Extending the method of exponential basis functions to problems with singularities

Abstract

Purpose -The purpose of this paper is to aim at extending the method of exponential basis functions (EBF) to solve a class of problems with singularities. Design/methodology/approach -In the procedure of EBF a summation of EBF satisfying the governing differential equation with unknown constant coefficients is considered for the solution. These coefficients are determined by the satisfaction of prescribed boundary conditions through a collocation approach. The applied basis functions are available in the case of linear partial differential equations (PDEs) with constant coefficients. Moreover, the method contributes to yield highly accurate results with ultra convergence rates for problems with smooth solution. This leads EBF to offer many advantages for a variety of engineering problems. However, owing to the global and smooth nature of the bases, the performance of EBF deteriorates in problems with singularities. In the present study, some exponential-like influence functions are developed, and a few of them are added to original bases. Findings -The new bases are capable of forming the constitutive terms of the asymptotic solution near the singularity points and alleviate the aforementioned limitation. The appealing feature of this method is that all the advantages of EBF such as its simplicity and efficiency are completely preserved. Research limitations/implications -In its current form, EBF can only solve PDEs with constant coefficients. Originality/value -Application of the method to some benchmark problems demonstrates its robustness over some other boundary approximation methods. This research may pave the road for future investigations corresponding to a wide range of practical engineering problems.

Simulating Fluid and Structure Interaction using Exponential Basis Functions

“…Since the formulation of the proposed method is based on using smooth bases, it cannot be directly applied to modelling of breaking waves in the tanks (which involve singularities in the solution). Appropriate singular bases may be introduced to the solution, as suggested by Mossaiby et al (2015).…”

Implementation of a generalized exponential basis functions method for linear and non‐linear problems

“…These functions absorb the effects of the singularities and leave the (rather) smooth parts to exponential basis functions. Such bases has been used in a variety of methods, such as collocation Trefftz methods and EBF . We employ this approach in Section 5 to solve a singular problem.…”

“…Because of the smooth and global nature of basis functions used in the proposed method, efficiency of the proposed method can be decreased in problems with singularities. However, we address this issue by adding a few singular functions to original bases (see for more information). This has been performed for a variety of methods, including the collocation Trefftz method and EBF itself .…”

“…However, we address this issue by adding a few singular functions to original bases (see for more information). This has been performed for a variety of methods, including the collocation Trefftz method and EBF itself . Figure illustrates the results obtained.…”

Coupling of 2D discretized Peridynamics with a meshless method based on classical elasticity using switching of nodal behaviour

“…EBF falls within the category of boundary meshless methods and shares some similarities with Trefftz-type methods (Chen et al , 2007). The application of EBF to different engineering problems can be found in different studies, such as in Movahedian et al (2013), Movahedian and Boroomand (2015a, 2015b), Hashemi et al (2013), Zandi et al (2012a), Shahbazi et al (2011), Abdollahi and Boroomand (2014), Mossaiby et al (2015) and Boroomand et al (2016), and the generalized format of EBF for dealing with nonlinear problems is investigated by Mossaiby et al (2015).…”