Abstract：This paper attempts to further extend the so-called Trefftz direct method(TDM)，which has been developed and applied to a wide variety of boundary value problems in recent years. In this approach，the complete system of solutions of the partial differential equations is used as weighting functions and a non-singular boundary integral equation is used as the starting formulation. Several relative problems are discussed here. They are the problems of necessary and sufficient conditions for Trefftz-type boundary integral equations；the relationship between the Trefftz method and the variational principle and the infrequently encountered but possible singular problems in the Trefftz method and the treatment in practice. Numerical results are presented to illustrate the computational efficiency of the approach and to demonstrate its advantages.
