Hybrid-Trefftz displacement elements for three-dimensional elastodynamics

In this paper the formulation of hybrid-Trefftz displacement finite elements for transient problems in three-dimensional elastic media is derived. The mathematical model is derived from the classical theory of elasticity. The governing domain and boundary equations are discretized in time using a wavelet basis expansion to yield a series of spectral problems which only depend on space. Displacements are the main approximation in the domain of the element (displacement model) and tractions are approximated on the essential boundaries (hybrid formulation). The displacement trial functions are constrained to satisfy exactly the domain equations (Trefftz constraint), and consist of one family of compression waves and two families of shear waves. The problem is reduced to solving an algebraic system whose unknowns are the weights of the displacement bases. Strains and stresses are derived from the displacement approximations through the compatibility and elasticity equations. The formulation is implemented as a new module in FreeHyTE, an open-source and user-friendly Trefftz platform. Numerical tests have been carried out with the new 3D FreeHyTE module implemented with the functions derived in this paper and the results are satisfactory.