Abstract
THE PURPOSE. The aim of the work is to find a method for mathematical modeling and analysis of inhomogeneous physical fields and the influence of internal structures on these fields. Solutions are sought in areas in which there are subdomains with already known behavior (“embedded” areas and embedded solutions). The goal is to find a modeling method that does not require a change in existing software and is associated only with the modification of the right-hand sides of the equations under consideration. METHODS. The proposed method of mathematical modeling is characterized by the use of characteristic functions for specifying the geometric location and shape of embedded areas, for specifying systems of embedded areas (for example, spherical fillings or turbulent vortices) without specifying them as geometric objects, for modifying the calculated differential equation within the embedded areas. RESULTS. A theorem is formulated and proved (in the form of a statement) that formalizes the essence of the proposed method and gives an algorithm for its application. This algorithm consists in a) representation of the differential equation of the problem in another analytical form; in this form, a term is added to the original differential equation (to its right-hand side), in the presence of which this equation gives a predetermined ("built-in") solution in the necessary regions and b) a representation of the desired solution (using the characteristic function) in the form in which this solution takes the form of either the desired function (in the main area) or the specified functions (in the embedded areas). Examples of calculations from two physical and technical areas - thermal conductivity and hydrodynamics are presented. The result of the work is also the calculation of a turbulent flow in a pipe, in which a system of ball vortices, the speed and direction of rotation of these vortices are specified. CONCLUSION. The proposed method makes it possible to simulate complex physical processes, including turbulence, has been tested, is quite simple and indispensable in cases where embedded structures can be specified only by software.
Highlights
Характеристические функции шаров зададим тем же образом, как и круг выше, а именно: i (r, z), где η – единичная функция Хевисайда и ti (r ri )2 (z zi )2 ; ri, zi – координаты центров шаровых вихрей
Заключение Предложенный метод позволяет задавать условия и решать достаточно сложные задачи математической физики, в которых расчетное поле включает области с известным поведением, отличающимся от неизвестного, искомого поведения основной среды
Summary
При этом упор делается не на внешние границы расчетной области, которые, как правило, достаточно просты, а на внутренние структуры, для которых При моделировании турбулентности можно задать систему вихрей или иных структур, вращающихся с заданными окружными скоростями в заданных направлениях и при этом движущихся с заданной скоростью. Рассмотрим задачу В, которая отличается от А только тем, что внутри области Ω (возможно, с выходом на ee границу) выделена некоторая подобласть Ω1, на которой вместо искомого решения u задана функция f , удовлетворяющая основному уравнению.
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