The boundary element method (BEM) for linear elasticity in its curent usage is based on the boundary integral equation for displacements. The stress field in the interior of the body is computed by differentiating the displacement field at the source point in the BEM formulation, via the strain field. However, at the boundary, this method gives rise to a hypersingular integral relation which becomes numerically intractable. A novel approach is presented here, where hyper-singular kernels for stresses on the boundary are made numerically tractable through the imposition of certain equilibrated displacement modes. Numerical results are also presented for benchmark problems, to illustrate the efficacy of the present approach. Solutions are compared to the commonly used boundary stress algorithm wherein the boundary stresses are computed from known boundary tractions, and derivatives of known displacements tangential to the boundary. An extension of this approach to solve linear elasticity problems using the traction boundary integral equation (TBIE) is also discussed.

On the evaluation of hyper-singular integrals arising in the boundary element method for linear elasticity

In this paper, a series of effective formulae of the boundary element method is presented. In these formulae, by using a new variable, two kernels are only of the weaker singularity of Lnr (where r is the distance between a source point and a field point). Hence, the singularities in the conventional displacement formulation and stress formulation at internal points are reduced respectively so that the “boundary-layer” effect which strongly degenerates the accuracy of stress calculation by using original formulae is eliminated. Also the direct evaluation of coefficients C (boundary tensor), which are difficult to calculate, is avoided. This method is used in elastoplastic analysis. The results of the numerical investigation demonstrate the potential advantages of this method.

An effective boundary method for the analysis of elastoplastic problems

The two most promising approaches to determine Stress Intensity Factor (SIF) developedover the past decade are the Symmetric Galerkin Boundary Element Method - Finite Element Method(SGBEM-FEM) based alternating method and the Extended Finite Element (XFEM) method. Thepurpose of this paper is to determine the SIFs for a number of 2-D crack problems by the two ap-proaches and measure their relative effectiveness in terms of accuracy, speed and computational re-sources.In the SGBEM-FEM alternating method, a finite element analysis is carried out on the un-crackedbody using the externally applied loading and next a boundary element analysis is performed byreversing the stresses found on the crack location from the finite element analysis, and the residualstresses on the boundary of the finite body are determined. The steps are repeated until convergenceis achieved where the residual stresses on the boundaries and traction on the crack surfaces are closeto zero.In the XFEM method, the mesh is created without considering the topology of the crack configura-tion and the discontinuities are handled by special discontinuity enrichment functions. The enrichmentfunctions increase the degrees of freedom and the regular stiffness matrix is augmented by additionalterms corresponding to the extra degrees of freedom but the increase in computational requirement isoffset by not having the burden of remeshing the finite elements.Both SGBEM-FEM alternating method and XFEM method are used to solve a number of crackproblems and the example cases clearly show the computational efficiency of the SGBEM-FEM al-ternating method over the XFEM method.

Comparison of SGBEM-FEM Alternating Method and XFEM Method for Determining Stress Intensity Factor for 2D Crack Problems

https://lib.dr.iastate.edu/cgi/viewcontent.cgi?article=11289&context=rtd

Application of displacement and traction boundary integral equations for fracture mechanics analysis

Nonlinear elastoplastic boundary element analysis (BEA) involves an algebraic subproblem requiring the solution of dense nonsymmetric matrix equations with an evolving right hand side vector. When multiple right hand side vectors are present, direct matrix triangular factorization techniques have been the compelling choice, amortizing the work of a single matrix factorization over the sequence of multiple fast ‘solutions’ of the resulting triangular systems. Recently, the superior performance of preconditioned Krylov equation solvers in linear BEA has also been documented. In this paper, the superior performance of preconditioned Krylov equation solvers is shown to be extendable to elastoplastic BEA. This is accomplished by exploiting the strategic reuse of the preconditioner, its factorization, and the Krylov vectors computed in the solution for the first right hand side vector, in the subsequent solution of matrix equations with multiple ‘nearby’ right hand side vectors. The details associated with this strategy are given, and the computer resources required in three dimensional elastoplastic BEA are used to quantify the computational efficiency associated with this new algorithm.

Preconditioned Krylov equation solvers in elastoplastic boundary element analysis

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

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