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Symmetric Galerkin boundary element method for quasibrittle fracture and frictional contact problems
 Comp. Mech
, 1993
"... This review concerns a methodology for solving numerically, to engineering purposes, boundary and initialboundary value problems by a peculiar approach characterized by the following features: the continuous formulation is centered on integral equations based on the combined use of singlelayer and ..."
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Cited by 57 (3 self)
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This review concerns a methodology for solving numerically, to engineering purposes, boundary and initialboundary value problems by a peculiar approach characterized by the following features: the continuous formulation is centered on integral equations based on the combined use of singlelayer and doublelayer sources, so that the integral operator turns out to be symmetric with respect to a suitable bilinear form; the discretization is performed either on a variational basis or by a Galerkin weighted residual procedure, the interpolation and weight functions being chosen so that the variables in the approximate formulation are generalized variables in Prager’s sense. As main consequences of the above provisions, symmetry is exhibited by matrices with a key role in the algebraized versions, some quadratic forms have a clear energy meaning, variational properties characterize the solutions and other results, invalid in traditional boundary element methods, enrich the theory underlying the computational applications. The present survey outlines recent theoretical and computational developments of the title methodology with particular reference to linear elasticity, elastoplasticity, fracture mechanics, timedependent problems, variational approaches, singular integrals, approximation issues, sensitivity analysis, coupling of boundary and finite elements, computer implementations. Areas and aspects which at present require further research are identified and comparative assessments are attempted with respect to traditional boundary integralelement methods. 1
Efficient Field Point Evaluation by Combined Direct and Hybrid Boundary Element Methods
, 1997
"... A new procedure for calculating field points with the boundary element method (BEM) is outlined. In the conventional BEM (CBEM) field points are computed as postprocessing by recurrently solving the boundary integral equation with known boundary data for every field point of interest. This procedu ..."
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A new procedure for calculating field points with the boundary element method (BEM) is outlined. In the conventional BEM (CBEM) field points are computed as postprocessing by recurrently solving the boundary integral equation with known boundary data for every field point of interest. This procedure is very timeconsuming. On the other hand, newly developed hybrid boundary element formulations (HBEM) based on variational principles compute field points from the known boundary solution by simply transforming it to the domain using fundamental solutions as field approximation. The advantage of the HBEM method is that the related system matrices are symmetric by construction but the computational effort to obtain the boundary solution is higher than in conventional boundary element methods. Thus, if symmetry of the system matrices has no priority, it is advantageous to compute the boundary solution with the CBEM and the field point solution with the HBEM. The additional implementation ...
Assessment of the spectral properties of the doublelayer potential matrix H, Boundary Elements and Other Mesh Reduction Methods
 Transactions on Modelling and Simulation, vol 54
, 2013
"... The doublelayer potential matrix H of the conventional, collocation boundary element method (CBEM) is singular, as referred to a static problem in a bounded continuum. This means that no rigid body displacements can be transformed between two different reference systems – and that unbalanced forces ..."
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The doublelayer potential matrix H of the conventional, collocation boundary element method (CBEM) is singular, as referred to a static problem in a bounded continuum. This means that no rigid body displacements can be transformed between two different reference systems – and that unbalanced forces are to be excluded from a consistent linear algebra contragradient transformation. The properties of HT may become quite informative, as they reflect the topology (concavities, notches, cracks, holes) of the discretized domain as well as material nonhomogeneities, which comes from the fact that local stress gradients can be represented by fundamental solutions only in a global sense. Symmetries and antisymmetries are also evidenced in N(HT) as well as the spectral properties related to simple polynomial solutions that one may propose as patch tests. In the usual implementations of the CBEM with real fundamental solutions, all eigenvalues λ of H are real, λ ∈ R, 0 ≥ λ < 1 for a bounded domain. This means that H is a contraction – a paramount mechanical feature that comes up naturally in the frame of a virtual work investigation of Kelvin’s (singular) fundamental solution and is resorted to in a simplified variational implementation of the boundary element method.
The boundary element method revisited
 Boundary Elements and Other Mesh Reduction Methods XXXII
, 2010
"... The collocation boundary element method is derived on the basis of the weightedresiduals 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 s ..."
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The collocation boundary element method is derived on the basis of the weightedresiduals 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 rigidbody 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 singlelayer and doublelayer 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.
Recent advances and emerging applications . . .
, 2011
"... was prepared after this workshop by the organizers and participants based on the presentations and discussions at the workshop. The paper aims to review the major research achievements in the last decade, the current status, and the future directions of the BEM in the next decade. The review starts ..."
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was prepared after this workshop by the organizers and participants based on the presentations and discussions at the workshop. The paper aims to review the major research achievements in the last decade, the current status, and the future directions of the BEM in the next decade. The review starts with a brief introduction to the BEM. Then, new developments in Green’s functions, symmetric Galerkin formulations, boundary meshfree methods, and variationally based BEM formulations are reviewed. Next, fast solution methods for efficiently solving the BEM systems of equations, namely, the
Sensitivity Analysis with the Simplified Hybrid Boundary Element Method
, 2005
"... The paper describes a formulation for computing design sensitivities required in inverse problems as well as in shape or material property optimization in the frame of the hybrid boundary element method. Implicit differentiation of the discretized boundary integrals is performed. The resulting integ ..."
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The paper describes a formulation for computing design sensitivities required in inverse problems as well as in shape or material property optimization in the frame of the hybrid boundary element method. Implicit differentiation of the discretized boundary integrals is performed. The resulting integrals present the same types of singularities of the basic hybrid formulation, which can be generally handled by means of wellestablished quadrature techniques, as developed for the conventional, collocation boundary element method. It is demonstrated that the spectral properties of the original matrices (as related to rigid body displacements) apply to the sensitivity matrices, thus yielding a general and efficient technique for the shape design sensitivity analysis of all structural quantities one may be interested in. The formulation is valid for threedimensional solids, in general. Multiplyconnected and unbounded domains may be handled, as well. Most important, however, is the outline of the sensitivity equations in the frame of the frequencydomain, simplified hybrid boundary element method, which with no loss of accuracy circumvents the necessity of the timeconsuming evaluation of a flexibility matrix. Moreover, it is shown that the sensitivity analysis of transient problems on the basis of a generalized mode superposition technique as well as of structures in free vibration can be performed efficiently. The formulations developed apply directly to Pian's hybrid finite element method, as a particular case. Comparisons with finite difference results are given in an academic example to demonstrate the effectiveness of the formulation.