An isogeometric extension of Trefftz method for elastostatics in two dimensions

Abstract

Summary In this paper, an approach to blend the Hybrid‐Trefftz Finite Element Method (HTFEM) and the Isogeometric Analysis (IGA) called the Isogeometric Trefftz (IGAT) method is presented. The structure of the isogeometric extension of the Trefftz method is formally the same as for its conventional counterpart, except the approximation of the boundary displacements and geometry that are carried out using the Non‐Uniform Rational B‐Splines (NURBS) instead of polynomials. In other words, only the element boundaries are approximated using NURBS basis while the Trefftz approximation is used in the interior of the elements. For that reason, IGAT can be ranked alongside recently developed Isogeometric Boundary Element Method (IGABEM), the NURBS‐Enhanced Finite Element Method (NEFEM), the Isogeometric Local Maximum Entropy (IGA‐LME) method, and the Isogeometrically enhanced Scaled‐Boundary element method (SBFEM), which all use NURBS approximation at the domain boundary only. Theoretical conjectures made in this paper are accompanied by three examples that show that IGAT leads to excellent results using only a few elements.

A hybrid-Trefftz finite element platform for solid and porous elastodynamics

“…Substituting approximations (8) and (12) into Eq (16) and taking into account that generalized displacements u n and velocities v n are independent of time, one gets,…”

“…In more recent studies, hybrid-Trefftz elements were developed for plates and shells [15] , including particular solutions for cracking modes [10] , and for the analysis of the eigenmodes of optical fibers [23] . Finally, Trefftz methods have also been combined with other methods, as for example with isogeometric analysis [12] for the solution of elastostatic problems.…”

An efficient isostatic mixed shell element for coarse mesh solution

“…In this regards, notable contributions are the hybrid Trefftz element for Mindlin-Reissner plates proposed by Cen et al, 23 the hybrid plates by Teixeira de Freitas and Tiago, 24 the plate and shell elements by Martins et al 25 and the FE for bending analysis by Rezaiee-Pajand and Karkon. 26,27 The Trefftz method has also been used by Horak et al 28 within isogeometric formulations.…”