Indirect Boundary Element Method Research Articles

Seismic Response of Utility Tunnel in Trapezoidal Backfilled Soil Under P Wave and Amplification Effect of Ground Motion

ABSTRACT For the seismic risk of utility tunnel in trapezoidal backfilled soil and its possible amplification effect of ground motion, the seismic response of utility tunnel in trapezoidal backfilled soil under P wave is studied by a high-precision indirect boundary element method (IBEM) with the idea of “partition fit.” The results show that the peak surface displacement appears on the backfill surface, and the softer the backfilled soil, the greater the peak surface displacement; the peak vertical displacement at oblique incidence is generally smaller than that at vertical incidence; the existence of utility tunnel can restrain the incidence wave to a certain extent, especially for the low-frequency wave; the peak displacement and stress of utility tunnel in the soft backfilled soil are all greater than that in stiff backfilled soil; the peak stress of utility tunnel at oblique incidence is greater than that at vertical incidence; the stress concentration occurs at the axillary corner; and the stress in the inner wall of utility tunnel is greater than that in the outer wall. The similar conclusions could be further confirmed from the time history analysis. The research results can provide some reference for the seismic design of utility tunnel in the backfill site and the evaluation of seismic performance of near surface pipeline, etc.

Numerical prediction and optimization of aerodynamic noise of straw crushers by considering the straw-crushing process

Straw crops are struck and broken into soft filaments by the high-speed rotating hammers of straw crushers, which disturb the internal airflow field and generate much noise during the operation of straw crushers. To accurately estimate and reduce the aerodynamic noise of straw crushers at the design stage, in this study, first, the coupling method of the discrete element method, bonded-particle model, and computational fluid dynamics were used to obtain the acoustics source data. Next, the Ffowcs Williams–Hawkins theory and the indirect boundary element method were used to predict the aerodynamic noise generated during the straw crushing process. The multi-island genetic algorithm was used to optimize the aerodynamic noise of straw crushers. The results indicate that the simulated and measured total sound pressure levels (TSPLs) at the outlet and inlet differed by 1.43 and 2.12 dB(A), respectively. Additionally, aerodynamic noise at the inlet appears to be primarily influenced by the sound pressure level at the excitation fundamental frequency, while noise at the outlet is primarily influenced by the sound pressure level at the double frequency. Higher sound pressure levels were mainly concentrated at the fundamental frequency and its lower harmonic frequencies, and the sound pressure level gradually decreased with the increase in the frequency. After optimization, the aerodynamic noise TSPL at the inlet decreased from 100.87 to 88.58 dB(A) and at the outlet decreased from 102.26 to 89.62 dB(A). This study provides a methodological reference for aerodynamic noise prediction and the design of low-noise straw crushers.

Three-dimensional broadband simulation of near-fault seismic ground motion considering complete source-path-site effect: Modeled by fast multipole indirect boundary element method

Efficient and accurate numerical methods are essential for analyzing seismic wave propagation and amplification in near-fault complex sites. Understanding the complete process of ground motion simulation, including fault rupture, path propagation, and near-surface complex site response, is crucial for studying earthquake damage mechanisms, seismic zoning, and designing large-scale engineering structures. In this study, we propose a fast multipole indirect boundary element method (FMIBEM) to achieve broadband and high-efficiency simulation, enabling a comprehensive analysis of the complete process involving seismic ground motion. The FMIBEM significantly reduces the computational and storage costs associated with 3D near-fault complex site seismic wave scattering problems to O(N). We validate the accuracy of the method by comparing it with the analytical solution. Compared to the conventional indirect boundary element method (IBEM), our proposed method reduces computational time by over 95 % and storage costs by nearly 90 %. FMIBEM greatly enhances the efficiency of the boundary element method for simulating seismic wave scattering in near-fault complex sites. To demonstrate the effectiveness of our approach, we apply the FMIBEM method to two typical 3D near-fault complex site examples of seismic wave scattering. The simulation results successfully capture both the local site amplification effect and the characteristic near-fault seismic features, such as the hanging wall effect, permanent displacement effect, and large velocity pulse.

3D isogeometric indirect BEM solution based on virtual surface sources on the boundaries of Helmholtz acoustic problems

AbstractA solution for 3D Helmholtz acoustic problems is introduced based on an indirect boundary element method (indirect BEM) coupled with isogeometric analysis (IGA). The novelty of this work arises from using virtual surface sources placed directly on the scatterer boundaries, producing robust results. These virtual surface sources are discretized by the same Non-Uniform Rational B-Splines (NURBS) approximating the scatterer CAD model. This allows modeling of general irregular geometries. The proposed solution has the same features of BEM approaches, which do not need any domain discretization or truncation boundaries at the far-field. It shows an additional merit by arranging the linear system of equations directly depending on a single coefficient matrix, consuming less computational time compared to other BEM methods. A Greville abscissae collocation scheme is proposed with offsets at $$C^0$$ C 0 -continuities. This collocation scheme allows for easy evaluation for both free-terms and normals at the collocation points. The performance of the proposed solution is discussed on 3D numerical exterior problems and compared against other BEM methods. Then, the practical interior muffler problem with internal extended thin tubes is studied and the obtained results are compared against other numerical methods in addition to the available experimental data, showing the capability of the proposed solution in handling thin-walled geometries.

The equivalent source method is an indirect boundary element method with an implicit Voronoi mesh

The Equivalent Source Method (ESM) consists of modeling the acoustical field radiated or scattered from an object by a set of radiating monopole sources located below its surface. While highly efficient for modeling the scattered field by a smooth object, ESM becomes less effective in representing acoustic diffraction from sharp edges. In this article, ESM is understood as an approximation to the single layer integral equation. In this framework, ESM belongs to the family of Indirect Boundary Element Methods (BEM) with an implicit surface mesh slightly below the object surface, and a zero order integration scheme on each element of the mesh. This implicit mesh is well approximated by the Voronoi diagram associated to the source distribution. This formalism establishes a strong link between these two widely used methods (ESM and BEM) and allows the construction of hybrid methods that can combine performances of both methods. On the first hand, a new hybrid method able to model edge diffraction while remaining as computationally efficient as ESM is developed. In this method, labeled “BEM Voronoi”, equivalent sources are positioned on the surface’s object rather than below it and singularities are handled with the integral formulation. On the other hand, a set of new hybrid methods which combines ESM with higher order integration scheme is constructed. Performances of the new methods developed in this paper are numerically compared to each other and to Finite Element simulations (used as a reference) for modeling scattering from an object excited by a plane wave. Each hybrid model is tested on multiple configurations involving two sets of boundary conditions (rigid and elastic objects respectively) successively immersed in two different external environments (water and air) and for both smooth and a sharp-edged objects. The results show that the hybrid method “BEM Voronoi” better handles edge diffraction than conventional ESM, at a much lower computational cost (implementation of BEM Voronoi is 50 times faster than conventional BEM implementation).

Nonlinear seismic response and damage evolution of a mountain tunnel: Multi-scale simulation by IBEM-FEM coupled method

Studying dynamic response and damage assessment of mountain tunnels is paramount in earthquake engineering. This paper proposed a novel method, the indirect boundary element method-finite element method (IBEM-FEM) coupled method, aimed to concurrently simulate the amplification effect of kilometer-scale mountain topography and the damage evolution in a centimeter-scale lining tunnel section. The proposed method involves two fundamental steps. Initially, the input motion on the truncation boundary within the mountain is accurately determined using the indirect boundary element method (IBEM). Subsequently, the nonlinear dynamic behavior of the tunnel structure and its adjacent area is analyzed by the finite element method (FEM) based on the results from the previous step. This study numerically simulates the seismic damage and failure mechanisms of a mountain tunnel considering three influence factors: incident wave types, incident wave intensities, and surrounding rock mass grades. The results indicate that mountain topography amplifies the seismic response, thereby affecting the dynamic behavior of the tunnel. Under the incidence of SV-waves and P-waves, the dynamic damage distribution patterns of the tunnel lining are significantly different. Moreover, as seismic wave intensity increases, the peak values of tunnel stress exhibit a non-monotonic increasing trend. The surrounding rock mass grade also significantly affects the dynamic response and damage distribution of the mountain tunnel. Overall, the proposed IBEM-FEM coupled method can effectively consider complex mountain topography and is applicable for evaluating the nonlinear seismic response of mountain tunnels.

A numerical scheme for heat transfer in fluid flowing in three-dimensional fractures

We present a numerical scheme using a finite volume method (FVM) to simulate the heat transfer in fluid flowing in a three-dimensional fracture in a rock mass. Heat conduction in the rock mass and heat exchange between the fluid and the rock mass are considered, but fracture deformation is not. This scheme could approximate the thermal energy extraction process from a geothermal system when the fractures have been established and the fluid pressure and temperature variations are not high enough to cause a significant change of the fracture aperture. The FVM was employed to solve the pressure and temperature of fluid with the same triangular control cells. The heat conduction in the rock mass was simulated with an indirect boundary element method (IBEM), which uses the triangular control cells of the FVM as the discretization boundary elements. The fluid pressure and temperature are coupled in two systems of equations and a sequential coupling iteration procedure was employed to solve the equations. This paper introduces the FVM for the fluid temperature and the sequential iteration procedure for temperature and pressure which was implemented in a code, and the numerical results agree well with an analytical solution for the temperature of fluid flowing through a rectangular fracture. The numerical scheme was then employed to simulate illustration examples of heat transfer between two well holes.

Numerical vibroacoustic analysis of multi-layered composite structure under hygrothermal loading using coupled FEM-IBEM micromechanical model

The vibroacoustic responses of multi-layer composite shallow shell panels subjected to harmonic mechanical excitation in hygrothermal environment is numerically investigated. A homogenized micromechanical finite element (FE) based on the higher-order mid-plane kinematics replicating quadratic function as well as the through the thickness stretching effect together with the indirect boundary element (IBE) scheme has been first time employed. The isoparametric Lagrangian element with zero-order Hermitian interpolation function (10 degrees of freedom per node) is used for structural model discretization to attain the hygro-thermo-elastic natural frequencies via Hamilton’s principle. The uniqueness of the model is the incorporation of the effective material properties under combined hygrothermal loading via a micromechanics approach. An IBE method is then implemented to attain structure-surrounding coupling and the Helmholtz wave equation is solved to compute the sound radiation responses. The effectiveness of the model is tested by converging it with the similar analytical/numerical results as well as the experimentally acquired data for vibrational as well as acoustic responses. The present scheme is further hold out for solving diverse numerical illustrations. The geometrical parameters, volume fraction of fiber, layup, and support conditions alongside the hygrothermal load is found to have significant influence on the hygro-thermo-acoustic characteristics. The functional inferences will assist in designing the imminent multi-layered structures that serve under the influence of severe hygrothermal loading during high-performance engineering applications.

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 amplification effect of river valley topography on the seismic response of a tunnel near a valley: simulated using the boundary element method

Many tunnels are inevitably built in or near valley terrains, which greatly amplifies the seismic response of the tunnel. This paper considers the interaction between the valley terrain and tunnels, proposes a high-precision indirect boundary element method (IBEM) for simulation, and verifies the correctness of the method. The impact of factors such as incident-wave type, incident-wave frequency, and distance between the tunnel and the valley on the seismic response of the tunnel is then analysed in detail. The numerical analysis demonstrates that the stress of the tunnel can be approximately doubled when the distance between the river valley and the tunnel is 10 m due to the scattering and coherence effects of seismic waves. However, when this distance is greater than 50 m, the presence of river valleys has little effect on the stress of the tunnel. The stress at the right shoulder of the tunnel and the displacement amplification effect on the valley slope is significant compared to other areas. The findings of this study provide theoretical guidance for the seismic fortification and disaster avoidance of tunnel structures near river valleys.

On the influence of soil flexibility in structural topology optimization

AbstractThis paper investigates for the first time the effect of soil flexibility in structural topology optimization. The case of a structure resting on an underlying soil is explored. The response of the soil is modeled using the Indirect Boundary Element Method (IBEM). The structure is modeled using Finite Element Method (FEM) and the soil domain is coupled by imposing continuity and equilibrium conditions at the soil‐structure interface. Original soil stress influence functions are derived to model voids in the soil. The optimization is carried out by using the Topology Optimization of Binary Structures (TOBS), a recently developed method that employs sequential integer linear programming and the branch‐and‐cut algorithm. Structural compliance minimization subject to a volume constraint is solved. Selected optimization problems are solved for various soil‐to‐structure flexibility ratios. Results show that the soil flexibility may have a significant impact in both the optimized topology and in the achievable compliance of the structure. Results considering voids on the soil, which increases the soil flexibility, affected the final topology of the structures as well as the optimized values.

APPLICATION OF DISCONTINUOUS SOLUTIONS IN BOUNDARY ELEMENT METHOD FOR THE BENDING PROBLEMS OF KIRCHHOFF PLATES WITH AN ARBITRARY CONTOUR

The discontinuous solutions represent a new direction for the indirect Boundary Element Method (BEM) in the indirect formulation. Discontinuous solutions represent those functions, which when crossing certain lines, the transverse deflection, the slope angle, the bending moment and the generalized shear force may have jumps. These can be used to solve important problems for which the existing methods do not have solutions or a satisfactory accuracy, such as: plates of arbitrary contour, presence of defects, mixed boundary conditions, contact problems, infinite domains etc. In this paper, we describe the methodology of application and the numerical implementation of discontinuous solutions for the bending problems of plates with an arbitrary contour in the classical theory (Kirchhoff). For this purpose, programming codes were developed in the Matlab language, which allowed to calculate the displacements and efforts in the plate. The obtained results were compared with the Finite Element Method (FEM) for different mesh densities.

Broadband seismic isolation effect of three-dimensional seismic metamaterials in a semi-infinite foundation: modelled by fast multipole indirect boundary element method

In this study, a fast multipole indirect boundary element method (FM-IBEM) is developed for studying the scattering of seismic waves by 3-D seismic metamaterials (SMs), and a 3-D rock soil composite seismic metamaterial (RSCSM) is proposed. Based on the indirect boundary element method, the fast multipole expansion technique is adopted to reduce the calculation time and memory requirements, and effectively solve the broadband scattering problem of seismic waves by large-scale SMs. The bandgap characteristics of three-dimensional SM are studied in detail, and the effects of the geometric and material parameters on the isolation effect of SMs are investigated. The results show that RSCSM can attenuate the ultralow frequency of a seismic wave, the bandgap width is 0.5–19 Hz, and the isolation effect is 37%. The cell size and material parameters significantly affect the bandgap characteristics of the RSCSM. This type of seismic metamaterial provides a new technique for seismic wave isolation and could be applied in engineering structures.

Design sensitivity analysis of three-dimensional acoustic problems using a fast multipole accelerated indirect BEM

In this paper, the design sensitivity formulations for three-dimensional acoustic problems are presented based on the indirect boundary element method (IBEM). The non-unique solution difficulty of the IBEM in solving exterior acoustic problems is avoided by applying a hybrid combination of single and double layer potentials. In order to accelerate the computational procedure easily, a set of non-singular indirect boundary integral equations (IBIEs) for the acoustic design sensitivity analysis are derived and then the fast multipole method (FMM) is employed to accelerate the matrix–vector products in the IBEM. The method is further extended to the design sensitivity analysis of thin-body acoustic problems above an infinite plane. Numerical examples are given to verify the effectiveness and accuracy of the method, and the simulations of thin-body acoustic problems show its potential in engineering applications.

Optimization of Electrode-Spacer Arrangement Using Simplex Algorithm

The present article presents the optimization of electric stress distribution over the live electrode of an axisymmetric electrode-spacer arrangement that is used in a Gas insulated substation (GIS) typically for 12–420 kV operations. The technique is based on a classical approach employing Simplex Algorithm which is used to solve constrained linear programming problems (LPPs). For this purpose, the objective function is obtained by applying multiple linear regression analysis by which a mathematical relationship between the maximum resultant electric stress <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$({E}_{\text {max}}^{R})$ </tex-math></inline-formula> over the surface of the live electrode and the critical dimensions affecting this stress. For this purpose, a set of 71 data is used for building the regression model, which is prepared by varying the critical dimensions within the constraints of the overall dimension of the system and calculating the value of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${E}_{\text {max}}^{R}$ </tex-math></inline-formula> over the surface of the live electrode by employing the indirect boundary element method (BEM).

IBEM-FEM coupling method for full process nonlinear ground motion simulation of near-fault sedimentary basins

Based on the concept of domain reduction, a two-step method combining the indirect boundary element method (IBEM) and the finite element method (FEM) was developed to simulate the full-process ground motion of a 3D (three-dimensional) near-fault complex site. In the first step, the seismic response of dislocation motion in semi-infinite space crust model was accurately solved using the IBEM, which can significantly reduce the dimensionality of the numerical model and easily achieve a computational frequency of 5 Hz on an ordinary workstation. In the second step, considering the inhomogeneity and nonlinearity of soft soils, the seismic wave scattering from a 3D local site was finely simulated using a FEM. The external seismic response obtained through IBEM is converted into equivalent load and transferred to the internal finite element domain, while the PML boundary is applied to absorb the pseudo reflected wave on the truncated boundary. Then, the reliability and accuracy of the method was demonstrated by comparing with the results of existing studies and the traditional IBEM. Further, the proposed coupled method was used to investigate the seismic response of a large near-faulted sedimentary basin. The numerical investigation focuses on four aspects: acceleration histories and response spectra, wave-field snapshot, duration of ground motion, and peak magnitudes and response spectrum of velocity. The results highlight the influence of local site soil non-linearity and medium impedance ratios on the analysis of near-fault seismic response. This research is essential for seismic zoning of complex sedimentary basin sites near faults, and also establishes a “bridge” for seismic design from faults to structures.

The scattering of seismic waves from saturated river valley with water layer: Modelled by indirect boundary element method

River valley topography will cause local amplification effect and significant spatial variation effect of ground motion. In this paper, an indirect boundary element method (IBEM) is proposed to solve seismic wave scattering in sedimentary valleys. A ``water layer - soft sedimentary layer - poroelastic halfspace'' model is established, and the effects of water layer, soil parameters of sedimentary layer and terrain on incident P and SV waves are discussed in detail. The numerical results demonstrate that the scattering characteristics are closely related to the depth of water layer, porosity, the location of observation points, et al. The surface displacement amplitude decreases with increasing water layer thickness, with a maximum reduction of 16.2%. Moreover, the peak ground acceleration (PGA) is significantly larger and the seismic response is stronger when the observation point is located outside the valley compared to the observation point inside the valley. The influence of porosity on Fourier spectrum and response spectrum curve is mainly reflected in the peak value and predominant frequency. The research results can provide a theoretical basis for the seismic fortification of the structures around the saturated sedimentary valley.

A benchmark study on room acoustic simulations with various material input complexities

This work investigates and evaluates the capabilities of an explicit finite-difference time-domain (FDTD), a fast multipole indirect boundary element method (FMBEM), and a ray-tracing (RT) solver in the context of room acoustic modeling and simulations. In room acoustic simulations, the wave-based FDTD and FMEBM methods are known for generating accurate results at low frequencies, while the RT technique is in principle more valid at higher frequencies. The numerical aspects of setting up the solutions for each solver are discussed. Special attention is given to the influence of material input data of various degrees of detail. Single/multiple frequency-independent/dependent materials are considered in the model setup. The modeling capabilities of the three solvers in handling material input with various complexities are analyzed. Numerical results are evaluated in both frequency and time domains. Room acoustic parameters, including the reverberation time (T_20), early decay time (EDT), clarity (C_80), and definition (D_50), are compared. These results are also compared with available measurement data. Last but not least, the computational efficiencies of the three solvers are briefly discussed.

Effect of Randomness of Parameters on Amplification of Ground Motion in Saturated Sedimentary Valley

Based on Biot’s theory and the indirect boundary element method (IBEM), the Monte Carlo method is utilized to generate random samples to calculate the displacement response of a saturated sedimentary valley under SV wave incidence. The purpose of this paper is to explore the effects of randomness of porosity, permeability coefficient, and geometric shape on the seismic amplification effect of saturated complex sites. It is shown that the change of media porosity in the saturated site with defined boundary has relatively little influence on the ground motion, and the influence of the permeability coefficient is slightly larger. While in the site with a random boundary, the influence of both the porosity and permeability coefficient are significant, which cannot be ignored. The conclusion plays an important guiding role in earthquake disaster prevention and mitigation, such as seismic risk analysis and earthquake microzonation in saturated sedimentary valleys.

Featured Cover

The cover image is based on the Research Article A novel high-order collocation indirect boundary element method based on the Leis formulation for three-dimensional high frequency exterior acoustic problems by Chao Zhang et al., https://doi.org/10.1002/nme.7138.