Boundary Element Program Research Articles

BEM Modeling for Stress Sensitivity of Nonlocal Thermo-Elasto-Plastic Damage Problems

The main objective of this paper is to propose a new boundary element method (BEM) modeling for stress sensitivity of nonlocal thermo-elasto-plastic damage problems. The numerical solution of the heat conduction equation subjected to a non-local condition is described using a boundary element model. The total amount of heat energy contained inside the solid under consideration is specified by the non-local condition. The procedure of solving the heat equation will reveal an unknown control function that governs the temperature on a specific region of the solid’s boundary. The initial stress BEM for structures with strain-softening damage is employed in a boundary element program with iterations in each load increment to develop a plasticity model with yield limit deterioration. To avoid the difficulties associated with the numerical calculation of singular integrals, the regularization technique is applicable to integral operators. To validate the physical correctness and efficiency of the suggested formulation, a numerical case is solved.

Impact of Tunneling on Adjacent Piles Based on the Kerr Foundation Model Considering the Influence of Lateral Soil

The Kerr foundation model simulates the interaction between piles and soil. Considering the impact of lateral soil displacement on adjacent piles, the lateral displacement and bending moment of the adjacent piles caused by shield tunnel excavation are calculated in detail. Additionally, the reactions of groups of piles are obtained by focusing on the shielding effect of the piles on the soil displacement caused by shield tunnel excavation. The validity of the solutions is verified by comparing the calculated results with the boundary element program GEPAN. Additionally, adjacent pile lateral displacement and bending moment are compared, with and without considering lateral soil effects. Furthermore, this study investigates the influence of various factors, such as soil spring stiffness, pile–tunnel distance, ground loss ratio, and pile diameter on the pile group’s lateral displacement and bending moment. The research findings indicate that increasing the soil spring stiffness or the horizontal distance between the pile and tunnel can reduce the lateral displacement and the bending moment of the pile. On the other hand, as the ground loss ratio gradually increases, the pile lateral displacement and bending moment will also increase. However, when the diameter of the pile grows, the lateral displacement reduces, while the bending moment increases.

A machine learning model for fast approximation of free-surface Green's function and its application

In potential flow theory, the accurate and efficient calculation of free-surface Green's functions is essential for solving hydrodynamic issues. Given the impressive performance of machine learning methods in nonlinear function fitting, the present study utilizes an effective machine learning model called StripeGF for numerical approximation. In this model, equidistant horizontal datum lines are arranged in the computational domain away from the singularity, and Green's function and its derivatives on each line are fitted by a multi-layer perceptron (MLP) with a single input. Based on the first-order ordinary differential equation (ODE) that they satisfy, the fourth-order Runge-Kutta method is used to solve the Green's function and its derivatives between adjacent lines. In the domain nearing the singularity, a double-input MLP is applied. Use the Romberg quadrature to create a double-precision data set for training and validation, the numerical results demonstrate that StripeGF outperforms all 4 comparison methods in terms of efficiency and has accuracy of at least 4 digits in more than 99.9% of all zones. The boundary element program improved by StripeGF is verified in the hydrodynamic calculation of S175, showing good accuracy and reliability.

Wave acoustic solutions in small rooms

This presentation will discuss the growing application of wave based solutions due to the increased processing power of desktop and distributed cloud cluster computing. The causes and mitigation of four types of potential acoustical distortion in small rooms, below and above 200 Hz, will be presented. Below 200 Hz, approaches to minimize modal effects and the speaker boundary interference response will be presented. We will discuss limitations of traditional dimensional ratio determinations, more accurate wave-based predictions, importance of speaker and listener locations, and prediction and measurement of low frequency absorption devices. Above 200 Hz, comb filtering from strong specular reflections and poor diffusion are potential problems. We will describe the necessary ratio of surface width to incident wavelength for a specular reflection and its effect on comb filtering. Finally, we describe the importance of creating a mixing room design and the use of broad bandwidth diffusive surfaces, which can be both measured and simulated with a virtual goniometer boundary element program called VIRGO, from a 3D CAD file topology of any shape. This offers the acoustician the ability to evaluate surface topologies that are of interest in a room design quickly and inexpensively.

Theoretical analysis of the barrier effect of embedded isolation piles on tunneling-induced vertical ground displacements

This paper proposes an elastic solution for evaluating tunneling-induced vertical ground displacements due to the restraint of embedded isolation piles. Based on the modified Loganathan-Poulos (L&P) formula and the general form of the Melan/Mindlin solution, the restrained tunneling-induced vertical ground displacements can be calculated according to the displacement compatibility condition at the pile-soil interface. The solution can consider the relative vertical slip between the pile shaft and the surrounding soil and the embedding of the pile tip into the supported soil. The method is verified through comparisons with existing analytical solutions, boundary element program and field-measured data. Parametric analysis indicates that the parameters associated with the isolation pile (the distance of the tunnel axis from the pile axis, the pile length and the relative pile axis stiffness) and the soil (Poisson's ratio, the friction angle and the ground volume loss) have important influences on the ground surface settlement troughs and the barrier efficiency. The results also show that the pile skin-soil shear stiffness, which characterizes the relative slip between the pile shaft and the surrounding soil, determines the barrier efficiency. When the pile-soil interface is completely bonded, the contact state of the pile tip with the supported soil also has a nonnegligible impact on the barrier efficiency.

Simplified solution for tunnel-soil-pile interaction in Pasternak’s foundation model

The current design practice for predicting the interaction mechanics for tunnel-soil-pile is generally based on Winkler’s foundation, which is subject to some important limitation, such as ignoring the continuity of the soil foundation. Furthermore, the current analytical studies are mostly employed the plane strain analyses and do not consider the influences of lateral soil displacements on pile behaviour. To improve the accuracy for the pile behaviour prediction induced by tunnelling, the analytical method should account for the effects of a number of parameters, such as the ground shearing displacements, and the influence of lateral soil displacements next to the pile. This paper focuses on a simplified solution based on Pasternak’s foundation model to predict the lateral displacements and internal forces of a single-pile and group-piles induced by tunnelling considering the effects of lateral soil displacements. First, the simplified solution of tunnel-soil-pile interaction, which reflects the influence of shearing displacements of foundation, is established on Pasternak’s foundation model. Second, the equivalent concentrated forces are supplied to the pile through the shear layer to consider the influence of lateral soils beside the pile. The validity of the solutions is verified by the boundary element program results, centrifuge test data, and field measurements. The calculated results are also compared with and without considering the effects of tunnel-soil-pile interaction. When the influences of lateral soil displacements are considered, the results are shown to be closer to the monitored in-situ data and the centrifuge test data. In addition, the influencing factors of a single-pile and group-piles displacements are also investigated, including the shear layer modulus, pile diameter, ground-loss ratio, pile-tunnel distance, and pile spacing. The influence of soil shear displacements on pile response cannot be ignored, and an error may occur when Winkler’s foundation model is used to solve this problem.

Numerical simulation of crack growth in piezoelectric structures by BEM

In this paper, a dual boundary element computer program is developed for numerical simulation of a crack propagating in piezoelectric plates under a combined quasi-static electric and mechanical loading. To determine the crack growth path, two fracture criteria are taken into account: the maximum of hoop stress intensity factor (HSIF) and hoop mechanical strain energy release rate (HMERR). By using the displacement extrapolation method, these fracture parameters of any small kinked crack branch are obtained and validated by comparing with the available analytical results. The critical fracture loads for some test specimens are numerically analyzed based on the maximum HMERR fracture criterion. Different electrical boundary conditions on the crack faces are considered and checked with the experimental data. Finally, one crack or a pair of cracks propagating in infinite or finite piezoelectric plates is numerically simulated. The influences of the loading conditions, the anisotropic fracture toughness and the interaction between the cracks on the crack growth paths are also studied. The comparisons with the exiting finite element results show the accuracy and efficiency of the present BEM program for numerical simulation of crack growth in piezoelectric materials.

A family of exponentially-gradient elements for numerical computation of singular boundary value problems

In this paper, a new family of single-parameter exponentially gradient elements (EG-elements) are introduced, which can be used in various numerical procedures such as boundary and finite element methods. These elements have the ability to accurately interpolate the unknown values in regions, where either high gradient or singularity of the unknown field occurs. The shape functions of two-dimensional EG-elements are high gradient at either a corner of the element or at an edge of the element. Another advantage of this element is that the regular quadratic shape functions are obtained as a special case of EG-elements by adjusting the single parameter of the element, which allows this element to be used as regular Lagrange quadratic element, where it is appropriate. Some mixed boundary value problems are solved with the use of EG-elements in a boundary element program to show the capability of these elements for capturing the solution with less number of elements and higher accuracy.

Stress characterization and support measures estimation around a coalmine tunnel passing through jointed rock masses: constraints from BEM simulation

In the present research, the stress characterization around an unsupported coalmine tunnel passing through jointed rock masses was ana-lyzed and effective support system was calculated by BEM numerical simulations. The distribution and magnitudes of major and minor principal stress contours, mean stress, differential stress, total displacement, maximum shear strain, maximum shear stress contours around the tunnel are simulated by using the examine2D software. It is reasonable to mention that examine2D is a plane strain boundary element program for calculation of stresses and displacements around underground and surface excavation in rock. Modeling results reveal that the major principal stress (σ1) was about 13 MPa at the immediate roof of the tunnel that ultimately increased to 20 MPa toward the left side and right side. Mean stress contour value was 12 MPa at the immediate roof and 15 MPa toward the both rib sides. The distribution contour value of differential stress at the roof and rib sides were 16 MPa and 23 MPa, respectively. The contour values of the strength factor around the tunnel ranged from 0.51 to 1.02, which specify that the loosening zone would be extended up to 1.53 m towards the roof and 1.25 m at the sidewalls. The thickness (1.53 m) of loosening zone can be classified as soft or poor rock mass. In the immediate roof, floor and the both rib sides of the tunnel, the Spalling Criterion values ranged from 2.7 to 8.0 MPa, which indicate no potential for rock-burst around the tunnel. However, flexible support would be required to accommodate the dilatancy deformation during development period. Finally, the stiff support would be required to provide a strong supporting reaction and to maintain the long-term stability of the tunnel.

Elimination of boundary effect for laminated overburden model in pillar stability analysis

The laminated overburden model (LaModel) has been widely used for pillar design and stability analysis. As a boundary element program, the LaModel program is sensitive to the boundary condition, which should be considered before creating the model. To eliminate the boundary effect in a LaModel pillar stability analysis, a suitable boundary buffer zone is needed around the model edge. The radius of influence (R) and the abutment load extent (D) are two major factors that affect the stresses and displacements calculated in LaModel. To determine the optimum buffer zone extent, a database of case histories was analyzed using the LaModel program. Values for R and D were varied until a buffer zone having negligible influence on the pillar stability factor (SF) of the active mining zone (AMZ) was determined.

Inverse analysis for interface fracture toughness of Ti coating film by laser spallation method

In the present paper, the interface fracture toughness between a Ti coating film and Al-alloy substrate is evaluated using a laser spallation method and a boundary element method. The fracture toughness can be estimated using inverse analyses by the boundary element method using a transfer function computed from the history of the displacement of the specimen. In the present study, an alternative boundary element program is developed for unsteady state vibration of an axi-symmetric solid body. The mode I interface fracture toughness between the Ti coating film and Al-alloy substrate is confirmed to be about 0.66 MPam1/2 from the present investigation.

OS1205-381 レーザースポレーション法によるTiコーティング膜の密着強度と界面破壊靭性値の評価

In the present study, the interface fracture toughness and adhesioln strength between a Ti coating film and Al-alloy substrate was evaluated using a laser spallation method. The fracture toughness and adhesion strength can be estimated using inverse analyses by the boundary element method using a transfer function computed from the history of the displacement of the specimen. In the present study, an alternative boundary element program is developed for unsteady state vibration of an axisymmetric solid body. The mode I interface fracture toughness between the Ti coating film and Al-alloy substrate is confirmed to be about 0.66 MPam^<1/2> from the present investigation.

Boundary Element Modeling of Crack Propagation Path in Brittle Matrix Composites Reinforced by Short Rectangular Particulates

Interaction between a crack and a rectangular particulate in a single edge notched beam is simulated using a boundary element program of automatic crack extension. The stress intensity factors of the growing crack are obtained from the J-integral. Three crack extension paths: deflecting around the particulate, growing along the interface and penetrating into the particulate are achieved in terms of the mismatch state of mechanical characteristics of matrix and particulate. The toughening is also given by the ratio of stress intensity factors. The results reveal that as stress shielding occurs, large toughening is obtained when the crack is approaching to a stiff and strong particulate weakly bonded to a relatively soft matrix; inversely the soft particulate is functional to attract the crack tip toward it, and obvious toughening arises when the tip is within this soft particulate. The present work intends to help for the design of particulate reinforced composites.

Analytical derivation and numerical experiments of degenerate scales for regular N-gon domains in two-dimensional Laplace problems

Degenerate scale of a regular N-gon domain is studied by using the boundary element method (BEM) and complex variables. Degenerate scale stems from either the non-uniqueness of BIE using the logarithmic kernel or the conformal radius of unit logarithmic capacity in the complex variables. Analytical formula and numerical results for the degenerate scale are obtained by using the conformal radius and boundary element program, respectively. Analytical formula of the degenerate scale contains the Gamma function for the Gamma contour which can be derived from the Schwarz–Christoffel mapping. Based on the dual BEM, the rank-deficiency (mathematical) mode due to the degenerate scale (mathematics) is imbedded in the left unitary vector for the influence matrices of weakly singular (U kernel) and strongly singular (T kernel) integral operators. On the other hand, we obtain the common right unitary vector corresponding to a rigid body mode (physics) in the influence matrices of strongly singular (T kernel) and hypersingular (M kernel) operators after using the singular value decomposition. To deal with the problem of non-unique solution, the constraint of boundary flux equilibrium instead of rigid body term, CHEEF and hypersingular BIE, is added to promote the rank of influence matrices to be full rank. Null field for the exterior domain and interior nonzero field are analytically derived and numerically verified for the normal scale while the interior null field and nonzero exterior field are obtained for the homogeneous Dirichlet problem in the case of the degenerate scale. It is found that the contour of nonzero exterior field for the degenerate scale using the BEM matches well with that of Schwarz–Christoffel transformation. Both analytical and numerical results agree well in the demonstrative examples of right triangle, square, regular 5-gon and regular 6-gon. It is straightforward to extend to general regular N-gon case.

Evaluation of Joining Strength of Silicon-Resin Interface at a Vertex in a Three-Dimensional Joint Structure

Portable electric devices such as mobile phones and portable music players have become compact and improved their performance. High-density packaging technology such as chip size package (CSP) and stacked-CSP is used for improving the performance of devices. CSP has a bonded structure composed of materials with different properties. A mismatch of material properties may cause a stress singularity, which leads to the failure of the bonding part in structures. In the present paper, stress analysis using the boundary element method and an eigenvalue analysis using the finite element method are used for evaluating the intensity of a singularity at a vertex in three-dimensional joints. A three-dimensional boundary element program based on the fundamental solution for two-phase isotropic materials is used for calculating the stress distribution in a three-dimensional joint. Angular function in the singular stress field at the vertex in the three-dimensional joint is calculated using an eigenvector determined from the eigenvalue analysis. The joining strength of interface in several kinds of sillicon-resin specimen with different triangular bonding areas is investigated analytically and experimentally. An experiment for debonding the interface in the joints is firstly carried out. Stress singularity analysis for the three-dimensional joints subjected to an external force for debonding the joints is secondly conducted. Combining results of the experiment and the analysis yields a final stress distribution for evaluating the strength of interface. Finally, a relationship of force for delamination in joints with different bonding areas is derived, and a critical value of the 3D intensity of the singularity is determined.

Boundary element formulation for the coupled stretching–bending analysis of thin laminated plates

The boundary integral equations for the coupled stretching–bending analysis of thin laminated plates involve an integral which will be singular when the field point approaches the source point. To avoid the singular problem occurring in the numerical programming, the boundary integral equations are modified in which the integrals of singular part are integrated analytically. The analytical solutions for the free term coefficients and singular integrals are obtained in explicit closed-form. By dividing the boundary into elements and using suitable interpolation polynomials for basic functions, the set of equations necessary for boundary element programming are written explicitly for regular nodes and corner nodes. The equations for the determination of displacements and stresses at internal points are also presented in this paper.

Simulation of linear inclusions with the BEM

In this paper a novel numerical approach is presented for the simulation of solids containing thin linear inclusions with the boundary element method (BEM). Assumptions are that the inclusions are in continuous contact with the surrounding solid and that they are only able to carry axial stresses, their bending stiffness is neglected. A practical application is the simulation of underground openings such as tunnels supported by rock bolts. To avoid an increase in the number of unknowns in the system of equations, the problem is solved iteratively. The iterative solution procedure is particularly suitable for solving problems containing a high number of inclusions and involving nonlinear material behaviour, where an iterative solution procedure has to be used anyway. The approach has been implemented into the boundary element program BEFE ++ . To verify the method a test example is presented, considering elastic and plastic material behaviour. The results are compared with those calculated by finite elements.

A simplified analysis method for the influence of tunneling on grouped piles

One of key issues of tunneling in urban areas is to assess the likely impact on adjacent piled buildings of tunnel construction. Simple and reliable predictions of tunneling-induced bending and axial stresses in pile foundations are important to the safety of tunneling. In this paper, a simple two-stage analysis method for determining the response of pile groups caused by tunneling was presented. At the first stage, an analytical solution proposed by Loganathan and Poulos [Loganathan, N., Poulos, H.G., 1998. Analytical prediction for tunneling-induced ground movement in clays. J. Geotech. Geoenviron. Eng., ASCE 124 (9), 846–856] is used to estimate the free-field vertical and lateral soil movements induced by tunneling. At the second stage, assuming no slippage at the soil-pile interface, the Winkler model is first adopted for simulating the pile-soil interaction, combined with finite difference method in the case of multi-layered soils. Then, shielding effect is considered for the interaction between two passive piles using a logarithmic attenuation function suggested by Randolph and Wroth [Randolph, M.F., Wroth, C.P., 1979. Analysis of the vertical deformation of pile groups. Géotechnique 29 (4), 423–439] for vertical response and Mindlin’s solution for lateral response. Finally, the response of a passive pile group due to tunneling is obtained by the superposition principle. Solutions obtained by the proposed approach for the analysis of single piles and piled groups subjected to ground movements induced by tunneling are compared with those using the boundary element program GEPAN. Comparisons are also made between the observed behavior of centrifuge model tests as well as field measurements and those computed by the proposed method. It is demonstrated that the present method can in general give a satisfactory prediction of the response of passive piles subjected to tunneling.

Advanced Boundary Element Analysis of Three-Dimensional Elastic Solids with Fiber Reinforcements

An advanced boundary element methodology for the analysis of three-dimensional fiber-reinforced elastic solids using the concept of fiber elements has been presented in this paper as an extension of the earlier work of Banerjee and co-workers. The previous simplified formulation was based on the assumption that the Poisson ratios of the matrix and the fiber are equal. However, this may not be a valid assumption for all values of elastic stiffness ratios and fiber to matrix volume ratios. Moreover, such restrictions do not allow for any future extensions to nonlinear analysis. Also, because of limiting computing power available at that time, their implementation was restricted to only a small number of fiber elements in a given analysis. The new algorithm proposed in this work does not make any such assumptions. In this algorithm, the fibers are semianalytically integrated and the resulting equations are back-substituted into the equations for the composite matrix without any further approximation. The present work has been implemented in a general purpose multiregion boundary element computer program and is capable of handling very large numbers of fiber elements in a given analysis. Several numerical examples are presented to validate the proposed method of fiber composite analysis and its applicability is demonstrated via practical engineering problems.

Modeling of the acoustic eigenproblem with sound absorption using boundary element method

This paper deals with determination of resonant frequencies for absorbant 3D acoustic cavities. The behaviour of the sound absorbing boundary can be described with a Robin condition as proposed by Rajakumar et al. [Int. J. Numer. Methods Eng., 36, 3957‐3072 (1993)]. This approach is inaccurate, especially for low frequencies because the absorption coefficient is assumed to be constant. We observed the acoustic admittance for foam and fibrous type materials varies linearly for low frequencies. The introduction of a new absorption coefficient allows to take into account this behavior in order to improve the accuracy in the determination of the first modes (typically less than 500 Hz in car interior). This formulation has been implemented in a boundary element program we have developed. The results are compared with those given by the finite element program ANSYS. Computations are carried out for rectangular parallelepiped and Sedan car interior.