A Trefftz collocation method (TCM) for three-dimensional linear elasticity by using the Papkovich-Neuber solutions with cylindrical harmonics

A Trefftz collocation method (TCM) is proposed for solving three-dimensional (3D) linear-elastic boundary value problems. By using the Papkovich-Neuber (P-N) general solutions, Trefftz trial functions are expressed in terms of cylindrical harmonics. Both non-singular and singular harmonic functions are included, facilitating the study of interior and exterior problems. To mitigate the problem of ill-conditioned functions, two steps are adopted: the first step is to introduce a characteristic length of the domains of interests into the Laplace equation, and the second step is to scale each column of the coefficient matrix in the established system of linear equations using another multi-scale characteristic length, letting each column have the equal norms. Several examples are presented to validate the proposed 3D Trefftz collocation method. The completeness of the trial functions, the effect of the scaling techniques, and the accuracy of solutions are also discussed.