Boundary element‐free method for fracture analysis of 2‐D anisotropic piezoelectric solids

This paper considers a 2‐D fracture analysis of anisotropic piezoelectric solids by a boundary element‐free method. A traction boundary integral equation (BIE) that only involves the singular terms of order 1/r is first derived using integration by parts. New variables, namely, the tangential derivative of the extended displacement (the extended displacement density) for the general boundary and the tangential derivative of the extended crack opening displacement (the extended displacement dislocation density), are introduced to the equation so that solution to curved crack problems is possible. This resulted equation can be directly applied to general boundary and crack surface, and no separate treatments are necessary for the upper and lower surfaces of the crack. The extended displacement dislocation densities on the crack surface are expressed as the product of the characteristic terms and unknown weight functions, and the unknown weight functions are modelled using the moving least‐squares (MLS) approximation. The numerical scheme of the boundary element‐free method is established, and an effective numerical procedure is adopted to evaluate the singular integrals. The extended ‘stress intensity factors’ (SIFs) are computed for some selected example problems that contain straight or curved cracks, and good numerical results are obtained. Copyright © 2006 John Wiley & Sons, Ltd.     

Antiplane crack analysis of a functionally graded material by a BIEM

Elastostatic analysis of an antiplane crack in a functionally graded material (FGM) is performed by using a hypersingular boundary integral equation method (BIEM). An exponential law is applied to describe the spatial variation of the shear modulus of the FGM. A Galerkin method is applied for the numerical solution of the hypersingular traction BIE. Both unidirectional and bidirectional material gradations are investigated. Stress intensity factors for an infinite and linear elastic FGM containing a finite crack subjected to an antiplane crack-face loading are presented and discussed. The influences of the material gradients and the crack orientation on the stress intensity factors are analyzed.     

Implementation of meshless LBIE method to the 2D non‐linear SG problem

In this paper the meshless local boundary integral equation (LBIE) method for numerically solving the non‐linear two‐dimensional sine‐Gordon (SG) equation is developed. The method is based on the LBIE with moving least‐squares (MLS) approximation. For the MLS, nodal points spread over the analyzed domain are utilized to approximate the interior and boundary variables. The approximation functions are constructed entirely using a set of scattered nodes, and no element or connectivity of the nodes is needed for either the interpolation or the integration purposes. A time‐stepping method is employed to deal with the time derivative and a simple predictor–corrector scheme is performed to eliminate the non‐linearity. A brief discussion is outlined for numerical integrations in the proposed algorithm. Some examples involving line and ring solitons are demonstrated and the conservation of energy in undamped SG equation is investigated. The final numerical results confirm the ability of method to deal with the unsteady non‐linear problems in large domains. Copyright © 2009 John Wiley & Sons, Ltd.     

Revisit of degenerate scales in the BIEM/BEM for 2D elasticity problems

The boundary integral equation method in conjunction with the degenerate kernel, the direct searching technique (singular value decomposition), and the only two-trials technique (2 × 2 matrix eigenvalue problem) are analytically and numerically used to find the degenerate scales, respectively. In the continuous system of boundary integral equation, the degenerate kernel for the 2D Kelvin solution in the polar coordinates is reviewed and the degenerate kernel in the elliptical coordinates is derived. Using the degenerate kernel, an analytical solution of the degenerate scales for the elasticity problem of circular and elliptical cases is obtained and compared with the numerical result. Further, the triangular case and square case were also numerically demonstrated.     

Accelerating boundary integral equation method using a special‐purpose computer

We present a new solution to accelerate the boundary integral equation method (BIEM). The calculation time of the BIEM is dominated by the evaluation of the layer potential in the boundary integral equation. We performed this task using MDGRAPE‐2, a special‐purpose computer designed for molecular dynamics simulations. MDGRAPE‐2 calculates pairwise interactions among particles (e.g. atoms and ions) using hardwired‐pipeline processors. We combined this hardware with an iterative solver. During the iteration process, MDGRAPE‐2 evaluates the layer potential. The rest of the calculation is performed on a conventional PC connected to MDGRAPE‐2. We applied this solution to the Laplace and Helmholtz equations in three dimensions. Numerical tests showed that BIEM is accelerated by a factor of 10–100. Our rather naive solution has a calculation cost of O(N² × N_iter), where N is the number of unknowns and N_iter is the number of iterations. Copyright © 2005 John Wiley & Sons, Ltd.     

A dual BIE approach for large‐scale modelling of 3‐D electrostatic problems with the fast multipole boundary element method

A dual boundary integral equation (BIE) formulation is presented for the analysis of general 3‐D electrostatic problems, especially those involving thin structures. This dual BIE formulation uses a linear combination of the conventional BIE and hypersingular BIE on the entire boundary of a problem domain. Similar to crack problems in elasticity, the conventional BIE degenerates when the field outside a thin body is investigated, such as the electrostatic field around a thin conducting plate. The dual BIE formulation, however, does not degenerate in such cases. Most importantly, the dual BIE is found to have better conditioning for the equations using the boundary element method (BEM) compared with the conventional BIE, even for domains with regular shapes. Thus the dual BIE is well suited for implementation with the fast multipole BEM. The fast multipole BEM for the dual BIE formulation is developed based on an adaptive fast multiple approach for the conventional BIE. Several examples are studied with the fast multipole BEM code, including finite and infinite domain problems, bulky and thin plate structures, and simplified comb‐drive models having more than 440 thin beams with the total number of equations above 1.45 million and solved on a PC. The numerical results clearly demonstrate that the dual BIE is very effective in solving general 3‐D electrostatic problems, as well as special cases involving thin perfect conducting structures, and that the adaptive fast multipole BEM with the dual BIE formulation is very efficient and promising in solving large‐scale electrostatic problems. Copyright © 2007 John Wiley & Sons, Ltd.     

An element‐free Galerkin method for 3D crack propagation simulation under complicated stress conditions

This study develops an element‐free Galerkin method based on the moving least‐squares approximation to trace three‐dimensional crack propagation under complicated stress conditions. The crack surfaces are modelled by a collection of planar triangles that are added when cracks propagate. The visibility criterion is adopted to treat the screening effect of the cracks on the influenced domain of a Gaussian point. Cracks are assumed to propagate in the perpendicular planes at crack front points when the strain energy release rates reach the material fracture toughness. This method is unique in that it uses a nonlinear contact iterative algorithm to consider contributions of crack surface interaction to the global equilibrium equations, so that crack opening, sliding and closing under complicated stress states can be efficiently modelled. Two numerical examples of three‐dimensional quasi‐static crack propagation were modelled with satisfactory results. Copyright © 2012 John Wiley & Sons, Ltd.     

A semi‐analytic collocation technique for steady‐state strongly nonlinear advection‐diffusion‐reaction equations with variable coefficients

The aim of this paper is to present a new semi‐analytic numerical method for strongly nonlinear steady‐state advection‐diffusion‐reaction equation (ADRE) in arbitrary 2‐D domains. The key idea of the method is the use of the basis functions which satisfy the homogeneous boundary conditions of the problem. Each basis function used in the algorithm is a sum of an analytic basis function and a special correcting function which is chosen to satisfy the homogeneous boundary conditions of the problem. The polynomials, trigonometric functions, conical radial basis functions, and the multiquadric radial basis functions are used in approximation of the ADRE. This allows us to seek an approximate solution in the analytic form which satisfies the boundary conditions of the initial problem with any choice of free parameters. As a result, we separate the approximation of the boundary conditions and the approximation of the ADRE inside the solution domain. The numerical examples confirm the high accuracy and efficiency of the proposed method in solving strongly nonlinear equations in an arbitrary domain.     

Half‐Metallic Behavior in 2D Transition Metal Dichalcogenides Nanosheets by Dual‐Native‐Defects Engineering

Two‐dimensional transition metal dichalcogenides (TMDs) have been regarded as one of the best nonartificial low‐dimensional building blocks for developing spintronic nanodevices. However, the lack of spin polarization in the vicinity of the Fermi surface and local magnetic moment in pristine TMDs has greatly hampered the exploitation of magnetotransport properties. Herein, a half‐metallic structure of TMDs is successfully developed by a simple chemical defect‐engineering strategy. Dual native defects decorate titanium diselenides with the coexistence of metal‐Ti‐atom incorporation and Se‐anion defects, resulting in a high‐spin‐polarized current and local magnetic moment of 2D Ti‐based TMDs toward half‐metallic room‐temperature ferromagnetism character. Arising from spin‐polarization transport, the as‐obtained T‐TiSe_1.8 nanosheets exhibit a large negative magnetoresistance phenomenon with a value of ?40% (5T, 10 K), representing one of the highest negative magnetoresistance effects among TMDs. It is anticipated that this dual regulation strategy will be a powerful tool for optimizing the intrinsic physical properties of TMD systems.     

A local multigrid X‐FEM strategy for 3‐D crack propagation

A multiscale method for 3‐D crack propagation simulation in large structures is proposed. The method is based on the extended finite element method (X‐FEM). The asymptotic behavior of the crack front is accurately modeled using enriched elements and no remeshing is required during crack propagation. However, the different scales involved in fracture mechanics problems can differ by several orders of magnitude and industrial meshes are usually not designed to account for small cracks. Enrichments are therefore useless if the crack is too small compared with the element size. To overcome this drawback, a project combining different numerical techniques was started. The first step was the implementation of a global multigrid algorithm within the X‐FEM framework and was presented in a previous paper (Eur. J. Comput. Mech. 2007; 16 :161–182). This work emphasized the high efficiency in cpu time but highlighted that mesh refinement is required on localized areas only (cracks, inclusions, steep gradient zones). This paper aims at linking the different scales by using a local multigrid approach. The coupling of this technique with the X‐FEM is described and computational aspects dealing with intergrid operators, optimal multiscale enrichment strategy and level sets are pointed out. Examples illustrating the accuracy and efficiency of the method are given. Copyright © 2008 John Wiley & Sons, Ltd.     

A crack propagation criterion based on ΔCTOD measured with 2D‐digital image correlation technique

The fatigue cracks growth rate of a forged HSLA steel (AISI 4130) was investigated using thin single edge notch tensile specimen to simulate the crack development on a diesel train crankshafts. The effect of load ratio, R, was investigated at room temperature. Fatigue fracture surfaces were examined by scanning electron microscopy. An approach based on the crack tip opening displacement range (ΔCTOD) was proposed as fatigue crack propagation criterion. ΔCTOD measurements were carried out using 2D‐digital image correlation techniques. J‐integral values were estimated using ΔCTOD. Under test conditions investigated, it was found that the use of ΔCTOD as a fatigue crack growth driving force parameter is relevant and could describe the crack propagation behaviour, under different load ratio R.     

Variation of stress intensity factor along the front of a 3D rectangular crack by using a singular integral equation method

In this paper a singular integral equation method is applied to calculate the distribution of stress intensity factor along the crack front of a 3D rectangular crack. The stress field induced by a body force doublet in an infinite body is used as the fundamental solution. Then, the problem is formulated as an integral equation with a singularity of the form of r ^–3. In solving the integral equation, the unknown functions of body force densities are approximated by the product of a polynomial and a fundamental density function, which expresses stress singularity along the crack front in an infinite body. The calculation shows that the present method gives smooth variations of stress intensity factors along the crack front for various aspect ratios. The present method gives rapidly converging numerical results and highly satisfied boundary conditions throughout the crack boundary.     

Evaluation of 2‐D Green's boundary formula and its normal derivative using Legendre polynomials,with an application to acoustic scattering problems

This paper presents the non‐singular forms, in a global sense, of two‐dimensional Green's boundary formula and its normal derivative. The main advantage of the modified formulations is that they are amenable to solution by directly applying standard quadrature formulas over the entire integration domain; that is, the proposed element‐free method requires only nodal data. The approach includes expressing the unknown function as a truncated Fourier–Legendre series, together with transforming the integration interval [a, b] to [‐1,1] ; the series coefficients are thus to be determined. The hypersingular integral, interpreted in the Hadamard finite‐part sense, and some weakly singular integrals can be evaluated analytically; the remaining integrals are regular with the limiting values of the integrands defined explicitly when a source point coincides with a field point. The effectiveness of the modified formulations is examined by an elliptic cylinder subject to prescribed boundary conditions. The regularization is further applied to acoustic scattering problems. The well‐known Burton–Miller method, using a linear combination of the surface Helmholtz integral equation and its normal derivative, is adopted to overcome the non‐uniqueness problem. A general non‐singular form of the composite equation is derived. Comparisons with analytical solutions for acoustically soft and hard circular cylinders are made. Copyright © 2001 John Wiley & Sons, Ltd.     

J‐integral in the description of fatigue crack growth rate induced by different ratios of torsion to bending loading in AlCu4Mg1

The paper presents the results of tests on fatigue crack growth under proportional torsion with bending in AlCu4Mg1 aluminium alloy. Specimens with rectangular cross‐sections and stress concentrator in the form of external one‐sided sharp notch were used. The tests were performed under the different ratios of torsion to bending moments. The results of the experimental tests have been described by a nonlinear formula based on ΔJ‐integral range. The tests have shown that the change of ratio of torsion to bending moments from 0.58 to 1.73 caused an increase in crack growth rate. It has been shown that at the constant loading and the change of stress ratio (R) from ‐ 1 to 0, the fatigue crack growth rate also increases.     

Extended finite element method for three‐dimensional crack modelling

An extended finite element method (X‐FEM) for three‐dimensional crack modelling is described. A discontinuous function and the two‐dimensional asymptotic crack‐tip displacement fields are added to the finite element approximation to account for the crack using the notion of partition of unity. This enables the domain to be modelled by finite elements with no explicit meshing of the crack surfaces. Computational geometry issues associated with the representation of the crack and the enrichment of the finite element approximation are discussed. Stress intensity factors (SIFs) for planar three‐dimensional cracks are presented, which are found to be in good agreement with benchmark solutions. Copyright © 2000 John Wiley & Sons, Ltd.     

The 3‐D elastodynamic boundary element method: semi‐analytical integration for linear isoparametric triangular elements

A semi‐analytical integration scheme is described in this paper which is designed to reduce the errors incurred when integrals with singular integrands are evaluated numerically. This new scheme can be applied to linear triangular elements for use in steady‐state elastodynamic BEM problems and is particularly useful for predicting displacement to high accuracy, close to surfaces for a spectrum of frequencies. The scheme involves the application of Taylor expansions to formulate the integrals into two parts. One part is regular and is evaluated numerically and the other part is singular but sufficiently simple to enable its transforma tion into a line integral. The line integral is solved numerically using Gauss–Legendre quadrature. This approach caters for all the integral types that appear in steady‐state elastodynamic boundary elements but, in particular, no special treatment is required for the evaluation of the Cauchy principal value singular integrals. Numerical tests are performed on a simple test‐problem for which a known analytical solution exists. The results obtained using the semi‐analytical approach are shown to be considerably more accurate than those obtained using standard quadrature methods. Copyright © 1999 John Wiley & Sons, Ltd.     

Self‐regular boundary integral equation formulations for Laplace's equation in 2‐D

The purpose of this work is to demonstrate the application of the self‐regular formulation strategy using Green's identity (potential‐BIE) and its gradient form (flux‐BIE) for Laplace's equation. Self‐regular formulations lead to highly effective BEM algorithms that utilize standard conforming boundary elements and low‐order Gaussian integrations. Both formulations are discussed and implemented for two‐dimensional potential problems, and numerical results are presented. Potential results show that the use of quartic interpolations is required for the flux‐BIE to show comparable accuracy to the potential‐BIE using quadratic interpolations. On the other hand, flux error results in the potential‐BIE implementation can be dominated by the numerical integration of the logarithmic kernel of the remaining weakly singular integral. Accuracy of these flux results does not improve beyond a certain level when using standard quadrature together with a special transformation, but when an alternative logarithmic quadrature scheme is used these errors are shown to reduce abruptly, and the flux results converge monotonically to the exact answer. In the flux‐BIE implementation, where all integrals are regularized, flux results accuracy improves systematically, even with some oscillations, when refining the mesh or increasing the order of the interpolating function. The flux‐BIE approach presents a great numerical sensitivity to the mesh generation scheme and refinement. Accurate results for the potential and the flux were obtained for coarse‐graded meshes in which the rate of change of the tangential derivative of the potential was better approximated. This numerical sensitivity and the need for graded meshes were not found in the elasticity problem for which self‐regular formulations have also been developed using a similar approach. Logarithmic quadrature to evaluate the weakly singular integral is implemented in the self‐regular potential‐BIE, showing that the magnitude of the error is dependent only on the standard Gauss integration of the regularized integral, but not on this logarithmic quadrature of the weakly singular integral. The self‐regular potential‐BIE is compared with the standard (CPV) formulation, showing the equivalence between these formulations. The self‐regular BIE formulations and computational algorithms are established as robust alternatives to singular BIE formulations for potential problems. Copyright © 2001 John Wiley & Sons, Ltd.