Explicit evaluation of hypersingular boundary integral equations for acoustic sensitivity analysis based on direct differentiation method

This paper presents a new set of boundary integral equations for three dimensional acoustic shape sensitivity analysis based on the direct differentiation method. A linear combination of the derived equations is used to avoid the fictitious eigenfrequency problem associated with the conventional boundary integral equation method when solving exterior acoustic problems. The strongly singular and hypersingular boundary integrals contained in the equations are evaluated as the Cauchy principal values and Hadamard finite parts for constant element discretization without using any regularization technique in this study. The present boundary integral equations are more efficient to use than the usual ones based on any other singularity subtraction technique and can be applied to the fast multipole boundary element method more readily and efficiently. The effectiveness and accuracy of the present equations are demonstrated through some numerical examples.     

线弹性断裂力学问题的扩展自然单元法

为了更加有效地求解线弹性断裂问题,提出了扩展自然单元法。该方法基于单位分解的思想,在自然单元法的位移模式中加入扩展项表征不连续位移场和裂纹尖端奇异场。通过水平集方法确定裂纹面和裂纹尖端区域,并基于虚位移原理推导了平衡方程的离散线性方程。由于自然单元法的形函数满足Kronecker delta函数性质,本质边界条件易于施加。混合模式裂纹的应力强度因子由相互作用能量积分方法计算。数值算例结果表明扩展自然单元法可以方便地求解线弹性断裂力学问题。     

Traction BIE solutions for flat cracks

The paper deals with the numerical solution techniques for the traction boundary integral equation (BIE), which describes the opening (and sliding) displacements of the surface of the traction loaded crack or arbitrary planform embedded in an elastic infinite body (buried crack problem). The traction BIE is a singular integral equation of the first kind for the displacement gradients. Its solution poses a number of numerical problems, such as the presence of derivatives of the unknown function in the integral equation, the modeling of the crack front displacement gradient singularity, and the regularization of the equation's singular kernels. All of the above problems have been addressed and solved. Details of the algorithm are provided. Numerical results of a number of crack configurations are presented, demonstrating high accuracy of the method.     

Hyper-singular boundary element method for elastoplastic fracture mechanics analysis with large deformation

In this paper a two-dimensional hyper-singular boundary element method for elastoplastic fracture mechanics analysis with large deformation is presented. The proposed approach incorporates displacement and the traction boundary integral equations as well as finite deformation stress measures, and general crack problems can be solved with single-region formulations. Efficient regularization techniques are applied to the corresponding singular terms in displacement, displacement derivatives and traction boundary integral equations, according to the degree of singularity of the kernel functions. Within the numerical implementation of the hyper-singular boundary element formulation, crack tip and corners are modelled with discontinuous elements. Fracture measures are evaluated at each load increment, using the J-integral. Several cases studies with different boundary and loading conditions have been analysed. It has been shown that the new singularity removal technique and the non-linear elastoplastic formulation lead to accurate solutions.     

The mixed-type integral equations for transient elastodynamic plane crack problems

In the present paper, by use of the boundary integral equation method and the techniques of Green fundamental solution and singularity analysis, the dynamic infinite plane crack problem is investigated. For the first time, the problem is reduced to solving a system of mixed-typed integral equations in Laplace transform domain. The equations consist of ordinary boundary integral equations along the outer boundary and Cauchy singular integral equations along the crack line. The equations obtained are strictly proved to be equivalent with the dual integral equations obtained by Sih in the special case of dynamic Griffith crack problem. The mixed-type integral equations can be solved by combining the numerical method of singular integral equation with the ordinary boundary element method. Further use the numerical method for Laplace transform, several typical examples are calculated and their dynamic stress intensity factors are obtained. The results show that the method proposed is successful and can be used to solve more complicated problems.     

Dual boundary element analysis of cracked plates: singularity subtraction technique

The present paper is concerned with the formulation of the singularity subtraction technique in the dual boundary element analysis of the mixed-mode deformation of general homogeneous cracked plates.The equations of the dual boundary element method are the displacement and the traction boundary integral equations. When the displacement equation is applied on one of the crack surfaces and the traction equation is applied on the other, general mixed-mode crack problems can be solved in a single region boundary element formulation, with both crack surfaces discretized with discontinuous quadratic boundary elements.The singularity subtraction technique is a regularization procedure that uses a singular particular solution of the crack problem to introduce the stress intensity factors as additional problem unknowns. The single-region boundary element analysis of a general crack problem restricts the availability of singular particular solutions, valid in the global domain of the problem. A modelling strategy, that considers an automatic partition of the problem domain in near-tip and far-tip field regions, is proposed to overcome this difficulty. After the application of the singularity subtraction technique in the near-tip field regions, regularized locally with the singular term of the Williams' eigenexpansion, continuity is restored with equilibrium and compatibility conditions imposed along the interface boundaries. The accuracy and efficiency of the singularity subtraction technique make this formulation ideal for the study of crack growth problems under mixed-mode conditions.     

A local boundary integral-based meshless method for Biot's consolidation problem

Traditional numerical techniques such as FEM and BEM have been successfully applied to the solutions of Biot's consolidation problems. However, these techniques confront some difficulties in dealing with moving boundaries. In addition, pre-designing node connectivity or element is not an easy task. Recently, developed meshless methods may overcome these difficulties. In this paper, a meshless model, based on the local Petrov–Galerkin approach with Heaviside step function as well as radial basis functions, is developed and implemented for the numerical solution of plane strain poroelastic problems. Although the proposed method is based on local boundary integral equation, it does not require any fundamental solution, thus avoiding the singularity integral. It also has no domain integral over local domain, thus largely reducing the computational cost in formulation of system stiffness. This is a truly meshless method. The solution accuracy and the code performance are evaluated through one-dimensional and two-dimensional consolidation problems. Numerical examples indicate that this meshless method is suitable for either regular or irregular node distributions with little loss of accuracy, thus being a promising numerical technique for poroelastic problems.     

A time‐dependent formulation of dual boundary element method for 3D dynamic crack problems

In this paper, the dual boundary element method in time domain is developed for three‐dimensional dynamic crack problems. The boundary integral equations for displacement and traction in time domain are presented. By using the displacement equation and traction equation on crack surfaces, the discontinuity displacement on the crack can be determined. The integral equations are solved numerically by a time‐stepping technique with quadratic boundary elements. The dynamic stress intensity factors are calculated from the crack opening displacement. Several examples are presented to demonstrate the accuracy of this method. Copyright © 1999 John Wiley & Sons, Ltd     

A comparison of integral formulations for the analysis of low Reynolds number flows

Integral formulations for the analysis of low Reynolds number flows have been developed over the past 25 years. These formulations can typically be categorized as being either direct or indirect and either velocity integral equations or traction integral equations. Depending on the boundary conditions imposed for a given problem, the resulting integral formulation will result in either a Fredholm integral equation of the first kind, second kind or mixed. In general, Fredholm integral equations of the first kind lead to unstable numerical schemes based upon discretization and can result in low-order accuracy. For most practical problems, the direct velocity integral equation results in a Fredholm equation of the first kind. Nevertheless, many researchers have used this integral formulation with great success and any potential matrix ill-conditioning seems to have little influence on the accuracy of the numerical solutions. In this research, three integral formulations are compared, namely, the direct velocity integral equation, the indirect velocity integral equation, and the direct traction integral equation. Three benchmark problems are chosen for which there are analytic solutions. Although the discretized matrix condition numbers are in some cases orders of magnitude larger for the direct velocity integral formulation, there is little to distinguish between the three methods in terms of accuracy. In fact, the results for the direct velocity integral equation were slightly more accurate in most cases for the three benchmark problems considered in this research.     

Hypersingular integral formulation of elastic wave scattering

A hypersingular boundary integral formulation for calculating two dimensional elastic wave scattering from thin bodies and cracks is described. The boundary integral equation for surface displacement is combined with the hypersingular equation for surface traction. The difficult part in employing the traction equation, the derivation of analytical formulas for the hypersingular integral by means of a limit to the boundary, is easily handled by means of symbolic computation. In addition, the terms containing an integrable logarithmic singularity are treated by a straightforward numerical method, bypassing the use of Taylor series expansions. Example wave scattering calculations for cracks and thin ellipses are presented.     

Evaluation of multiple stress singularity orders of a V-notch by the boundary element method

A new technique concerning the evaluation of multiple stress singularity orders of a V-notch with the boundary element method is proposed. For linear elastic material, the asymptotic displacement field in a V-notch tip region is expressed as a series expansion with respect to the radial coordinate from the V-notch tip. The series expansion of the asymptotic field is then substituted into the boundary integral equation established on the V-notch. By boundary element discretization, the boundary integral equation is transformed into an eigen-equation with respect to the stress singularity orders. Hence the eigen-values, which are the singularity orders, can be obtained simultaneously by solving the eigen-equation. An evident feature of the present method is that the real and complex singularity orders together with the non-singularity ones can be obtained at the same time.     

A new boundary integral equation method for cracked 2-D anisotropic bodies

By using integration by parts to the traditional boundary integral formulation, a traction boundary integral equation for cracked 2-D anisotropic bodies is derived. The new traction integral equation involves only singularity of order 1/r and no hypersingular term appears. The dislocation densities on the crack surface are introduced and the relations between stress intensity factors and dislocation densities near the crack tip are induced to calculate the stress intensity factors. The boundary element method based on the new equation is established and the singular interpolation functions are introduced to model the singularity of the dislocation density (in the order of ) for crack tip elements. The proposed method can be directly used for the 2-D anisotropic body containing cracks of arbitrary geometric shapes. Several numerical examples demonstrate the validity and accuracy of BEM based on the new boundary integral equation.     

Numerical operational methods for time-dependent linear problems

A general and systematic discussion on the use of the operational method of Laplace transform for numerically solving complex time-dependent linear problems is presented. Application of Laplace transform with respect to time on the governing differential equations as well as the boundary and initial conditions of the problem reduces it to one independent of time, which is solved in the transform domain by any convenient numerical technique, such as the finite element method, the finite difference method or the boundary integral equation method. Finally, the time domain solution is obtained by a numerical inversion of the transformed solution. Eight existing methods of numerical inversion of the Laplace transform are systematically discussed with respect to their use, range of applicability, accuracy and computational efficiency on the basis of some framework vibration problems. Other applications of the Laplace transform method in conjunction with the finite element method or the boundary integral equation method in the areas of earthquake dynamic response of frameworks, thermaliy induced beam vibrations, forced vibrations of cylindrical shells, dynamic stress concentrations around holes in plates and viscoelastic stress analysis are also briefly described to demonstrate the generality and advantages of the method against other known methods.     

Numerical treatment of finite part integrals in 2-D boundary element analysis with application infracture mechanics

For the solution of problems in fracture mechanics by the boundary element method usually the subregion technique is employed to decouple the crack surfaces. In this paper a different procedure is presented. By using the displacement boundary integral equation on one side of the crack surface and the hypersingular traction boundary integral equation on the opposite side, one can renounce the subregion technique.An essential point when applying the traction boundary integral equation is the treatment of the thus arising hypersingular integrals. Two methods for their numerical computation are presented, both based on the finite part concept. One may either scale the integrals properly and use a specific quadrature rule, or one may apply the definition formula for finite part integrals and transform the resulting regular integrals into the usual element coordinate system afterwards. While the former method is restricted to linear or circular approximations of the boundary geometry, the latter one allows for arbitrary curved (e.g. isoparametric) elements. Two numerical examples are enclosed to demonstrate the accuracy of the two boundary integral equations technique compared with the subregion technique.     

A new variable‐order singular boundary element for two‐dimensional stress analysis

A new variable‐order singular boundary element for two‐dimensional stress analysis is developed. This element is an extension of the basic three‐node quadratic boundary element with the shape functions enriched with variable‐order singular displacement and traction fields which are obtained from an asymptotic singularity analysis. Both the variable order of the singularity and the polar profile of the singular fields are incorporated into the singular element to enhance its accuracy. The enriched shape functions are also formulated such that the stress intensity factors appear as nodal unknowns at the singular node thereby enabling direct calculation instead of through indirect extrapolation or contour‐integral methods. Numerical examples involving crack, notch and corner problems in homogeneous materials and bimaterial systems show the singular element's great versatility and accuracy in solving a wide range of problems with various orders of singularities. The stress intensity factors which are obtained agree very well with those reported in the literature. Copyright © 2002 John Wiley & Sons, Ltd.     

Variation of stress intensity factor and crack opening displacement of semi-elliptical surface crack

In this paper a singular integral equation method is applied to calculate the stress intensity factor along crack front of a 3D surface crack. Stress field induced by body force doublet in a semi infinite body is used as a fundamental solution. Then the problem is formulated as an integral equation with a singularity of the form of r ^-3. In solving the integral equations, the unknown functions of body force densities are approximated by the product of a polynomial and a fundamental density function; that is, the exact density distribution to make an elliptical crack in an infinite body. The calculation shows that the present method gives the smooth variation of stress intensity factors along the crack front and crack opening displacement along the crack surface for various aspect ratios and Poisson's ratio. The present method gives rapidly converging numerical results and highly satisfactory boundary conditions throughout the crack boundary.     

基于X-SBFEM的裂纹体非网格重剖分耦合模型研究

扩展有限元法利用了非网格重剖分技术,但需要基于裂尖解析解构造复杂的插值基函数,计算精度受网格疏密和插值基函数等因素影响。比例边界有限元法则在求解无限域和裂尖奇异性问题优势明显,两者衔接于有限元法理论内,可建立一种结合二者优势的断裂耦合数值模型。该文从虚功原理出发,利用位移协调与力平衡机制,提出了一种断裂计算的新方法X-SBFEM,达到了扩展有限元模拟裂纹主体、比例边界有限元模拟裂尖的目的。在数值算例中,通过边裂纹和混合型裂纹的应力强度因子计算,并与理论解对比,验证了该方法的准确性和有效性。     

A boundary element model for near surface contact stresses of rough surfaces

This study proposes a boundary element based near surface elastic stress prediction model for rough surface point contacts. The contact body is considered as a half space, whose boundary is divided into the rough finite field and the smooth infinite field. The integral kernel singularities in the displacement boundary integral equation are eliminated through coordinate transformation and the approach of rigid body motion. For the integral of the nearly singular stress kernels, a numerical scheme that combines the distance transformation and a subdivision technique is devised to improve the numerical accuracy and efficiency. The computational results are compared to closed-form solutions to assess the model accuracy. Using a three-dimensional sinusoidal roughness profile, the near surface stress concentrations induced by surface asperities are demonstrated. It is shown the exclusion of roughness profile in stress computation leads to significant errors in both the locations and the magnitudes of near surface stress concentrations.     

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.     

Displacement gradients in BEM formulation for small strain plasticity

In the present work, we propose a consistent derivation of the regularized integral representations for displacement gradients within the boundary element formulation for solution of small strain plasticity problems. The regularization is carried out for both the domain and boundary integrals before any discretization and approximation of integral densities. Two kinds of the integral representations are derived with the leading singularity of the integral kernels being either hypersingularity or strong singularity. Both the two- and three-dimensional problems are considered simultaneously.