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 [1] 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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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.