 By Tania G.B. DeFigueiredo

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 power idea and linear elasticity. it truly is thought of a boundary aspect version as the ultimate imperative equation contains a few boundary integrals, whose evaluate calls for a boundary discretization. moreover, 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 point formula relies on Green's formulation (potential difficulties) and on Somigliana's identification (elas­ ticity), or then again throughout the weighted residual approach. 2 The multi-field variational precept which generates the formula in­ volves 3 self sustaining variables. For strength difficulties, those are the capability within the area and the capability and its general by-product at the boundary. in relation to elasticity, those variables are displacements within the area and displacements and tractions at the boundary. as a result, by way of analogy with the assumed displacement hybrid finite point version, ini­ tially proposed through Tong  in 1970, it may be known as a hybrid displacement version. the ultimate approach 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 simply appearing integrations alongside the boundary.

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Additional info for A New Boundary Element Formulation in Engineering

Example text

4. Two different types of elements will be developed in this chapter. For the first one of linear geometry, herein called constant element, the boundary variables - ii and ij - are assumed to have a constant value on each element. The second type of element is the isoparametric quadratic for which the geometry and the boundary variables - ii and ij - are interpo- lated through Lagrange polynomials, as it is usually done in the classical boundary element method [40,64, 65J. 51). Consequently the matrices F for these two types of elements only differ because the boundary geometry is assumed to vary differently in each case.

The integral to be evaluated is then: .. 6. 6): the first one corresponds to the straight part of rj , herein the semi-circumference re of radius E:. = . {lr l' hm _ fj+f. 16) and finally it can be written as follows: . {lr (1) 1-a ar l' = hm fj e-+O In -:r' i --:- n rJ . dr(x) }+hm e-+O {J In (-:-1) ---:-1-a ari n f. 17) If one uses the notation shown below: {fr (1) 1-a ar dr( (1) 1-a ar Ie = . {lr -i . I = hm _ In -:- ---:- hm In -:- ---:- e-+O e-+O r' fj f. 19) n j n the integral Ii can be evaluated by adding the results from the two previous integrals, i.

44), the functional I HP can be written as IHP(U,u,ij). = lr[ ~2 u q dr - lr[ ij (u - 1. 46) where I HP is a hybrid functional which depends on u, u and ij, 28 u is the potential defined in the domain, u and ij are, respectively, potential and flux, both defined on the boundary, q is the prescribed value of normal flux i valid on fq, = 1,2 for two-dimensional domains and i = 1,2,3 in three dimensions. Einstein's summation convention for repeated indices is implied. 46) gives the final form of the multi-field functional and is the starting point for the model proposed in this work. 