Indirect Boundary Integral Equation Method Research Articles

Are the direct and indirect BEM/BIEMs equivalent?

For a long time, direct and indirect BEM/BIEMs were always unified and seemed equivalent in real implementation for ensuring the solution. Regarding the solution space and the numerical aspects in linear algebra, it is interesting to find that they are not fully equivalent. To demonstrate this finding, antiplane shear problems containing the elliptical rigid inclusion and hole as well as rigid-line inclusion and line crack were given to demonstrate the non-equivalence between the direct and indirect BEM/BIEMs. It is interesting to find that nonunique boundary densities may yield the unique field solution in both analytical derivation and numerical implementation. The mechanism was analytically studied and numerically performed by using the degenerate kernel and the singular value decomposition technique, respectively. Besides, the Fredholm alternative theorem was employed to examine the solvability condition in both the indirect boundary integral equation method and indirect boundary element method. For continuous system and discrete system, we can explain why the boundary density does not match the exact solution but the field solution is always acceptable when setting ξ0 = 0 in the beginning of the derivation of BIE. Two approaches, direct and indirect BEM/BIEMs, show the non-equivalence of solution in case of degenerate boundary and degenerate scale.

The IBIEM solution to the scattering of P1 waves by an arbitrary shaped cavity embedded in a fluid-saturated double-porosity half-space

SUMMARY The double-porosity saturated medium is widespread in the Earth's crust, rocks and man-made materials. In this paper, we developed the indirect boundary integral equation method abbreviated as IBIEM to investigate the elastic waves scattering characteristics in a fluid-saturated double-porosity half-space with an arbitrary shaped cavity inside. First, the Helmholtz decomposition was utilized to derive the 2-D Green's functions for plane waves in a double-porosity half-space under fluid saturation. Then the scattered wavefield is constructed by line sources of compressive and shear waves located at a virtual surface near the cavity surface. After the verification of numerical accuracy, the scattering of plane P1 wave by an arbitrary shaped cavity in a 2-D fluid-saturated double-porosity half-space is solved and analysed. Numerical results show that: the displacement, hoop stress and pore pressure amplitudes are strongly dependent on incidence frequencies, porosities and boundary drainage conditions, etc. Displacement amplitudes usually reach their maximums under the dry condition, while hoop stress amplitudes obtain their peak values under the impermeable condition. The presence of matrix pore pressure and fracture pore pressure enhances the energy fluctuations of double-porosity media, which produces more complex seismic response.

3D coupled FEM-IBIEM for non-linear seismic response of a sedimentary basin using equivalent-linearized soil model

Based on an equivalent-linearized dynamic constitutive model of the soil, the coupled finite element method - indirect boundary integral equation method (FEM-IBIEM) is developed to carry out the non-linear seismic response of a three-dimensional (3-D) sedimentary basin. The indirect boundary integral equation method (IBIEM), which can significantly reduce the solution dimension, is employed to solve the exterior response of the sedimentary basin. Due to the inhomogeneity and nonlinearity of the soft soil, the finite element method (FEM) is used to analyze the response of the sedimentary basin. Based on the verification of numerical accuracy, the scattering issue in a sedimentary basin with linearly varying shear wave velocity under incident P and SV waves is investigated. Numerical results show that: the coherence effect increases with the increase of the incident frequency, and the amplification of surface displacement and acceleration is distinct. Due to the soil-nonlinearity, the peak ground acceleration (PGA) at a specific location on the basin surface under P waves decreases by approximately 51%, more significant than the decrease under SV waves. The effect of the soil-nonlinearity on the acceleration response spectrums is a decrease of the acceleration amplitude and an increase in the predominant period.

Dynamic Response of Twin Circular Unlined Tunnels Subjected to Blasting P Waves

The dynamic response of twin circular unlined tunnels is studied by indirect boundary integral equation method (IBIEM). The twin tunnels are assumed to lie in an unbounded elastic space subjected to blasting P waves. The influence of incident angle on the distribution of DSCF around the twin tunnels is analyzed. Besides, the influences of normalized wave number αa and normalized distance d* (ratio of spacing distance to tunnel radius) on the reference DSCF, that is, the peak DSCF for different incident angles with an assurance rate of 95%, are discussed. Results show that 1) the IBIEM is a high-precision method analyzing the DSCF around twin circular unlined tunnels induced by blasting P waves; 2) the incident angle θ0, αa, and d* all have significant influences on the distribution of DSCF; 3) for a constant d*, the peak DSCF around the right tunnel is slightly greater than the left one; 4) the reference DSCF decreases exponentially with the increasing of αa or d*, and the corresponding fitting functions are proposed.

A three-dimensional indirect boundary integral equation method for the scattering of seismic waves in a poroelastic layered half-space

A new indirect boundary integral equation method (IBIEM) is developed to solve the seismic wave scattering problem in a three-dimensional fluid-saturated layered half-space. Based on Biot's theory, the Green's functions of inclined circular loads in porous elastic layered half-space are firstly deduced. According to the single-layer potential theory, when establishing the scattered wave field, the uniform surface loads and fluid sources are located directly on the boundary surfaces of the scatterers, so as to avoid determining the optimal location of fictitious wave sources surface. At the same time, the radiation condition of wave in semi-infinite layered media can be accurately realized by using the dynamic Green's function of concentrated load, and the computational memory can be greatly reduced. The scattering of seismic waves by a canyon topography in saturated layered half-space is examined in detail. The numerical results indicate that with the increase of the porosity of the overlying soil, the amplitude of surface displacement can be amplified by 1∼6 times, and the amplification effect is more significant near the corner of the canyon, which can be attributed to the superposition of wave diffraction effect and resonance amplification effect of saturated soil layer.

A three-dimensional indirect boundary integral equation method for modeling elastic wave scattering in a layered half-space

A new indirect boundary integral equation method (IBIEM) is proposed in this study to solve three-dimensional (3-D) elastic wave scattering by heterogeneities in a multi-layered half-space, employing Green’s function of distributed loads on equivalent circular elements, thus avoiding the element discretization on layer interfaces. The proposed method enables the fictitious loads to be directly distributed on the surfaces of scatterer and the weak singularity to be tackled by analytical integration. Also, the radiation condition in the semi-infinite layered medium can be satisfied accurately, and the memory requirements can also be greatly reduced, especially for a large number of layers or gradient medium. The numerical accuracy was verified through comparisons with existing results and the numerical convergence was also confirmed. The results clearly demonstrate the simplicity and effectiveness of the method, and also reveals the complicated scattering characteristics in a layered half-space that are dominated by the resonant properties of the layered medium.

Seismic response of a 3-D canyon in a multilayered TI half-space modelled by an indirect boundary integral equation method

Seismic response of a 3-D canyon in a multilayered TI half-space modelled by an indirect boundary integral equation method

Erratum to: Development and Application of Two-Dimensional Numerical Tank using Desingularized Indirect Boundary Integral Equation Method

Erratum to: Development and Application of Two-Dimensional Numerical Tank using Desingularized Indirect Boundary Integral Equation Method

비특이화 간접경계적분방정식 방법을 이용한 부유식 구조물의 유체동역학적 거동에 대한 주파수영역 해석

비특이화 간접경계적분방정식 방법을 이용한 부유식 구조물의 유체동역학적 거동에 대한 주파수영역 해석

The scattering of plane P, SV waves by twin lining tunnels with imperfect interfaces embedded in an elastic half-space

A viscous-slip interface model is employed to simulate the contact between the tunnels lining and the surrounding rock, and the scattering of P, SV waves by twin shallowly buried lining tunnels is investigated with the indirect boundary integral equation method (IBIEM). The amplification effect of the dynamic stress concentration of the lining and the surface displacement near the tunnels is examined. It is evident that the slipping-stiffness coefficient and viscosity coefficient at the lining-surrounding rock interface have a significant influence on the dynamic stress distribution and the nearby surface displacement response of the lining tunnel, while the influence characteristics strongly depend on the incident wave type, frequency and angle. Under the incidence of low frequency wave, as a whole, with the increase of the sliding stiffness, the hoop stress increases gradually for plane P and SV waves; while in the resonance frequency (the incident wave frequency is consistent with the natural frequency of the soil column above the tunnels), specially for high-frequency band, the dynamic stress concentration effect is more significant for smaller sliding stiffness. With the increase of viscosity coefficient, the dynamic stress concentration factor inside the lining gradually decreases. Also, the tunnels with viscous-slip interfaces have a more significant amplification effect on the nearby surface displacement amplitude. Moreover, the hoop stress of the twin tunnels may be obviously larger than that of single tunnel in most cases. The dynamic analysis of the underground structure under the actual strong dynamic loading should consider the influence of the slip effect between the lining and surrounding rock interface.

Development and Application of Two-Dimensional Numerical Tank using Desingularized Indirect Boundary Integral Equation Method

In this study, a two-dimensional fully nonlinear transient wave numerical tank was developed using a desingularized indirect boundary integral equation method. The desingularized indirect boundary integral equation method is simpler and faster than the conventional boundary element method because special treatment is not required to compute the boundary integral. Numerical simulations were carried out in the time domain using the fourth order Runge-Kutta method. A mixed Eulerian-Lagrangian approach was adapted to reconstruct the free surface at each time step. A numerical damping zone was used to minimize the reflective wave in the downstream region. The interpolating method of a Gaussian radial basis function-type artificial neural network was used to calculate the gradient of the free surface elevation without element connectivity. The desingularized indirect boundary integral equation using an isolated point source and radial basis function has no need for information about the element connectivity and is a meshless method that is numerically more flexible. In order to validate the accuracy of the numerical wave tank based on the desingularized indirect boundary integral equation method and meshless technique, several numerical simulations were carried out. First, a comparison with numerical results according to the type of desingularized source was carried out and confirmed that continuous line sources can be replaced by simply isolated sources. In addition, a propagation simulation of a 2nd-order Stokes wave was carried out and compared with an analytical solution. Finally, simulations of propagating waves in shallow water and propagating waves over a submerged bar were also carried and compared with published data.

The indirect boundary integral equation method for the broadband scattering of plane P, SV and Rayleigh waves by a hill topography

The indirect boundary integral equation method (IBIEM) is developed to solve the broadband scattering of plane P, SV and Rayleigh waves by a hill topography, incorporated with the technique of domain-partition construction. It illustrates that the semi-analytical IBIEM can accurately and steadily solve the wave scattering around a steep or a gentle hill both for low and high frequency waves. Numerical results indicate that the seismic response characteristics of a hill strongly depend on the incident angle, wave frequency, the medium stiffness and height-width ratio of the hill. In general, the steep hill has more significant amplification effect on the surface displacement amplitude of the hill, and the displacement peak usually located at the top of the hill for vertically incident SV waves but can also appear near the hill bottom for high-frequency P waves. Multiple standing wave points can be observed for high-frequency wave and steep hill due to the interface effect of scattered waves. Moreover, the high hill has significant shielding effect on the Rayleigh waves. In practice, the notable amplification effect and obvious spatial variation of ground motion must be taken into account in the anti-seismic design of engineering structures in mountainous area.

The Diffraction of Rayleigh Waves by Twin Circular Cavities in a Poroelastic Half-Space

ABSTRACTThe diffraction of Rayleigh waves by twin circular cavities in a poroelastic half-space is investigated using the indirect boundary integral equation method (IBIEM). To satisfy the boundary conditions on the free surface, Green’s functions of compressional and shear wave sources in a poroelastic half-space are derived based on Biot’s theory. It is verified that the IBIEM has great accuracy and numerical stability. Then, the influences of the drainage boundary condition, incident wave frequency, porosity, and cavity spacing on the dynamic responses are investigated by numerical examples. The results show that the amplification effect of the twin cavities is greater in the case of undrained boundary conditions, lower frequencies, and smaller cavity spacings. The porosity of the saturated medium also influences the dynamic responses because the velocity of the Rayleigh wave will change with the porosity of the saturated medium.

Indirect Boundary Integral Equation Method for the Cauchy Problem of the Laplace Equation

In this paper, we examine the Cauchy problem of the Laplace equation. Motivated by the incompleteness of the single-layer potential function method, we investigate the double-layer potential function method. Through the use of a layer approach to the solution, we devise a numerical method for approximating the solution of the Cauchy problem, which are well known to be highly ill-posed in nature. The ill-posedness is dealt with Tikhonov regularization, whilst the optimal regularization parameter is chosen by Morozov discrepancy principle. Convergence and stability estimates of the proposed method are then given. Finally, some examples are given for the efficiency of the proposed method. Especially, when the single-layer potential function method does not give accurate results for some problems, it is shown that the proposed method is effective and stable.

Wave propagation in a 3D fully nonlinear NWT based on MTF coupled with DZ method for the downstream boundary

Wave propagation in a three-dimensional (3D) fully nonlinear numerical wave tank (NWT) is studied based on velocity potential theory. The governing Laplace equation with fully nonlinear boundary conditions on the moving free surface is solved using the indirect desingularized boundary integral equation method (DBIEM). The fourth-order predictor-corrector Adams-Bashforth-Moulton scheme (ABM4) and mixed Eulerian-Lagrangian (MEL) method are used for the time-stepping integration of the free surface boundary conditions. A smoothing algorithm, B-spline, is applied to eliminate the possible saw-tooth instabilities. The artificial wave speed employed in MTF (multi-transmitting formula) approach is investigated for fully nonlinear wave problem. The numerical results from incorporating the damping zone (DZ), MTF and MTF coupled DZ (MTF+DZ) methods as radiation condition are compared with analytical solution. An effective MTF+DZ method is finally adopted to simulate the 3D linear wave, second-order wave and irregular wave propagation. It is shown that the MTF+DZ method can be used for simulating fully nonlinear wave propagation very efficiently.

True and Spurious Eigensolutions of an Elliptical Membrane by Using the Nondimensional Dynamic Influence Function Method

In this paper, we employ the nondimensional dynamic influence function (NDIF) method to solve the free vibration problem of an elliptical membrane. It is found that the spurious eigensolutions appear in the Dirichlet problem by using the double-layer potential approach. Besides, the spurious eigensolutions also occur in the Neumann problem if the single-layer potential approach is utilized. Owing to the appearance of spurious eigensolutions accompanied with true eigensolutions, singular value decomposition (SVD) updating techniques are employed to extract out true and spurious eigenvalues. Since the circulant property in the discrete system is broken, the analytical prediction for the spurious solution is achieved by using the indirect boundary integral formulation. To analytically study the eigenproblems containing the elliptical boundaries, the fundamental solution is expanded into a degenerate kernel by using the elliptical coordinates and the unknown coefficients are expanded by using the eigenfunction expansion. True and spurious eigenvalues are simultaneously found to be the zeros of the modified Mathieu functions of the first kind for the Dirichlet problem when using the single-layer potential formulation, while both true and spurious eigenvalues appear to be the zeros of the derivative of modified Mathieu function for the Neumann problem by using the double-layer potential formulation. By choosing only the imaginary-part kernel in the indirect boundary integral equation method (BIEM) to solve the eigenproblem of an elliptical membrane, spurious eigensolutions also appear at the same position with those of NDIF since boundary distribution can be lumped. The NDIF method can be seen as a special case of the indirect BIEM by lumping the boundary distribution. Both the analytical study and the numerical experiments match well with the same true and spurious solutions.

Numerical simulation of fully nonlinear sloshing waves in three-dimensional tank under random excitation

Based on the fully nonlinear velocity potential theory, the liquid sloshing in a three dimensional tank under random excitation is studied. The governing Laplace equation with fully nonlinear boundary conditions on the moving free surface is solved using the indirect desingularized boundary integral equation method (DBIEM). The fourth-order predictor-corrector Adams-Bashforth-Moulton scheme (ABM4) and mixed Eulerian-Lagrangian (MEL) method are used for the time-stepping integration of the free surface boundary conditions. A smoothing scheme, B-spline curve, is applied to both the longitudinal and transverse directions of the tank to eliminate the possible saw-tooth instabilities. When the tank is undergoing one dimensional regular motion of small amplitude, the calculated results are found to be in very good agreement with linear analytical solution. In the simulation, the normal standing waves, travelling waves and bores are observed. The extensive calculation has been made for the tank undergoing specified random oscillation. The nonlinear effect of random sloshing wave is studied and the effect of peak frequency used for the generation of random oscillation is investigated. It is found that, even as the peak value of spectrum for oscillation becomes smaller, the maximum wave elevation on the side wall becomes bigger when the peak frequency is closer to the natural frequency.

Wave Propagation in a Three-Dimensional, Fully-Nonlinear Numerical Wave Tank Based on Multi-transmitting Formula Coupled Damping Zone Method

Abstract The objective of this work is to develop a novel numerical model solving the coupled internal sloshing and external sea-keeping interaction for small/medium liquefied natural gas (LNG) carrier. Currently, the work focuses on simulating wave propagation in a three-dimensional (3D) fully-nonlinear numerical wave tank (NWT) to model the sea. When simulating the nonlinear wave propagation through an unbounded domain in the time domain, it is necessary to truncate the computational domain artificially into a finite domain in order to reduce computational costs. Thus, non-reflecting radiation boundary condition is required for the truncated surface, however, there is no exact non-reflecting condition in existence. In this work, wave propagation in a 3D fully nonlinear NWT is studied based on the fully-nonlinear velocity potential theory. The governing Laplace equation with fully nonlinear boundary conditions on the moving free surface is solved using the indirect desingularized boundary integral equation method (DBIEM). The fourth-order predictor-corrector Adams-Bashforth-Moulton scheme (ABM4) and mixed Eulerian-Lagrangian (MEL) method are used for the time-stepping integration of the free surface boundary conditions. A smoothing algorithm, B-spline, is applied to eliminate the possible saw-tooth instabilities. An effective multi-transmitting formula coupled damping zone (MTF+DZ) radiation condition is employed to minimize wave reflection on the truncated surface. The numerical results are compared with analytical solutions. It is shown that MTF+DZ method can be used for simulating fully-nonlinear, irregular wave propagation.

Diffraction of plane P waves around an alluvial valley in poroelastic half-space

Diffraction of plane P waves around an alluvial valley of arbitrary shape in poroelastic half-space is investigated by using an indirect boundary integral equation method. Based on the Green’s functions of line source in poroelastic half-space, the scattered waves are constructed using the fictitious wave sources close to the interface of the valley and the density of fictitious wave sources are determined by boundary conditions. The precision of the method is verified by the satisfaction extent of boundary conditions, and the comparison between the degenerated solutions and available results in single-phase case. Finally, the nature of diffraction of plane P waves around an alluvial valley in poroelastic half-space is investigated in detail through numerical examples.

Permeance computation using indirect boundary integral equation method

An approach using indirect boundary integral equation method is proposed to determine the permeance between ferromagnetic poles in axisymmetric and three-dimensional magnetic systems. A generalised mathematical model is given for both types of magnetic systems. It consists of Fredholm integral equations of the first kind with respect to fictitious magnetic charge density sought in the form of simple layer potential. The system of boundary integral equations is solved using the method of mechanical quadratures. The approach is implemented in its own computer code. Results are presented for axisymmetric poles of electromagnets (cylinders, cones and frustum cones) and for a three-dimensional clapper-type system. Comparisons with known formulas are made and their accuracies are estimated. The approach presented is useful at the stage of preliminary design of magnetic systems. It is also applicable to computation of capacitances and electrical conductances.