Topological sensitivity of the objective function defined on morphing boundaries of two-dimensional heat conduction problems

A topological sensitivity expression is presented for the objective function defined as a boundary integral of temperature and heat flux on the morphing boundaries which are newly generated during the topology optimization process. Since the objective function is defined by using only the boundary quantities, the boundary element method (BEM) is used for the heat conduction analyses of two-dimensional media. A level set-based topology optimization method is assumed to obtain the optimum shape in two-dimensional heat conduction problems. The shape of the domain being morphed during the updating process of optimization can be extracted from the iso-surface of the level set function. The distribution of the level set function is obtained by solving the evolution equation of the level set function defined over a fixed design domain. The derivative of the level set function with respect to fictitious time is assumed to be proportional to the topological sensitivity of the objective function. In this paper, the objective function is considered to be defined on the newly generated morphing boundaries in the optimization process. The topological sensitivity for this case is derived and is verified through a numerical demonstration.