Study of a distinctly optimal solution in topology optimization based on continuous adjoint method for the natural convection problem

Abstract

The topology optimization is becoming a highly promising technique in engineering with the development of the additive layer manufacturing technology. Compared to the structural design, however, there are necessities to further improve its methodology in the field of fluid flow and heat transfer, such as the suppression of the gray region. In this study, a topology optimization technique for a heat transfer problem is developed based on the finite volume method, adopting the continuous adjoint method. For the objective of minimizing the difference between the temperature fields and desired temperature, adjoint equations and sensitivity field are derived from the primal equations, which are the continuity, momentum, and energy equations considering Boussinesq approximation. Filtering and projection techniques are implemented to obtain a distinctly optimal structure by eliminating gray elements. A gradual variation of steepness parameter value, consisting of exponential and linear functions, is proposed in the projection process to ensure numerical stability. The suggested algorithm consists of two-step: i) to consider the consistency of the initial condition, which estimate sensitivity fields only, and ii) to obtain a distinctly optimal solution, which update the design variables using an optimizer. Topology optimizations are conducted for a benchmark case of natural convection problem. Optimal performance and the level of constraints satisfaction are evaluated corresponding to the parameter values of filtering and projection. As a result, a guidance of handling parameter values is suggested for natural convection problem. Finally, the physical aspects of the generated optimal structure for the objective function are discussed.