A Semi-Analytical Finite Element Method for Three-Dimensional Contact Problems with Axisymmetric Geometry

A combination of conventional two-dimensional finite elements and a Fourier series expansion in the tangential direction is shown to be efficient for modelling an elastic three-dimensional frictionless contact problem of geometrically axisymmetric bodies under non-axisymmetric external loads. For the solution procedure, the governing partial differential equations and contact conditions on the contact surface are formulated into an equivalent minimization problem with constraints. It is shown that the resulting objective function can be expressed as the sum of decoupled contributions from each harmonic. A modified simplex method is used to solve this quadratic programming problem. An example problem, motivated from a modular prosthesis design for artificial joint replacements, has shown the significant computational efficiency of this approach as compared to a conventional full three-dimensional finite element method.