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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.

We present in this paper a modiÿed form of the REP gradient recovery process recently published.1 This new form is not only cheaper but has a much improved performance-which equals and occasionally exceeds the performance of the SPR (Superconvergent Patch Recovery) method. 2-4The comparisons are based on the robustness test originally proposed by Babu ska et al.5-8 This test is described brie y in a manner suitable for those who ÿnd some aspects of modern mathematics di cult to follow.The result and comparison of the tests for various repeatable patches with regular and irregular element distribution are made with SPR and the 'old' form of REP and are based on the general error estimator introduced by Zienkiewicz and Zhu in 1987.

SUMMARYA stabilized version of the finite point method (FPM) is presented. A source of instability due to the evaluation of the base function using a least square procedure is discussed. A suitable mapping is proposed and employed to eliminate the ill-conditioning effect due to directional arrangement of the points. A step by step algorithm is given for finding the local rotated axes and the dimensions of the cloud using local average spacing and inertia moments of the points distribution. It is shown that the conventional version of FPM may lead to wrong results when the proposed mapping algorithm is not used.It is shown that another source for instability and non-monotonic convergence rate in collocation methods lies in the treatment of Neumann boundary conditions. Unlike the conventional FPM, in this work the Neumann boundary conditions and the equilibrium equations appear simultaneously in a weight equation similar to that of weighted residual methods. The stabilization procedure may be considered as an interpretation of the finite calculus (FIC) method. The main difference between the two stabilization procedures lies in choosing the characteristic length in FIC and the weight of the boundary residual in the proposed method. The new approach also provides a unique definition for the sign of the stabilization terms. The reasons for using stabilization terms only at the boundaries is discussed and the two methods are compared.Several numerical examples are presented to demonstrate the performance and convergence of the proposed methods.

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