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

By Tania G.B. DeFigueiredo

1. 1 The Hybrid Displacement Boundary aspect 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 really is thought of a boundary point version as the ultimate vital equation contains a few boundary integrals, whose evaluate calls for a boundary discretization. in addition, all of the unknowns are boundary vari­ ables. The version is totally new; it differs from the classical boundary point formula ·in how it is generated and therefore within the fi­ nal equations. A generalized variational precept is used as a foundation for its derivation, while the traditional boundary point formula is predicated on Green's formulation (potential difficulties) and on Somigliana's identification (elas­ ticity), or however throughout the weighted residual approach. 2 The multi-field variational precept which generates the formula in­ volves 3 self sufficient variables. For strength difficulties, those are the aptitude within the area and the aptitude and its general spinoff at the boundary. in relation to elasticity, those variables are displacements within the area and displacements and tractions at the boundary. as a result, through analogy with the assumed displacement hybrid finite point version, ini­ tially proposed by way of Tong [1] in 1970, it may be referred to as a hybrid displacement version. the ultimate process 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 by way of in basic terms acting integrations alongside the boundary.

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101 ) 43 The symmetry of the flexibility type matrix F has then been proved and so has the symmetry of the stiffness matrix K. 1 Introduction The procedures required for the numerical solution. of the equations presented in the previous chapter will be explained in the following sections. Although only the two dimensional case will be considered throughout this chapter, similar considerations could be used for three dimensional problems. 69) respectively. 80) which produces the solution of the problem.

E. 78). 51) in Xl direction, i. e. 13). 88) From that relationship it is possible to state that, if the matrix F is symmetric then the matrix K is also symmetric. Therefore the aim here is to prove the symmetry of F. 4, respectively. 91) where u*' is the fundamental solution at a field point x corresponding to a concentrated unit source at source point i (i = 1,2, ... ,N. 2). Similarly q*i is the fundamental normal flux at point x due to a concentrated unit source at j (j = 1,2, ... , N). 2). 93) and where u*i is the fundamental solution corresponding to flux q*i.

This situation occurs when evaluating the main diagonal terms of F, namely hi. 11), i. 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.

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