كليدواژه :
روش توابع پايه نمايي , شبه وارون , جداسازي مقدار تكين , معادلات ديفرانسيل پاره اي , پياده سازي بهينه
چكيده فارسي :
در دهه هاي اخير روش هاي بدون شبكه مورد توجه محققان قرار گرفته اند. هزينه بالاي توليد شبكه المان بندي، چه در بعد محاسباتي و چه در بعد نيروي انساني متخصص يكي از مهم ترين دلايل اين امر به شمار مي رود. روش توابع پايه نمايي يكي از اين روش ها است كه در چند سال اخير جهت حل انواع معادلات ديفرانسيل پاره اي در مسائل مختلف علوم مهندسي با موفقيت به كار رفته است. در اين مقاله پياده سازي اين روش روي بسترهاي مختلف نرم افزاري مورد بحث قرار گرفته و كارايي نسبي آنها با يكديگر مقايسه مي شود. نتايج نشان مي دهد با پياده سازي مناسب مي توان خطاي ناشي از حل عددي را به شدت كاهش داد. كارايي نسبي انجام حل با استفاده از زبان هاي برنامه نويسي معمول مانند C++ در مقايسه با بسته هاي نرم افزاري رياضي همواره يكي از سوالات رايج هنگام استفاده از اين بسته هاي نرم افزاري است. در اين تحقيق نشان داده مي شود در صورت پياده سازي بهينه روش توابع پايه نمايي اين نسبت بين 2/5 تا 6 متغير است.
چكيده لاتين :
Despite the success and versatility of mesh based methods --finite element method in particular- there has been a growing demand in last decades towards the development and adoption of methods which can eliminate using the mesh, i.e. the so called meshless or mesh-free methods. Difficulties in the 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 have been the main motivation for the implementation of new researches. Apart from these, the human required expertise can never be completely omitted from the analysis process. However, the problem is much more pronounced in 3D problems. To this end, many meshless methods have been developed in recent years among which 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 their application 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 basic 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 on a set of the so called T-complete bases for their approximation of the field variables. 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 the determination of the basis, 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 been recently extended to more general cases of linear and also non-linear problems with variable coefficients. The relative performance of usual programming languages such as C++ in comparison with mathematical software packages -like Mathematica and/or Matlab- is one of the major questions when using such packages to develop new numerical methods. This can affect the interpretation of the 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. C++ and Mathematica programming have been examined as a representative of different software platforms. The exponential basis function method is implemented in each platform, using various options available. Results show that with a proper implementation, the numerical error of the method can be decreased considerably. Regarding the results of this research, optimal implementations of C++ and Mathematica platforms, error ratio is between 2.5 and 6, respectively.
