A New Boundary Element Formulation in Engineering by Tania G.B. DeFigueiredo PDF

By Tania G.B. DeFigueiredo

ISBN-10: 354054030X

ISBN-13: 9783540540304

ISBN-10: 3642845045

ISBN-13: 9783642845048

1. 1 The Hybrid Displacement Boundary point version This paintings is worried with the derivation of a numerical version for the answer of boundary-value difficulties in capability concept and linear elasticity. it truly is thought of a boundary aspect version as the ultimate imperative equation consists of a few boundary integrals, whose overview calls for a boundary discretization. in addition, the entire unknowns are boundary vari­ ables. The version is totally new; it differs from the classical boundary aspect formula ·in how it is generated and accordingly within the fi­ nal equations. A generalized variational precept is used as a foundation for its derivation, while the normal boundary aspect formula relies on Green's formulation (potential difficulties) and on Somigliana's identification (elas­ ticity), or however in the course of the weighted residual process. 2 The multi-field variational precept which generates the formula in­ volves 3 self sufficient variables. For power difficulties, those are the capability within the area and the aptitude and its common by-product at the boundary. with regards to elasticity, those variables are displacements within the area and displacements and tractions at the boundary. hence, via analogy with the assumed displacement hybrid finite point version, ini­ tially proposed by means of Tong [1] in 1970, it may be known as a hybrid displacement version. the ultimate process of equations to be solved is identical to that present in a stiffness formula. The stiffness matrix for this version is symmetric and will be evaluated through purely appearing integrations alongside the boundary.

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Additional info for A New Boundary Element Formulation in Engineering

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25) where mi and Vi are functions of u. 27) between these two fields on the boundary needs to be introduced as a subsidiary condition. 27) can be introduced into the variational expression by using a Lagrange multiplier, A. 30) Notice that the new functional I2 has three independent field variabIes and the modified principle is no longer a minimum principle but a stationary principle. The Euler equations can then be obtained as the stationary conditions of I2 . 29), namely u, it and A. It can therefore be written as follows: c5I2 (u, it, A) = 10 [~(mi c5Vi + Vi c5mi) - b c5u] dO.

50) ~. These fundamental solutions are singular when the points x and ~ co- incide, i. e. their values tend to infinity at the source point. They are the only fundamental solutions which enter in the formulation. 2 Approximation for the Domain Variable Herein, and in the next four subsections, the matrix notation instead of the indicial notation will be used. In this manner, the final system of equations is obtained in matrix form which is simple and convenient. The potential field at a point x inside the domain {l - u{ x) - will be approximated as a series of products of fundamental solutions and unknown parameters.

The domain n and the infinite domain noo have the same properties, and consequently, the same mass density p. 58) vanishes in consequence of the kind of approximation used for the potential in the domain. 3). 3). 58) vanishes. 59). 71) 37 There are no singularities in matrices Land Q, consequently the limiting process is not necessary; they can be simply written in terms of the integral over the actual boundary, as indicated. 72) with respect to u, it and ij; to be more precise, with respect to the unknown parameters - Ii, Ui and which define those qi - variables.

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A New Boundary Element Formulation in Engineering by Tania G.B. DeFigueiredo

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