Boundary stress calculation—a comparison study

SUMMARYThe boundary integral representation of second-order derivatives of the primary function involves secondorder (hypersingular) and third-order (supersingular) derivatives of the Green's function. By defining these highly singular integrals as a difference of boundary limits, interior minus exterior, the limiting values are shown to exist. With a Galerkin formulation, coincident and edge-adjacent supersingular integrals are separately divergent, but the sum is finite, while the individual hypersingular integrals are finite. Moreover, the cancellation of the supersingular divergent terms only requires a continuous interpolation of the surface potential, and there is no continuity requirement on the surface flux. The algorithm is efficient, the non-singular integrals vanish and the singular integrals are computed entirely analytically, and accurate values are obtained for smooth surfaces. However, it is shown that a

Section: Cancellationmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%

Evaluation of supersingular integrals: second‐order boundary derivatives

Moore

Gray

Kaplan

2006

“…The literature is substantial, see for example References [11,[25][26][27][28][29]. The approach employed herein [12,13] exploits the definition of the integral equations as boundary limits, and therefore falls into the class of methods based upon boundary integral representations.…”

Section: Gradient Evaluationmentioning

confidence: 99%

“…A standard boundary integral representation of the gradient involves second-order derivatives of the Green's function, and moreover requires a complete integration over the boundary (see Reference [11] for a discussion of boundary integral gradient evaluation methods). Herein we show that the boundary limit approach in References [12,13] is advantageous, in that all of the complexity of the axisymmetric kernel functions disappears: the kernels for gradient evaluation are no more difficult than for the simple two-dimensional Laplace equation.…”

Section: Introductionmentioning

confidence: 99%

Galerkin boundary integral analysis for the axisymmetric Laplace equation

Gray

Garzon

Mantič

et al. 2006

SUMMARYThe boundary integral equation for the axisymmetric Laplace equation is solved by employing modified Galerkin weight functions. The alternative weights smooth out the singularity of the Green's function at the symmetry axis, and restore symmetry to the formulation. As a consequence, special treatment of the axis equations is avoided, and a symmetric-Galerkin formulation would be possible. For the singular integration, the integrals containing a logarithmic singularity are converted to a non-singular form and evaluated partially analytically and partially numerically. The modified weight functions, together with a boundary limit definition, also result in a simple algorithm for the post-processing of the surface gradient.

“…Hypersingular boundary integral equations (HBIEs) have diverse important applications and are the subject of considerable current research (see References [30][31][32][33]). HBIEs, for example, have been employed for the evaluation of boundary stresses [34][35][36], in wave scattering (e.g. Reference [37]), in fracture mechanics (e.g.…”

Section: Introductionmentioning

confidence: 99%

The meshless standard and hypersingular boundary node methods—applications to error estimation and adaptivity in three‐dimensional problems

Chati

Paulino

Mukherjee

2001

SUMMARYThe standard (singular) boundary node method (BNM) and the novel hypersingular boundary node method (HBNM) are employed for the usual and adaptive solutions of three-dimensional potential and elasticity problems. These methods couple boundary integral equations with moving least-squares interpolants while retaining the dimensionality advantage of the former and the meshless attribute of the latter. The 'hypersingular residuals', developed for error estimation in the mesh-based collocation boundary element method (BEM) and symmetric Galerkin BEM by Paulino et al., are extended to the meshless BNM setting. A simple 'a posteriori' error estimation and an e ective adaptive reÿne-ment procedure are presented. The implementation of all the techniques involved in this work are discussed, which includes aspects regarding parallel implementation of the BNM and HBNM codes. Several numerical examples are given and discussed in detail. Conclusions are inferred and relevant extensions of the methodology introduced in this work are provided.