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

The steady-state elastodynamic boundary element method is an efficient method that can be used in the modelling of vibration systems. The ability to locate natural frequencies and predict displacements close to these frequencies requires the use of high order elements and accurate integration schemes. The possible unboundedness of the displacements at or close to a natural frequency highlights poor numerical conditioning of the algebraic equations resulting from the discretization process, thus making accurate integration schemes a necessity. This paper presents a semi-analytical integration scheme that can be applied to quadratic subparametric triangular elements. The scheme involves subdividing the triangular elements into four triangular subelements. The quadratic shape functions of the original element can then be represented in terms of the linear shape functions of each subelement. A semi-analytical scheme is applied to the integrals involving the linear shape functions of the subelements. Taylor expansions are utilized in the scheme 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 numerically using Gauss-Legendre quadrature. The scheme can handle all integrals appearing in the steady-state elastodynamic BEM with good accuracy. In addition, the Cauchy principal value singular integrals can be dealt with without special treatment. The new scheme is tested by considering integration over two test elements and by application to simple test-problems for which analytical solutions are known.