On The Relaxed Continuity Approach For The Selfregular Traction-Bie.

Abstract

The ‘relaxed continuity’ hypothesis adopted on the self-regular traction-BIE is
investigated for bidimensional problems. The self-regular traction-BIE, a fully regular
equation, is derived from Somigliana stress identity, which contains hypersingular integrals.
Due to the presence of hypersingular integrals the displacement field is required to achieve
C1,􀁄 Hölder continuity. This condition is not met by the use of standard conforming elements,
based on C0 interpolation functions, which only provide a piecewise C1,􀁄 continuity. Thus, a
relaxed continuity hypothesis is adopted, allowing the displacement field to be C1,􀁄 piecewise
continuous at the vicinity of the source point. The self-regular traction-BIE makes use of the
displacement tangential derivatives, which are not part of the original BIE. The tangential
derivatives are obtained from the derivative of the element interpolation functions. Therefore,
two possible sources of error, which are the discontinuity of the displacement gradients at
inter-element nodes and the approximation of the displacement tangential derivatives, are
introduced. In order to establish the dominant error, non-conforming elements are
implemented since they satisfy the continuity requirement at each collocation point. Standard
Gaussian integration scheme is applied in the evaluation of all integrals involved. Quadratic,
cubic and quartic isoparametric boundary elements are employed. Some numerical results
are presented comparing the accuracy of conforming and non-conforming elements on the
self-regular traction-BIE and highlighting the dominant error.