توسعه توابع شکل و توابع پایه‌ی شعاعی جدید هنکل کروی در بهینه‌سازی توپولوژی‌ سازه به روش سطح تراز

در این مقاله، از توابع پایه هنکل کروی جهت بهینه سازی توپولوژی سازه با استفاده از روش سطح تراز استفاده شده است. توابع پیشنهادی، ترکیبی از میدان توابع بسل نوع اول و دوم و همچنین میدان توابع چند جمله‌ای در فضای مختلط و برگرفته از توابع پایه‌ی شعاعی هستند. با استفاده از توابع هنکل کروی، وابستگی تابع مجموعه سطوح تراز به مکان و زمان از یکدیگر جدا گشته و این سبب تبدیل ‌شدن معادله دیفرانسیل با مشتقات جزئی همیلتونژاکوبی به یک معادله دیفرانسیل معمولی می شود. بدین طریق، مشکلات ناشی از حل معادله دیفرانسیل با مشتقات جزئی برطرف شده و در نتیجه نیازی به مقداردهی مجدد تابع مجموعه سطوح تراز در فرایند بهینه‌سازی نمی‌باشد. در ادامه، جهت افزایش سرعت و دقت همگرایی در ایجاد طرح بهینه، توابع شکل هنکل کروی جایگزین توابع شکل کلاسیک لاگرانژ می‌شود. توابع شکل پیشنهادی علاوه بر ارضای خاصیت دلتای کرونکر و افراز واحد، بی‌نهایت مشتق‌پذیر بوده همچنین توابع پیشنهادی به لحاظ دارا بودن هر سه میدان توابع چند جمله‌ای، بسل نوع اول و دوم در فضای مختلط می‌تواند در بهبود دقت و سرعت همگرایی موثر باشند در حالی که در توابع شکل کلاسیک لاگرانژ تنها میدان توابع چند جمله ای اغنا می شود. در انتها چندین مثال عددی جهت بررسی عملکرد توابع پایه‌ی شعاعی هنکل کروی و توابع شکل هنکل کروی بیان ‌شده است.

Development of Novel Spherical Hankel Shape and Radial Basis Functions in Structural Topology Optimization by Level Set Method

Shape and topology optimization have become one of the main researches that is widely used in engineering fields. The purpose of topology optimization is to find an appropriate (optimal) distribution of materials in the design domain so that the shape and number of voids is optimized and the objective function is minimized or maximized. In recent decades, noticeable researches and various topology optimization methods were proposed. The level set method is being used successfully in structural shape and topology optimization. This method is an implicit method for moving interior and exterior boundaries, while these boundaries may join together during the process and new voids may be formed. The structural boundary is illustrated by the zero level set and nonzero in the domain. In the above context, the level set function is used as a switch to distinguish between the two domains present in the computing space. This way of illustration has an important feature by which the domain boundaries can be combined together or divided. By using the solution of HamiltonJacobi equation resulting from this function, the domain rsquo;s boundary starts to move. The control over movement of this boundary is done by velocity vector of HamiltonJacobi equation. Now, in order to use this method in topology optimization, it is sufficient to establish a relationship between velocity vector of HamiltonJacobi and shape derivation, which is used for optimizing objective function. It is possible to use standard level set for structural topology optimization.In this paper, the spherical Hankel basis functions are used to optimize the structural topology using the level set method. The proposed functions are a combination of the first and second kind of Bessel functions fields as well as the polynomial ones in complex space and are derived from radial basis functions. Using the spherical Hankel functions, the dependence of the function of the level set method on the space and time is separated, which results in the transformation of the HamiltonJacobian partial differential equation into a conventional differential equation. In this way, the difficulties arising from solving partial differential equations are eliminated, and thus there is no need to reset the function of the level set method in the optimization process. Further, in order to increase the speed and precision of convergence in creating an optimal design, the classic Lagrange shape functions are replaced with the spherical Hankel ones. The proposed shape functions have some properties such as infinite piecewise continuity, the Kronecker delta property, and the partition of unity. Moreover, since they satisfy all three polynomial fields and the first and second kind of Bessel ones in the complex space, they can be effective in improving the accuracy and speed of convergence, while the classic Lagrange shape functions are able to satisfy only the polynomial function fields. Finally, several numerical examples are presented to study the performance of the spherical Hankel radial basis and shape functions.