Evaluation of singular integrals in the symmetric Galerkin boundary element method

The implementation of the symmetric Galerkin boundary element method (SGBEM) involves extensive work on the evaluation of various integrals, ranging from regular integrals to hypersingular integrals. In this paper, the treatments of weak singular integrals in the time domain are reviewed, and analytical evaluations for the spatial double integrals which contain weak singular terms are derived. A special scheme on the allocation of Gaussian integration points for regular double integrals in the SGBEM is developed to improve the efficiency of the Gauss–Legendre rule. The proposed approach is implemented for the two-dimensional elastodynamic problems, and two numerical examples are presented to verify the accuracy of the numerical implementation.     

A mechanistic insight into the shaft resistance of vertically vibrating piles

A mechanism for the dynamic shaft resistance of vertically vibrating single end-bearing and infinitely long piles is proposed through the use of elastodynamics. The generalized expression for the shaft resistance is obtained by means of Hamilton's principle. The shaft resistance is divided analytically into three parts, namely, the Winkler spring, the membrane and the participating mass of the surrounding soil. In light of the contributions of these three components of shaft resistance, a physical model is presented for the nature of single piles subjected to dynamic loading. The three components are obtained as explicit forms. The first is linear to the pile settlement. The second is related to the derivative of the second order of the pile settlement. The third involves the participating mass of the surrounding soil. The effect of the properties of the soil-pile system on the dynamic responses of the characteristic coefficients of the shaft resistance is investigated. The results indicate that the Winkler spring contributes the most to the shaft resistance. The contributions of the membrane and the participating mass of the surrounding soil are negligible for cases of high incident frequency.     

The improved element-free Galerkin method for two-dimensional elastodynamics problems

In this paper, we derive an improved element-free Galerkin (IEFG) method for two-dimensional linear elastodynamics by employing the improved moving least-squares (IMLS) approximation. In comparison with the conventional moving least-squares (MLS) approximation function, the algebraic equation system in IMLS approximation is well-conditioned. It can be solved without having to derive the inverse matrix. Thus the IEFG method may result in a higher computing speed. In the IEFG method for two-dimensional linear elastodynamics, we employed the Galerkin weak form to derive the discretized system equations, and the Newmark time integration method for the time history analyses. In the modeling process, the penalty method is used to impose the essential boundary conditions to obtain the corresponding formulae of the IEFG method for two-dimensional elastodynamics. The numerical studies illustrated that the IEFG method is efficient by comparing it with the analytical method and the finite element method.     

非局部弹性薄板侵彻问题的非线性分析

采用有限变形理论分析了非局部弹性薄板侵彻问题。利用张量分析、积分变换、微分方程和特殊函数的数学工具,求出相关问题的运动方程,得到不相关问题的解答。     

Shape reconstruction of three-dimensional flaw from backscattering data

Three dimensional Born and Kirchhoff inverse scattering methods are modified to convenient forms for a cylindrical specimen that includes three dimensional defect. One aluminum cylinder with flaw model is prepared and ultrasonic measurements are carried out. The measurement area in the modified methods is restricted in the plane perpendicular to the axis of cylindrical specimen. That is to say that the methods are modified to convenient form to use measured waveforms in the x₁–x₂ plane. The measured wave data are fed into the inversion method and cross-sectional images are obtained. Then, three dimensional shape reconstruction of flaw model in aluminum specimen is performed by piling up the cross-sectional images.     

Application of hierarchical matrices to boundary element methods for elastodynamics based on Green's functions for a horizontally layered halfspace

This paper presents the application of hierarchical matrices to boundary element methods for elastodynamics based on Green's functions for a horizontally layered halfspace. These Green's functions are computed by means of the direct stiffness method; their application avoids meshing of the free surface and the layer interfaces. The effectiveness of the methodology is demonstrated through numerical examples, indicating that a significant reduction of memory and CPU time can be achieved with respect to the classical boundary element method. This allows increasing the problem size by one order of magnitude. The proposed methodology therefore offers perspectives to study large scale problems involving three-dimensional elastodynamic wave propagation in a layered halfspace, with possible applications in seismology and dynamic soil–structure interaction.     

Fundamental solutions for variable density two-dimensional elastodynamic problems

Fundamental solutions in the form of free-space Green's functions are developed for a class of two-dimensional, variable density elastodynamic problems. These fundamental solutions are evaluated by means of a coordinate transformation based on conformal mapping in conjunction with wave decomposition, which allows for both vertical and lateral heterogeneities, and can be used within the context of a boundary integral equation formulation analogous to that originally proposed by Cruse and Rizzo (J Math Anal Appl 22 (1968) 244). Finally, a numerical example serves to illustrate the methodology developed herein.     

两端简支圆柱体的轴对称振动

本文用三维弹性动力学理论研究圆柱体在两端简支约束下的轴对称自由振动问题。给出位移振型和应力振型的解析表达式，求得固有频率的精确解。研究结果表明：在圆柱体的径向还存在着不同的半波数的位移振型，这种性质在初等理论中是无法反映的     

Computation of the stresses in a moving reference system in a half-space due to a traversing time-varying concentrated load

An analytical approach is employed to investigate the transient and steady-state stresses in an isotropic, homogeneous half-space subjected to moving concentrated loads with subsonic speeds. Applying the Stokes–Helmholtz resolution to the Navier’s equation of motion for the half-space results in a system of wavetype partial differential equations. Based on the new moving coordinate system, a modified system of partial differential equations is obtained. Applying a concurrent two-sided and one-sided Laplace transformation, this system is modified to a system of ordinary differential equations, the solutions of which are obtained with respect to boundary conditions. The transformed transient stresses can be inverted by the Cagniard–de Hoop method. Special properties of Laplace transformation yield the steady-state stresses through an analytical approach. Numerical examples are presented to illustrate the methodology. Final results revealed the importance of considering the stresses related to the initial stages of the loading.     

Numerical modelling of acoustic–elastodynamic coupled problems by stabilized boundary element techniques

In this work, an efficient, flexible, accurate and stable algorithm to numerically model interacting acoustic–elastodynamic sub-domains is described. Stabilized time-domain boundary element techniques are considered to discretize each sub-domain of the model and proper numerical expressions on acoustic–elastodynamic interfaces are presented. Moreover, stabilized iterative coupling procedures are adopted and different time and space sub-domain discretizations are allowed, improving the robustness and versatility of the methodology. At the end of the paper, numerical results are presented, illustrating the potentialities of the proposed formulation.