Boundary element analysis of Kirchhoff plates with direct evaluation of hypersingular integrals

The typical Boundary Element Method (BEM) for fourth‐order problems, like bending of thin elastic plates, is based on two coupled boundary integral equations, one strongly singular and the other hypersingular. In this paper all singular integrals are evaluated directly, extending a general method formerly proposed for second‐order problems. Actually, the direct method for the evaluation of singular integrals is completely revised and presented in an alternative way. All aspects are dealt with in detail and full generality, including the evaluation of free‐term coefficients. Numerical tests and comparisons with other regularization techniques show that the direct evaluation of singular integrals is easy to implement and leads to very accurate results. Copyright © 1999 John Wiley & Sons, Ltd.     

Dual boundary integral equations for exterior problems

A dual integral formulation for the interior problem of the Laplace equation with a smooth boundary is extended to the exterior problem. Two regularized versions are proposed and compared with the interior problem. It is found that an additional free term is present in the second regularized version of the exterior problem. An analytical solution for a benchmark example in ISBE is derived by two methods, conformal mapping and the Poisson integral formula using symbolic software. The potential gradient on the boundary is calculated by using the hypersingular integral equation except on the two singular points where the potential is discontinuous instead of failure in ISBE benchmarks. Based on the matrix relations between the interior and exterior problems, the BEPO2D program for the interior problem can be easily reintegrated. This benchmark example was used to check the validity of the dual integral formulation, and the numerical results match the exact solution well.     

Multinode finite element based on boundary integral equations

A finite element constructed on the basis of boundary integral equations is proposed. This element has a flexible shape and arbitrary number of nodes. It also has good approximation properties. A procedure of constructing an element stiffness matrix is demonstrated first for one-dimensional case and then for two-dimensional steady-state heat conduction problem. Numerical examples demonstrate applicability and advantages of the method. © 1998 John Wiley & Sons, Ltd.     

Galerkin formulation and singularity subtraction for spectral solutions of boundary integral equations

A new spectral Galerkin formulation is presented for the solution of boundary integral equations. The formulation is carried out with an exact singularity subtraction procedure based on analytical integrations, which provides a fast and precise way to evaluate the coefficient matrices. The new Galerkin formulation is based on the exact geometry of the problem boundaries and leads to a non-element method that is completely free of mesh generation. The numerical behaviour of the method is very similar to the collocation method; for Dirichlet problems, however, it leads to a symmetric coefficient matrix and therefore requires half the solution time of the collocation method. © 1998 John Wiley & Sons, Ltd.     

Tangential derivative of singular boundary integrals with respect to the position of collocation points

This paper investigates the evaluation of the sensitivity, with respect to tangential perturbations of the singular point, of boundary integrals having either weak or strong singularity. Both scalar potential and elastic problems are considered. A proper definition of the derivative of a strongly singular integral with respect to singular point perturbations should accommodate the concomitant perturbation of the vanishing exclusion neighbourhood involved in the limiting process used in the definition of the integral itself. This is done here by esorting to a shape sensitivity approach, considering a particular class of infinitesimal domain perturbations that ‘move’ individual points, and especially the singular point, but leave the initial domain globally unchanged. This somewhat indirect strategy provides a proper mathematical setting for the analysis. Moreover, the resulting sensitivity expressions apply to arbitrary potential-type integrals with densities only subjected to some regularity requirements at the singular point, and thus are applicable to approximate as well as exact BEM solutions. Quite remarkable is the fact that the analysis is applicable when the singular point is located on an edge and simply continuous elements are used. The hypersingular BIE residual function is found to be equal to the derivative of the strongly singular BIE residual when the same values of the boundary variables are substituted in both SBIE and HBIE formulations, with interesting consequences for some error indicator computation strategies. © 1998 John Wiley & Sons, Ltd.     

Circulant preconditioners for ill-conditioned boundary integral equations from potential equations

In this paper, we consider solving potential equations by the boundary integral equation approach. The equations so derived are Fredholm integral equations of the first kind and are known to be ill-conditioned. Their discretized matrices are dense and have condition numbers growing like O(n) where n is the matrix size. We propose to solve the equations by the preconditioned conjugate gradient method with circulant integral operators as preconditioners. These are convolution operators with periodic kernels and hence can be inverted efficiently by using fast Fourier transforms. We prove that the preconditioned systems are well conditioned, and hence the convergence rate of the method is linear. Numerical results for two types of regions are given to illustrate the fast convergence. © 1998 John Wiley & Sons, Ltd.     

Solution of non‐linear boundary integral equations in complex geometries with auxiliary integral subtraction

The boundary integral equation that results from the application of the reciprocity theorem to non‐linear or non‐homogeneous differential equations generally contains a domain integral. While methods exist for the meshless evaluation of these integrals, mesh‐based domain integration is generally more accurate and can be performed more quickly with the application of fast multipole methods. However, polygonalization of complex multiply‐connected geometries can become a costly task, especially in three‐dimensional analyses. In this paper, a method that allows a mesh‐based integration in complex domains, while retaining a simple mesh structure, is described. Although the technique is intended for the numerical solution of more complex differential equations, such as the Navier–Stokes equations, for simplicity the method is applied to the solution of a Poisson equation, in domains of varying complexity. It is shown that the error introduced by the auxiliary domain subtraction method is comparable to the discretization error, while the complexity of the mesh is significantly reduced. The behaviour of the error in the boundary solution observed with the application of the new method is analogous to the behaviour observed with conventional cell‐based domain integration. Copyright © 2002 John Wiley & Sons, Ltd.     

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.     

On the efficient computation of the stress components near a closed boundary in plane elasticity by using classical complex boundary integral equations

Complex boundary integral equations (Fredholm‐type regular or Cauchy‐type singular or even Hadamard–Mangler‐type hypersingular) have been used for the numerical solution of general plane isotropic elasticity problems. The related Muskhelishvili and, particularly, Lauricella–Sherman equations are famous in the literature, but several more extensions of the Lauricella–Sherman equations have also been proposed. In this paper it is just mentioned that the stress and displacement components can be very accurately computed near either external or internal simple closed boundaries (for anyone of the above equations: regular or singular or hypersingular, but with a prerequisite their actual numerical solution) through the appropriate use of the even more classical elementary Cauchy theorem in complex analysis. This approach has been already used for the accurate numerical computation of analytic functions and their derivatives by Ioakimidis, Papadakis and Perdios (BIT 1991; 31 : 276–285), without applications to elasticity problems, but here the much more complicated case of the elastic complex potentials is studied even when just an appropriate non‐analytic complex density function (such as an edge dislocation/loading distribution density) is numerically available on the boundary. The present results are also directly applicable to inclusion problems, anisotropic elasticity, antiplane elasticity and classical two‐dimensional fluid dynamics, but, unfortunately, not to crack problems in fracture mechanics. Brief numerical results (for the complex potentials), showing the dramatic increase of the computational accuracy, are also displayed and few generalizations proposed. Copyright © 2000 John Wiley & Sons, Ltd.     

Evaluation of supersingular integrals: second‐order boundary derivatives

The boundary integral representation of second‐order derivatives of the primary function involves second‐order (hypersingular) and third‐order (supersingular) derivatives of the Green's function. By defining these highly singular integrals as a difference of boundary limits, interior minus exterior, the limiting values are shown to exist. With a Galerkin formulation, coincident and edge‐adjacent supersingular integrals are separately divergent, but the sum is finite, while the individual hypersingular integrals are finite. Moreover, the cancellation of the supersingular divergent terms only requires a continuous interpolation of the surface potential, and there is no continuity requirement on the surface flux. The algorithm is efficient, the non‐singular integrals vanish and the singular integrals are computed entirely analytically, and accurate values are obtained for smooth surfaces. However, it is shown that a (continuous) linear interpolation is not appropriate for evaluation at boundary corners. Published in 2006 by John Wiley & Sons, Ltd.     

On the Cauchy principal value of the surface integral in the boundary integral equation of 3D elasticity

A simple demonstration of the existence of the Cauchy principal value (CPV) of the strongly singular surface integral in the Somigliana Identity at a non-smooth boundary point is presented. First a regularization of the strongly singular integral by analytical integration of the singular term in the radial direction in pre-image planes of smooth surface patches is carried out. Then it is shown that the sum of the angular integrals of the characteristic of the tractions of the Kelvin fundamental solution is zero, a formula for the transformation of angles between the tangent plane of a suface patch and the pre-image plane at smooth mapping of the surface patch being derived for this purpose.     

On non‐linear transformations for accurate numerical evaluation of weakly singular boundary integrals

This paper presents a study of the performance of the non‐linear co‐ordinate transformations in the numerical integration of weakly singular boundary integrals. A comparison of the smoothing property, numerical convergence and accuracy of the available non‐linear polynomial transformations is presented for two‐dimensional problems. Effectiveness of generalized transformations valid for any type and location of singularity has been investigated. It is found that weakly singular integrals are more efficiently handled with transformations valid for end‐point singularities by partitioning the element at the singular point. Further, transformations which are excellent for CPV integrals are not as accurate for weakly singular integrals. Connection between the maximum permissible order of polynomial transformations and precision of computations has also been investigated; cubic transformation is seen to be the optimum choice for single precision, and quartic or quintic one, for double precision computations. A new approach which combines the method of singularity subtraction with non‐linear transformation has been proposed. This composite approach is found to be more accurate, efficient and robust than the singularity subtraction method and the non‐linear transformation methods. Copyright © 2001 John Wiley & Sons, Ltd.     

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.     

Numerical solution of integral equations with logarithmic-, Cauchy- and Hadamard-type singularities

This study presents an extension of the piecewise quadratic polynomial technique to solve singular integral equations with logarithmic- and Hadamard-type singularities. For completeness and continuity, the evaluation of the weights for logarithmic-, Cauchy- and Hadamard-type singularities are given explicitly. Numerical results concerning logarithmic- and Hadamard-type singular integral equations are compared with the known solutions. The solution of Cauchy-type singular integral equations was discussed in detail in a previous study by Miller and Keer. © 1998 John Wiley & Sons, Ltd.     

Regularized integral equations and fast high‐order solvers for sound‐hard acoustic scattering problems

This text introduces the following: (1) new regularized combined field integral equations (CFIE‐R) for frequency‐domain sound‐hard scattering problems; and (2) fast, high‐order algorithms for the numerical solution of the CFIE‐R and related integral equations. Similar to the classical combined field integral equation (CFIE), the CFIE‐R are uniquely‐solvable integral equations based on the use of single and double layer potentials. Unlike the CFIE, however, the CFIE‐R utilize a composition of the double‐layer potential with a regularizing operator that gives rise to highly favorable spectral properties—thus making it possible to produce accurate solutions by means of iterative solvers in small numbers of iterations. The CFIE‐R‐based fast high‐order integral algorithms introduced in this text enable highly accurate solution of challenging sound‐hard scattering problems, including hundred‐wavelength cases, in single‐processor runs on present‐day desktop computers. A variety of numerical results demonstrate the qualities of the numerical solvers as well as the advantages that arise from the new integral equation formulation. Copyright © 2012 John Wiley & Sons, Ltd.     

A wideband fast multipole accelerated boundary integral equation method for time‐harmonic elastodynamics in two dimensions

This article presents a wideband fast multipole method (FMM) to accelerate the boundary integral equation method for two‐dimensional elastodynamics in frequency domain. The present wideband FMM is established by coupling the low‐frequency FMM and the high‐frequency FMM that are formulated on the ingenious decomposition of the elastodynamic fundamental solution developed by Nishimura's group. For each of the two FMMs, we estimated the approximation parameters, that is, the expansion order for the low‐frequency FMM and the quadrature order for the high‐frequency FMM according to the requested accuracy, considering the coexistence of the derivatives of the Helmholtz kernels for the longitudinal and transcendental waves in the Burton–Muller type boundary integral equation of interest. In the numerical tests, the error resulting from the fast multipole approximation was monotonically decreased as the requested accuracy level was raised. Also, the computational complexity of the present fast boundary integral equation method agreed with the theory, that is, Nlog N, where N is the number of boundary elements in a series of scattering problems. The present fast boundary integral equation method is promising for simulations of the elastic systems with subwavelength structures. As an example, the wave propagation along a waveguide fabricated in a finite‐size phononic crystal was demonstrated. Copyright © 2012 John Wiley & Sons, Ltd.     

A high‐order approach for modelling transient wave propagation problems using the scaled boundary finite element method

A high‐order time‐domain approach for wave propagation in bounded and unbounded domains is proposed. It is based on the scaled boundary FEM, which excels in modelling unbounded domains and singularities. The dynamic stiffness matrices of bounded and unbounded domains are expressed as continued‐fraction expansions, which leads to accurate results with only about three terms per wavelength. An improved continued‐fraction approach for bounded domains is proposed, which yields numerically more robust time‐domain formulations. The coefficient matrices of the corresponding continued‐fraction expansion are determined recursively. The resulting solution is suitable for systems with many DOFs as it converges over the whole frequency range, even for high orders of expansion. A scheme for coupling the proposed improved high‐order time‐domain formulation for bounded domains with a high‐order transmitting boundary suggested previously is also proposed. In the time‐domain, the coupled model corresponds to equations of motion with symmetric, banded and frequency‐independent coefficient matrices, which can be solved efficiently using standard time‐integration schemes. Numerical examples for modal and time‐domain analysis are presented to demonstrate the increased robustness, efficiency and accuracy of the proposed method. Copyright © 2013 John Wiley & Sons, Ltd.     

Efficient solution of time‐domain boundary integral equations arising in sound‐hard scattering

We consider the efficient numerical solution of the three‐dimensional wave equation with Neumann boundary conditions via time‐domain boundary integral equations. A space‐time Galerkin method with C^∞‐smooth, compactly supported basis functions in time and piecewise polynomial basis functions in space is employed. We discuss the structure of the system matrix and its efficient parallel assembly. Different preconditioning strategies for the solution of the arising systems with block Hessenberg matrices are proposed and investigated numerically. Furthermore, a C++ implementation parallelized by OpenMP and MPI in shared and distributed memory, respectively, is presented. The code is part of the boundary element library BEM4I. Results of numerical experiments including convergence and scalability tests up to a thousand cores on a cluster are provided. The presented implementation shows good parallel scalability of the system matrix assembly. Moreover, the proposed algebraic preconditioner in combination with the FGMRES solver leads to a significant reduction of the computational time. Copyright © 2015 John Wiley & Sons, Ltd.     

Application of sigmoidal transformations to weakly singular and near‐singular boundary element integrals

Accurate numerical determination of line integrals is fundamental to reliable implementation of the boundary element method. For a source point distant from a particular element, standard Gaussian quadrature is adequate, as well as being the technique of choice. However, when the integrals are weakly singular or nearly singular (source point near the element) this technique is no longer adequate. Here a co‐ordinate transformation technique, based on sigmoidal transformations, is introduced to evaluate weakly singular and near‐singular integrals. A sigmoidal transformation has the effect of clustering the integration points towards the endpoints of the interval of integration. The degree of clustering is governed by the order of the transformation. Comparison of this new method with existing co‐ordinate transformation techniques shows that more accurate evaluation of these integrals can be obtained. Based on observations of several integrals considered, guidelines are suggested for the order of the sigmoidal transformations. Copyright © 1999 John Wiley & Sons, Ltd.     

Implementation of regularized isogeometric boundary element methods for gradient‐based shape optimization in two‐dimensional linear elasticity

The present work addresses shape sensitivity analysis and optimization in two‐dimensional elasticity with a regularized isogeometric boundary element method (IGABEM). Non‐uniform rational B‐splines are used both for the geometry and the basis functions to discretize the regularized boundary integral equations. With the advantage of tight integration of design and analysis, the application of IGABEM in shape optimization reduces the mesh generation/regeneration burden greatly. The work is distinct from the previous literatures in IGABEM shape optimization mainly in two aspects: (1) the structural and sensitivity analysis takes advantage of the regularized form of the boundary integral equations, eliminating completely the need of evaluating strongly singular integrals and jump terms and their shape derivatives, which were the main implementation difficulty in IGABEM, and (2) although based on the same Computer Aided Design (CAD) model, the mesh for structural and shape sensitivity analysis is separated from the geometrical design mesh, thus achieving a balance between less design variables for efficiency and refined mesh for accuracy. This technique was initially used in isogeometric finite element method and was incorporated into the present IGABEM implementation. Copyright © 2016 John Wiley & Sons, Ltd.