Trefftz methods with cracklets and their relation to BEM and MFS. (October 2018)

• Record Type: Journal Article
• Authors: Alves, Carlos J.S.; Martins, Nuno F.M.; Valtchev, Svilen S.
• Is Part Of: Engineering analysis with boundary elements. Volume 95(2018)
• Page Start: 93
• Page End: 104
• Publication Date: 2018-10
• Subjects: Trefftz method -- Cracklets -- Boundary element method -- Method of fundamental solutions; 65N38 -- 65N80; Boundary element methods -- Periodicals; Engineering mathematics -- Periodicals; Équations intégrales de frontière, Méthodes des -- Périodiques; Mathématiques de l'ingénieur -- Périodiques; Boundary element methods; Engineering mathematics; Periodicals; 620.00151
• DOI: 10.1016/j.enganabound.2018.06.021
• Languages: English
• ISSNs: 0955-7997

Abstract

Abstract: In this paper we consider Trefftz methods which are based on functions defined by single layer or double layer potentials, integrals of the fundamental solution, or their normal derivative, on cracks. These functions are called cracklets, and satisfy the partial differential equation, as long as the crack support is not placed inside the domain. A boundary element method (BEM) interpretation is to consider these cracks as elements of the original boundary, in a direct BEM approach, or elements of an artificial boundary, in an indirect BEM approach. In this paper we consider the cracklets just as basis functions in Trefftz methods, as the method of fundamental solutions (MFS). We focus on the 2D Laplace equation, and establish some comparisons and connections between these methods with cracklets and standard approaches like the BEM, indirect BEM, and the MFS. Namely, we propose the enrichment of the MFS basis with the cracklets. Several numerical simulations are presented to test the performance of the methods, in particular comparing the results with the MFS and the BEM.