Abstract
A simple Trefftz-type finite element method (TFEM) is proposed for solving certain potential problems in orthotropic media. The body force, which is induced by internal sources or sinks, may produce domain integrals in the standard Trefftz finite element formulation. This will make the advantage only-boundary integration of TFEM lose entirely. To overcome this difficulty, the dual reciprocity method (DRM) is employed to transfer the original problem into a homogeneous one. Then, a particular solution (PS) Trefftz-type finite element model is established based on the modified functional. Three benchmark examples are investigated by the proposed approach and compared with the analytical solutions.
Highlights
A wide range of problems in physics and engineering such as heat transfer, fluid flow motion, flow in porous media, shaft torsion, electrostatics and magnetostatics can come down to the solution of potential problems
To solve the orthotropic potential problem governed by Eqs. (1)-(3) using type finite element method (TFEM) approach, the solution domain W has to be divided into a number of elements as done in the conventional finite element method (FEM)
We presented a particular solution Trefftz-type finite element approach for solving certain potential problems with “body force” in plane orthotropic materials
Summary
A wide range of problems in physics and engineering such as heat transfer, fluid flow motion, flow in porous media, shaft torsion, electrostatics and magnetostatics can come down to the solution of potential problems. Powerful methods to pursue numerical solutions are mainly based on the finite element method (FEM) and boundary element method (BEM) Among these methods, the Trefftz-type finite element method (TFEM), originally developed by Jirousek and Leon (1977), has recently received great attention. Due to the fact that no domain integrals are involved in the formulation of TFEM, the Trefftz-type elements are less sensitive to mesh distortion in practical applications This feature has been investigated by Jirousek and Wroblewski (1995); Jirousek and Qin (1996); Choi et al (2006); Cen et al (2011); Wang et al (2012) using different 4-node quadrilateral elements. Motivated by the strength of TFEM and the popularity of DRM, this paper makes an effort in dealing with potential problems with “body force” in orthotropic media In this methodology, Trefftz functions proposed by Wang et al (2012) are employed to construct the intra-element field.
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