This work is concerned with the shape optimal design of an obstacle
immersed in the Stokes-Brinkmann fluid, which is also coupled with a
thermal model in the bounded domain. The shape optimal problem is
formulated and analyzed based on the framework of the continuous adjoint
method, with the advantage that the cost of computing the gradients and
sensitivities is independent of the number of design variables. Then,
the velocity method is utilized to describe the domain deformation, and
the structure of Eulerian derivative with respect to the shape of the
variable domain for a cost functional is established by applying the
differentiability of a minimax problem based on the function space
parametrization technique. Moreover, an iterative algorithm is proposed
to optimize the boundary of the obstacle in order to reduce the total
dissipation energy. Finally, numerical examples are presented to
illustrate the feasibility and effectiveness of our method.