Semi-analytical integration of sub-parametric elements used in the BEM forthree-dimensional elastodynamics

The steady-state elastodynamic boundary element method is an efficient meth od that can be used in the modelling of vibration systems. The ability to l ocate natural frequencies and predict displacements close to these frequenc ies requires the use of high order elements and accurate integration scheme s. The possible unboundedness of the displacements at or close to a natural frequency highlights poor numerical conditioning of the algebraic equation s resulting from the discretization process, thus making accurate integrati on schemes a necessity. This paper presents a semi-analytical integration s cheme that can be applied to quadratic subparametric triangular elements. T he scheme involves subdividing the triangular elements into four triangular subelements. The quadratic shape functions of the original element can the n be represented in terms of the linear shape functions of each subelement. A semianalytical scheme is applied to the integrals involving the linear s hape functions of the subelements. Taylor expansions are utilized in the sc heme presented to enable the formulation of the integrals into regular and singular parts. Standard numerical schemes are applied to the regular part. The singular part can be transformed into a line integral and evaluated nu merically using Gauss-Legendre quadrature. The scheme can handle all integr als appearing in the steady-state elastodynamic BEM with good accuracy. In addition, the Cauchy principal value singular integrals can be dealt with w ithout special treatment. The new scheme is tested by considering integrati on over two test elements and by application to simple test-problems for wh ich analytical solutions are known. (C) 1999 Elsevier Science Ltd. All righ ts reserved.