A new approach to non-homogeneous hyperbolic boundary value problems using hybrid-Trefftz stress finite elements

Abstract

A new approach to the solution of non-homogeneous hyperbolic boundary value problems is casted here using the hybrid-Trefftz stress/flux elements. Similarly to the Dual Reciprocity Method, the technique adopted in this paper uses a Trefftz-compliant set of functions to approximate the complementary solution of the problem and an additional trial basis to approximate its particular solution. However, the particular and complementary solutions are obtained here in a single step, and not sequentially, as typical of the Dual Reciprocity Method. The trial functions used for both particular and complementary solutions are merged together in the same basis and offered full flexibility to combine so as to recover the enforced equations in the best possible way. This option enables Trefftz-compliant functions to compensate for deficiencies of the particular solution basis, meaning that accurate total solutions can be obtained with relatively poor particular solution approximations. The formulation preserves the Hermitian, sparse and localized structure that typifies the matrix of coefficients of hybrid-Trefftz finite elements and avoids the drawbacks of the collocation procedures that arise in the Dual Reciprocity Method. Moreover, all terms of the matrix of coefficients are reduced to boundary integral expressions provided the particular solution trial functions satisfy the static problem obtained after discarding both non-homogeneous and time derivative terms from the governing equation.