Non-linear boundary element formulation applied to contact analysis using tangent operator

This work presents a non-linear boundary element formulation applied to analysis of contact problems. The boundary element method (BEM) is known as a robust and accurate numerical technique to handle this type of problem, because the contact among the solids occurs along their boundaries. The proposed non-linear formulation is based on the use of singular or hyper-singular integral equations by BEM, for multi-region contact. When the contact occurs between crack surfaces, the formulation adopted is the dual version of BEM, in which singular and hyper-singular integral equations are defined along the opposite sides of the contact boundaries. The structural non-linear behaviour on the contact is considered using Coulomb's friction law. The non-linear formulation is based on the tangent operator in which one uses the derivate of the set of algebraic equations to construct the corrections for the non-linear process. This implicit formulation has shown accurate as the classical approach, however, it is faster to compute the solution. Examples of simple and multi-region contact problems are shown to illustrate the applicability of the proposed scheme.     

3D non-linear time domain FEM–BEM approach to soil–structure interaction problems

Dynamic soil–structure interaction is concerned with the study of structures supported on flexible soils and subjected to dynamic actions. Methods combining the finite element method (FEM) and the boundary element method (BEM) are well suited to address dynamic soil–structure interaction problems. Hence, FEM–BEM models have been widely used. However, non-linear contact conditions and non-linear behavior of the structures have not usually been considered in the analyses. This paper presents a 3D non-linear time domain FEM–BEM numerical model designed to address soil–structure interaction problems. The BEM formulation, based on element subdivision and the constant velocity approach, was improved by using interpolation matrices. The FEM approach was based on implicit Green's functions and non-linear contact was considered at the FEM–BEM interface. Two engineering problems were studied with the proposed methodology: the propagation of waves in an elastic foundation and the dynamic response of a structure to an incident wave field.     

Comparison of the basic and the discontinuity formulation of the 3D-dual boundary element method

The dual boundary element method (Dual BEM) has established as a numerical approach for solving arbitrary 3D-crack problems in linear elastostatics. In the case of symmetrically loaded cracks — especially traction-free cracks — often the more efficient displacement discontinuity method (DDM) is used, because one obtains a reduced system of algebraic equations.

In our paper we will show that the discontinuity method is just a special formulation of the basic Dual BEM and can be applied to arbitrary boundary value problems on the crack. We will present a numerical example for an unsymmetrically loaded crack and discuss which combinations of boundary conditions on the crack surfaces lead to a reduced system of algebraic equations. The savings in memory and computing time compared to the basic formulation of the Dual BEM will be quantified and illustrated by the numerical simulation of 3D crack propagation.     

A dual BIE approach for large‐scale modelling of 3‐D electrostatic problems with the fast multipole boundary element method

A dual boundary integral equation (BIE) formulation is presented for the analysis of general 3‐D electrostatic problems, especially those involving thin structures. This dual BIE formulation uses a linear combination of the conventional BIE and hypersingular BIE on the entire boundary of a problem domain. Similar to crack problems in elasticity, the conventional BIE degenerates when the field outside a thin body is investigated, such as the electrostatic field around a thin conducting plate. The dual BIE formulation, however, does not degenerate in such cases. Most importantly, the dual BIE is found to have better conditioning for the equations using the boundary element method (BEM) compared with the conventional BIE, even for domains with regular shapes. Thus the dual BIE is well suited for implementation with the fast multipole BEM. The fast multipole BEM for the dual BIE formulation is developed based on an adaptive fast multiple approach for the conventional BIE. Several examples are studied with the fast multipole BEM code, including finite and infinite domain problems, bulky and thin plate structures, and simplified comb‐drive models having more than 440 thin beams with the total number of equations above 1.45 million and solved on a PC. The numerical results clearly demonstrate that the dual BIE is very effective in solving general 3‐D electrostatic problems, as well as special cases involving thin perfect conducting structures, and that the adaptive fast multipole BEM with the dual BIE formulation is very efficient and promising in solving large‐scale electrostatic problems. Copyright © 2007 John Wiley & Sons, Ltd.     

Boundary element analysis of two-dimensional elastoplastic contact problems

A general boundary element formulation for contact problems, capable of dealing with local elastoplastic effects and friction, is presented. Both conforming and non-conforming problems may be analysed. The contact problem is solved by means of a direct constraint technique, in which compatibility and equilibrium conditions are directly enforced in the general system of equations. The contact areas are modelled with linear interpolation functions, and quadratic interpolation functions are used everywhere else. Elastoplasticity is solved by a BEM initial strain approach The Von Mises yield criterion with its associated flow rule is adopted. Both perfectly plastic and work hardening materials are studied in the proposed formulation.

An incremental loading technique is proposed, which allows accurate development of the loading history of the problem. The non-linear nature of these problems demands the use of an iterative procedure, to determine the correct frictional conditions at every node of the contact area and the value of the plastic strains at selected points where local yielding may have occurred. Several numerical examples are presented to demonstrate the efficiency of the proposed formulation.     

Critical study of existing solutions for a penny-shaped interface crack, comparing with a new boundary element solution allowing for frictionless contact

A numerical study of the fundamental problem of a pressurized penny-shaped crack at the interface of two dissimilar half spaces is carried out allowing for the possibility of frictionless contact between crack faces. A new, highly accurate axi-symmetric formulation of the boundary element method (BEM) for the solution of elastic contact problems is employed. The correctness and accuracy of available predictions of different kinds for several key characteristics of the solution of this problem are checked. First, comparison of the BEM results for the near-tip contact length shows a very good agreement with some existing predictions. Second, the global solution obtained by BEM is compared with existing asymptotic solutions, obtained with both the open and the frictionless contact models. BEM results show that at the closest neighborhood to the crack tip the global solution of the problem is governed by the first term of the asymptotic solution of the frictionless contact model (up to a distance of the order of a fraction of the near-tip contact length). After a small transition region, in an adjacent surrounding zone whose extent is almost independent of the near-tip contact length, the global solution of the problem is governed by the first term of the asymptotic solution of the open model. As a result of the comparison presented, the regions in which the classical fracture parameters, stress intensity factor (SIF) and energy release rate, can be accurately obtained from the global numerical solution of a crack of this kind have been determined. Third, BEM results and previous estimations show certain discrepancies with a recently published closed form solution of the near-tip contact length and the mode II SIF of the frictionless contact model. A new closed form expression of this mode II SIF, derived from the asymptotic solution of the open model, is proposed in this paper.     

Stress analyses around holes in composite laminates using boundary element method

A three-dimensional (3D) boundary element method (BEM) is developed for the analysis of composite laminates with holes. Instead of using Kelvin-type Green's functions of anisotropic infinite space, 3D layered Green's functions with the materials of each layer being generally anisotropic, derived recently in the Fourier transform domain, are implemented into a 3D BEM formulation. A novel numerical algorithm is designed to calculate layered Green's functions efficiently. It should be noted that since layered Green's functions satisfy exactly the continuity conditions along the interfaces and top and bottom free surfaces a priori, the model becomes truly 2D and discretization is only needed along the hole surface and prescribed traction and/or displacement boundaries. To test the validity and accuracy of the proposed method, the present layered BEM formulation is applied to the problem of an infinite anisotropic plate with a circular hole where the analytical solution is available. It is found that even with a very coarse mesh, the present BEM can predict the hoop stress very accurately along the hole surface. The BEM formulation is then applied to analyze two composite laminates (90/0)_s and (−45/45)_s, under a remote in-plane strain, that have been studied previously with different approaches. For the (90/0)_s case, the hoop stresses along the hole surface predicted by the present layered BEM formulation are in very close agreement with the previous results. For the (−45/45)_s case, however, it is found that a nearly converged solution (less than 5% convergence by doubling the mesh) by the present method is at significant variance with the previous ones that are lack-of-convergence checks. It can be expected that for designing the bolted joints of composites with many layers, a computational tool developed based on the present techniques would be robust and offer a much better solution with regard to accuracy, versatility and design cycle time.     

A fast multipole boundary element method for 2D multi-domain elastostatic problems based on a dual BIE formulation

A new fast multipole formulation for the hypersingular BIE (HBIE) for 2D elasticity is presented in this paper based on a complex-variable representation of the kernels, similar to the formulation developed earlier for the conventional BIE (CBIE). A dual BIE formulation using a linear combination of the developed CBIE and HBIE is applied to analyze multi-domain problems with thin inclusions or open cracks. Two pre-conditioners for the fast multipole boundary element method (BEM) are devised and their effectiveness and efficiencies in solving large-scale problems are discussed. Several numerical examples are presented to study the accuracy and efficiency of the developed fast multipole BEM using the dual BIE formulation. The numerical results clearly demonstrate the potentials of the fast multipole BEM for solving large-scale 2D multi-domain elasticity problems. The method can be applied to study composite materials, functionally-graded materials, and micro-electro-mechanical-systems with coupled fields, all of which often involve thin shapes or thin inclusions.     

An analysis of solidification problems by the boundary element method

The problem of interest in this paper is the calculation of the motion of the solid–liquid interface and the time-dependent temperature field during solidification of a pure metal. An iterative implicit algorithm has been developed for this purpose using the boundary element method (BEM) with time-dependent Green's functions and convolution integrals. The BEM approach requires discretization of only the surface of the solidifying body. Thus, the numerical method closely follows the physics of the problems and is intuitively very appealing. The formulation and the numerical scheme presented here are general and can be applied to a broad range of moving boundary problems. Emphasis is given to two-dimensional problems. Comparison with existing semi-analytical solutions and other numerical solutions from the literature reveals that the method is fast, accurate and without major time step limitations.     

A boundary element sensitivity formulation for contact problems using the implicit differentiation method

In this paper a sensitivity formulation using the boundary element method (BEM), for problems involving contact is presented. The proposed formulation is based on the implicit differentiation method (IDM), where the boundary integral equations are differentiated analytically with respect to the design variables. In the proposed formulation the design variables are defined in terms of the normal gap between the contact bodies. The analysis demonstrates that the proposed method is accurate and robust, as it does not resolve the whole system. The proposed method can be used for evaluating the sensitivities in any shape-optimisation problem involving contact.     

Analytical integration of the moments in the diagonal form fast multipole boundary element method for 3-D acoustic wave problems

A diagonal form fast multipole boundary element method (BEM) is presented in this paper for solving 3-D acoustic wave problems based on the Burton-Miller boundary integral equation (BIE) formulation. Analytical expressions of the moments in the diagonal fast multipole BEM are derived for constant elements, which are shown to be more accurate, stable and efficient than those using direct numerical integration. Numerical examples show that using the analytical moments can reduce the CPU time by a lot as compared with that using the direct numerical integration. The percentage of CPU time reduction largely depends on the proportion of the time used for moments calculation to the overall solution time. Several examples are studied to investigate the effectiveness and efficiency of the developed diagonal fast multipole BEM as compared with earlier p³ fast multipole method BEM, including a scattering problem of a dolphin modeled with 404,422 boundary elements and a radiation problem of a train wheel track modeled with 257,972 elements. These realistic, large-scale BEM models clearly demonstrate the effectiveness, efficiency and potential of the developed diagonal form fast multipole BEM for solving large-scale acoustic wave problems.     

Fast FE–BEM algorithms for orthotropic frictional contact

This paper presents a new and efficient methodology for solving 3D frictional contact problems considering an orthotropic friction law. The contact methodology is based on a proposed augmented Lagrangian formulation for orthotropic frictional contact problems, and a new discrete contact operator, which allows to reduce the number of unknowns in a Newton‐like algorithm that accelerates the attainment of the solution. A fast Uzawa scheme is also proposed on the basis of the Steffensen's method. Both algorithms prove to be very robust and efficient to solve orthotropic frictional contact problems. The proposed formalism has the advantage of being very compact and valid for both the FEM and the BEM. Numerical results are given to demonstrate the validity of the formulation and algorithms proposed. Copyright © 2013 John Wiley & Sons, Ltd.     

A new fast multipole boundary element method for solving large‐scale two‐dimensional elastostatic problems

A new fast multipole boundary element method (BEM) is presented in this paper for large‐scale analysis of two‐dimensional (2‐D) elastostatic problems based on the direct boundary integral equation (BIE) formulation. In this new formulation, the fundamental solution for 2‐D elasticity is written in a complex form using the two complex potential functions in 2‐D elasticity. In this way, the multipole and local expansions for 2‐D elasticity BIE are directly linked to those for 2‐D potential problems. Furthermore, their translations (moment to moment, moment to local, and local to local) turn out to be exactly the same as those in the 2‐D potential case. This formulation is thus very compact and more efficient than other fast multipole approaches for 2‐D elastostatic problems using Taylor series expansions of the fundamental solution in its original form. Several numerical examples are presented to study the accuracy and efficiency of the developed fast multipole BEM formulation and code. BEM models with more than one million equations have been solved successfully on a laptop computer. These results clearly demonstrate the potential of the developed fast multipole BEM for solving large‐scale 2‐D elastostatic problems. Copyright © 2005 John Wiley & Sons, Ltd.     

A hybrid micro–macro BEM formulation for micro-crack clusters in elastic components

Local analysis schemes capable of detailed representations of the micro-features of a problem are integrated with a macro-scale BEM technique capable of handling complex finite geometries and realistic boundary conditions. The micro-scale effects are introduced into the macro-scale BEM analysis through an augmented fundamental solution obtained from an integral equation representation of the micro-scale features. The proposed hybrid micro-macro BEM formulation allows decomposition of the complete problem into two sub-problems, one residing entirely at the micro-level and the other at the macro-level. This allows for investigations of the effects of the micro-structural attributes while retaining the macro-scale geometric features and actual boundary conditions for the component or structure under consideration. As a first attempt, elastic fracture mechanics problems with interacting cracks at close spacings are considered. The numerical results obtained from the hybrid BEM analysis establish the accuracy and effectiveness of the proposed micro–macro computational scheme for this class of problems. The proposed micro–macro BEM formulation can easily be extended to investigate the effects of other micro-features (e.g. interfaces, short or continuous fibre reinforcements, voids and inclusions, in the context of linear elasticity) on macroscopic failure modes observed in structural components.     

The first-kind and the second-kind boundary integral equation systems for solution of frictionless contact problems

An original approach to the numerical solution of displacement boundary integral equation (BIE) and traction hypersingular boundary integral equation (HBIE) by the boundary element method (BEM) for contact problems is given. The main point is to show, how the contact conditions are used to formulate the first-kind and the second-kind BIE systems in the case of frictionless two-body elastic contact. The solution of the first-kind BIE is performed by symmetric Galerkin BEM; the second-kind BIE is solved by an appropriate collocation BEM. The contact problem in itself is solved by the method of subsequent approximations of contact region. Both forms of BIE system are compared in several numerical examples. This comparison is made for different kinds of contact problem. The major emphasis is put on the evaluation of contact pressure. The obtained results are compared with referenced numerical and with the analytical ones.     

A rigid surface boundary element for soil-structure interaction analysis in the direct time domain

Many soil-structure interaction problems involve studies of single or multiple rigid bodies of arbitrary shape and soil media. The commonly used boundary element methods implement the equations of the rigid body in a form that depends on the particulars of the geometry and requires partitioning and condensation of the associated algebraic system of equations. The present work employs the direct time domain B-Spline BEM for 3D elastodynamic analysis and presents an efficient implementation of rigid bodies of arbitrary shape in contact with, or embedded in, elastic media. The formulation of a rigid surface boundary element introduced herein is suitable for direct superposition in the BEM system of algebraic equations. Consequently, solutions are computed in a single analysis step, eliminating, thus, the need for partitioning of the system of equations. Computational efficiency is also achieved due to the extremely sparse form of the associated coefficient matrices. The proposed element can be used for the modeling of single or multiple rigid bodies of arbitrary shape within the framework of the BEM method. The efficiency and general nature of the proposed element is demonstrated through applications related to the dynamic analysis of rigid surface and embedded foundations and their interaction with embedded rigid bodies of arbitrary shape. Received 17 December 1999     

Improving the performance of the boundary element method with time-dependent fundamental solutions by the use of a wavelet expansion in the time domain

The object of this paper is a wavelet-based formulation of the boundary element method (BEM) for diffusion problems, characterized by time-dependent fundamental solution. While the BEM is a well known and often used technique, its time-dependent formulation for diffusion problems is very rarely used in practical applications, due to the high computational cost which characterizes it. Here, a new formulation is proposed, which, through the use of the wavelet expansion of the time behaviour of the boundary elements, is characterized by a lower CPU time consumption when compared with the standard BEM diffusion formulation. The problem to be solved is transformed into an algebraic system (of higher dimension) and its solution gives the time domain behaviour of the desired quantities; in this way, the time stepping procedure is avoided. Together with the formulation, the analysis of the computational cost, and two examples are given in the paper. Copyright © 2006 John Wiley & Sons, Ltd.     

Three-dimensional magnetic field analysis method using scalarpotential formulated by boundary element method

A computational method for magnetic fields formulated by a BEM (boundary element method) has been developed. In the method, a reduced scalar potential is selected as an unknown variable to simplify the calculation of the boundary conditions. Its use requires a high numerical accuracy of the potential gradient. Conventional BEM does not provide this, because numerical element integration for the singular kernal causes a large error. To overcome this difficulty, a highly accurate numerical integration scheme is proposed based on the BEM, and it is applied to magnetic field problems. Calculation results for a spherical permeable material in a problem proposed by the Institute of Electrical Engineers of Japan (the problem of a magnetic field generated by a coil) agreed with the exact solution and the experimental data within 5%     

Creeping analysis with variable temperature applying the boundary element method

This paper presents the implementation of two creep rate equations in the boundary element method (BEM) to predict the creep strain rate at different temperatures and stresses. The creep analysis is basically applied to metals, but particular attention is focused on the numerical evaluation of creeping components subject to variable temperature. The constitutive model used in the numerical calculation is confined to standard power law creep equations. The direct formulation of the BEM is applied to secondary creep problems in a two-dimensional analysis. Numerical results are compared with solutions obtained from references to illustrate the accuracy of the proposed formulation, where a single test was simulated under the mentioned criteria.