A Simple Multi-Source-Point Trefftz Method for Solving Direct/Inverse SHM Problems of Plane Elasticity in Arbitrary Multiply-Connected Domains

In this paper, a generalized Trefftz method in plane elasticity is developed, for solving problems in an arbitrary multiply connected domain. Firstly, the relations between Trefftz basis functions from different source points are discussed, by using the binomial theorem and the logarithmic binomial theorem. Based on these theorems, we clearly explain the relation between the T-Trefftz and the F-Trefftz methods, and why the traditional T-Trefftz method, which uses only one source point, cannot successfully solve problems in a multiply connected domain with genus larger than 1. Thereafter, a generalized Trefftz method is proposed, which uses logarithmic and negative power series from multiple source points, and positive power series from only one source point, as complex potentials. In addition, a characteristic length for each source point is used to scale the Trefftz basis functions, in order to resolve the ill-posedness of Trefftz methods. For direct problems, no further regularization techniques are used, because the coefficient matrix of the system of linear equations to be solved is already well-conditioned, by using characteristic lengths to scale the Trefftz basis functions. Inverse problems in plane elasticity, wherein both tractions as well as displacements, or, both strains as well as displacements, are prescribed at a part of the boundary, and the data at the other part of the boundary and in the domain have to be solved for, are also considered. These problems are of importance in Structural Health Monitoring (SHM). For inverse problems where noises are present, a very simple regularization method is used, to mitigate the inherent ill-posed nature of inverse problems. By several numerical examples, we show that this generalized Trefftz method can successfully solve both direct/inverse problems in simply as well as multiply connected domains. Therefore, we consider this multi-source-point multi-characteristic-length-scale Trefftz method to be simple, general as well as very useful. And the essential idea of how to construct basis functions from multiple source points can be used to develop other Trefftz methods, as well as special Trefftz Voronoi Cell Finite Elements, with circular, elliptical, or arbitrary shaped voids and rigid/flexible inclusions.