The Hybrid Boundary Element Method

Dumont, Ney Augusto

doi:10.1007/978-3-662-21908-9_8

Cited by 28 publications

(29 citation statements)

References 5 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%

“…since, for a sufficiently refined boundary mesh, the displacements homogeneous governing eqn (1), whenever available [7], can be approximated accurately enough by nodal displacement and traction force parameters Then, making use of eqns (14, 17), a convenient way of expressing eqn (13) is…”

Section: Numerical Discretizationmentioning

confidence: 99%

“…There is an uncountable number of alternatives of formulations and implementations, some of them based on sound principles [7,8], many of them contributing to the richness of the method [1], but unfortunately some of them just misleading. The present contribution is not a review paper and, on the contrary, makes just a few references to the technical literature.…”

Section: Introductionmentioning

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%

Section: Numerical Discretizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The boundary element method revisited

Dumont¹

2010

Boundary Elements and Other Mesh Reduction Methods XXXII

Self Cite

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“…Rigid body motion is included in terms of functions u r is multiplied by in principle arbitrary constants C sm ∈ R n r ×n * , where n r is the number of rigid body displacements (r.b.d.) of the discretized problem, as dealt with formally in Definition 1, introduced in Section 4 [7,10]. The fundamental solutions σ conventional boundary element method, given as…”

Section: Stress and Displacement Assumptionsmentioning

confidence: 99%

“…Although neither conceptually nor formally necessary, the following approximation may render all subsequent equations simpler and more elegant [7]. Given a sufficiently refined boundary mesh, the displacements u p i and the traction forces t p i related to an arbitrary particular solution of the non-homogeneous governing eqn (1), whenever available, can be approximated accurately enough by nodal displacement parameters…”

Section: Boundary Approximation Of the Particular Solutionmentioning

confidence: 99%

The expedite boundary element method

Dumont¹,

Aguilar²

2011

Boundary Elements and Other Mesh Reduction Methods XXXIII

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The present developments combine the variationally-based, hybrid boundary element method with a consistent formulation of the conventional, collocation boundary element method in order to establish a computationally less intensive procedure, although not necessarily less accurate, for large-scale, two-dimensional and three-dimensional problems of potential and elasticity, including timedependent phenomena. Both the double-layer and the single-layer potential matrices, H and G, whose evaluation usually requires dealing with singular and improper integrals, are obtained in an expedite way that circumvents almost any numerical integration -except for a few regular integrals in the case of H. A few numerical examples are shown to assess the applicability of the method, its computational effort and some convergence issues.

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Hybrid boundary element formulation for acoustic wave propagation

Wagner

Gaul

2002

Proc. Appl. Math. Mech.

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The so-called hybrid stress boundary element method (HSBEM) is introduced in a frequency domain formulation for the computation of acoustic radiation and scattering in closed and infinite domains. Different from other boundary element formulations, the HSBEM is based on an extended Hellinger-Reissner variational principle and leads to a Hermitian, frequency-dependent stiffness equation. Due to this, the method is very well suited for treating fluidstructure interaction problems since the effort for the coupling the structure, discretized by a finite elements, and the fluid, discretized by the HSBEM is strongly reduced.To arrive at a boundary integral formulation, the field variables are separated into boundary variables, which are approximated by piecewise polynomial functions, and domain variables, which are approximated by a superposition of singular fundamental solutions weighed by source strength. This approximation cancels the domain integral over the equation of motion in the hybrid principle and leads to a boundary integral formulation, incorporating singular integrals. Comparing to previous results published by the authors, new considerations concerning the interpretation of singular contributions in the stiffness matrix for exterior domain problems are communicated here.

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The Hybrid Boundary Element Method

Cited by 28 publications

References 5 publications

The boundary element method revisited

The boundary element method revisited

The expedite boundary element method

Hybrid boundary element formulation for acoustic wave propagation

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