Keywords: thin plate, finite element, boundary integral equations, complex variables.

The analysis of complex structural problems using standard numerical models, based on domain discretisation, can imply high computation costs as a result of the large number of variables involved. In some contexts significant savings are possible using boundary elements and renouncing the flexibility and simplicity which characterise most finite elements. A further possibility consists of coupling finite elements with boundary elements, however it requires an interface finite element computer code with more complicated boundary element codes. Indeed boundary element models can be used in an alternative way, directly designing finite elements ready to include in the library of the available computer codes. In this approach the boundary integral description of the domain fields replaces the usual polynomial interpolation of the same fields, avoiding the usual restrictions on number and location of interpolation nodes, element shapes and quality of approximations. So doing, the typical accuracy of the boundary element approximation is combined with the flexibility of the finite element methods, obtaining finite elements with flexible shapes and arbitrary number of nodes. So far this approach has been adopted only for the development of finite elements for plane elasticity problems [1]. In this paper a new thin plate finite element is designed exploiting the Galerkin boundary integral approach as a discretisation tool. More specifically the discrete forms of the weighted boundary integral equations associated with static and kinematic sources are used to evaluate the strain energy transferred on the boundary of the finite element. In this case the stiffness matrix descends from some influence matrices whose entries are obtained by the double (single) boundary integration of the product between a fundamental solution and two (one) polynomial shape functions. As the simultaneous presence of high order shape functions and articulated fundamental solutions makes the accurate and efficient evaluation of the above mentioned boundary integrals difficult, an analytical integration technique based on complex variables and specific integration rules is also used to minimise the computational costs [2]. The Gauss transformations, used to deactivate the singularities, allow the reduction of the number of types of prime integrals, making the entire integration process easier. After a brief introductory description of the integral formulation used in the symmetric boundary element analysis of thin plates, the paper reports the procedure followed the definition of the strain energy in terms of boundary variables and then describes its evaluation by using the boundary integral equations weighted on the boundary of the finite element. The complex variable procedure used to evaluate the integral coefficients is then presented, discussing the main computational tasks which occur in the development of a computer code based on the proposed thin plate finite element.

1

V.E. Bulgakov, M.V. Bulgakova, "Multinode finite element based on boundary integral equations", Int. J. Numer. Meth. Engng., 43, 533-548, 1998. doi:10.1002/(SICI)1097-0207(19981015)43:3<533::AID-NME440>3.0.CO;2-V

2

M. Mazza, M. Aristodemo, "Costruzione efficiente di modelli BEM simmetrici di lastre di Kirchhoff", GIMC 2008 - XVII Convegno Italiano di Meccanica Computazionale.

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