Iterative algorithms for impressed cathodic protection systems

A non‐linear cathodic protection problem arising from corrosion engineering is considered. The objective is to present some iterative methods and study the convergence. We show both numerically and theoretically that the Newton–Raphson iteration is monotonically lower‐convergent and a proposed combined iteration is alternative‐convergent. In the combined case, the exact solution locates between two subsequent iterative solutions. Boundary element methods are employed for the numerical calculation. Copyright © 2000 John Wiley & Sons, Ltd.     

Using the boundary element method to solve rolling contact problems

A new approach to steady-state rolling, with and without force transmission, based on the boundary element method is presented. The proposed formulation solves the problem in a more general way than semi-analytical methods, with which it shares some approximations. The robustness and accuracy of the proposed method is reflected in the comparative analysis of the results obtained for three different types of rolling problems involving identical, dissimilar and tyred cylinders, respectively.     

An alternative collocation boundary element method for static and dynamic problems

A collocation boundary element formulation is presented which is based on a mixed approximation formulation similar to the Galerkin boundary element method presented by Steinbach (SIAM J Numer Anal 38:401–413, 2000) for the solution of Laplace’s equation. The method is also applicable to vector problems such as elasticity. Moreover, dynamic problems of acoustics and elastodynamics are included. The resulting system matrices have an ordered structure and small condition numbers in comparison to the standard collocation approach. Moreover, the employment of Robin boundary conditions is easily included in this formulation. Details on the numerical integration of the occurring regular and singular integrals and on the solution of the arising systems of equations are given. Numerical experiments have been carried out for different reference problems. In these experiments, the presented approach is compared to the common nodal collocation method with respect to accuracy, condition numbers, and stability in the dynamic case.     

A semi-analytic boundary element method for parabolic problems

A new semi-analytic solution method is proposed for solving linear parabolic problems using the boundary element method. This method constructs a solution as an eigenfunction expansion using separation of variables. The eigenfunctions are determined using the dual reciprocity boundary element method. This separation of variables-dual reciprocity method (SOV-DRM) allows a solution to be determined without requiring either time-stepping or domain discretisation. The accuracy and computational efficiency of the SOV-DRM is found to improve as time increases. These properties make the SOV-DRM an attractive technique for solving parabolic problems.     

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.     

A boundary element strategy for elastostatic inverse problems involving uncertain boundary conditions

In this paper, a convenient strategy is developed to find solutions for a class of uncertain‐boundary‐value problems by the Boundary Element Method (BEM). Such problems are ill‐posed, but ill‐conditioning of the associated algebraic systems of equations can be controlled to a large extent, and useful boundary data can be obtained despite ill‐conditioning. Interior data of not only sufficient quantity, but also of good quality at good locations contribute to good solutions. Our strategy permits the condition number of the algebraic systems, as a function of interior‐data locations, to be monitored, such that measured data from displacement sensors and/or strain sensors, at locations found to be good ones for the problem at hand, can be used. The present strategy is based upon the concept of a Green's‐function library through partitioning of the BEM algebraic system. Algebraic systems are solved using least squares via Singular Value Decomposition (SVD). The library idea takes advantage of modern data storage and retrieval technology and permits the process of repeated trials, in order to determine good data sensor locations, to be done quickly and efficiently. Several numerical examples are given to demonstrate the strategy. Some examples examine the consequences of errors in measured data. Copyright © 1999 John Wiley & Sons, Ltd.     

Out-of-core solver for large,multi-zone boundary element matrices

In this paper an out-of-core solver is developed for a three-dimensional elastostatic boundary element program. This program includes multi-regions which enables a large model to be divided into a number of homogeneous sub-models, each with their own material properties. The system matrix produced by multi-regions is sparse, blocked and unsymmetric in character and so reduces disc space and solution time. The solver presented in this paper utilizes efficiently the structure of this matrix by holding only non-zero parts of this matrix in a sequential, unformatted direct access file, reading in as much of the file as possible into the working space, performing Gauss elimination, then writing part of the matrix back to file. The maximum utilization of working space is particularly important on vector machines as vector activity is maximized whilst record read/write is minimized. The effect of multi-regions on CPU is demonstrated on both the CONVEX and CRAY machines.     

Plastic,viscoplastic and creep fracture problems with the boundary element method

The present paper shows the applicability of the dual boundary element method to analyse plastic, viscoplastic and creep behaviours in fracture mechanics problems. Several models with a crack, including a square plate, a holed plate and a notched plate, are analysed. Special attention is taken when the discretization of the domain is performed. In fact, for the plasticity and viscoplasticity cases, only the region susceptible to yielding was discretized, whereas the creep case required the discretization of the whole domain. The proposed formulation is presented as an alternative technique to study these kinds of nonlinear problems. Results from the present formulation are compared to those of the well‐established finite element technique, and they are in good agreement. Important fracture mechanic parameters like K_I, K_II, J‐integrals and C‐integrals are also included. In general, the results, for the plastic, viscoplastic and creep cases, exhibit that the highest stress concentrations are in the vicinity of the crack tip and they decrease as the distance from the crack tip is increased.     

Iterative solution of systems of equations in the dual reciprocity boundary element method for the diffusion equation

In this paper the diffusion equation is solved in two-dimensional geometry by the dual reciprocity boundary element method (DRBEM). It is structured by fully implicit discretization over time and by weighting with the fundamental solution of the Laplace equation. The resulting domain integral of the diffusive term is transformed into two boundary integrals by using Green's second identity, and the domain integral of the transience term is converted into a finite series of boundary integrals by using dual reciprocity interpolation based on scaled augmented thin plate spline global approximation functions. Straight line geometry and constant field shape functions for boundary discretization are employed. The described procedure results in systems of equations with fully populated unsymmetric matrices. In the case of solving large problems, the solution of these systems by direct methods may be very time consuming. The present study investigates the possibility of using iterative methods for solving these systems of equations. It was demonstrated that Krylov-type methods like CGS and GMRES with simple Jacobi preconditioning appeared to be efficient and robust with respect to the problem size and time step magnitude. This paper can be considered as a logical starting point for research of iterative solutions to DRBEM systems of equations. © 1998 John Wiley & Sons, Ltd.     

A multidomain boundary element method for two equation turbulence models

The paper deals with the multidomain Boundary Element Method (BEM) for modelling 2D complex turbulent flow using low Reynolds two equation turbulence models. While the BEM is widely accepted for laminar flow this is the first case, where this method is applied for a complex flow problems using k–ε turbulence model. The integral boundary domain equations are discretised using mixed boundary elements and a multidomain method also known as subdomain technique. The resulting system matrix is overdetermined, sparse, block banded and solved using fast iterative linear least squares solver. The simulation of turbulent flow over a backward step is in excellent agreement with the finite volume method using the same turbulent model.     

Analysis of symmetric domains in advanced applications with the boundary element method

The strategy in the boundary element method for the analysis of symmetric domains that does not require the modelling of contours corresponding to the axes of symmetry is investigated for a number of advanced applications. These applications include: treatment of domain loadings, two-dimensional time domain transient elastodynamics, and the analysis of probabilistic problems in elastostatics with a random geometric configuration. Both symmetric as well as arbitrary loadings acting on the symmetric objects are considered. A number of case studies are presented to provide comparisons of computer memory and CPU time requirements for the analyses of the entire object versus the analyses of only the symmetric portion of the object. The numerical data presented clearly demonstrates the advantages of developing efficient symmetric formulations for advanced applications.     

T-stress solutions for two-dimensional crack problems in anisotropic elasticity using the boundary element method

The importance of a two‐parameter approach in the fracture mechanics analysis of many cracked components is increasingly being recognized in engineering industry. In addition to the stress intensity factor, the T stress is the second parameter considered in fracture assessments. In this paper, the path‐independent mutual M‐integral method to evaluate the T stress is extended to treat plane, generally anisotropic cracked bodies. It is implemented into the boundary element method for two‐dimensional elasticity. Examples are presented to demonstrate the veracity of the formulations developed and its applicability. The numerical solutions obtained show that material anisotropy can have a significant effect on the T stress for a given cracked geometry.     

A three-dimensional shape optimization for transient acoustic scattering problems using the time-domain boundary element method

We develop a three-dimensional shape optimization (SO) framework for the wave equation with taking the unsteadiness into account. Resorting to the adjoint variable method, we derive the shape derivative (SD) with respect to a deformation (perturbation) of an arbitrary point on the target surface of acoustic scatterers. Successively, we represent the target surface with non-uniform rational B-spline patches and then discretize the SD in term of the associated control points (CPs), which are useful for manipulating a surface. To solve both the primary and adjoint problems, we apply the time-domain boundary element method (TDBEM) because it is the most appropriate when the analysis domain is the ambient air and thus infinitely large. The issues of the severe computational cost and instability of the TDBEM are resolved by exploiting the fast and stable TDBEM proposed by the present authors. Instead, since the TDBEM is mesh-based and employs the piecewise-constant element for space, we introduce some approximations in evaluating the discretized SD from the two solutions of TDBEM. By regarding the evaluation scheme as the computation of the gradient of the objective functional, given as the summation of the absolute value of the sound pressure over the predefined observation points, we can solve SO problems with a gradient-based non-linear optimization solver. To assess the developed SO system, we performed several numerical experiments from the perspective of verification and application with satisfactory results.     

Stochastic spline fictitious boundary element method in elastostatic problems with random fields

Mathematical formulation and computational implementation of the stochastic spline fictitious boundary element method (SFBEM) are presented for the analysis of plane elasticity problems with material parameters modeled with random fields. Two sets of governing differential equations with respect to the means and deviations of structural responses are derived by including the first order terms of deviations. These equations, being in similar forms to those of deterministic elastostatic problems, can be solved using deterministic fundamental solutions. The calculation is conducted with SFBEM, a modified indirect boundary element method (IBEM), resulting in the means and covariances of responses. The proposed method is validated by comparing the solutions obtained with Monte Carlo simulation for a number of example problems and a good agreement of results is observed.     

A fast convergent iterative boundary element method on PVM cluster

This paper reports a fast convergent boundary element method on a Parallel Virtual Machine (PVM) (Geist et al., PVM: Parallel Virtual Machine, A Users' Guide and Tutorial for Networked Parallel Computing. MIT Press, Cambridge, 1994) cluster using the SIMD computing model (Single Instructions Multiple Data). The method uses the strategy of subdividing the domain into a number of smaller subdomains in order to reduce the size of the system matrix and to achieve overall speedup. Unlike traditional subregioning methods, where equations from all subregions are assembled into a single linear algebraic system, the present scheme is iterative and each subdomain is handled by a separate PVM node in parallel. The iterative nature of the overall solution procedure arises due to the introduction of the artificial boundaries. However, the system equations for each subdomain is now smaller and solved by direct Gaussian elimination within each iteration. Furthermore, the boundary conditions at the artificial interfaces are estimated from the result of the previous iteration by a reapplication of the boundary integral equation for internal points. This method provides a consistent mechanism for the specification of boundary conditions on artificial interfaces, both initially and during the iterative process. The method is fast convergent in comparison with other methods in the literature. The achievements of this method are therefore: (a) simplicity and consistency of methodology and implementation; (b) more flexible choice of type of boundary conditions at the artificial interfaces; (c) fast convergence; and (d) the potential to solve large problems on very affordable PVM clusters. The present parallel method is suitable where (a) one has a distributed computing environment; (b) the problem is big enough to benefit from the speedup achieved by coarse-grained parallelisation; and (c) the subregioning is such that communication overhead is only a small percentage of total computation time.     

A dynamic algorithm for integration in the boundary element method

The discretization of the boundary in boundary element method generates integrals over elements that can be evaluated using numerical quadrature that approximate the integrands or semi-analytical schemes that approximate the integration path. In semi-analytical integration schemes, the integration path is usually created using straight-line segments. Corners formed by the straight-line segments do not affect the accuracy in the interior significantly, but as the field point approaches these corners large errors may be introduced in the integration. In this paper, the boundary is described by a cubic spline on which an integration path of straight-line segments is dynamically created when the field point approaches the boundary. The algorithm described improves the accuracy in semi-analytical integration schemes by orders of magnitude at insignificant increase in the total solution time by the boundary element method. Results from two indirect BEM and a direct BEM formulation in which the unknowns are approximated by linear and quadratic Lagrange polynomial and a cubic Hermite polynomial demonstrate the versatility of the described algorithm. © 1998 John Wiley & Sons, Ltd.     

The dual boundary contour method for two-dimensional crack problems

This paper concerns the dual boundary contour method for solving two-dimensional crack problems. The formulation of the dual boundary contour method is presented. The crack surface is modeled by using continuous quadratic boundary elements. The traction boundary contour equation is applied for traction nodes on one of the crack surfaces and the displacement boundary contour equation is applied for displacement nodes on the opposite crack surface and noncrack boundaries. The direct calculation of the singular integrals arising in displacement BIEs is addressed. These singular integrals are accurately evaluated with potential functions. The singularity subtraction technique for determining the stress intensity factor K_I, K_II and the T-term are developed for mixed mode conditions. Some two-dimensional examples are presented and numerical results obtained by this approach are in very good agreement with the results of the previous papers. This revised version was published online in August 2006 with corrections to the Cover Date.     

Exact integration in the boundary element method for two-dimensional elastostatic problems

This paper derives the exact integrations for the integrals in the boundary element analysis of two-dimensional elastostatics. For facilitation, the derivation is based on the simple forms of the fundamental functions by taking constant, discontinuous linear and discontinuous quadratic elements as examples. The efficiency and accuracy of the derived exact integrations are verified against five benchmark problems; the results indicate that the derived exact integrations significantly reduces the CPU time for forming the matrices of the boundary element analysis and solving the internal displacements.     

hp Fast multipole boundary element method for 3D acoustics

A fast multipole boundary element method (FMBEM) extended by an adaptive mesh refinement algorithm for solving acoustic problems in three‐dimensional space is presented in this paper. The Collocation method is used, and the Burton–Miller formulation is employed to overcome the fictitious eigenfrequencies arising for exterior domain problems. Because of the application of the combined integral equation, the developed FMBEM is feasible for all positive wave numbers even up to high frequencies. In order to evaluate the hypersingular integral resulting from the Burton–Miller formulation of the boundary integral equation, an integration technique for arbitrary element order is applied. The fast multipole method combined with an arbitrary order h‐p mesh refinement strategy enables accurate computation of large‐scale systems. Numerical examples substantiate the high accuracy attainable by the developed FMBEM, while requiring only moderate computational effort at the same time. Copyright © 2016 John Wiley & Sons, Ltd.