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By Tania G.B. DeFigueiredo

1. 1 The Hybrid Displacement Boundary aspect version This paintings is worried with the derivation of a numerical version for the answer of boundary-value difficulties in strength conception and linear elasticity. it truly is thought of a boundary point version as the ultimate vital equation consists of a few boundary integrals, whose assessment 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 for this reason within the fi­ nal equations. A generalized variational precept is used as a foundation for its derivation, while the traditional boundary point formula relies on Green's formulation (potential difficulties) and on Somigliana's id (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 sustaining variables. For capability difficulties, those are the aptitude within the area and the capability and its common by-product at the boundary. when it comes to elasticity, those variables are displacements within the area and displacements and tractions at the boundary. consequently, via analogy with the assumed displacement hybrid finite point version, ini­ tially proposed via Tong [1] in 1970, it may be known as a hybrid displacement version. the ultimate method of equations to be solved is the same to that present in a stiffness formula. The stiffness matrix for this version is symmetric and will be evaluated through purely acting integrations alongside the boundary.

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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. It can be represented as follows: u{x) = u*T-y X f fl. 53) '1N 31 in which N is the number of boundary points to be chosen in each particular case. These correspond to the points where the boundary variables will be evaluated. The components of both vectors u and that there is an infinite domain properties as n.

Is called the fundamental solution at It is usually denoted by x, is called the field point. The point ~, u·(~,x). The point where the concentrated unit source is applied, is defined as the source point. Source: The source is the entity that generates the potential field. It varies according to the physical problem under consideration. For example; in problems concerning gravitational, electric or temperature fields the source is a mass, an electric charge or a heat source. 48) This fundamental solution corresponds to the Newtonian potential for three-dimensional problems or the logarithmic potential for two-dimensional problems.

The q*i function defines the value of the fluxes of the fundamental solution at a point x corresponding to a source point at j. 3. 8) on =----,-- is the normal derivative of ri with respect to the outward normal n. ij becomes: J;i =- p 471"2 . 9) over elements which include neither point i nor point j is not a difficult problem. ble scheme. In the limit, however, when f is equal to zero, this integral becomes singular when either the source point i of u*i or the source point j of q*i coincides with the field point x.

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