Optimal implementation of exponential basis functions method on different software platforms and performance comparison

Despite the success and versetality of mesh based methods and the finite element method in particular, there has been a growing demand in last decades towards the development and adoption of methods which eliminate the mesh, i.e. the so called meshless or meshfree methods. The difficulties in generation of high quality meshes, in terms of computational cost, technical problems such as serial nature of the mesh generation process and the urge of parallel processing for today’s huge problems has been the main motivation for researches conducted on this subject. Apart from these, the human expertise required can never be completely omitted from the process. The problem is much more pronounced in 3D problems. To this end, many meshless methods have been developed in recent years where, among others, SPH, EFG, MLPG, RKPM, FPM and RBF-based methods could be named. The exponential basis functions method (EBF) is one of these methods which has been successfully employed in various engineering problems, ranging from heat transfer and various plate theories to classical and non-local elasticity and fluid dynamics. The method uses a linear combination of exponential basis functions to approximate the field variables. It is shown that these functions have very good approximation capabilities and using them guarantees a high convergence rate. These exponential bases are chosen such that they satisfy the homogenous form of the differential equation. This leads to an algebraic characteristic equation in terms of exponents of basis functions. From this point of view, this method may be categorized as an extension to the well-known Trefftz family of methods. These methods rely for their approximation of the field variables on a set of the so called T-complete bases. These bases should satisfy the homogenous form of the governing equation. They have been used with various degrees of success in a wide range of problems. The main drawback of these methods however lies in determination of the bases, which should be found for every problem. This problem has been reduced to the solution of the algebraic characteristic equation in the exponential basis functions method. The method is readily applicable to linear, constant coefficient operators, and has recently been extended to more general cases of variable coefficient linear and also non-linear problems. The relative performance of usual programming languages like C++ to mathematical software packages like Mathematica and/or Matlab is one of the major questions when using such packages to develop new numerical method, as this can affect the interpretation of performance of newly developed methods compared to established ones. In this paper the implementation of the exponential basis functions method on various software platforms has been discussed. We examine C++ and Mathematica programming as a representative of different software platforms. On each platform we implement the exponential basis function method using various options available. The relative performance of these implementations is thoroughly investigated. The results show that with a proper implementation, the numerical error of the method can also be decreased considerably. In this research we show that using optimal implementations of on both platforms, this ratio is between 2.5 and 6.