Download e-book for kindle: A New Boundary Element Formulation in Engineering by Tania G.B. DeFigueiredo

By Tania G.B. DeFigueiredo

ISBN-10: 354054030X

ISBN-13: 9783540540304

ISBN-10: 3642845045

ISBN-13: 9783642845048

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 strength idea and linear elasticity. it really is thought of a boundary point version as the ultimate quintessential equation comprises 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 point 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 is predicated on Green's formulation (potential difficulties) and on Somigliana's id (elas­ ticity), or on the other hand throughout the weighted residual procedure. 2 The multi-field variational precept which generates the formula in­ volves 3 self sustaining variables. For power difficulties, those are the aptitude within the area and the capability and its general spinoff at the boundary. on the subject of elasticity, those variables are displacements within the area and displacements and tractions at the boundary. consequently, by means of analogy with the assumed displacement hybrid finite aspect version, ini­ tially proposed through Tong [1] in 1970, it may be referred to as a hybrid displacement version. the ultimate procedure 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 via merely appearing integrations alongside the boundary.

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Case 4: Integration over an element which includes both points i and j. 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.

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. The non-singular integral to be evaluated in this case will be denoted by Ik. ,. 3. This integral can be computed numerically by using a Gauss- Legendre quadrature formula.

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.

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

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