Numerical implementation of local effects due to two-dimensional discontinuous loads using special elements based on boundary integrals

In this paper, special-purpose elements are developed for solving local effects caused by discontinuous loads such as concentrated forces, line loads and patch loads applied in plane elastic structures. During the derivation of the special-purpose elements, the interior displacement and stress fields are composed of two parts: (1) the homogeneous solution part, which is represented by a linear combination of fundamental solutions at a number of source points outside the element domain; and (2) the particular load-dependent part, which is analytically represented by suitable local solutions. Meanwhile the independent frame displacements defined over the element boundary are approximated by conventional shape functions. The linkage between the two independent fields is established through use of a newly constructed hybrid variational functional, in which discontinuous loads are treated as generalized body forces. Using the property of delta function, the domain integral associated with discontinuous loads in the variational functional can be removed. The advantage of such special-purpose elements is that a large element, independent of the location of discontinuous loads, can be used to avoid the requirement of mesh refinement in the vicinity of the area with local loads. Numerical experiments are carried out to verify the special-purpose elements and to investigate their effectiveness in terms of mesh reduction and accuracy.