Stress intensity factor computation using the method of fundamental solutions

In this paper, we study the application of the method of fundamental solutions to the computation of stress intensity factors in linear elastic fracture mechanics. The displacements are approximated by linear combinations of the fundamental solutions of the Cauchy–Navier equations of elasticity and the leading terms for the displacement near the crack tip. The applicability of two formulations of the method is demonstrated on two mode I crack problems, where it is shown that accurate approximations for the stress intensity factors can be obtained with relatively few degrees of freedom. Parts of this work were undertaken while the first author was a Visiting Professor in the Department of Mathematical and Computer Sciences, Colorado School of Mines, Golden, Colorado 80401, U.S.A.     

Stress intensity factors for cracked triangular cross‐section thin‐walled tubes

For one kind of finite‐boundary crack problems, the cracked equilateral triangular cross‐section tube, an analytical and very simple method to determine the stress intensity factors has been proposed based on a new concept of crack surface widening energy release rate and the principle of virtual work. Different from the classical crack extension energy release rate, the crack surface widening energy release rate can be defined by the G*‐integral theory and expressed by stress intensity factors. This energy release rate can also be defined easily by the elementary strength theory for slender structures and expressed by axial strains and loads. These two forms of crack surface widening energy release rate constitute the basis of a new analysis method for cracked tubes. From present discussions, a series of stress intensity factors are derived for cracked equilateral triangular cross‐section tubes. Actually, the present method can also be applied to cracked polygonal tubes.     

Methods of fundamental solutions for time‐dependent heat conduction problems

A time‐dependent heat conduction problem can be solved by the method of fundamental solutions using the fundamental solution to the modified Helmholtz equation or the fundamental solution to the heat equation. This paper presents solutions using both formulations in terms of initial and boundary conditions. Such formulations enable calculation of errors and variance, which indicates sensitivities of solutions to uncertainties in initial and boundary conditions. Both errors and variance of solutions to three test problems by the two methods of fundamental solutions are used to compare performances of the methods. Copyright © 2005 John Wiley & Sons, Ltd.     

Error estimation of stress intensity factors for mixed‐mode cracks

This paper presents an a posteriori error estimator for mixed‐mode stress intensity factors in plane linear elasticity. A surface integral over an arbitrary crown is used for the separate calculation of the combined mode's stress intensity factors. The error in the quantity of interest is based on goal‐oriented error measures and estimated through an error in the constitutive relation. Copyright © 2006 John Wiley & Sons, Ltd.     

The method of fundamental solutions for axisymmetric potential problems

In this paper, we investigate the application of the Method of Fundamental Solutions (MFS) to two classes of axisymmetric potential problems. In the first, the boundary conditions as well as the domain of the problem, are axisymmetric, and in the second, the boundary conditions are arbitrary. In both cases, the fundamental solutions of the governing equations and their normal derivatives, which are required in the formulation of the MFS, can be expressed in terms of complete elliptic integrals. The method is tested on several axisymmetric problems from the literature and is also applied to an axisymmetric free boundary problem. Copyright © 1999 John Wiley & Sons, Ltd.     

A direct method for calculating the stress intensity factor in BEM

The proposed algorithm employs singular crack tip elements in which the stress intensity factor appears as a degree of freedom. The additional degrees of freedom are compensated by constraint conditions which originate from imposing continuity across elements and a contour integration formula. The two benchmark problems indicate the proposed algorithm can accurately predict the stress intensity factor and the distribution of the primary and secondary variables in fracture problems.     

Numerical solution of two backward parabolic problems using method of fundamental solutions

This work is devoted to a numerical algorithm based on the method of fundamental solutions (MFS) for solving two backward parabolic problems with different boundary conditions, one with nonlocal Dirichlet boundary conditions, and second one with Robin type boundary conditions. The initial temperature distribution will be identified from the final temperature distribution, which appear in some applied subjects. The Tikhonov regularization method with the L-curve criterion for choosing the regularization parameter is adopted for solving the resulting matrix equation which is highly ill-conditioned. Two numerical examples are provided to show the high efficiency of the suggested method.     

The method of fundamental solutions for steady-state groundwater flow problems

The method of fundamental solutions is a meshless method. Only boundary collocation points are needed during the whole solution process. It has the merits of mathematical simplicity, ease of programming, high solution accuracy, and others. In this paper, the method of fundamental solutions is applied to simulate 2D steady-state groundwater flow problems. The principle of superposition is used during the whole solution process. Numerical results are compared with the multiquadrics method and the mixed finite element method as well as analytical solutions. It is shown that the method of fundamental solutions is promising in dealing with steady groundwater flow 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     

The method of fundamental solutions for problems in static thermo-elasticity with incomplete boundary data

An inverse problem in static thermo-elasticity is investigated. The aim is to reconstruct the unspecified boundary data, as well as the temperature and displacement inside a body from over-specified boundary data measured on an accessible portion of its boundary. The problem is linear but ill-posed. The uniqueness of the solution is established but the continuous dependence on the input data is violated. In order to reconstruct a stable and accurate solution, the method of fundamental solutions is combined with Tikhonov regularization where the regularization parameter is selected based on the L-curve criterion. Numerical results are presented in both two and three dimensions showing the feasibility and ease of implementation of the proposed technique.     

An accurate scheme for mixed‐mode fracture analysis of functionally graded materials using the interaction integral and micromechanics models

The interaction integral is a conservation integral that relies on two admissible mechanical states for evaluating mixed‐mode stress intensity factors (SIFs). The present paper extends this integral to functionally graded materials in which the material properties are determined by means of either continuum functions (e.g. exponentially graded materials) or micromechanics models (e.g. self‐consistent, Mori–Tanaka, or three‐phase model). In the latter case, there is no closed‐form expression for the material‐property variation, and thus several quantities, such as the explicit derivative of the strain energy density, need to be evaluated numerically (this leads to several implications in the numerical implementation). The SIFs are determined using conservation integrals involving known auxiliary solutions. The choice of such auxiliary fields and their implications on the solution procedure are discussed in detail. The computational implementation is done using the finite element method and thus the interaction energy contour integral is converted to an equivalent domain integral over a finite region surrounding the crack tip. Several examples are given which show that the proposed method is convenient, accurate, and computationally efficient. Copyright © 2003 John Wiley & Sons, Ltd.     

3D J‐integral evaluation using the computation of line and surface integrals

In the analysis of fracture mechanics of structures using three‐dimensional (3D) J‐integral, an integral evaluation of line and surface is required. However, because surface integral evaluation requires the calculation of the second derivative of displacement field and commercial finite element codes cannot calculate it, then this portion of the integral is neglected in some research. In this paper, a method for computing 3D J‐integral is presented using finite element analysis. In the analysis, the second derivative evaluation of displacement field is employed. The method is implemented in calculating the J‐integral of some 3D cracks and results are compared to well‐known reference values. The results show that the method is reliable and is suitable for applications in engineering. The portion of 3D J‐integral, namely the surface integral value is investigated and it is shown that neglecting this portion can introduce considerable error in the final results.     

The method of fundamental solutions for inverse source problems associated with the steady‐state heat conduction

This paper presents the use of the method of fundamental solutions (MFS) for recovering the heat source in steady‐state heat conduction problems from boundary temperature and heat flux measurements. It is well known that boundary data alone do not determine uniquely a general heat source and hence some a priori knowledge is assumed in order to guarantee the uniqueness of the solution. In the present study, the heat source is assumed to satisfy a second‐order partial differential equation on a physical basis, thereby transforming the problem into a fourth‐order partial differential equation, which can be conveniently solved using the MFS. Since the matrix arising from the MFS discretization is severely ill‐conditioned, a regularized solution is obtained by employing the truncated singular value decomposition, whilst the optimal regularization parameter is determined by the L‐curve criterion. Numerical results are presented for several two‐dimensional problems with both exact and noisy data. The sensitivity analysis with respect to two solution parameters, i.e. the number of source points and the distance between the fictitious and physical boundaries, and one problem parameter, i.e. the measure of the accessible part of the boundary, is also performed. The stability of the scheme with respect to the amount of noise added into the data is analysed. The numerical results obtained show that the proposed numerical algorithm is accurate, convergent, stable and computationally efficient for solving inverse source problems in steady‐state heat conduction. Copyright © 2006 John Wiley & Sons, Ltd.     

The method of fundamental solutions for inverse 2D Stokes problems

A numerical scheme based on the method of fundamental solutions is proposed for the solution of two-dimensional boundary inverse Stokes problems, which involve over-specified or under-specified boundary conditions. The coefficients of the fundamental solutions for the inverse problems are determined by properly selecting the number of collocation points using all the known boundary values of the field variables. The boundary points of the inverse problems are collocated using the Stokeslet as the source points. Validation results obtained for two test cases of inverse Stokes flow in a circular cavity, without involving any iterative procedure, indicate the proposed method is able to predict results close to the analytical solutions. The effects of the number and the radius of the source points on the accuracy of numerical predictions have also been investigated. The capability of the method is demonstrated by solving different types of inverse problems obtained by assuming mixed combinations of field variables on varying number of under- and over-specified boundary segments.     

Stress intensity factors computation for bending plates with extended finite element method

The modelization of bending plates with through‐the‐thickness cracks is investigated. We consider the Kirchhoff–Love plate model, which is valid for very thin plates. Reduced Hsieh–Clough–Tocher triangles and reduced Fraejis de Veubeke–Sanders quadrilaterals are used for the numerical discretization. We apply the eXtended Finite Element Method strategy: enrichment of the finite element space with the asymptotic bending singularities and with the discontinuity across the crack. The main point, addressed in this paper, is the numerical computation of stress intensity factors. For this, two strategies, direct estimate and J‐integral, are described and tested. Some practical rules, dealing with the choice of some numerical parameters, are underlined. Copyright © 2012 John Wiley & Sons, Ltd.     

Time-dependent fundamental solutions for homogeneous diffusion problems

This paper describes the applications of the method of fundamental solutions (MFS) for 1-, 2- and 3-D diffusion equations. The time-dependent fundamental solutions for diffusion equations are used directly to obtain the solution as a linear combination of the fundamental solution of the diffusion operator. The proposed scheme is free from the conventionally used Laplace transform or the finite difference scheme to deal with the time derivative of the governing equation. By properly placing the field points and the source points at a given time level, the solution is advanced in time until steady state solutions are reached. Test results obtained for 1-, 2- and 3-D diffusion problems show good comparisons with the analytical solutions and some with the MFS based on the modified Helmholtz fundamental solutions, thus the demonstration present numerical scheme of MFS with the space–time unification has been demonstrated as a promising mesh-free numerical tool to solve homogeneous diffusion problem.     

Solution of potential flow problems by the modified method of fundamental solutions: Formulations with the single layer and the double layer fundamental solutions

This paper describes an application of the recently proposed modified method of fundamental solutions (MMFS) to potential flow problems. The solution in two-dimensional Cartesian coordinates is represented in terms of the single layer and the double layer fundamental solutions. Collocation is used for the determination of the expansion coefficients. This novel method does not require a fictitious boundary as the conventional method of fundamental solutions (MFS). The source and the collocation points thus coincide on the physical boundary of the system. The desingularised values, consistent with the fundamental solutions used, are deduced from the direct boundary element method (BEM) integral equations by assuming a linear shape of the boundary between the collocation points. The respective values of the derivatives of the fundamental solution in the coordinate directions, as required in potential flow calculations, are calculated indirectly from the considerations of the constant potential field. The normal on the boundary is calculated by parametrisation of its length and the use of the cubic radial basis functions with the second-order polynomial augmentation. The components of the normal are calculated in an analytical way. A numerical example of potential flow around a two-dimensional circular region is presented. The results with the new MMFS are compared with the results of the classical MFS and the analytical solution. It is shown that the MMFS gives better accuracy for the potential, velocity components (partial derivatives of the potential), and absolute value of the velocity as compared with the classical MFS. The results with the single layer fundamental solution are more accurate than the results with the double layer fundamental solution.     

Determination of stress intensity factors of 2D fracture mechanics problems through a new semi‐analytical method

This study presents a novel development of a new semi‐analytical method with diagonal coefficient matrices to model crack issues. Accurate stress intensity factors based on linear elastic fracture mechanics are extracted directly from the semi‐analytical method. In this method, only the boundaries of problems are discretized using specific subparametric elements and higher‐order Chebyshev mapping functions. Implementing the weighted residual method and using Clenshaw–Curtis numerical integration result in diagonal Euler's differential equations. Consequently, when the local coordinates origin is located at the crack tip, the stress intensity factors can be determined directly without further processing. In order to present infinite stress at the crack tip, a new form of nodal force function is proposed. Validity and accuracy of the proposed method is fully demonstrated through four benchmark problems, which are successfully modeled using a few numbers of degrees of freedom. The numerical results agree very well with the analytical solution, experimental outcomes and the results from existing numerical methods available in the literature.