The Effect of Essential Boundary Conditions on the Convergence of Iterative Equation Solvers in BEM

Rencis, Joseph J.; Mann, Kimberly C.; Urekew, Timothy J.

doi:10.1007/978-3-642-79654-8_446

Cited by 2 publications

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“…According to the above definition of fundamental solution, the domain integral on the left-hand side of eqn (5) is actually evaluated as (6,7), one obtains the modified expression of the Somigliana's identity, (9) which is used to evaluate displacements im u (and, subsequently, stresses) at a domain point m for prescribed forces i b , i t , and boundary displacements i u . The term in brackets vanishes only if i b and i t are in equilibrium, which is not necessarily true when one is dealing with approximations.…”

Section: From a Variational To A Consistent Weighted-residuals Statementmentioning

confidence: 99%

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The boundary element method revisited

Dumont¹

2010

Boundary Elements and Other Mesh Reduction Methods XXXII

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The collocation boundary element method is derived on the basis of the weighted-residuals statement. Only the static case is addressed, as it already involves all relevant conceptual issues. The present outline brings to discussion some relevant aspects and implementation issues of the method that should belong in any text book. It is shown that, if the boundary element method is consistently formulated, an inherent error term -related to arbitrary rigid-body displacements -is naturally taken into account and has no influence on the resultant matrix equation, with traction force parameters that are always in balance independently of mesh discretization. The constitutive matrices of the method -the single-layer and double-layer potential matrices G and H -present some spectral properties that are per se interesting but that also have applicability consequences. The matrix G is rectangular, if consistently obtained. For and adequately formulated problem, the solution of the resultant matrix equation is always possible (and unique) whether directly or approximated in terms of equivalent nodal forces. The effects of body forces, whenever transformable to boundary actions, may be expressed in terms of the boundary interpolation functions, which renders the final matrix equation more elegant and speeds up calculations in no detriment to accuracy. There is a novel proposition for the interpolation of traction forces along curved boundaries, with results that may be only slightly improved, as compared to the classical procedure, but that simplifies numerical computation and adds to the consistency of the method in terms of patch test assessments. The conceptual and numerical developments are illustrated by means of a few examples.

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Section: From a Variational To A Consistent Weighted-residuals Statementmentioning

confidence: 99%

“…However, there is an in principle uncontrolled amount of rigid-body displacements implicit in the term  im u of eqn (7), so that the good conditioning of A cannot be assured beforehand [9].…”

Section: On the Numerical Solution Of The Problemmentioning

confidence: 99%