An iterative algorithm for singular Cauchy problems for the steady state anisotropic heat conduction equation

In this paper, we propose an alternating iterative algorithm to solve a singular Cauchy problem for the anisotropic heat conduction equation. The numerical algorithm is based on the boundary element method (BEM), modified to take into account the form of the singularity, without substantially increasing the amount of computation involved. Two test examples, the first with a singularity caused by an abrupt change in the boundary conditions and the second with a singularity caused by a sharp re-entrant corner, are investigated. The numerical results obtained confirm that provided an appropriate stopping regularization criterion is imposed, the iterative BEM is efficient in dealing with the difficulties arising from both the instabilities produced by the boundary condition formulation and the slow rate of convergence of standard numerical methods around the singular point.     

Boundary element method for the Cauchy problem in linear elasticity

In this paper, the iterative algorithm proposed by Kozlov et al. [Comput Maths Math Phys 32 (1991) 45] for obtaining approximate solutions to ill-posed boundary value problems in linear elasticity is analysed. The technique is then numerically implemented using the boundary element method (BEM). The numerical results obtained confirm that the iterative BEM produces a convergent and stable numerical solution with respect to increasing the number of boundary elements and decreasing the amount of noise added into the input data. An efficient stopping regularizing criterion is given and in addition, the accuracy of the iterative algorithm is improved by using a variable relaxation procedure. Analytical formulae for the integration constants resulting from the direct application of the BEM for an isotropic linear elastic medium are also presented.     

Relaxation procedures for an iterative MFS algorithm for two-dimensional steady-state isotropic heat conduction Cauchy problems

We investigate two algorithms involving the relaxation of either the given Dirichlet data (boundary temperatures) or the prescribed Neumann data (normal heat fluxes) on the over-specified boundary in the case of the alternating iterative algorithm of Kozlov et al. [26] applied to two-dimensional steady-state heat conduction Cauchy problems, i.e. Cauchy problems for the Laplace equation. The two mixed, well-posed and direct problems corresponding to each iteration of the numerical procedure are solved using a meshless method, namely the method of fundamental solutions (MFS), in conjunction with the Tikhonov regularization method. For each direct problem considered, the optimal value of the regularization parameter is chosen according to the generalized cross-validation (GCV) criterion. The iterative MFS algorithms with relaxation are tested for Cauchy problems associated with the Laplace operator in various two-dimensional geometries to confirm the numerical convergence, stability, accuracy and computational efficiency of the method.     

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.     

Solution of identification inverse problems in elastodynamics using semi-analytical sensitivity computation

The purpose of this work is to study a class of inverse problems that arises in solid mechanics areas such as quantitative non-destructive testing (QNDT) or shape optimization. The technique is based on the boundary integral equations (BIEs) used in the classical boundary element method (BEM), which are differentiated semi-analytically with respect to variations of the boundary geometry and used in an iterative search algorithm. The extension of this strategy is presented here for the case of elasticity in dynamics using the displacement or singular BIE, which allows to apply this strategy to QNDT problems based on vibrations or ultrasonics.The central point is the evaluation of the capability of solving numerically a QNDT problem such as the location and characterization of cavity and inclusion-type defects by measuring the dynamic response at an accessible boundary of the specimen. To test this capability, comprehensive convergence tests are made for the badness of the initial guess, the amount of supplied measurements, and simulated errors on measurements, geometry, elastic constants and frequency.     

An alternating iterative algorithm for the Cauchy problem in anisotropic elasticity

The alternating iterative algorithm proposed by Kozlov et al. [An iterative method for solving the Cauchy problem for elliptic equations. USSR Comput Math Math Phys 1991;31:45–52] for obtaining approximate solutions to the Cauchy problem in two-dimensional anisotropic elasticity is analysed and numerically implemented using the boundary element method (BEM). The ill-posedness of this inverse boundary value problem is overcome by employing an efficient regularising stopping. The numerical results confirm that the iterative BEM produces a convergent and stable numerical solution with respect to increasing the number of boundary elements and decreasing the amount of noise added into the input data.     

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.     

Analytical integral algorithm in the BEM for orthotropic potential problems of thin bodies

There exist nearly singular integrals for boundary layer effect problem and thin body effect problem in the boundary element method (BEM). A new completely analytical integral algorithm is proposed and applied to evaluate the nearly singular integrals in the BEM for two-dimensional orthotropic potential problems of thin bodies. The completely analytical integral formulas are derived with integration by parts for the linear boundary interpolation. The present algorithm applies these analytical formulas to deal with the nearly singular integrals. The unknown potentials and fluxes at boundary nodes are firstly calculated accurately and then the physical quantities at the interior points are computed. Two benchmark numerical examples on heat conduction demonstrate that the present algorithm can handle thin structures with the thickness-to-length ratio down to 1.E−08. This indicates that the BEM is especially accurate and efficient for numerical analysis of thin body problems.     

Efficient boundary element analysis of sharp notched plates

The present paper further develops the boundary element singularity subtraction technique, to provide an efficient and accurate method of analysing the general mixed-mode deformation of two-dimensional linear elastic structures containing sharp notches. The elastic field around sharp notches is singular. Because of the convergence difficulties that arise in numerical modelling of elastostatic problems with singular fields, these singularities are subtracted out of the original elastic field, using the first term of the Williams series expansion. This regularization procedure introduces the stress intensity factors as additional unknowns in the problem; hence extra conditions are required to obtain a solution. Extra conditions are defined such that the local solution in the neighbourhood of the notch tip is identical to the Williams solution; the procedure can take into account any number of terms of the series expansion. The standard boundary element method is modified to handle additional unknowns and extra boundary conditions. Analysis of plates with symmetry boundary conditions is shown to be straightforward, with the modified boundary element method. In the case of non-symmetrical plates, the singular tip-tractions are not primary boundary element unknowns. The boundary element method must be further modified to introduce the boundary integral stress equations of an internal point, approaching the notch-tip, as primary unknowns in the formulation. The accuracy and efficiency of the method is demonstrated with some benchmark tests of mixed-mode problems. New results are presented for the mixed-mode analysis of a non-symmetrical configuration of a single edge notched plate.     

Non‐singular boundary integral formulations for plane interior potential problems

In this article, a non‐singular formulation of the boundary integral equation is developed to solve smooth and non‐smooth interior potential problems in two dimensions. The subtracting and adding‐back technique is used to regularize the singularity of Green's function and to simplify the calculation of the normal derivative of Green's function. After that, a global numerical integration is directly applied at the boundary, and those integration points are also taken as collocation points to simplify the algorithm of computation. The result indicates that this simple method gives the convergence speed of order N ^?3 in the smooth boundary cases for both Dirichlet and mix‐type problems. For the non‐smooth cases, the convergence speed drops at O(N ^?1/2) for the Dirichlet problems. Copyright © 2001 John Wiley & Sons, Ltd.     

A method of fundamental solutions without fictitious boundary

This paper proposes a novel meshless boundary method called the singular boundary method (SBM). This method is mathematically simple, easy-to-program, and truly meshless. Like the method of fundamental solutions (MFS), the SBM employs the singular fundamental solution of the governing equation of interest as the interpolation basis function. However, unlike the MFS, the source and collocation points of the SBM coincide on the physical boundary without the requirement of introducing fictitious boundary. In order to avoid the singularity at the origin, this method proposes an inverse interpolation technique to evaluate the singular diagonal elements of the MFS coefficient matrix. The SBM is successfully tested on a benchmark problems, which shows that the method has a rapid convergence rate and is numerically stable.     

Unsteady Fully-Developed Flow in a Curved Pipe

It is shown that the boundary layer which develops from rest in a loosely coiled pipe of circular cross-section, following the imposition of a constant pressure gradient, terminates in singular behaviour at the inside bend after a finite time. This singularity of the boundary-layer equations is interpreted as an eruption of boundary-layer fluid into the interior or core flow. This result complements earlier work by Stewartson et al. [1] who consider the steady inlet flow to a curved pipe at high Dean number. In that case a singularity also develops, now at a finite distance from the entrance at the inside bend, which is again interpreted in terms of a boundary-layer collision or eruption.     

Investigation of the three-dimensional boundary layer separation criteria by a minimum principle

Existing separation criteria for the steady state three-dimensional boundary layers are at best heuristic. In this paper, the contemporary singular separation criterion has been reconsidered in the light of a set of transformed boundary layer equations. First a rigorous proof for the existence of the singularities of the boundary layer equations is provided. Based on these results the theory of boundary layer separation is developed both when the prevalant singularity in the equation of the skin-friction lines is or is not removable. It is shown that if the differential equation of the skin-friction lines has a removable singularity then the locus of such singular points on the body surface can be determined solely on the basis of the external conditions.     

A general algorithm for the numerical evaluation of nearly singular boundary integrals of various orders for two- and three-dimensional elasticity

 A general algorithm of the distance transformation type is presented in this paper for the accurate numerical evaluation of nearly singular boundary integrals encountered in elasticity, which, next to the singular ones, has long been an issue of major concern in computational mechanics with boundary element methods. The distance transformation is realized by making use of the distance functions, defined in the local intrinsic coordinate systems, which plays the role of damping-out the near singularity of integrands resulting from the very small distance between the source and the integration points. By taking advantage of the divergence-free property of the integrals with the nearly hypersingular kernels in the 3D case, a technique of geometric conversion over the auxiliary cone surfaces of the boundary element is designed, which is suitable also for the numerical evaluation of the hypersingular boundary integrals. The effects of the distance transformations are studied and compared numerically for different orders in the 2D case and in the different local systems in the 3D case using quadratic boundary elements. It is shown that the proposed algorithm works very well, by using standard Gaussian quadrature formulae, for both the 2D and 3D elastic problems. Received: 20 November 2001 / Accepted: 4 June 2002 The work was supported by the Science Foundation of Shanghai Municipal Commission of Education.     

Direct evaluation of singular integrals in elastoplastic analysis by the boundary element method

This paper presents a new method for the direct and accurate evaluation of strongly singular integrals in the sense of Cauchy principal values and weakly singular integrals over quadratic boundary elements in three-dimensional stress analysis and quadratic internal cells in two-dimensional elastoplastic analysis by the boundary element method. A quadratic triangle polar co-ordinate transformation technique is applied to reduce the order of singularity of the singular integrals. Next, a form of Stokes' theorem is introduced in order to remove the singularity in the Cauchy principal value integrals; therefore, the evaluation of these integrals can be carried out by standard Gaussian quadrature. Numerical examples of 2-D elastoplastic problems and a 3-D elastic problem show the effectiveness and efficiency of the method.     

Dynamic node adaptive strategy for nearly singular problems on large domains

Many physical phenomena develop singular or nearly singular behavior in localized regions, e.g. boundary layers or blowup solutions. Using uniform grids for such problems becomes computationally prohibitive as the solution approaches singularity. For these problems, adaptive methods may be preferred over uniform grids methods. In large computational domains, because of the ill conditioning due to the large domain of the partial differential equation (PDE) problem, the existing node adaptive strategies perhaps encounter difficulty in detecting nearly singular regions. In this paper, we are interested in solving PDE problems on large domains, whose solution presents rapid variations or high gradients in some local regions of the domain. Our main purpose is to introduce a dynamic algorithm which finds regions with rapid variations and performs a local node adaptive strategy only in these nearly singular regions. In this algorithm, a step by step scheme is applied by using collocation points and thin plate spline radial basis functions. In spite of using local node adaptive strategy, the global solution exists in the whole computational domain. Another advantage of the new algorithm is its ability to keep the condition number and the required memory under control. The new algorithm is applied for two problems in two dimensions and the obtained results confirm the accuracy and efficiency of the proposed method.     

Announcements

A new finite element method is devised for the numerical solution of elliptic boundary value problems with geometrical singularities. In it, the singularity is eliminated form the computational domain in an exact fashion. This is in contrast to other common methods, such as those which use a refined mesh in the singularity region, or those which use special singular finite elements. In them, the singularity is treated as a part of the numerical scheme. The new method is illustrated on an elliptic differential equation in a domain containing a re-entrant corner. Numerical experiments show that the new method yields result which are generally much more accurate than those obtained by using the standard finite element method with mesh refinement in the singularity region. Both methods require about the same computing time.     

Convergence acceleration of iterative algorithms. Applications to thin shell analysis and Navier–Stokes equations

This work deals with the convergence acceleration of iterative nonlinear methods. Two convergence accelerating techniques are evaluated: the Modified Mininal Polynomial Extrapolation Method (MMPE) and the Padé approximants. The algorithms studied in this work are iterative correctors: Newton’s modified method, a high-order iterative corrector presented in Damil et al. (Commun Numer Methods Eng 15:701–708, 1999) and an original algorithm for vibration of viscoelastic structures. We first describe the iterative algorithms for the considered nonlinear problems. Secondly, the two accelerating techniques are presented. Finally, through several numerical tests from the thin shell theory, Navier–Stokes equations and vibration of viscoelastic shells, the advantages and drawbacks of each accelerating technique is discussed.     

Self-consistent linearization of non-linear BEM formulations with quadratic convergence

In this work, a general technique to obtain the self-consistent linearization of non-linear formulations of the boundary element method (BEM) is presented. In the incremental-iterative procedure required to solve the non-linear problem the convergence is quadratic, being the solution obtained from the consistent tangent operator. This technique is applied to non-linear BEM formulations for plates where two independent problems are discussed: the plate bending and the stretching problem. For both problems an equilibrium equation is written in terms of strains and internal forces and then the consistent tangent operator is derived by applying the Newton–Raphson’s scheme. The Von Mises criterion is adopted to govern the elasto-plastic material behaviour checked at points along the plate thickness, although the presented formulations can be used with any non-linear model. Numerical examples are presented showing the accuracy of the results as well as the high convergence rate of the iterative procedure.     

Developments of the method of difference potentials for linear elastic fracture mechanics problems

Recently, the method of difference potentials has been extended to linear elastic fracture mechanics. The solution was calculated on a grid boundary belonging to the domain of an auxiliary problem, which must be solved multiple times. Singular enrichment functions, such as those used within the extended finite element method, were introduced to improve the approximation near the crack tip leading to near‐optimal convergence rates. Now, the method is further developed by significantly reducing the computation time. This is achieved via the implementation of a system of basis functions introduced along the physical boundary of the problem. The basis functions form an approximation of the trace of the solution at the physical boundary. This method has been proven efficient for the solution of problems on regular (Lipschitz) domains. By introducing the singularity into the finite element space, the approximation of the crack can be realised by regular functions. Near‐optimal convergence rates are then achieved for the enriched formulation. A solution algorithm using the fast Fourier transform is provided with the aim of further increasing the efficiency of the method.