical results. In addition, this extrinsic EFEM also possesses higher convergence rate than the conventional FEM. More importantly, the formulation of this extrinsic EFEM can be formulated quite easily without adding the extra nodes. Therefore, the present extrinsic EFEM can be regarded as a competitive alternative to the traditional finite element approach in dealing with the Helmholtz equation in relatively high frequency ranges