This paper presents a new general method for the direct evaluation of Cauchy principal value integrals in several dimensions, which is an issue of major concern in any boundary element method analysis in applied mechanics. It is shown that the original Cauchy principal value integral can be transformed into an element-by-element sum of regular integrals, each one expressed in terms of intrinsic (local) coordinates. The actual computation can be performed by standard quadrature formulae and can be easily included in any existing computer code. The numerical results demonstrate the accuracy and efficiency of the method, along with its insensitivity to the mesh pattern. This new method has full generality and, therefore, can be applied in any field of applied mechanics. Moreover, there are no restrictions on the numerical implementation, as the singular integrals may be defined on surface elements or internal cells of any order and type.

A General Algorithm for Multidimensional Cauchy Principal Value Integrals in the Boundary Element Method

A new general purpose boundary element method for domains with cracks has been recently developed. This technique avoids the use of a multi-domain decomposition by including an additional integral equation expressing the boundary condition on the crack. The principal requirement of this technique is the analytic determination of certain hypersingular integrals of the Green's function which arise from this equation. In order to establish the applicability of this method for fracture, these integrals are evaluated herein for the Kelvin solution of the three-dimensional Navier equations of linear elasticity. Numerical results for fracture problems using the single-domain boundary element analysis are also presented.

Hypersingular integrals in boundary element fracture analysis

A new meshless method for solving nonlinear boundary value problems, based on the local boundary integral equation (LBIE) method and the moving least squares approximation, is proposed in the present paper. The total formulation and a rate formulation are developed for the implementation of the present method. The present method does not need domain and boundary elements to deal with the volume and boundary integrals, which will cause some difficulties for the conventional boundary element method (BEM) or the field/boundary element method (FBEM), as the volume integrals are inevitable in dealing with nonlinear boundary value problems. This is the same for the element free Galerkin (EFG) method which also needs element-like cells in the entire domain to evaluate volume integrals. The “companion fundamental solution” introduced in Zhu, Zhang and Atluri (1998) is used so that no derivatives of the shape functions are needed to construct the stiffness matrix for the interior nodes, as well as for those nodes with no parts of their local boundaries coinciding with the global boundary of the domain of the problem, where essential boundary conditions are specified. It is shown that the satisfaction of the essential as well as natural boundary conditions is quite simple, and algorithmically very efficient, in the present nonlinear LBIE approach. Numerical examples are presented for several problems, for which exact solutions are available. The present method converges fast to the final solution with reasonably accurate results for both the unknown variable and its derivatives. No post processing procedure is required to compute the derivatives of the unknown variable (as in the conventional FBEM), since the solution from the present method, using the moving least squares approximation, is already smooth enough. The numerical results in these examples show that high rates of convergence for the Sobolev norms ∥·∥0 and ∥·∥1 are achievable, and that the values of the unknown variable and its derivatives are quite accurate.

A meshless local boundary integral equation (LBIE) method for solving nonlinear problems

A review is presented for analysis of problems in engineering & the sciences, with the use of the meshless local Petrov-Galerkin (MLPG) method. The success of the meshless methods lie in the local nature, as well as higher order continuity, of the trial function approximations, high adaptivity and a low cost to prepare input data for numerical analyses, since the creation of a finite element mesh is not required. There is a broad variety of meshless methods available today; however the focus is placed on the MLPG method, in this paper. The MLPG method is a fundamental base for the derivation of many meshless formulations, since the trial and test functions can be chosen from different functional spaces. In the last decade, a broad community of researchers and scientists contributed to the development and implementation of the MLPG method in a wide range of scientific disciplines. This paper first presents the basics and principles of the MLPG method, the meshless local approximation techniques for trial and test functions, applications to elasticity and elastodynamics, plasticity, fracture and crack analysis, heat transfer and fluid flow, coupled problems involving multiphase materials, and techniques for increasing the accuracy and computational effectiveness. Various applications to 2-D planar problems, axisymmetric problems, plates and shells or 3-D problems are included. An increased number of published papers in literature in the recent years can be considered as a measure of the growing research activity in the general scope of the MLPG method, and thus, several trends and ideas for future research interest are also outlined.

Applications of the MLPG Method in Engineering &Sciences: A Review

Three different truly Meshless Local Petrov-Galerkin (MLPG) methods are developed for solving 3D elasto-static problems. Using the general MLPG concept, these methods are derived through the local weak forms of the equilibrium equations, by using different test functions, namely, the Heaviside function, the Dirac delta function, and the fundamental solutions. The one with the use of the fundamental solutions is based on the local unsymmetric weak form (LUSWF), which is equivalent to the local boundary integral equations (LBIE) of the elasto-statics. Simple formulations are derived for the LBIEs in which only weakly-singular integrals are included for a simple numerical implementation. A novel definition of the local 3D sub-domain is presented, which enables the numerical integrations to be performed in an accurate and efficient way, based on a truly meshless implementation. The augmented radial basis functions (RBF) and the moving least squares (MLS) are chosen to construct the shape functions, for the three MLPG methods. Numerical examples are included to demonstrate that the present methods are very promising for solving the elastic problems, as compared to the traditional Galerkin Finite Element Method. keyword: Meshless Local Petrov-Galerkin approach (MLPG), Local Boundary Integral Equations (LBIE), Radial Basis Functions (RBF), Moving Least Squares (MLS).

Meshless Local Petrov-Galerkin (MLPG) approaches for solving 3D Problems in elasto-statics

The field-boundary-element method naturally admits the solution algorithm in the incompressible regimes of fully developed plastic flow. This is not the case with the generally popular finite-element method, without further modifications to the method such as reduced integration or a mixed method for treating the dilatational deformation. The analyses by the field-boundary-element method for geometric and material non-linear problems are generally carried out by an incremental algorithm, where the velocities (or displacement increments) on the boundary are treated as the primary variables and an initial strain iteration method is commonly used to obtain the state of equilibrium. For problems such as buckling and diffused tensile necking, involving very large strains, such a solution scheme may not be able to capture the bifurcation phenomena, or the convergence will be unacceptably slow when the post-bifurcation behaviour needs to be analysed. To avoid this predicament, a full tangent stiffness field-boundary-element formulation which takes the initial stress–velocity gradient (displacement gradient) coupling terms accurately into account is presented in this paper. Here, the velocity field both inside and on the boundary are treated as primary variables. The large strain plasticity constitutive equation employed is based on an endochronic model of combined isotropic/kinematic hardening plasticity using the concepts of material director triad and the associated plastic spin. A generalized mid-point radial return algorithm is presented for determining the objective increments of stress from the computed velocity gradients. Numerical results are presented for problems of diffuse necking, involving very large strains and plastic instability, in initially perfect elastic–plastic plates under tension. These results demonstrate the clear superiority of the full tangent stiffness algorithm over the initial strain algorithm, in the context of the integral equation formulations for large strain plasticity.

A full tangent stiffness field-boundary-element formulation for geometric and material non-linear problems of solid mechanics

Novel displacement gradient boundary element method for elastic stress analysis with high accuracy

A Note on the Efficiency of the Boundary Element Method for Inelastic Axisymmetric Problems With Large Strains

New integral representations for deformation (velocity) gradients in elastic or elasto-plastic solids undergoing small or large deformations are presented. Compared to the cases wherein direct differentiation of the integral representations for displacements (or velocities) were carried out to obtain displacement (or velocity) gradients (which gave rise to hyper-singularity when the source point was taken to the boundary), the present integral representations have lower order singularities which are quite tractable from a numerical point of view. Moreover, the present representations, allow the source point to be taken in the limit, to the boundary, without any difficulties. This obviates a need for a two tier system of evaluation of deformation gradients in the interior of the domain on one hand and at the boundary of the domain on the other. It is expected that the present formulations would yield more accurate and stable deformation gradients in problems dominated by geometric and material nonlinearities.

Non-hyper-singular integral-representations for velocity (displacement) gradients in elastic/plastic solids (small or finite deformations)

A Novel Displacement Gradient Boundary Element Method for Elastic Stress Analysis With High Accuracy

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