Displacements approach with external variables only for multi-domain analysis via symmetric BEM

In the present paper a new displacement method, defined as external variables one, is proposed inside the multidomain symmetric Boundary Element formulation. This method is a natural evolution of the displacement approach with interface variables in the multidomain symmetric BEM analysis. Indeed, the strategy employed has the advantage of considering only the kinematical quantities of the free boundary nodes and the algebraic operators involved show symmetry and very small dimensions. The proposed approach is characterized by strong condensation of the mechanical and kinematical boundary nodes variables of the macro-elements. All the domain quantities, such as tractions and stresses, displacements and strains, are computed through the Somigliana Identities in a subsequent phase. Some examples are shown using the calculus code Karnak.sGbem, by which it was possible to make some comparisons with analytical solutions andothe rapproaches to show the effectiveness of the method propose

Multidomain SBEM analysis for two dimensionalelastoplastic-contact problems

The Symmetric Boundary Element Method based on the Galerkin hypotheses has found application in the nonlinear analysis of plasticity and contact-detachment problems, but dealt with separately. In this paper we wants to treat these complex phenomena together. This method works in structures by introducing a subdivision into sub-structures, distinguished into macroelements, where elastic behaviour is assumed, and bem-elements, where it is possible for plastic strains to occur. In all the sub-structures, elasticity equations are written and regularity conditions in weighted (weak) form and/or in nodal (strong) form between boundaries have to be introduced, to attain the solving equation system

Active macro-zones algorithm via multidomain SBEM for strain-hardening elastoplastic analysis

In this paper a strategy to perform strain-hardening elastoplastic analysis by using the Symmetric Boundary Element Method (SBEM) for multi-domain type problems is shown. The procedure has been developed inside Karnak.sGbem code by introducing an additional module

Computational aspects in 2D SBEM analysis with domain inelastic actions

The Symmetric Boundary Element Method, applied to structures subjected to temperature and inelastic actions, shows singular domain integrals. 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. 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

Energetic criterion of the error evaluation in the analysis via SGBEM

The Symmetric Galerkin Boundary Element Method (sGbem) is assuming more and more an effective role in the solving problems of mechanics in different fields of engineering [1]. The presence of symmetric and defined in sign algebraic operators make such Method more competitive in comparison to the formulation for collocation. The present work has as objective the improvement of the response in the process of analysis of the system where a first discretizazion has been operated, by using a strategy that allows to operate a estimate of the error. On the base of such estimate it is possible to operate a new discretizazion of the boundary through adaptive procedures

Internal spring distribution for quasi brittle fracture via Symmetric Boundary Element Method

In this paper the symmetric boundary element formulation is applied to the fracture mechanics problems for quasi brittle materials. The basic aim of the present work is the development and implementation of two discrete cohesive zone models using Symmetric Galerkin multi-zone Boundary Elements Method. The non-linearity at the process zone of the crack will be simulated through a discrete distribution of nodal springs whose generalized (or weighted) stiffnesses are obtainable by the cohesive forces and relative displacements modelling. This goal is reached coherently with the constitutive relation that describes the interaction between mechanical and kinematical quantities along the process zone. The cracked body is considered as a solid having a “particular” geometry whose analysis is obtainable through the displacement approach employed in (Panzeca et al., 2000; 2002-b) by some of the present authors in the ambit of the Symmetric Galerkin Boundary Elements Method (SGBEM). In this approach the crack edge nodes are considered distinct and the analysis is performed by evaluating all the equation system coefficients in closed form (Guiggiani, 1991; Gray, 1998; Panzeca et al., 2001; Terravecchia, 2006)

Boundary discretization based on the residual energy using the SGBEM

The paper has as objective the estimation of the error in the structural analysis performed by using the displacement approach of the Symmetric Galerkin Boundary Element Method (SGBEM) and suggests a strategy able to reduce this error through an appropriate change of the boundary discretization. The body, characterized by a domain X and a boundary C , is embedded inside a complementary unlimited domain X1nX bounded by a boundary C+. In such new condition it is possible to perform a separate valuation of the strain energies in the two subdomains through the computation of the work, defined generalized, obtained as the product among nodal and weighted quantities on the actual boundary C and on the complementary boundary C+. In order to reduce the error in the analysis phases, the scattered energy has been computed as generalized work in each boundary element of C+ and an adequate node number has been introduced inside the boundary elements where this generalized work is higher. This strategy, made in a recursive way, has shown effectiveness whether in the convergence proofs of some mechanical and kinematical quantities or in computing the percentage error obtained as ratio between the scattered work in X1nX and the total work, both expressed in terms of generalized quantities