Exponential basis functions in solution of static and time harmonic elastic problems in a meshless style

Abstract

SUMMARYIn this paper, exponential basis functions (EBFs) are used in a boundary collocation style to solve engineering problems whose governing partial differential equations (PDEs) are of constant coefficient type. Complex-valued exponents are considered for the EBFs. Two-dimensional elasto-static and time harmonic elasto-dynamic problems are chosen in this paper. The solution procedure begins with first finding a set of appropriate EBFs and then considering the solution as a summation of such EBFs with unknown coefficients. The unknown coefficients are determined by the satisfaction of the boundary conditions through a collocation method with the aid of a consistent and complex discrete transformation technique. The basis and various forms of the transformation have been addressed and discussed. We shall propose several strategies for selection of EBFs with the aid of the basis explained for the transformation. While using the transformation, the number of EBFs should not necessarily be equal to (or less than) the number of boundary information data. A library of EBFs has also been presented for further use. The effect of body forces is included in the solution via construction of particular solution by the use of the discrete transformation and another series of EBFs. A number of sample problems are solved to demonstrate the capabilities of the method. It has been shown that the time harmonic problems with high wave number can be solved without much effort. The method, categorized in meshless methods, can be applied to many other problems in engineering mechanics and general physics since EBFs can easily be found for almost all problems with constant coefficient PDEs.

Exponential basis functions in solution of problems with fully incompressible materials: A mesh-free method

“…In this reference, the authors have suggested two simple strategies in this regard; one with mathematical basis and another heuristically based on numerical experiments. As shown in [11], the obtained accuracy with either of these approaches is satisfactory in all the cases studied.…”

“…For more studies on the applications of the transformation used above the reader may refer to [8][9][10][11]. Note that the pressure does not appear as a separate variable during the solution process.…”

“…The solution accuracy may differ by changing the grid. A detailed discussion on the selection of a and b is out of the scope of this paper and the reader may refer to [11] for further information. In this reference, the authors have suggested two simple strategies in this regard; one with mathematical basis and another heuristically based on numerical experiments.…”

“…The readers may refer to [11] for normalization with respect to material constants appearing in part of V j as ðt j Þ k . Note that with defining scaling factor s j , the coefficients c j in (53) are now related to the coefficients C j in (54) as…”

“…In this section, the way that the boundary conditions are imposed through a collocation approach is described [11]. To this end, by considering u in its general form (53) and applying a collocation approach at M boundary points one may write the following relation…”