Elastic Boundary Element Method Research Articles

AstroSeis: A 3D Boundary Element Modeling Code for Seismic Wavefields in Irregular Asteroids and Bodies

Abstract We developed a 3D elastic boundary element method computer code, called AstroSeis, to model seismic wavefields in a body with an arbitrary shape, such as an asteroid. Besides the AstroSeis can handle arbitrary surface topography, it can deal with a liquid core in an asteroid model. Both the solid and liquid domains are homogenous in our current code. For seismic sources, we can use single forces or moment tensors. The AstroSeis is implemented in the frequency domain, and the frequency-dependent Q can be readily incorporated. The code is in MATLAB (see Data and Resources), and it is straightforward to set up the model to run the code. The frequency-domain calculation is advantageous to study the long-term elastic response of a celestial body due to a cyclic force, such as the tidal force, with no numerical dispersion issue suffered by many other methods requiring volume meshing. Our AstroSeis has been benchmarked with other methods such as normal-mode summation and the direct solution method. This open-source AstroSeis will be a useful tool to study the interior and surface processes of asteroids.

PyFrac: A planar 3D hydraulic fracture simulator

Fluid driven fractures propagate in the upper earth crust either naturally or in response to engineered fluid injections. The quantitative prediction of their evolution is critical in order to better understand their dynamics as well as to optimize their creation. We present an open-source Python implementation of a hydraulic fracture growth simulator based on the implicit level set algorithm originally developed by Peirce & Detournay (2008). This algorithm couples a finite discretization of the fracture with the use of the near tip asymptotic solutions of a steadily propagating semi-infinite hydraulic fracture. This allows to resolve the multi-scale processes governing hydraulic fracture propagation accurately, even on relatively coarse meshes. We present an overview of the mathematical formulation, the numerical scheme and the details of our implementation. A series of problems including a radial hydraulic fracture verification test, the propagation of a height contained hydraulic fracture, the lateral spreading of a magmatic dyke and an example of fracture closure are presented to demonstrate the capabilities, accuracy and robustness of the implemented algorithm. Program summaryProgram title:PyFracCPC Library link to program files:http://dx.doi.org/10.17632/gv7yy9mmwj.1Licensing provisions: GPLv3Programming language: PythonNature of problem: Simulation of the propagation and closure of a planar three-dimensional hydraulic fracture driven by the injection of a Newtonian fluid in a material having heterogeneous fracture toughness under a non-uniform in-situ stress field.Solution method: The fully coupled hydro-mechanical moving boundary problem is solved combining a finite volume scheme for lubrication flow with a boundary element method for elasticity. The algorithm couples a finite scale discretization of the fracture with the near-tip asymptotic solution of a steadily moving hydraulic fracture. The fracture front is tracked via a level set approach using a fast marching method.

A state space boundary element method for elasticity of functionally graded materials

PurposeThe state space method (SSM) is good at analyzing the interfacial physical quantities in laminated materials, while it has difficulty in calculating the mechanical quantities of interior points, which can be easily evaluated by the boundary element method (BEM). However, the material has to be divided into many subdomains when the traditional BEM is applied to analyze the functionally graded material (FGM), so that the computational amount will be increased enormously. This study aims to couple these two methods to strengthen their advantages and overcome their disadvantages.Design/methodology/approachHerein, a state space BEM in which the SSM is coupled by the BEM is proposed to analyze the elasticity of FGMs, where one BEM domain is set and the others belong to SSM domains. The discretized elements occur only on the boundary of the BEM domain and at the interfaces between different SSM domains. In SSM domains, the horizontal interfaces of FGMs are discretized by linear elements and the variables along the vertical direction are yielded by the precise integration method.FindingsThe accuracy of the proposed method is verified by comparing the present results with the ones from the finite element method (FEM). It is found that the present method can provide accurate displacements and stresses in the FGMs by fewer freedom degrees in comparison with the FEM. In addition, the present method can provide continuous interfacial stresses at the interfaces between different material domains, while the interfacial stresses by the FEM are discontinuous.Originality/valueThe system equations of the state space BEM are built by combining the boundary integral equation with the state equations according to the continuity conditions at the interfaces. The mechanical parameters of any inner point can be evaluated by the boundary integral equation after the unknowns on the boundaries and interfaces are determined by the system equation.

Improvement of squeezing ground prediction and monitoring capabilities for Mount Isa Mines Northern 3500 orebody

This paper investigates readily incorporable improvements to the identification of squeezing ground conditions and the determination of the depth of movement for Mount Isa Copper Operations’ Northern 3500 (MICO N3500) orebody. Numerical modelling using the boundary element method (BEM) indicated that the current use of elastic material properties to calculate the rockwall condition factor and predict tunnel squeezing is appropriate, as no apparent benefit was gained by employing elastoplastic material properties or including the tunnel void geometry. Further analysis using the finite element method indicated that complex, large-scale models would be required to make meaningful improvement on the elastic BEM model. However, due to large time and computational overheads such models are not practical within the site’s engineering workflow. In conjunction with numerical modelling, a prototype monitoring device was also developed to address the problem of rigid instrument shear and inspection camera obstruction when determining the depth of movement in a squeezing rockmass. The reverse downhole probe device will allow distance measurement and a live video feed to be returned to the excavation surface by taking advantage of stable ground beyond the excavation zone of influence and using a thin, flexible cable to transmit power and data.

접착이음의 계면덧살에 대한 응력특이성 해석

AbstractThe stress singularity for the excess adhesive on interface of adhensively bonded joint was investigated by using the 2-dimensional elastic boundary element method (BEM). To establish a reasonable strength evaluation method and a fracture criterion for the excess adhesive of interface in adhensively bonded joint, it is necessary to evaluate fracture parameters with various bonding conditions. Under the variations of adhensively bonded thickness (h) and diameter (d) for the excess adhesive, a stress analysis was peformed, and from the results, the stress singularity index (  ) and the stress singularity factor (  ) were calculated. The variations have a great influences on the stress singularity for the excess adhesive of interface in adhensively bonded joint, and the  is reduced as the “h” and “d” increase. Key Words : Adhensively bonded joint(접착이음), Boundary element method(경계요소법), Excess adhesive(접착덧살), Stress singularity index(응력특이성지수), Stress singularity factor(응력특이성계수), Strength evaluation(강도평가), Fracture parameter(파괴인자)

Multiple circular nano-inhomogeneities and/or nano-pores in one of two joined isotropic elastic half-planes

The paper considers the problem of multiple interacting circular nano-inhomogeneities or/and nano-pores located in one of two joined, dissimilar isotropic elastic half-planes. The analysis is based on the solutions of the elastostatic problems for (i) the bulk material of two bonded, dissimilar elastic half-planes and (ii) the bulk material of a circular disc. These solutions are coupled with the Gurtin and Murdoch model of material surfaces [Gurtin ME, Murdoch AI. A continuum theory of elastic material surfaces. Arch Ration Mech Anal 1975;57:291–323; Gurtin ME, Murdoch AI. Surface stress in solids. Int J Solids Struct 1978;14:431–40.]. Each elastostatic problem is solved with the use of complex Somigliana traction identity [Mogilevskaya SG, Linkov AM. Complex fundamental solutions and complex variables boundary element method in elasticity. Comput Mech 1998;22:88–92]. The complex boundary displacements and tractions at each circular boundary are approximated by a truncated complex Fourier series, and the unknown Fourier coefficients are found from a system of linear algebraic equations obtained by using a Taylor series expansion. The resulting semi-analytical method allows one to calculate the elastic fields everywhere in the half-planes and inside the nano-inhomogeneities. Numerical examples demonstrate that (i) the method is effective in solving the problems with multiple nano-inhomogeneities, and (ii) the elastic response of a composite system is profoundly influenced by the sizes of the nano-features.

Virtual boundary element-integral collocation method for the plane magnetoelectroelastic solids

This paper presents a virtual boundary element—integral collocation method (VBEM) for the plane magnetoelectroelastic solids, which is based on the basic idea of the virtual boundary element method for elasticity and the fundamental solutions of the plane magnetoelectroelastic solids. Besides sharing all the advantages of the conventional boundary element method (BEM) over domain discretization methods, it avoids the computation of singular integral on the boundary by introducing the virtual boundary. In the end, several numerical examples are performed to demonstrate the performance of this method, and the results show that they agree well with the exact solutions. The method is one of the efficient numerical methods used to analyze megnatoelectroelastic solids.

Virtual boundary element-equivalent collocation method for the plane magnetoelectroelastic solids

This paper presents a virtual boundary element-equivalent collocation method (VBEM) for the plane magnetoelectroelastic solids, which is based on the fundamental solutions of the plane magnetoelectroelastic solids and the basic idea of the virtual boundary element method for elasticity. Besides all the advantages of the conventional boundary element method (BEM) over domain discretization methods, this method avoids the computation of singular integral on the boundary by introducing the virtual boundary. In the end, several numerical examples are performed to demonstrate the performance of this method, and the results show that they agree well with the exact solutions. So the method is one of the efficient numerical methods used to analyze megnatoelectroelastic solids.

Virtual boundary element integral method for 2-D piezoelectric media

This paper presents a virtual boundary element integral method for 2-D piezoelectric media, which is based on the fundamental solutions of 2-D piezoelectric media, the principle of superposition and the basic idea of the virtual boundary element method for elasticity. Besides all the advantages of the conventional boundary element method (BEM) over domain discretization methods, this method avoids the computation of singular integral on the boundary by introducing the virtual boundary. In the end, several numerical examples are given to demonstrate the performance of this method, and the results show that they agree well with the exact solutions.

Deforming Mesh for Computational Aeroelasticity Using a Nonlinear Elastic Boundary Element Method

A nonlinear elastic boundary element method (NBEM) approach is developed as an innovative deforming mesh generator for computational aeroelastic simulation. The computational fluid dynamics (CFD) mesh is assumed to be embedded in an infinite nonlinear elastic medium of a hardening material, leading to the formulation of a pseudononlinear elastostatic problem. Whereas the CFD surface mesh is treated as a boundary element model and the CFD flowfield grid as domain sample points, the NBEM approach solves Navier's equations using a particular solution scheme that removes the requirement of the domain integral in the conventional NBEM formulation. The NBEM approach has a unified feature that is applicable to all mesh systems, including unstructured, multiblock structured, and overset grids. An optimization strategy is employed to determine the optimum hardening material properties by minimizing the mesh distortion in the viscous region where grid orthogonality must be preserved. Three test cases are performed to demonstrate the robustness and effectiveness of the NBEM approach for deforming mesh generation.

Deforming mesh for computational aeroelasticity using a nonlinear elastic boundary element method

Deforming mesh for computational aeroelasticity using a nonlinear elastic boundary element method

An axisymmetric boundary integral model for incompressible linear viscoelasticity: application to the micropipette aspiration contact problem.

The micropipette aspiration test has been used extensively in recent years as a means of quantifying cellular mechanics and molecular interactions at the microscopic scale. However, previous studies have generally modeled the cell as an infinite half-space in order to develop an analytical solution for a viscoelastic solid cell. In this study, an axisymmetric boundary integral formulation of the governing equations of incompressible linear viscoelasticity is presented and used to simulate the micropipette aspiration contact problem. The cell is idealized as a homogeneous and isotropic continuum with constitutive equation given by three-parameter (E, tau 1, tau 2) standard linear viscoelasticity. The formulation is used to develop a computational model via a "correspondence principle" in which the solution is written as the sum of a homogeneous (elastic) part and a nonhomogeneous part, which depends only on past values of the solution. Via a time-marching scheme, the solution of the viscoelastic problem is obtained by employing an elastic boundary element method with modified boundary conditions. The accuracy and convergence of the time-marching scheme are verified using an analytical solution. An incremental reformulation of the scheme is presented to facilitate the simulation of micropipette aspiration, a nonlinear contact problem. In contrast to the halfspace model (Sato et al., 1990), this computational model accounts for nonlinearities in the cell response that result from a consideration of geometric factors including the finite cell dimension (radius R), curvature of the cell boundary, evolution of the cell-micropipette contact region, and curvature of the edges of the micropipette (inner radius a, edge curvature radius epsilon). Using 60 quadratic boundary elements, a micropipette aspiration creep test with ramp time t* = 0.1 s and ramp pressure p*/E = 0.8 is simulated for the cases a/R = 0.3, 0.4, 0.5 using mean parameter values for primary chondrocytes. Comparisons to the half-space model indicate that the computational model predicts an aspiration length that is less stiff during the initial ramp response (t = 0-1 s) but more stiff at equilibrium (t = 200 s). Overall, the ramp and equilibrium predictions of aspiration length by the computational model are fairly insensitive to aspect ratio a/R but can differ from the half-space model by up to 20 percent. This computational approach may be readily extended to account for more complex geometries or inhomogeneities in cellular properties.

An effective method for finding values on and near boundaries in the elastic BEM

Recent research has paid great attention to the exact calculation of physical values on and near the boundaries of engineering problems. In the boundary element method (BEM), this problem demands special attention because various kinds of singularity may occur when these values are calculated. Despite extensive research, this problem has not yet been solved well. In the present paper, the elimination of various kinds of singular integrals in displacement boundary integral equations and their derivative forms is studied in detail. Firstly, a modified treatment which uses rigid body displacement solutions is introduced to calculate displacements near the boundary. Then derivative formulations of the displacement boundary integral equations are deduced, which form the basis of an investigation of the related hypersingular and nearly singular integrals. Similarly, a modified treatment, which uses the unit displacement derivative solutions together with some general numerical techniques, is proposed for the calculation of displacement derivatives on and near the boundary considered. Finally, strains and stresses are obtained from the calculated displacement derivatives by the use of the compatibility and constitutive equations. Numerical examples show that the proposed method is simple, regular and accurate in treating most singular or nearly singular integrals in the elastic BEM.

Complex fundamental solutions and complex variables boundary element method in elasticity

The CVBEM for elasticity problems in its basic formulations (direct, indirect and displacement discontinuity (DD) ones) is presented. It is based on the use of complex fundamental solutions. The complex boundary integral equations (CBIEs) arising from the basic CVBEM formulations are considered. It was shown that the developed theory includes the main CBIEs obtained by using the different approaches. Besides, new real and complex integral equations were obtained. The revealed links between real variables BEM (RVBEM) and CVBEM serve to mutual enrichment of these two approaches.

Analysis of Stress Singularity on Ceramics/Metal Bonded Joints

With increasing use of ceramics/metal bonded joints, It is required to analyze the residual stress distribution and stress singularity at an interface edge for its bonded joints. In this paper, the residual stress distribution and stress singularity index of the ceramics/metal bonded joints were analyzed by using 2-dimensional elastic boundary element method. The variations of residual stress distribution and stress singularity index are studied with changes for the combinations of ceramics/metal bodned joints.

Traction-based Completed Adjoint Double Layer Boundary Element Method in elasticity

This paper is concerned with a traction-based Completed Adjoint Double Layer Boundary Element Method to solve for the surface traction of a system of rigid particles embedded in an elastic matrix. The main feature of the method is a single layer representation of the displacement field, which leads to a system of second-kind integral equations for the traction field, the extreme eigenvalue of which could be deflated, allowing iterative solution strategies to be effectively applied. The method is therefore most suitable for large-scale simulations of particulate solids. The method is benchmarked against some known analytic solutions, including the difficult stress singularity problems at sharp edges. The effectiveness of the method in dealing with a large number of inclusions is also demonstrated with an elongational deformation problem involving up to 25 inclusions.

The h‐, p‐ and hp‐versions of the BEM in elasticity: Numerical results

AbstractThe paper investigates the convergence of the h‐, p‐ and hp‐versions of a variational symmetric boundary element method (BEM) in plane elasticity by numerical experiments. The study discusses mixed boundary value problems on polygonal domains, i.e. problems for which the exact solution is analytic except in a finite number of points. The convergence of the error in energy norm is displayed for all versions of the BEM. All results are also compared with those obtained by the corresponding finite element methods. The theoretically predicted asymptotic convergence rates for all versions of the method can be observed in the numerical experiments. A comparison of computer times is given as well.

Stochastic boundary element method in elasticity

The stochastic boundary element method is developed to analyzeelasticity problems with random material and/or geometrical parameters and ran-domly perturbed boundaries. Based on the first-order Taylor series expansion, theboundary integration equations concerning the mean and deviation of the displace-ments are derived, respectively. It is found that the randomness of material param-eters is equivalent to a random body force, so the mean and covariance matricesof unknown boundary displacements and tractions can be obtained. Furthermore,the mean and covariance of displacements and stresses at inner points can also beobtained. Numerical examples show that the proposed stochastic boundary elementmethod gives satisfactory solutions, as compared with those obtained by theoreticalanalysis or other numerical methods.

Stochastic boundary element method in elasticity

The stochastic boundary element method is developed to analyze elasticity problems with random material and/or geometrical parameters and randomly perturbed boundaries. Based on the first-order Taylor series expansion, the boundary integration equations concerning the mean and deviation of the displacements are derived, respectively. It is found that the randomness of material parameters is equivalent to a random body force, so the mean and covariance matrices of unknown boundary displacements and tractions can be obtained. Furthermore, the mean and covariance of displacements and stresses at inner points can also be obtained. Numerical examples show that the proposed stochastic boundary element method gives satisfactory solutions, as compared with those obtained by theoretical analysis or other numerical methods.

Completed double layer boundary element method in elasticity

This paper reports an indirect boundary element method in elasticity that is most suitable to deal with particulate solids. The method involves a distribution of a double layer potential and, after a suitable completion and deflation, is amenable to iterative solution techniques. It can therefore accommodate a large number of particles with complex geometries. Convergence of the method is significantly improved by the introduction of a simple domain decomposition to solve the system of equations. The method is illustrated by the translating sphere problem, the load transfer problem between two spheres at near contact, and the shear deformation of a cluster of 125 spheres initially in a simple cubic array.