Dual Boundary Element Method Research Articles

On the path independence and invariant of the J-integral for a slant crack and rigid-line inclusion using degenerate kernels and the dual BEM

The J-integral and stress intensity factor (SIF) are two major parameters in linear elastic fracture mechanics (LEFM) for the fracture criterion. In this paper, we focus on the J-integral of the slant crack and the slant rigid-line inclusion under the remote anti-plane shear. By employing the degenerate kernel, the path independence of J-integral is analytically demonstrated by using the elliptic coordinates. The positive and negative J-integrals are also analytically derived and numerically implemented by using the dual BEM for the crack and the rigid-line inclusion, respectively. It is interesting to find that the J-integral is not an invariant by using different observer systems but is one component of the vector of the first order tensor. Transformation law of the J-integral with respect to different observers is analytically proved and numerically demonstrated. Finally, the tensor property of order one is examined.

Deterministic fatigue crack-growth simulations for a railway axle by Dual Boundary Element Method

The maintenance of railway axles requires an optimization of their inspection intervals in order to reduce costs. Such components can endure more than 30 years of service life and are subjected to scheduled Non Destructive Testing (NDT) inspections. It is therefore important to assess the influence of the severe in-service loads on the propagation of cracks in the axles. This manuscript reports deterministic fracture calculations, for the simulation of cracks propagating through a railway axle, by the Dual Boundary Element Method. Once Stress Intensity Factors along the crack front are obtained, the residual fatigue life estimates were calculated through a user made spreadsheet, taking into account the complex loading conditions of the axles. The presented results support the application of the crack propagation simulation for the determination of axle NDT intervals under different loading conditions.

A new DBEM for solving crack problems in arbitrary dissimilar materials

In this paper, a new and efficient Dual Boundary Element Method (DBEM) named Dual Boundary-Interface Integral Equation Method (DBIIEM) is presented to solve crack problems in objects composed of arbitrary number of different materials. The conventional DBEM has been widely used in crack analysis and proved to be one of the most effective BEM for solving crack problems, however it cannot be directly employed to solve more complicated multi-medium crack problems. The presented DBIIEM can remedy this deficiency by dually collocating the displacement boundary-interface integral equation on one of the multi-medium crack surface and traction boundary-interface integral equation on the other. The newly derived displacement and traction boundary-interface integral equations are naturally fit for solving multi-medium elasticity problems, and independent each other, therefore they can be employed together in multi-medium crack analysis. The established DBIIEM inherits the advantage of single -region formulation of DBEM for solving more complicated crack problems. Complex stress intensity factors which characterize the oscillatory singularity of crack tips located at the interface crack are evaluated. Several numerical examples are given to verify correctness of the presented method.

Wave scattering by inverted trapezoidal porous boxes using dual boundary element method

A numerical model is developed based on the dual boundary element method (DBEM) to solve the problem of surface gravity waves past multiple floating inverted trapezoidal porous boxes under the framework of linear potential flow theory. The continuity of velocity and semi-empirical quadratic pressure drop equation is adopted on the porous boundary to incorporate the effect of incident wave amplitude on the energy loss coefficient. Various results by an array of inverted trapezoidal porous boxes are discussed in the form of scattering coefficients (wave reflection, transmission, and energy loss coefficients), and force coefficients (horizontal force, vertical force, and moment). The verification of the numerical model is carried out by an independently developed multi-domain boundary element method (MBEM) code, and also with results available in the literature. The study reveals that, in most cases, the long waves exhibit insignificant hindrance whereas intermediate waves delineate the harmonic oscillations and deep water waves feel maximal hindrance by inverted trapezoidal porous boxes. The wave damping of KL≥0.95 is obtained for structural width B/λ≥0.66 by a pair of identical inverted trapezoidal porous boxes. The angle of porous wings is helpful to alter the scattering coefficients and also helps in the fine-tuning of force coefficients. The angle of partially open wings within 300≤θ≤600 is suggested along with structural porosity μ=0.25 for effective constriction of incident waves. The parametric analysis will be useful in the meticulous design of cost-effective breakwaters.

Study on the stress intensity factor and the double-degeneracy mechanism in the BEM/BIEM for anti-plane shear problems

The crack and the rigid-line inclusion problems under the anti-plane shear are the special case of the elliptic hole and elliptic rigid inclusion, respectively. We revisit this kind of problem by using the degenerate kernels in terms of the elliptic coordinates. The stress intensity factor (SIF) is also addressed. Three ways are employed to determine the SIF. One is the extrapolation approach for the boundary or interior displacement near the tip. Another is the extrapolation approach for the boundary stress or interior stress near the tip. The other is the J-integral enclosing the crack tip. It is interesting to find that a rigid-line inclusion case yields a singular influence matrix due to the degenerate scale of length four in the log kernel. However, double-degeneracy including degenerate scale and degenerate boundary may still result in a singular matrix even though the dual BEM/BIEM is employed. The mechanism is well explained thanks to the introduction of degenerate kernel. Without the introduction of degenerate kernel, the mechanism of the double-degeneracy problem in the BIEM can’t be clearly examined. By using the degenerate kernel, the SIF can be easily determined for the crack or the rigid-line inclusion under the anti-plane shear. The path independence of the J-integral can be derived analytically. The reciprocal relation for the SIF between a crack and a rigid-line inclusion with respect to opposite loading is also addressed. In the numerical implementation, the SIF can be determined using the dual BEM. It is worth noting that BEM shows the advantage that the obtained boundary displacement or boundary stress can be directly used to obtain the more accurate SIF for the crack or the rigid inclusion, respectively.

Tetrahedral meshing for a slanted semi-elliptical surface crack at a solid cylinder

This technical note demonstrates the use of tetrahedral meshing for modelling a slanted semi-elliptical surface crack at a solid cylinder by finite element method (FEM). For a relatively small ratio of the crack depth over cylinder diameter, Modes I and II (in-plane) stress intensity factors (SIFs) at the corner points by FEM were observed to be in good agreements with those by the dual boundary element method (DBEM). Effect of the crack aspect ratio, crack size and loading mode on the corner point singularity was noted. The tearing loading mode was found to promote a greater corner point singularity.

Wave interaction with multiple wavy porous barriers using dual boundary element method

The gravity wave interaction with horizontally staggered multiple thin wavy porous barriers is analyzed under the assumption of potential flow theory. The dual boundary element method (DBEM) is used to solve the boundary value problem. The accuracy and reliability of the DBEM solutions are ascertained by comparing with known results in the literature and with an independently developed multi-domain boundary element method (MBEM) code. The study is focused on the scattering coefficients and vertical forces on an array of wavy barriers. The numerical results demonstrate that wave transmission by a horizontally staggered porous barrier is much better compared to a single horizontal plate. It is recommended to use plate porosity of 0.05 to 0.20, if transmission coefficient of less than 0.50 is needed, especially for relative water depth k0h > 2. With an increase in the number of ripples and ripple amplitude on the porous plates, it is possible to further reduce wave transmission by 5% to 10%. Also, an increase in the inclination of the plate from 0 to 10 degrees can reduce the wave transmission and increases the wave reflection to 5% to 10%. Further, a reduction of about 10% to 30% in the vertical force on the wavy plate is possible for k0h ≥ 1 with an increase in the relative ripple amplitude, the number of ripples and increased plate rotation. This essentially will help in the reduction of pile penetration length into the sea bed soil. The results of the present study can be used for the optimal design of wavy plate as a wave barrier for offshore and coastal engineering applications.

DBEM computation of T-stress and mixed-mode SIFs using interaction integral technique

In this paper, dual boundary element method (DBEM) is combined with interaction integral (I-integral) to evaluate stress intensity factors (SIFs) and T-stress simultaneously for mixed-mode crack analyzing. DBEM is proved to be one of the most efficient boundary element methods (BEM) for solving crack problems with boundary-only discretization, however its current implementation is in lack of efficient methods to evaluate fracture parameters. The available geometrical extrapolation method and J-integral technique for calculating fracture parameters in combination with DBEM are imprecise and difficult to be implemented in mix-mode crack analysis. The presented combined DBEM and I-integral approach can be easily used to extract SIFs and T-stress for mixed-mode cracks, which does not need the decomposition of elastic field into symmetric and antisymmetric components. The fundamental basis lies in the introduction of auxiliary fields for some proper defined single mode crack, then fracture parameters are calculated by evaluating the formulated I-integral in terms of DBEM solution and auxiliary elastic fields. In this paper, the basis of DBEM is illustrated at first, then the explicit expressions for auxiliary fields of displacement gradients are derived, then an automatic scheme for generating the path of I-integral is presented. Several numerical examples are given to validate correctness of the presented method.

An enriched dual boundary element method formulation for linear elastic crack propagation

This study presents an eXtended Boundary Element Method (XBEM) formulation for simulating the linear elastic crack growth in two-dimensional domains. A displacement approximation enrichment based on the Williams first-order expansions is used to represent the near-tip square root behavior predicted by the linear elastic fracture mechanics. This strategy also enables the direct evaluation of the Stress Intensity Factors (SIFs) after a crack tip tying constraint is enforced at the crack tip to accommodate the additional enrichment parameters. Additionally, discontinuous functions are embedded into the displacement approximation of elements intercepted by cracks to avoid remeshing of these elements. For the discontinuous enrichment, new equations are provided by imposing displacement continuity conditions at the crack mouth. Shifted enrichment functions are used to preserve the physical meaning of the nodal parameters and to reduce the singularity order of the integral kernels containing the enrichment terms in the XBEM formulation. Furthermore, an enriched traction approximation is proposed to apply concentrated forces and support points along the boundary. When considering the XBEM formulation, changes in the conventional BEM are minimal, which facilitates the implementation of the extended approach into existing codes. Three numerical examples demonstrate the application of the proposed XBEM formulation in fracture modeling.

Iterative dual BEM solution for water wave scattering by breakwaters having perforated thin plates

This study develops an iterative dual BEM (Boundary Element Method) model for water wave scattering by breakwaters having perforated thin plates, which avoids setting artificial sub-domains near the thin plates and thus is very convenient to write general computer codes. In order to directly consider the effect of wave height on the wave energy dissipation by perforated plates, a quadratic (nonlinear) pressure drop condition is imposed on the plates. Then, iterative calculations are conducted to solve the nonlinear boundary condition in the dual BEM model. Also, an analytical method to calculate the integrals of kernel function, which is generated from the hypersingular boundary integral equation in the dual BEM model, is introduced to enhance the calculation efficiency and accuracy. Numerical results show that the iterative calculations in the newly developed dual BEM solution converge rapidly. As examples, wave scattering by several typical breakwaters involving perforated thin plates are solved using the iterative dual BEM model. The corresponding numerical results of the reflection and transmission coefficients are in excellent agreement with the predictions by the analytical solutions. The numerical results also agree reasonably well with experimental data in literature. The present iterative dual BEM model is valuable for analyzing the hydrodynamic performance of breakwaters with perforated thin plates in preliminary engineering design.

The Kriging integration method applied to the boundary element analysis of Poisson problems

A novel efficient technique is presented for the evaluation of domain integrals that appear in the boundary element method (BEM). Herein, the source term is approximated with the use of radial basis functions, as in the dual reciprocity BEM. The proposed technique, called the Kriging Integration Method (KIM), comprises the use of the Simple Kriging Method in non-overlapping patches for obtaining the weights of the integration points located inside. As it is necessary to compute the integrals of the covariance function prior to obtaining these weights, this can be efficiently realized using the Cartesian Transformation Method. The domain integrals over all the generated partitions are then computed and added to obtain the value of the whole-domain integral. Using KIM, it is possible to evaluate approximately weakly singular domain integrals over simply or multiply connected domains without applying any transformation or regularization method to the singular integrand. The numerical results obtained in several 2D potential problems demonstrate that this integration scheme is as accurate as both the dual reciprocity method and RIM and less time consuming than the RIM.

Cohesive crack propagation analysis using a dipole BEM formulation with tangent operator

This work deals with the use of Boundary Element Method (BEM) formulations in fracture problems in a 2D approach. BEM is a suitable method for solving this type of problem, since the absence of a domain mesh translates into a more efficient modeling of regions with high stress concentration. Besides, reducing the dimensionality of the mesh results in a convenient remeshing process. Regarding the nonlinear BEM formulation, an alternative to the classic dual formulation is used, with the introduction of an initial stress field to represent the cohesive zone, based on the concept of dipoles. This formulation is particularly interesting because it is able to represent mathematically the presence of the fracture process zone (FPZ) with only three algebraic equations (related to stress correction) per source point located in the crack path. In contrast, dual BEM formulation requires four algebraic equations (displacements and forces) per source point. As for the effects relevant to the nonlinear system, two distinct iterative solving algorithms are used, Constant Operator (CO) and Tangent Operator (TO). Only the elastic stiffness of the structure in the search of equilibrium point is considered in the first one, while the latter takes into account the mechanical degradation properties. Some examples are presented to validate the use of the Dipole BEM/TO formulation in crack propagation analysis and also in multiple crack analysis. Finally, this formulation proves to be a powerful alternative to other classic methods, considering the results presented.

Statistical inference of the equivalent initial flaw size for assembled plate structures with the dual boundary element method

The statistical inference of the Equivalent Initial Flaw Size Distribution (EIFSD) is developed for the first time using the Dual Boundary Element Method (DBEM) for assembled shear deformable plate structures. As part of this inference procedure, Bayesian updating is employed to enable the continuous refinement of the EIFSD via data obtained by many simulated routine inspections of a stiffened panel from a fleet of aircraft. Fatigue crack growth is modelled using an incremental crack growth procedure that only requires modelling of the boundary of the 2.5D structure with line elements and requires no remeshing during crack growth simulations. Stochastic Kriging is employed to account for the stochastic nature of fatigue crack growth and to offset the high computational cost associated with modelling complex built-up structures. To demonstrate the efficiency of the proposed inference methodology, a numerical example featuring a stiffened panel subjected to complex loading in the form of combined tension and bending is presented. Once the EIFSD has been inferred, it can be used to optimise the intervals between routine aircraft inspections via the use of reliability analysis techniques as part of a combined reliability-EIFS approach. It is demonstrated that the proposed methodology offers the capability to reduce the costs associated with inspections.

An improved singular curved boundary integral evaluation method and its application in dual BEM analysis of two- and three-dimensional crack problems

In this paper, an improvement is made to an efficient direct method for numerical evaluation of high order singular curved boundary integrals. Then this improved singular integral evaluation method is employed to solve the strongly and hypersingular integrals involved in the dual boundary element method (Dual BEM), which combine the use of displacement and traction boundary integral equations to solve crack problems in a single domain formulation. The singular integral evaluation method is carried out based on a parameter plane expansion and radial integral approach, this paper proposed a new strategy for treating the singular radial integral, which plays a vital role in this method. In isoparametric coordinate system, the singular curved boundary integral is mapped into a singular square plane integral in intrinsic coordinates, then the radial integration method (RIM) is employed to transform the singular square plane integral into a regular line integral over the contour of intrinsic square plane and a singular radial integral over the path from source point to the contour of intrinsic square plane. A singularity isolation technique is utilized to divide the singular radial integral into two parts, the regular radial integral can be evaluated normally using Gauss quadrature and the singular radial that can be evaluated analytically by expanding the non-singular part of the integrand function into a power series. Compared with conventional local interpolation approach to deal with the singular radial integral, the newly proposed method has a more rigorous mathematical derivation, and can achieve more stable and precise results. Based on the successful implementation of direct evaluation of singular boundary integrals, Dual BEM is successfully applied to solve two- and three-dimensional elastic crack problems including straight and curved crack paths with continuous or discontinuous elements. Two different approaches, geometrical extrapolation method and J-integral method are used in the evaluation of stress intensity factors. Several numerical examples are given to validate effectiveness of the presented method.

Dual boundary element method for elastoplastic fracture mechanics of shear deformable plate

A dual boundary element method formulation for elastoplastic fracture mechanics of shear deformable plate is reported in this work. The formulation can be viewed as an extension to the two-dimensional problem. The boundaries of crack surfaces are modeled using discontinuous elements of quadratic elements to fulfill the continuity requirements of displacements and tractions at collocation points for displacement and traction integral equations. Rectangular element of 9-nodes quadrilateral is used to discretize the domain. Semi-discontinuous rectangular elements are placed along the boundary including the crack surfaces in order to force the cell nodes to be internal points. Two examples are presented. The first example isa centre-cracked plate subjected to tensile, bending and combined of the two loads. The second example is rectangular plate with double edge symmetric cracks subjected to combined tensile and bending load. J-type integrals are calculated.

Analytical and numerical studies for solving Steklov eigenproblems by using the boundary integral equation method/boundary element method

The theory of boundary eigensolutions is developed for boundary value problems. It is general for boundary value problem. Steklov-Poincaré operator maps the values of a boundary condition of the solution of the Laplace equation in a domain to the values of another boundary condition. The eigenvalue is imbedded in the Dirichlet to Neumann (DtN) map. The DtN operator is called the Steklov operator. In this paper, we study the Steklov eigenproblems by using the dual boundary element method/boundary integral equation method (BEM/BIEM). First, we consider a circular domain. To analytically derive the eigensolution of the above shape, the closed-form fundamental solution of the 2D Laplace equation, ln(r), is expanded into degenerate kernel by using the polar coordinate system. After the boundary element discretization of the BIE for the Steklov eigenproblem, it can be transformed to a standard linear eigenequation. Problems can be effectively solved by using the dual BEM. Finally, we consider the annulus. Not only the Steklov problem but also the mixed Steklov eigenproblem for an annular domain has been considered.

A Generation of Special Triangular Boundary Element Shape Functions for 3D Crack Problems

This paper focuses on tackling the two drawbacks of the dual boundary element method (DBEM) when solving crack problems with a discontinuous triangular element: low accuracy of the calculation of integrals with singularity and crack front element must be utilized to model the square-root property of displacement. In order to calculate the integrals with higher order singularity, the triangular elements are segmented into several subregions which consist of subtriangles and subpolygons. The singular integrals in those subtriangles are handled by the singularity subtraction technique in the integration space and can be regularized and accurately calculated. For the nearly singular integrals in those subpolygons, the element subdivision technique is employed to improve the calculation accuracy. In addition, considering the location of the crack front in the element, special crack front elements are constructed based on a 6-node discontinuous triangular element, in which the displacement extrapolation method is introduced to obtain the stress intensity factors (SIFs) without consideration of orthogonalization of the crack front mesh. Several numerical results are investigated to fully verify the validation of the presented approach.

Fracture Dynamic Analysis of Cracked Reissner Plates Using the Boundary Element Method

This work presents a Dual Boundary Element Method/Dual Reciprocity Boundary Element Method formulation for the dynamic analysis of fractured shear deformable plates. Equations for the Dual Boundary Element Method, including the direct boundary and the traction boundary equation for Reissner plates, were used for crack modeling. The Dual Reciprocity Boundary Element Method was used to treat both domain integrals related to distributed forces and inertial terms. Proposed formulation considers the rotary inertia of the plate. A general crack modeling strategy is presented. The J-Integral is used to evaluate the Dynamic Stress Resultant Intensity Factors. Test problems, including comparisons to Finite Element Method solutions, are presented. Results demonstrate, proposed formulation is a reliable method for dynamic analysis of shear deformable cracked plate bending problems.

The stress intensity factor assessment in three-dimensional problems by the displacement fitting technique and the dual Boundary Element Method

This work presents an extension of the displacement fitting technique for the assessment of stress intensity factors (SIFs) of three-dimensional linear elastic fracture problems using the dual Boundary Element Method. The developed framework accounts for higher-order terms of the asymptotic displacement solution near crack front. The number and location of points surrounding the crack front are properly defined in order to accurately evaluate the SIFs. Three-dimensional benchmarks demonstrate the efficiency of the proposed framework. Moreover, two different fracture criteria illustrate the influence of SIFs values with respect to the crack propagation angle and equivalent factors calculations. The proposed higher-order technique has demonstrated superior performance in comparison with the conventional displacement fitting technique.

Dynamic Fracture Analysis of Plates Loaded in Tension and Bending Using the Dual Boundary Element Method

The purpose of this paper is to solve dynamic fracture problems of plates under both tension and bending using the boundary element method (BEM). The dynamic problems were solved in the Laplace-transform domain, which avoided the calculation of the domain integrals resulting from the inertial terms. The dual boundary element method, in which both displacement and traction boundary integral equations are utilized, was applied to the modelling of cracks. The dynamic fracture analysis of a plate under combined tension and bending loads was conducted using the BEM formulations for the generalized plane stress theory and Mindlin plate bending theory. Dynamic stress intensity factors were estimated based on the crack opening displacements.