By Tania G.B. DeFigueiredo

1. 1 The Hybrid Displacement Boundary point version This paintings is anxious with the derivation of a numerical version for the answer of boundary-value difficulties in power concept and linear elasticity. it truly is thought of a boundary aspect version as the ultimate critical equation comprises a few boundary integrals, whose assessment 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 for that reason 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 on the other hand 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 strength difficulties, those are the capability within the area and the capability and its basic by-product at the boundary. in terms of elasticity, those variables are displacements within the area and displacements and tractions at the boundary. for that reason, through analogy with the assumed displacement hybrid finite aspect version, ini tially proposed via Tong [1] in 1970, it may be referred to 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 by means of in basic terms appearing integrations alongside the boundary.

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G T and LT are non-singular square matrices and can then be inverted. 75), i. e. 79) In the case of Laplace's equation, the equilibrium requires that the integral of the fluxes along the boundary be zero and this is satisfied in a variational sense. 6; the vector Q is a consistent load vector. Notice that matrix K is evaluated by carrying out integrations along the boundary only and that the unknown u is also calculated only on the boundary. The other primary unknowns - ~ and q - can be obtained from the boundary potentials.

E. Iii = lim f . 28) 56 Direct computation of this integral is not possible and physical considerations will then be applied to compute the main diagonal of the matrix F. The property that the fluxes are null for a constant potential field will be used. Consider, for instance, that the boundary potentials are constant all over the boundary and equal to 1. 29) 1 Applying the property mentioned above, one can say that for u = I, Q = o. 32) and a simple relationship has been obtained giving the values of the diagonal coefficients Iii in terms of the off-diagonal terms Iij.

4. i and q*i r~ are continuous functions on that interval, 51 i. e. when the point x moves along the element. It is then possible to write f. ,. ,. ari df + r' r' an . ) .!.. ) .!. 11) iq ri ri an ir: ri ri an P { 411"2 k=l,k'li,k'li [In irk -0 The four terms on the right-hand-side of the previous equation correspond to the four different types of integrals to be evaluated. Each one of these cases will be studied in detail in what follows. Case 1: Integration over an element which includes neither source point i nor source point j.