Initial strain formulation without internal cells for elastoplastic analysis by triple‐reciprocity BEM

In general, internal cells are required to solve elastoplasticity problems using a conventional boundary element method (BEM). However, in this case, the merit of BEM, which is the easy method of preparation of data, is lost. The conventional multiple‐reciprocity boundary element method (MRBEM) cannot be used to solve the elastoplasticity problems because the distribution of initial strain or initial stress cannot be determined analytically. In this paper, we show that two‐dimensional elastoplasticity problems can be solved without the use of internal cells, by using the triple‐reciprocity boundary element method. An initial strain formulation is adopted and the initial strain distribution is interpolated using boundary integral equations. A new computer programme was developed and applied to several problems. Copyright © 2001 John Wiley & Sons, Ltd.     

A numerical algorithm for nonlinear dynamic problems based on BEM

A semi-analytical time-integration procedure for the integration of discretized dynamic mechanical systems is presented. This method utilizes the advantages of the boundary element method (BEM), well known from quasi-static field problems. Motivated by these spatial formulations, the present dynamic method is based on influence functions in time, and gives exact solutions in the linear time-invariant case. Similar to domain-type BEM’s for nonlinear field problems, the method is extended for different nonlinear dynamic systems having nonclassical damping and time-varying mass. The numerical stability and accuracy of the semi-analytical method is discussed in two steps for the nonclassical damping and for the nonlinear restoring forces, e.g. of the Duffing type. The damped Duffing oscillator and a linear oscillator with time-varying mass are used as representative model problems. For a nonlinear rotordynamic system, a comparison is given to other conventionally used time integration procedures, which shows the efficiency of the present method.     

A boundary element method without internal cells for solving viscous flow problems

In this paper, a new boundary element method without internal cells is presented for solving viscous flow problems, based on the radial integration method (RIM) which can transform any domain integrals into boundary integrals. Due to the presence of body forces, pressure term and the non-linearity of the convective terms in Navier–Stokes equations, some domain integrals appear in the derived velocity and pressure boundary-domain integral equations. The body forces induced domain integrals are directly transformed into equivalent boundary integrals using RIM. For other domain integrals including unknown quantities (velocity product and pressure), the transformation to the boundary is accomplished by approximating the unknown quantities with the compactly supported fourth-order spline radial basis functions combined with polynomials in global coordinates. Two numerical examples are given to demonstrate the validity and effectiveness of the proposed method.     

A direct complementarity approach for the elastoplastic analysis of plane stress and plane strain structures

This paper presents a direct complementarity approach for carrying out the elastoplastic analysis of plane stress and plane strain structures. Founded on a traditional finite‐step formulation, our approach, however, avoids the typically cumbersome implementation of iterative predictor–corrector procedures associated with the ubiquitous return mapping algorithm. Instead, at each predefined step, the governing formulation—cast in its most natural mathematical programming format known as a mixed complementarity problem—is directly solved by using a complementarity solver run from within a mathematical modeling system. We have chosen the industry‐standard General Algebraic Modeling System/PATH mixed complementarity problem solver that is called from within the General Algebraic Modeling System environment. We consider both von Mises and Tresca materials, with perfect or hardening (kinematic and isotropic) behaviors. Our numerical tests, five (benchmark) examples of which are presented in this paper, have been carried out using models constructed from the mixed finite element of Capsoni and Corradi (Comput. Methods Appl. Mech. Eng. 1997; 141 :67–93), which beneficially offers a locking‐free behavior and coarse‐mesh accuracy. The results indicate, in addition to an isochoric locking‐free behavior, good accuracy and the ability to circumvent the difficult singularity problem associated with the corners of Tresca yield surfaces. Copyright © 2012 John Wiley & Sons, Ltd.     

BEM approach to inverse heat conduction problems

In the paper the inverse problem of the estimation of the temperature and heat flux on the surface of a heat conducting body is considered. Since the problem belongs to the ill-posed, the method of solving the boundary probelem as well as the method of stabilizing the results of calculations are required. The boundary element method is applied to solve the boundary problem whereas combined ‘future steps’ and the regularization method is applied to obtain stable results. A numerical example is included.     

Transient dynamic elastoplastic analysis by the time-domain BEM formulation

This paper presents a time-domain BEM formulation applied to the solution of transient dynamic elastoplastic problems. The initial stress approach is adopted to solve the elastoplastic problem. Linear time variation is assumed for the displacements and initial stress components whereas traction components are assumed to have a constant time variation. Boundary discretization employs linear elements and the part of the domain where plastic deformation is expected to occur is discretized by employing linear triangular cells. Time integrals are computed analytically, boundary integrals are computed numerically and the domain integrals are computed by following a semi-analytical procedure. A numerical example is presented and the results are compared with another BEM formulation.     

A stable 3D energetic Galerkin BEM approach for wave propagation interior problems

We consider 3D interior wave propagation problems with vanishing initial and mixed boundary conditions, reformulated as a system of two boundary integral equations with retarded potentials. These latter are then set in a weak form, based on a natural energy identity satisfied by the solution of the differential problem, and discretized by the energetic Galerkin boundary element method. Numerical results are presented and discussed in order to show the stability and accuracy of the proposed technique.     

Coupled BEM/FEM approach for nonlinear soil/structure interaction

A general coupled boundary element/finite element formulation is presented for the investigation of dynamic soil/structure interaction including nonlinearities. It is applied to investigate the transient inelastic response of structures coupled with a halfspace. The structure itself and the surrounding soil in the near field are modeled with finite elements. In this part of the model inhomogeneities and an elastoplastic material behavior with hardening effects can be taken into account. The remaining soil region, i.e. the elastic halfspace, is discretized with the boundary elements. Thus wave radiation to infinity is included in the model. In representative examples it is shown that the methodology is computationally powerful and can be used efficiently for the nonlinear analyses of complex soil/structure interaction problems.     

A second order cone complementarity approach for the numerical solution of elastoplasticity problems

In this paper we present a new approach for solving elastoplastic problems as second order cone complementarity problems (SOCCPs). Specially, two classes of elastoplastic problems, i.e. the J ₂ plasticity problems with combined linear kinematic and isotropic hardening laws and the Drucker-Prager plasticity problems with associative or non-associative flow rules, are taken as the examples to illustrate the main idea of our new approach. In the new approach, firstly, the classical elastoplastic constitutive equations are equivalently reformulated as second order cone complementarity conditions. Secondly, by employing the finite element method and treating the nodal displacements and the plasticity multiplier vectors of Gaussian integration points as the unknown variables, we obtain a standard SOCCP formulation for the elastoplasticity analysis, which enables the using of general SOCCP solvers developed in the field of mathematical programming be directly available in the field of computational plasticity. Finally, a semi-smooth Newton algorithm is suggested to solve the obtained SOCCPs. Numerical results of several classical plasticity benchmark problems confirm the effectiveness and robustness of the SOCCP approach.     

A self-regularized approach for rank-deficient systems in the BEM of 2D Laplace problems

The Laplace problem subject to the Dirichlet or Neumann boundary condition in the direct and indirect boundary element methods (BEM) sometimes both may result in a singular or ill-conditioned system (some special situations) for the interior problem. In this paper, the direct and indirect BEMs are revisited to examine the uniqueness of the solution by introducing the Fichera’s idea and the self-regularized technique. In order to construct the complete range of the integral operator in the BEM lacking a constant term in the case of a degenerate scale, the Fichera’s method is provided by adding the constraint and a slack variable to circumvent the problem of degenerate scale. We also revisit the Fredholm alternative theorem by using the singular value decomposition (SVD) in the discrete system and explain why the direct BEM and the indirect BEM are not indeed equivalent in the solution space. According to the relation between the SVD structure and Fichera’s technique, a self-regularized method is proposed in the matrix level to deal with non-unique solutions of the Neumann and Dirichlet problems which contain rigid body mode and degenerate scale, respectively, at the same time. The singularity and proportional influence matrices of 3 by 3 are studied by using the property of the symmetric circulant matrix. Finally, several examples are demonstrated to illustrate the validity and the effectiveness of the self-regularized method.     

Application of the multiple reciprocity BEM to nonlinear potential problems

In this paper types of nonlinear potential problems are discussed and some of these problems are solved by the boundary element method. In order to avoid domain discretization the multiple reciprocity method is used. Solution of the problem is expressed in terms of a series which involves the higher order fundamental solutions and Laplacians of source terms. Convergence criteria as well as numerical examples are included.     

A BEM approach to static and dynamic analysis of plates with internal supports

A boundary element approach is developed for the static and dynamic analysis of Kirchhoff's plates of arbitrary shape which, in addition to the boundary supports, are also supported inside the domain on isolated points (columns), lines (walls) or regions (patches). All kinds of boundary conditions are treated. The supports inside the domain of the plate may yield elastically. The method uses the Green's function for the static problem without the internal supports to establish an integral representation for the solution which involves the unknown internal reactions and inertia forces within the integrand of the domain integrals. The Green's function is established numerically using BEM. Subsequently, using an effective Gauss integration for the domain integrals and a BEM technique for line integrals a system of simultaneous, in general, nonlinear algebraic equations is obtained which is solved numerically. Several examples for both the static and dynamic problem are presented to illustrate the efficiency and the accuracy of the proposed method.     

BEM solution of two dimensional unilateral contact problems with friction by a new approach

Recently, a new numerical resolution method has been introduced for frictionless unilateral contact problems which reduces the resolution of the unilateral problem to the resolution of a nonlinear equation. This reduction is based on a reformulation of the unilateral problem, which at the same time governs the usual unknowns and extra unknowns characterising the contact zone position. The supplementary equations needed by the introduction of these extra unknowns are given by writing the contact force and the gap between the body and its support vanish simultaneously on the contact zone boundary. The boundaries of the bodies in contact are supposed regular in the contact zone vicinity. The aim of the present paper is to extend this resolution method to unilateral contact problems with Coulomb friction.     

An energy approach to space–time Galerkin BEM for wave propagation problems

In this paper we consider Dirichlet or Neumann wave propagation problems reformulated in terms of boundary integral equations with retarded potential. Starting from a natural energy identity, a space–time weak formulation for 1D integral problems is briefly introduced, and continuity and coerciveness properties of the related bilinear form are proved. Then, a theoretical analysis of an extension of the introduced formulation for 2D problems is proposed, pointing out the novelty with respect to existing literature results. At last, various numerical simulations will be presented and discussed, showing unconditional stability of the space–time Galerkin boundary element method applied to the energetic weak problem. Copyright © 2009 John Wiley & Sons, Ltd.     

BEM for crack-hole problems in thermopiezoelectric materials

The boundary element formulation for analysing interaction between a hole and multiple cracks in piezoelectric materials is presented. Using Green's function for hole problems and variational principle, a boundary element model (BEM) for a 2-D thermopiezoelectric solid with cracks and holes has been developed and used to calculate stress intensity factors of the crack-hole problem. In BEM, the boundary condition on the hole circumference is satisfied a priori by Green's function, and is not involved in the boundary element equations. The method is applicable to multiple-crack problems in both finite and infinite solids. Numerical results for stress and electric displacement intensity factors at a particular crack tip in a crack-hole system of piezoelectric materials are presented to illustrate the application of the proposed formulation.     

Multi-domain BEM for two-dimensional problems of elastodynamics

An advanced implementation of the boundary element technique for the periodic and transient dynamic analyses of two-dimensional elastic or visco elastic solids of arbitrary shape and connectivity is presented. For transient dynamic analysis the problem is first solved in the Laplace transform space and then the time domain solutions are obtained by numerical inversion of transformed domain solutions. The present analysis is capable of treating very large, multi-domain problems by substructuring and satisfying the equilibrium and compatibilities at the interfaces. With the help of this substructuring capability, problems related to the layered media and soil–structure interaction can all be analysed. This paper also introduces a new type of element called ‘Enclosing Element’, which has been developed and used to model the infinitely extending boundaries of a half-space or a layered medium. A number of numerical examples are presented, and through comparisons with available analytical and numerical results, the accuracy, stability and efficiency of the present analysis are established.     

A bone adaptation integrated approach using BEM

An algorithm for the mathematical representation of bone adaptation is proposed. This model can be used at bone tissue level (local level) as well as at whole bone level (global level). The boundary element method is used for the numerical analysis of trabecular bone tissue together with the remodeling algorithm presented by (Fridez P., Rakotomanana L., Terrier A., Leyvraz P.F., Three dimensional model of bone external adaptation, Comput Methods Biomech Biomed Eng, 2, 189–196, 1998, [1]). Some numerical examples are given to show the versatility and power of the algorithm here discussed. Additionally, the method converges rapidly to the solution, which is one of the main advantages of the proposed numerical scheme.     

A new boundary element technique without domain integrals for elastoplastic solids

A simple idea is proposed to solve boundary value problems for elastoplastic solids via boundary elements, namely, to use the Green's functions corresponding to both the loading and unloading branches of the tangent constitutive operator to solve for plastic and elastic regions, respectively. In this way, domain integrals are completely avoided in the boundary integral equations. Though a discretization of the region where plastic flow occurs still remains necessary to account for the inhomogeneity of plastic deformation, the elastoplastic analysis reduces, in essence, to a straightforward adaptation of techniques valid for anisotropic linear elastic constitutive equations (the loading branch of the elastoplastic constitutive operator may be viewed formally as a type of anisotropic elastic law). Numerical examples, using J₂‐flow theory with linear hardening, demonstrate that the proposed method retains all the advantages related to boundary element formulations, is stable and performs well. The method presented is for simplicity developed for the associative flow rule; however, a full derivation of Green's function and boundary integral equations is also given for the general case of non‐associative flow rule. It is shown that in the non‐associative case, a domain integral unavoidably arises in the formulation. Copyright © 2005 John Wiley & Sons, Ltd.     

A parallel domain decomposition BEM for diffusion problems with surface reactions

A parallel domain decomposition boundary element method (BEM) is developed for the solution of three-dimensional multispecies diffusion problems. The chemical species are uncoupled in the interior of the domain but couple at the boundary through a nonlinear surface reaction equation. The method of lines is used whereby time is discretized using the finite difference method and space is discretized using the boundary element method. The original problem is transformed into a sequence of nonhomogeneous modified Helmholtz equations. A Schwarz Neumann–Neumann iteration scheme is used to satisfy interfacial boundary conditions between subdomains. A segregated solver based on a quasi-predictor–corrector time integrator is used to satisfy the nonlinear boundary conditions on the reactive surfaces. The accuracy and parallel efficiency of the method is demonstrated through a benchmark problem.     

Three-step multi-domain BEM solver for nonhomogeneous material problems

A three-step solution technique is presented for solving two-dimensional (2D) and three-dimensional (3D) nonhomogeneous material problems using the multi-domain boundary element method. The discretized boundary element formulation expressed in terms of normalized displacements and tractions is written for each sub-domain. The first step is to eliminate internal variables at the individual domain level. The second step is to eliminate boundary unknowns defined over nodes used only by the domain itself. And the third step is to establish the system of equations according to the compatibility of displacements and equilibrium of tractions at common interface nodes. Discontinuous elements are utilized to model the traction discontinuity across corner nodes. The distinct feature of the three-step solver is that only interface displacements are unknowns in the final system of equations and the coefficient matrix is blocked sparse. As a result, large-scale 3D problems can be solved efficiently. Three numerical examples for 2D and 3D problems are given to demonstrate the effectiveness of the presented technique.