Closed form coefficients in the symmetric boundary element approach

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The Symmetric Boundary Element Method, applied to structures subjected to temperature and inelastic actions, shows singular domain integrals.\ud In the present paper the strong singularity involved in the domain integrals of the stresses and tractions is removed and, by means of a limiting operation, this traction is evaluated on the boundary. First the weakly singular domain integral in the Somigliana Identity of the displacements is regularized and the singular integral is transformed into a boundary one using the Radial Integration Method; subsequently, using the differential operator applied to the displacement field, the Somigliana Identity of the tractions inside the body is obtained and through a limit operation its expression is evaluated on the boundary. The latter operation makes it possible to substitute the strongly singular domain integral in a strongly singular boundary one, defined as a Cauchy Principal Value, with which the related free term is associated. The expressions thus obtained for the displacements and the tractions, in which domain integrals are substituted by boundary integrals, were utilized in the Galerkin approach, for the evaluation in closed form of the load coefficients connected to domain inelastic actions.\ud This strategy makes it possible to evaluate the load coefficients avoiding considerable difficulties due to the geometry of the solid analyzed; the obtained coefficients were implemented in the Karnak.sGbem calculus code

Section: Present Approachmentioning

confidence: 98%

“…These terms were obtained in closed form and implemented in the Karnak.sGbem calculus code [19] according to a strategy that works in the substructuring approach [20,21]. The strategy for evaluation in closed form was shown by Terravecchia [22].…”

Section: D Sbem Analysis With Domain Inelastic Actions 185mentioning

confidence: 99%

Computational aspects in 2D SBEM analysis with domain inelastic actions

Panzeca

Terravecchia

Zito

2009

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“…It could be objected that the assemblage between bem-elements having large dimensions and bemelements having small dimensions involves numerical instability in the analysis. This does not happen because all the coefficients of the equation system are obtained in closed form [29]. This characteristic is a special quality of the SGBEM in comparison with other BEM methodologies.…”

mentioning

confidence: 98%

“…These difficulties have been overcome within the SGBEM approach by using different techniques. We can mention the strategies of Bonnet [27], Frangi and Novati [28], Panzeca et al [26], Terravecchia [29] and Holzer [30].…”

mentioning

confidence: 99%

Active macro‐zone approach for incremental elastoplastic‐contact analysis

Cucco

Terravecchia

Zito

2013

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The symmetric boundary element method, based on the Galerkin hypotheses, has found an application in the nonlinear analysis of plasticity and in contact-detachment problems, but both dealt with separately. In this paper, we want to treat these complex phenomena together as a linear complementarity problem.A mixed variable multidomain approach is utilized in which the substructures are distinguished into macroelements, where elastic behavior is assumed, and bem-elements, where it is possible that plastic strains may occur. Elasticity equations are written for all the substructures, and regularity conditions in weighted (weak) form on the boundary sides and in the nodes (strong) between contiguous substructures have to be introduced, in order to attain the solving equation system governing the elastoplastic-contact/detachment problem. The elastoplasticity is solved by incremental analysis, called for active macro-zones, and uses the well-known concept of self-equilibrium stress field here shown in a discrete form through the introduction of the influence matrix (self-stress matrix). The solution of the frictionless contact/detachment problem was performed using a strategy based on the consistent formulation of the classical Signorini equations rewritten in discrete form by utilizing boundary nodal quantities as check elements in the zones of potential contact or detachment. the integral equations satisfying the boundary and domain conditions, the Signorini equations regarding the contact/detachment conditions rewritten as LCP, and the constitutive relations for rate-independent plasticity.

“…The computation of the hypersingular integrals which appears in the context of the SGBEM has been the subject of a significant number of works which include closed-form evaluations of the integrals involved in the computation of matrix elements or regularization methods [8][9][10][12][13][14][15]. These methods are often limited in their field of application (use of properties of the elasticity tensor or restriction to the plane case), except for the generalization of SGBEM by the Fourier method [13] which transforms the singularity by Fourier transform into a problem of convergence at infinite of the Fourier transform and seems quite general.…”

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confidence: 99%

A general regularization of the hypersingular integrals in the symmetric Galerkin boundary element method

Bonnet¹

2009

SUMMARYThe symmetric Galerkin boundary element method is used to solve boundary value problems by keeping the symmetric nature of the matrix obtained after discretization. The matrix elements are obtained from a double integral involving the double derivative of Green's operator, which is highly singular. The paper presents a regularization of the hypersingular integrals which depend only on the properties of Green's tensor. The method is presented in the case of Laplace's operator, with an example of application. The case of elasticity is finally addressed theoretically, showing an easy extension to any case of anisotropy.