Boundary element analysis of three-dimensional mixed-mode cracks via the interaction integral

A three-dimensional boundary element method (BEM) implementation of the interaction integral methodology for the numerical analysis of mixed-mode three-dimensional cracks is presented in this paper. The interaction integral is evaluated from a domain representation naturally compatible with the BEM, since stresses, strains and derivatives of displacements at internal points can be evaluated using their appropriate boundary integral equations. Special emphasis is put in the selection of the auxiliary function that represents the virtual crack advance in the domain integral. This is found to be a key feature to obtain reliable results at the intersection of crack fronts with free surfaces. Several examples are analysed to demonstrate the efficiency and accuracy of the implementation.     

A boundary collocation method for cracked plates

A boundary collocation method is developed for analyzing cracked thin plates. Complex stress functions which satisfy the equilibrium equations of an infinite domain having a single crack are first derived. As the functions have also satisfied the stress singularity at the crack tips, it is only necessary to enforce the stress functions to satisfy the boundary conditions along the edges of the plates and the surfaces of the cracks, if there is more than one crack. This is achieved by the collocation least square approach. The unknown coefficients of the stress functions having been determined, the stress intensity factors can then be computed according to the related formulae. Examples of rectangular and circular plates with a different number of cracks and under different loadings are used to demonstrate the accuracy, versatility and advantages of the method.     

Fracture analysis of plane piezoelectric/piezomagnetic multiphase composites under transient loading

The transient response of cracked composite materials made of piezoelectric and piezomagnetic phases, when subjected to in-plane magneto-electro-mechanical dynamic loads, is addressed in this paper by means of a mixed boundary element method (BEM) approach. Both the displacement and traction boundary integral equations (BIEs) are used to develop a single-domain formulation. The convolution integrals arising in the time-domain BEM are numerically computed by Lubich’s quadrature, which determines the integration weights from the Laplace transformed fundamental solution and a linear multistep method. The required Laplace-domain fundamental solution is derived by means of the Radon transform in the form of line integrals over a unit circumference. The singular and hypersingular BIEs are numerically evaluated in a precise and efficient manner by a regularization procedure based on a simple change of variable, as previously proposed by the authors for statics. Discontinuous quarter-point elements are used to properly capture the behavior of the extended crack opening displacements (ECOD) around the crack-tip and directly evaluate the field intensity factors (stress, electric displacement and magnetic induction intensity factors) from the computed nodal data. Numerical results are obtained to validate the formulation and illustrate its capabilities. The effect of the combined application of electric, magnetic and mechanical loads on the dynamic field intensity factors is analyzed in detail for several crack configurations under impact loading.     

Structural shape optimisation using boundary elements and the biological growth method

A numerical evolutionary procedure for the structural optimisation for stress reduction of two-dimensional structures is presented in this paper. The proposed procedure couples the biological growth method (BGM) with the boundary element method (BEM). The boundary-only intrinsic characteristic of BEM together with its accuracy in the boundary displacement and stress solutions make BEM especially attractive for solving shape-optimisation problems. Two formulations of BEM are used in this work: the standard for two-dimensional elastostatics for the stress analysis and the dual reciprocity method (DRM), which is used to model the swelling or shrinking of the material. Two examples are analysed to illustrate the proposed methodology and to demonstrate its versatility and robustness.     

Boundary point method for linear elasticity using constant and quadratic moving elements

Based on the boundary integral equations and stimulated by the work of Young et al. [J Comput Phys 2005;209:290–321], the boundary point method (BPM) is a newly developed boundary-type meshless method enjoying the favorable features of both the method of fundamental solution (MFS) and the boundary element method (BEM). The present paper extends the BPM to the numerical analysis of linear elasticity. In addition to the constant moving elements, the quadratic moving elements are introduced to improve the accuracy of the stresses near the boundaries in the post processing and to enhance the analysis for thin-wall structures. Numerical tests of the BPM are carried out by benchmark examples in the two- and three-dimensional elasticity. Good agreement is observed between the numerical and the exact solutions.     

Estimation of wave responses from subvertical macrofracture systems using a grid characteristic method

This work investigates the formation and propagation of scattered waves forming a response of crack structures on the seismologic record. The initial impulse is a flat wave front propagating into the depth of the medium. The paper considers the periodic structure of the scattered wave response from a system (cluster) of subvertical cracks. The procedures of estimation of geometrical parameters of such crack structures are based on numerical experiments. We use a grid-characteristic method to calculate the triangular grids with the formulation of boundary conditions at the interface of the medium and the cracks, as well as on the borders of the region of integration with the characteristic properties of the governing equations of the hyperbolic type. This numerical method makes it possible to build the most correct numerical algorithms on the boundaries of the integration domain and on the surfaces of the media division (interfaces) and to take into account the domain of the solution’s dependence and the physics of the problem (propagation of disturbances on the characteristic directions). This is why this method is perhaps the most suitable one for the numerical solution of dynamic problems with a distinct wave character in geologically substantially inhomogeneous media, in particular, for the considered problem of seismic waves interaction with fissured structures.     

Boundary element solution for the Cauchy problem in linear elasticity using singular value decomposition

In this paper the singular value decomposition (SVD), truncated at an optimal number, is analysed for obtaining approximate solutions to ill-conditioned linear algebraic systems of equations which arise from the boundary element method (BEM) discretisation of an ill-posed boundary value problem in linear elasticity. The regularisation parameter, namely the optimal truncation number, is chosen according to the discrepancy principle. The numerical results obtained confirm that the SVD+BEM produces a convergent and stable numerical solution with respect to decreasing the mesh size discretisation and the amount of noise added into the input data.     

A Matlab library for solving quasi-static volume conduction problems using the boundary element method

The boundary element method (BEM) is commonly used in the modeling of bioelectromagnetic phenomena. The Matlab language is increasingly popular among students and researchers, but there is no free, easy-to-use Matlab library for boundary element computations. We present a hands-on, freely available Matlab BEM source code for solving bioelectromagnetic volume conduction problems and any (quasi-)static potential problems that obey the Laplace equation. The basic principle of the BEM is presented and discretization of the surface integral equation for electric potential is worked through in detail. Contents and design of the library are described, and results of example computations in spherical volume conductors are validated against analytical solutions. Three application examples are also presented. Further information, source code for application examples, and information on obtaining the library are available in the WWW-page of the library: (http://biomed.tkk.fi/BEM).     

Elastodynamic fields near running cracks by finite elements

A finite element method is developed for the computation of elastodynamic stress intensity factors at a rapidly moving crack tip. The method is restricted to bodies whose surfaces and two-material interfaces are either parallel to the direction of propagation or are sufficiently remote. The crack tip starts to move at the instant that it is struck by an incident wave. The finite element grid moves undeformed with the crack tip. The main result consists in the fact that the method of non-singular calibrated crack tip elements, in which the stress-intensity factor is determined from its statically calibrated ratio to the crack opening displacement in an adjacent node, is extended to dynamic problems with moving cracks, for both in-plane and anti-plane motions. The dependence of the calibration ratio on the crack tip velocity is established from previously developed analytical solutions for the near-tip displacement fields. Numerical results compare favorably with known analytical solutions for cracks moving in an infinite solid. The grid motion causes an apparent asymmetric additional damping matrix.     

Analysis of three-dimensional anisotropic heat conduction problems on thin domains using an advanced boundary element method

In this paper, an advanced boundary element method (BEM) is developed for solving three-dimensional (3D) anisotropic heat conduction problems in thin-walled structures. The troublesome nearly singular integrals, which are crucial in the applications of the BEM to thin structures, are calculated efficiently by using a nonlinear coordinate transformation method. For the test problems studied, promising BEM results with only a small number of boundary elements have been obtained when the thickness of the structure is in the orders of micro-scales (10^?6), which is sufficient for modeling most thin-walled structures as used in, for example, smart materials and thin layered coating systems. The advantages, disadvantages as well as potential applications of the proposed method, as compared with the finite element method (FEM), are also discussed.     

Meshfree Particle Methods in the Framework of Boundary Element Methods for the Helmholtz Equation

In this paper, we study electromagnetic wave scattering from periodic structures and eigenvalue analysis of the Helmholtz equation. Boundary element method (BEM) is an effective tool to deal with Helmholtz problems on bounded as well as unbounded domains. Recently, Oh et al. (Comput. Mech. 48:27–45, 2011) developed reproducing polynomial boundary particle methods (RPBPM) that can handle effectively boundary integral equations in the framework of the collocation BEM. The reproducing polynomial particle (RPP) shape functions used in RPBPM have compact support and are not periodic. Thus it is not ideal to use these RPP shape functions as approximation functions along the boundary of a circular domain. In order to get periodic approximation functions, we consider the limit of the RPP shape function as its support is getting infinitely large. We show that the basic approximation function obtained by the limit of the RPP shape function yields accurate solutions of Helmholtz problems on circular, or annular domains as well as on the infinite domains.     

A fast quasi-multiple medium method for 3-D bem calculation of parasitic capacitance

A quasi-multiple medium (QMM) method is proposed to accelerate the boundary element method (BEM) for the 3-D parasitic capacitance calculation. In the QMM method, a homogeneous dielectric is decomposed into a number of fictitious medium blocks, each with the same permittivity of original medium. By the localization character of BEM, the QMM method makes great sparsity to the coefficient matrix of the overall discretized BEM equations. Then, using storing technique of sparse matrix and iterative equation solvers, the sparsity is explored to greatly reduce CPU time and memory usage of BEM computation. The computational complexity of the QMM accelerated BEM for a single-medium model problem is analyzed, and it is concluded as O(N), if the number of iterations is bounded. Numerical results verify the theoretical analysis and show the accelerating efficiency of the QMM method for calculation of 3-D parasitic capacitance.     

A BEM-based domain meshless method for the analysis of Mindlin plates with general boundary conditions

In this paper, a BEM-based domain meshless method is developed for the analysis of moderately thick plates modeled by Mindlin’s theory which permits the satisfaction of three physical conditions along the plate boundary. The presented method is achieved using the concept of the analog equation of Katsikadelis. According to this concept, the original governing differential equations are replaced by three uncoupled Poisson’s equations with fictitious sources under the same boundary conditions. The fictitious sources are established using a technique based on BEM and approximated by radial basis functions series. The solution of the actual problem is obtained from the known integral representation of the potential problem. Thus, the kernels of the boundary integral equations are conveniently established and evaluated. The presented method has the advantages of the BEM in the sense that the discretization and integration are performed only on the boundary, and consequently Mindlin plates with general boundary conditions can be analyzed without difficulty. To illustrate the effectiveness, applicability as well as accuracy of the method, numerical results of various example problems are presented.     

Postbuckling analysis of plates on an elastic foundation by the boundary element method

The postbuckling behavior of plates on an elastic foundation is studied by using the boundary element method (BEM). A new fundamental solution of lateral deflection is derived through the resolution theory of a differential operator, and a set of boundary element formulae in incremental form is presented. By using these formulae, the BEM solution procedure becomes relatively simple. The results of a number of numerical examples are compared with existing solutions and good agreement is observed. It shows that the proposed method is effective for solving the postbuckling problems of plates with arbitrary shape and various boundary conditions.     

Dual reciprocity boundary element method in Laplace domain applied to anisotropic dynamic crack problems

In this paper the dual reciprocity boundary element method in the Laplace domain for anisotropic dynamic fracture mechanic problems is presented. Crack problems are analyzed using the subregion technique. The dynamic stress intensity factors are computed using traction singular quarter-point elements positioned at the tip of the crack. Numerical inversion from the Laplace domain to the time domain is achieved by the Durbin method. Numerical examples of dynamic stress intensity factor evaluation are considered for symmetric and non-symmetric problems. The influence of the number of Laplace parameters and internal points in the solution is investigated.     

Hybrid laplace transform/weighting function scheme for transient multidimensional conservation equations

A hybrid Laplace transform/weighting function scheme is developed for solving time-dependent multidimensional conservation equations. The new method removes the time derivatives from the governing differential equations using the Laplace transform and solves the associated equation with the weighting function scheme. The similarity transform method is used to treat the complex coefficient system of the equations, which allows the simplest form of complex number functions to be obtained, and then to use the partial fractions method or a numerical method to invert the Laplace transform and transform the functions to the physical plane. Three different examples have been analyzed by the present method. The present method solutions are compared in tables with the exact solutions and those obtained by the other numerical methods. It is found that the present method is a reliable and efficient numerical tool.     

A MLPG(LBIE) approach in combination with BEM

The local boundary integral equation (LBIE) approach is a promising meshless method, recently proposed as an effective alternative to the boundary element method (BEM), for solving non-homogeneous, anisotropic and non-linear problems. Since the LBIE method utilizes in its weak form fundamental solutions as test functions, it can be considered as one of the six meshless local Petrov–Galerkin (MLPG) methods proposed by Atluri and coworkers. This explains the use of the initials MLPG(LBIE) in the title of the present paper. This work addresses a coupling of a new MLPG(LBIE) method, recently proposed by the authors for elastodynamic problems, and the BEM. Because both methods conclude to a final system of linear equations expressed in terms of nodal displacement and tractions, their combination is accomplished directly with no further transformations as it happens in other MLPG/BEM formulations as well as in typical hybrid finite element method/BEM schemes. The coupling approach is demonstrated for static and frequency domain elastodynamic problems. Three representative examples are provided in order to illustrate the achieved accuracy of the proposed here MLPG(LBIE)/BE methodology.     

Particle Monte Carlo and lattice-Boltzmann methods for simulations of gas-particle flows

A numerical solution concept is presented for simulating the transport and deposition to surfaces of discrete, small (nano-)particles. The motion of single particles is calculated from the Langevin equation by Lagrangian integration under consideration of different forces such as drag force, van der Waals forces, electrical Coulomb forces and not negligible for small particles, under stochastic diffusion (Brownian diffusion). This so-called particle Monte Carlo method enables the computation of macroscopic filter properties as well the detailed resolution of the structure of the deposited particles. The flow force and the external forces depend on solutions of continuum equations, as the Navier-Stokes equations for viscous, incompressible flows or a Laplace equation of the electrical potential. Solutions of the flow and potential fields are computed here using lattice-Boltzmann methods. Essential advantage of these methods are the easy and efficient treatment of three-dimensional complex geometries, given by filter geometries or particle covered surfaces. A number of numerical improvements, as grid refinement or boundary fitting, were developed for lattice-Boltzmann methods in previous studies and applied to the present problem. The interaction between the deposited particle layer and the fluid field or the external forces is included by recomputing of these fields with changed boundaries. A number of simulation results show the influence of different effects on the particle motion and deposition.     

Constructing multi-looped surfaces by loop joining

The objective of this research is to develop a surface joining method for constructing multi-looped surfaces. Based on this method, surfaces can be self-joined or inter-joined on loops to construct multi-looped surfaces. A loop is a continuous and closed boundary on a surface. By constructing one or several single-looped B-spline surfaces, multi-looped surfaces are constructed through loop joining of boundary curve segments. Loop joining is completed by merging the control points on the surface boundaries of B-spline surfaces. With the merging of data structures simultaneously, the consistencies of geometry and topology are kept in the surface model.The proposed method for constructing multi-looped surfaces has the advantages that it is conceptually explicit, simple for user understanding and easy for system programming, and there is no computation of surface/surface intersections involved, thus no numerical solutions for nonlinear equations are required.     

Performance of the BEM solution in 3D acoustic wave scattering

A fixed cylindrical circular cavity and a cylindrical circular column of fluid of infinite length submerged in a homogeneous fluid medium, and subjected to a pressure point source, for which closed form solutions are known, are used to assess the performance of constant, linear and quadratic boundary elements in the analysis of acoustic scattering.This aim is accomplished by evaluating the error committed by the boundary element method (BEM) for a wide range of frequencies and wave numbers. First, the position of dominant BEM errors in the frequency versus spatial wave number domains are identified and related to the natural modes of vibration of the cylindrical circular inclusion. Then, the errors that occur by using constant, linear and quadratic elements are compared when the inclusion is modelled with the same number of nodes (i.e. maintaining computational cost). Finally, the importance of the position of the nodal points inside discontinuous boundary elements is analysed.