The Trefftz methods for 3D biharmonic equation using directors and in-plane biharmonic functions

Because the complete set of Trefftz functions for the 3D biharmonic equation is not yet well established, a multiple-direction Trefftz method (MDTM) and an in-plane biharmonic functions method (IPBFM) are deduced in the paper. Inspired by the Trefftz method for the 2D biharmonic equation, a novel MDTM incorporates planar directors into the 2D like Trefftz functions to solve the 3D biharmonic equation. These functions being a series of biharmonic polynomials of different degree, automatically satisfying the 3D biharmonic equation, are taken as the bases to expand the solution. Then, we derive a quite large class solution of the 3D biharmonic equation in terms of 3D harmonic functions, and 2D biharmonic functions in three sub-planes. The 2D biharmonic functions are formulated as the Trefftz functions in terms of the polar coordinates for each sub-plane. Introducing a projective variable, we can obtain the projective type general solution for the 3D Laplace equation, which is used to generate the 3D Trefftz type harmonic functions. Several numerical examples confirm the efficiency and accuracy of the proposed MDTM and IPBFM.