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 capability conception and linear elasticity. it's thought of a boundary aspect version as the ultimate crucial equation consists of a few boundary integrals, whose review calls for a boundary discretization. in addition, 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 therefore within the fi nal equations. A generalized variational precept is used as a foundation for its derivation, while the normal 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 strategy. 2 The multi-field variational precept which generates the formula in volves 3 self sustaining variables. For capability difficulties, those are the aptitude within the area and the capability and its general by-product at the boundary. relating to 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 via Tong  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 by way of basically acting integrations alongside the boundary.
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Extra info for A New Boundary Element Formulation in Engineering
30) Notice that the new functional I2 has three independent field variabIes and the modified principle is no longer a minimum principle but a stationary principle. The Euler equations can then be obtained as the stationary conditions of I2 . 29), namely u, it and A. It can therefore be written as follows: c5I2 (u, it, A) = 10 [~(mi c5Vi + Vi c5mi) - b c5u] dO. e. ) = [mi j >'l5u dr + dO + jq (>. 37) are functions of the inde- pendent variable u. 40) q->. 42), in other words, the Lagrange multiplier >.
9) • Equilibrium within the domain is satisfied hy the following mass production equation ami b 0 -+ = aXi mO where b is the mass production density defined in O. 1). 11). 1). 10) is equivalent to Laplace's equation or to Poisson's equation depending on b being equal to zero or otherwise. e. 22) when b = o. 15), can be used to define the potential problem. 25) where mi and Vi are functions of u. 27) between these two fields on the boundary needs to be introduced as a subsidiary condition. 27) can be introduced into the variational expression by using a Lagrange multiplier, A.
The domain n and the infinite domain noo have the same properties, and consequently, the same mass density p. 58) vanishes in consequence of the kind of approximation used for the potential in the domain. 3). 3). 58) vanishes. 59). 71) 37 There are no singularities in matrices Land Q, consequently the limiting process is not necessary; they can be simply written in terms of the integral over the actual boundary, as indicated. 72) with respect to u, it and ij; to be more precise, with respect to the unknown parameters - Ii, Ui and which define those qi - variables.
A New Boundary Element Formulation in Engineering by Tania G.B. DeFigueiredo