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Evaluation of Supersingular Integrals: Second-Order Boundary Derivatives...

by Matthew N Moore, Leonard J Gray, Theodore Kaplan
Publication Type
Journal
Journal Name
International Journal for Numerical Methods in Engineering
Publication Date
Page Numbers
1930 to 1947
Volume
69
Issue
9

The boundary integral representation of second-order derivatives of the primary function involves second-order (hypersingular) and third-order (supersingular) derivatives of the Green's function. By
defining these highly singular integrals as a difference of boundary limits, interior minus exterior, the limiting values are shown to exist. With a Galerkin formulation, coincident and edge-adjacent
supersingular integrals are separately divergent, but the sum is finite, while the individual hypersingular integrals are finite. Moreover, the cancellation of the supersingular divergent terms only requires a continuous interpolation of the surface potential, and there is no continuity requirement on the surface flux. The algorithm is efficient, the nonsingular integrals vanish and the singular integrals are computed entirely analytically, and accurate values are obtained for smooth surfaces. However, it is shown that a (continuous) linear interpolation is not appropriate for evaluation at boundary corners.